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b/_images/fcb2cefa233355c15605817e37826c6ed3e69137306d11d3bf0e009a8f290296.png deleted file mode 100644 index bf05c6075..000000000 Binary files a/_images/fcb2cefa233355c15605817e37826c6ed3e69137306d11d3bf0e009a8f290296.png and /dev/null differ diff --git a/_sources/tutorials/W3D1_TimeSeriesAndNaturalLanguageProcessing/student/W3D1_Tutorial2.ipynb b/_sources/tutorials/W3D1_TimeSeriesAndNaturalLanguageProcessing/student/W3D1_Tutorial2.ipynb index b69e95d1a..35ff008aa 100644 --- a/_sources/tutorials/W3D1_TimeSeriesAndNaturalLanguageProcessing/student/W3D1_Tutorial2.ipynb +++ b/_sources/tutorials/W3D1_TimeSeriesAndNaturalLanguageProcessing/student/W3D1_Tutorial2.ipynb @@ -378,7 +378,7 @@ "\n", "In classical transformer systems, a core principle is encoding and decoding. We can encode an input sequence as a vector (that implicitly codes what we just read). And we can then take this vector and decode it, e.g., as a new sentence. So a sequence-to-sequence (e.g., sentence translation) system may read a sentence (made out of words embedded in a relevant space) and encode it as an overall vector. It then takes the resulting encoding of the sentence and decodes it into a translated sentence.\n", "\n", - "In modern transformer systems, such as GPT, all words are used parallelly. In that sense, the transformers generalize the encoding/decoding idea. Examples of this strategy include all the modern large language models (such as GPT)." + "In modern transformer systems, such as GPT, all words are used in parallel. In that sense, the transformers generalize the encoding/decoding idea. Examples of this strategy include all the modern large language models (such as GPT)." ] }, { diff --git a/projects/ComputerVision/data_augmentation.html b/projects/ComputerVision/data_augmentation.html index 90ab553a7..414908acf 100644 --- a/projects/ComputerVision/data_augmentation.html +++ b/projects/ComputerVision/data_augmentation.html @@ -1763,8 +1763,8 @@

Cutout

Mixup#

Mixup is a data augmentation technique that combines pairs of examples via a convex combination of the images and the labels. Given images \(x_i\) and \(x_j\) with labels \(y_i\) and \(y_j\), respectively, and \(\lambda \in [0, 1]\), mixup creates a new image \(\hat{x}\) with label \(\hat{y}\) the following way:

-
-(128)#\[\begin{align} +
+(128)#\[\begin{align} \hat{x} &= \lambda x_i + (1 - \lambda) x_j \\ \hat{y} &= \lambda y_i + (1 - \lambda) y_j \end{align}\]
diff --git a/projects/modelingsteps/Example_Deep_Learning_Project.html b/projects/modelingsteps/Example_Deep_Learning_Project.html index 9d9f19ff5..420982139 100644 --- a/projects/modelingsteps/Example_Deep_Learning_Project.html +++ b/projects/modelingsteps/Example_Deep_Learning_Project.html @@ -2067,33 +2067,33 @@

Build model -
Epoch [100/500], Step [1/2], Loss: 0.8369, Accuracy: 71.51%
+
Epoch [100/500], Step [1/2], Loss: 0.8982, Accuracy: 72.67%
 ------------------------------------------
-Epoch [100/500], Step [2/2], Loss: 0.9079, Accuracy: 67.83%
+Epoch [100/500], Step [2/2], Loss: 0.9187, Accuracy: 68.80%
 ------------------------------------------
 

 

-
Epoch [200/500], Step [1/2], Loss: 0.6369, Accuracy: 77.52%
+
Epoch [200/500], Step [1/2], Loss: 0.6901, Accuracy: 77.13%
 ------------------------------------------
-Epoch [200/500], Step [2/2], Loss: 0.5581, Accuracy: 81.40%
+Epoch [200/500], Step [2/2], Loss: 0.6060, Accuracy: 79.26%
 ------------------------------------------
 

 

-
Epoch [300/500], Step [1/2], Loss: 0.5327, Accuracy: 80.23%
+
Epoch [300/500], Step [1/2], Loss: 0.5478, Accuracy: 81.20%
 ------------------------------------------
-Epoch [300/500], Step [2/2], Loss: 0.4692, Accuracy: 85.08%
+Epoch [300/500], Step [2/2], Loss: 0.4853, Accuracy: 83.91%
 ------------------------------------------
 

 

-
Epoch [400/500], Step [1/2], Loss: 0.3940, Accuracy: 87.98%
+
Epoch [400/500], Step [1/2], Loss: 0.4423, Accuracy: 83.53%
 ------------------------------------------
-Epoch [400/500], Step [2/2], Loss: 0.4626, Accuracy: 82.56%
+Epoch [400/500], Step [2/2], Loss: 0.5842, Accuracy: 78.88%
 ------------------------------------------
 

 

-
Epoch [500/500], Step [1/2], Loss: 0.3307, Accuracy: 90.50%
+
Epoch [500/500], Step [1/2], Loss: 0.3525, Accuracy: 89.15%
 ------------------------------------------
-Epoch [500/500], Step [2/2], Loss: 0.3613, Accuracy: 87.60%
+Epoch [500/500], Step [2/2], Loss: 0.3686, Accuracy: 87.02%
 ------------------------------------------
 

 

@@ -2123,7 +2123,7 @@ 
Build model
-
Test Accuracy of the model on the 172 test moves: 84.302%
+
Test Accuracy of the model on the 172 test moves: 79.070%
 

 

 

@@ -2137,7 +2137,7 @@ 
Build model
-../../_images/78566d46f1f05e3965d54f9bb1efec22378c1d3f0317e8f5faf8b50b2220d972.png
+../../_images/64058d37f289e3656e6d296491b14d5cfc7641127067f41d0c6a3f6a91b7e179.png
 

 

 
The errors vary each time the model is run, but a common error seems to be that head scratching is predicted from some other movements that also involve arms a lot: throw/catch, hand clapping, phone talking, checking watch, hand waving, taking photo. If we train the model longer, these errors tend to go away as well. For some reason, crossed legged sitting is sometimes misclassified for crawling, but this doesn’t always happen.

@@ -2225,10 +2225,10 @@ 
Step 8: Modeling completion
-
73.83720930232558
+
71.51162790697676
 

 

-../../_images/3d16ae3899ad056181e31c4171b56e0d6efa3a29eb3ddf32ed9cf0fdc143a92b.png
+../../_images/4d22edbf348f43256d1e465ca1629b5d492e11651a4230519554fc5c37f7d2f0.png
 

 

 
That is some pretty good performance based on only 6 / 24 joints!

@@ -2274,32 +2274,32 @@ 
Step 9: Model evaluation
*** FITTING: Left Leg
 

 

-
limb performance: 72.67%
+
limb performance: 78.49%
 
 *** FITTING: Right Leg
 

 

-
limb performance: 70.93%
+
limb performance: 66.28%
 
 *** FITTING: Left Arm
 

 

-
limb performance: 66.86%
+
limb performance: 56.40%
 
 *** FITTING: Right Arm
 

 

-
limb performance: 38.95%
+
limb performance: 40.12%
 
 *** FITTING: Torso
 

 

-
limb performance: 79.65%
+
limb performance: 78.49%
 
 *** FITTING: Head
 

 

-
limb performance: 50.00%
+
limb performance: 47.67%
 

 

 

@@ -2353,44 +2353,44 @@ 
Step 9: Model evaluation
*** FITTING: limbs only
 

 

-
performance: 76.16%
+
performance: 79.65%
 

 

-
performance: 68.02%
+
performance: 62.21%
 

 

-
performance: 75.58%
+
performance: 80.81%
 

 

-
performance: 73.26%
+
performance: 69.77%
 

 

-
performance: 70.35%
+
performance: 74.42%
 

 

-
performance: 85.47%
-median performance: 74.42%
+
performance: 84.30%
+median performance: 77.03%
 
 *** FITTING: limbs+torso+head
 

 

-
performance: 81.98%
+
performance: 74.42%
 

 

-
performance: 81.98%
+
performance: 71.51%
 

 

-
performance: 86.63%
+
performance: 78.49%
 

 

-
performance: 79.07%
+
performance: 78.49%
 

 

-
performance: 79.65%
+
performance: 75.00%
 

 

-
performance: 86.05%
-median performance: 81.98%
+
performance: 83.14%
+median performance: 76.74%
 

 

 

diff --git a/projects/modelingsteps/ModelingSteps_10_DL.html b/projects/modelingsteps/ModelingSteps_10_DL.html
index bf8f2a7fb..641012d84 100644
--- a/projects/modelingsteps/ModelingSteps_10_DL.html
+++ b/projects/modelingsteps/ModelingSteps_10_DL.html
@@ -58,7 +58,7 @@
 const thebe_selector_output = ".output, .cell_output"
 
     
-    
+    
     
     
     
@@ -1451,7 +1451,7 @@ 
Step 10: publishing the model#

 

 

-

+

 

 
Guiding principles:

 
    
diff --git a/projects/modelingsteps/ModelingSteps_1through2_DL.html b/projects/modelingsteps/ModelingSteps_1through2_DL.html
index 953b3c187..0a9a270e7 100644
--- a/projects/modelingsteps/ModelingSteps_1through2_DL.html
+++ b/projects/modelingsteps/ModelingSteps_1through2_DL.html
@@ -58,7 +58,7 @@
 const thebe_selector_output = ".output, .cell_output"
 
     
-    
+    
     
     
     
@@ -1515,11 +1515,11 @@ 
    Objectives#
    
 
    
 
    
-
    
+
    
 

 

 

-

+

 

 
 
@@ -1746,7 +1746,7 @@ 
Disclaimer#

 

 

-

+

 

 
 
@@ -1759,7 +1759,7 @@ 
Step 1: Finding a phenomenon and a question to ask about it#

 

 

-

+

 

 
 

@@ -1843,7 +1843,7 @@ 
Example projects step 1
 
 

-

+

 

 
 

@@ -1917,7 +1917,7 @@ 
Step 2: Understanding the state of the art & background#

 

 

-

+

 

 
 

@@ -2059,7 +2059,7 @@ 
Example projects step 2
<IPython.core.display.Markdown object>
 

 

-

+

 

 
Here you will do a literature review. For the projects, do not spend too much time on this. A thorough literature review could take weeks or months depending on your prior knowledge of the field…

 
The important thing for your project here is not to exhaustively survey the literature but rather to learn the process of modeling. 1-2 days of digging into the literature should be enough!

diff --git a/projects/modelingsteps/ModelingSteps_3through4_DL.html b/projects/modelingsteps/ModelingSteps_3through4_DL.html
index be6614909..17bf57f8c 100644
--- a/projects/modelingsteps/ModelingSteps_3through4_DL.html
+++ b/projects/modelingsteps/ModelingSteps_3through4_DL.html
@@ -60,7 +60,7 @@
     
     
     
-    
+    
     
     
     
@@ -1464,7 +1464,7 @@ 
Step 3: Determining the basic ingredients#

 

 

-

+

 

 
 

@@ -1731,7 +1731,7 @@ 
Example projects step 3
<IPython.core.display.Markdown object>
 

 

-

+

 

 
 

@@ -1801,7 +1801,7 @@ 
Step 4: Formulating specific, mathematically defined hypotheses#

 

 

-

+

 

 
 

@@ -1921,7 +1921,7 @@ 
Example projects step 4
<IPython.core.display.Markdown object>
 

 

-

+

 

 
 

diff --git a/projects/modelingsteps/ModelingSteps_5through6_DL.html b/projects/modelingsteps/ModelingSteps_5through6_DL.html
index 34dd7de13..63200085d 100644
--- a/projects/modelingsteps/ModelingSteps_5through6_DL.html
+++ b/projects/modelingsteps/ModelingSteps_5through6_DL.html
@@ -58,7 +58,7 @@
 const thebe_selector_output = ".output, .cell_output"
 
     
-    
+    
     
     
     
@@ -1441,7 +1441,7 @@ 
Step 5: Selecting the toolkit#

 

 

-

+

 

 
Once you have completed Steps 1-4 to your satisfaction, you are now ready to model. You have a specific question, a goal in mind, and precise hypotheses expressed in mathematical language. All these components will empower you to chose an appropriate modeling approach.

 
In selecting the right toolkit, i.e. the right mathematics, computer science, engineering, or physics, etc approaches, you should consider the following important rules:

@@ -1511,7 +1511,7 @@ 
Step 6: Planning / drafting the model#

 

 

-

+

 

 
Planning the model involves thinking about the general outline of the model, its components and how they might fit together. You want to draw a model diagram, make some sketches and formalize necessary equations. This step will thus outline a plan of implementation. Once you have that plan, this will hugely facilitate the actual implementation of the model in computer code.

 
Your model will have:

diff --git a/projects/modelingsteps/ModelingSteps_7through9_DL.html b/projects/modelingsteps/ModelingSteps_7through9_DL.html
index 5abc0e98c..5771dd0a2 100644
--- a/projects/modelingsteps/ModelingSteps_7through9_DL.html
+++ b/projects/modelingsteps/ModelingSteps_7through9_DL.html
@@ -58,7 +58,7 @@
 const thebe_selector_output = ".output, .cell_output"
 
     
-    
+    
     
     
     
@@ -1458,7 +1458,7 @@ 
Step 7: Implementing the model#

 

 

-

+

 

 
This is the step where you finally start writing code! Separately implement each box, icon, or flow relationship identified in Step 6. Test each of those model components separately! (This is called a unit test). Unit testing ensures that each model components works are expected/planned.

 
Guiding principles:

@@ -1525,7 +1525,7 @@ 
Step 8: Completing the model#

 

 

-

+

 

 
Determing what you’re done modeling is a hard question. Referring back to your original goals will be crucial. This is also where a precise question and specific hypotheses expressed in mathematical relationships come in handy.

 
Note: you can always keep improving our model, but at some point you need to decide that it is finished. Once you have a model that displays the properties of a system you are interested in, it should be possible to say something about your hypothesis and question. Keeping the model simple makes it easier to understand the phenomenon and answer the research question.

@@ -1575,7 +1575,7 @@ 
Step 9: testing and evaluating the model#

 

 

-

+

 

 
Every models needs to be evaluated quantitatively. There are many ways to achieve that and not every model should be evaluated in the same way. Ultimately, model testing depends on what your goals are and what you want to get out of the model, e.g. qualitative vs quantitative fit to data.

 
Guiding principles:

diff --git a/projects/modelingsteps/TrainIllusionDataProjectDL.html b/projects/modelingsteps/TrainIllusionDataProjectDL.html
index 0d6766352..ae84e8a30 100644
--- a/projects/modelingsteps/TrainIllusionDataProjectDL.html
+++ b/projects/modelingsteps/TrainIllusionDataProjectDL.html
@@ -1772,8 +1772,8 @@ 
Question\(N\) neurons and \(M\) trials for each of 3 motion conditions: no self-motion, slowly accelerating self-motion and faster accelerating self-motion.

-

-(126)#\[\begin{align}
+

+(126)#\[\begin{align}
 N &= 40 \\
 M &= 400
 \end{align}\]

@@ -1824,7 +1824,7 @@ 
Background
-../../_images/691998358ad0b84a3aa3e260513e493dea28f2884db29c3b72d2ae13d752d828.png
+../../_images/12309e1b0f5f8b2678fe9ce4f50df80f08c0a5c46dd3ef0d18aefa6382df374b.png
 

 

 
Blue is the no-motion condition, and produces flat average spike counts across the 3 s time interval. The orange and green line do show a bell-shaped curve that corresponds to the acceleration profile. But there also seems to be considerable noise: exactly what we need. Let’s see what the spike trains for a single trial look like:

@@ -1836,9 +1836,9 @@ 
Background
-../../_images/86d5ecbd5807f4033da9d40d5f7cda9fe5964ddc6b6f1329e5e61217afe379ff.png
-../../_images/54885e65427d84d7b01f9a435a4e143802b2c70c468ce6468848a91c2138d1ea.png
-../../_images/d8aea21b3d80d298463d9b5b90da9a832b7a4cfa982949836b3ac13c8ae399b7.png
+../../_images/6fbd1e5b4f33d1bd8d6047d2263785e28305ea8ee5c46874e97cb634148224a4.png
+../../_images/434b95f0afef8e61b449992591137c1368f3dace3f520f30e5de25a90e62bc3d.png
+../../_images/c66b47652a1108d84fc48ff6084ec8a491fc75c66614eccf023d769bb9d1889e.png
 

 

 
You can change the trial number in the bit of code above to compare what the rasterplots look like in different trials. You’ll notice that they all look kind of the same: the 3 conditions are very hard (impossible?) to distinguish by eye-balling.

@@ -1986,7 +1986,7 @@ 
Model implementation
 

 

-../../_images/657b75caf157f5e8a44e7e14cd2f691d4f1560628aa3d1d8a2e085eeec1c3356.png
+../../_images/f0e8f6d99d4b7ea50d65e7ac08504675500847058d0a74bb06e228699b609634.png
 

 

 
We asked for 8 cross validations, which show up as the blue dots in the graph (two have the same accuracy). Prediction accuracy ranges from 56% to 72%, with the average at 65%, and the orange line is the median. Given the noisy data, that is not too bad actually.

@@ -2037,7 +2037,7 @@ 
Model implementation
 

 

-../../_images/657b75caf157f5e8a44e7e14cd2f691d4f1560628aa3d1d8a2e085eeec1c3356.png
+../../_images/f0e8f6d99d4b7ea50d65e7ac08504675500847058d0a74bb06e228699b609634.png
 

 

 
This is the exact same figure as before, so our function classifyMotionFromSpikes() also works as intended.

@@ -2174,7 +2174,7 @@ 
Model evaluation & testing
-../../_images/92c8a721ca3f1972276d94a0fb7c12964f0ce72c90ab24090445dd10e9060044.png
+../../_images/f04470b0ce7dd4df80a6d3eb32bc7a026ddd3df7c45c65a45cf960fa4954d5aa.png
 

 

 
Well, that’s interesting! The logistic regression doesn’t do a perfect job, but there is information in these results.

diff --git a/projects/modelingsteps/TrainIllusionModelingProjectDL.html b/projects/modelingsteps/TrainIllusionModelingProjectDL.html
index e3143f74d..5830ed819 100644
--- a/projects/modelingsteps/TrainIllusionModelingProjectDL.html
+++ b/projects/modelingsteps/TrainIllusionModelingProjectDL.html
@@ -1553,8 +1553,8 @@ 
Selected toolkitDrift-Diffusion Model (DDM) because it is a well-established framework that allows us to model decision making in the case of 2 alternative choices (here: self-motion vs. other train motion).

 
For our purposes simplest equation looks something like this:

-

-(127)#\[\begin{align}
+

+(127)#\[\begin{align}
 \dot e = \frac{de}{dt}= -c \cdot e + v \, ,
 \end{align}\]

 
where \(e\) is the accumulated evidence and \(v\) is our vestibular input already containing the noise (so we don’t need to add more noise?). \(c\) is the leakage constant, i.e., \(c=0\) means perfect integration; \(c=1\) means no integration (perfect leakage).

@@ -1630,7 +1630,7 @@ 
1. Vestibular signal generator
Text(0, 0.5, 'vestibular signal (a.u.)')
 

 

-../../_images/f6ab9f77e2837ddf28ee5f76c8cb41df931cd5789762da1f6effcf890edabc16.png
+../../_images/6b176037749651980a9a02a512bd69e994655b36cab66cb2d1a25f8d9af0c528.png
 

 

 
@@ -1671,7 +1671,7 @@ 
2. Integrator (DDM mechanism)
-../../_images/3a10e03b56d5ecdbbd326f9f6a3bfe98d118669c377b69497bff4dde15d0c0cf.png
+../../_images/ee0219c6b1af7da294c7b8282f6483798df6b41b8798f2663e1a8b3c19d3e74f.png
@@ -1821,7 +1821,7 @@

Model evaluation & testing -../../_images/44f2df9fa9aecdb09d84a11587e25f6add12dd7d32d158eb0853acd7ab29f906.png +../../_images/7acdaa9daf0507c86354349586991516c97f08178468e1d51655fc8892312fe9.png

There seems to be some parameter redundancy, i.e., we could chose different parameter combinations to make the model do something sensible…

@@ -1842,7 +1842,7 @@

Model evaluation & testing -
 Motion detected for no-motion: 31.5% and motion: 61.6%
+
 Motion detected for no-motion: 31.1% and motion: 61.1%
@@ -1880,7 +1880,7 @@

Model evaluation & testing -../../_images/c81c8f6a77c176e76725be01bc6deec7a048dfc2f57487287143f02c94740461.png +../../_images/8a1a0f9de4ef832c52f09191ecc0214d8b6996df8d6613698f36e29e05832d63.png

Our hypothesis of linear increase of illusion strength with noise only holds true in a limited range of noise… It’s monotonic but saturating of course…

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84, "kei": 84, "intut": 84, "corpu": 84, "dot": 84, "product": 84, "multihead": 84, "q": [84, 97], "k": 84, "v": 84, "i": [84, 94], "ii": 84, "complex": 84, "posit": [84, 85], "positionalencod": 84, "bias": [84, 94], "probabl": 84, "mask": 84, "problem": [84, 97], "approach": [84, 94], "hint": 84, "field": 84, "robust": 85, "gpt": [85, 88], "osf": 85, "context": [85, 101], "extend": 85, "binari": 85, "likelihood": 85, "light": 85, "weight": 85, "break": 85, "origin": 85, "textattack": 85, "issu": 85, "seri": [86, 87], "fasttext": [87, 89], "homonym": 87, "analog": [87, 93], "feed": 87, "llm": 88, "Is": 88, "pre_token": 88, "practic": 88, "chines": 88, "english": 88, "nlg": 88, "sota": 88, "todai": 88, "tomorrow": 88, "conclus": [88, 94], "around": 88, "larg": 88, "multilingu": 89, "thinking2": 90, "multimod": 91, "strategi": 91, "tumor": 91, "still": 91, "isn": 91, "enough": 91, "forrest": 91, "gump": 91, "pull": 91, "unsupervis": 92, "melani": 93, "mitchel": 93, "un": 94, "allow": 94, "independ": 94, "introduc": 94, "dsprite": 94, "schemat": 94, "directli": 94, "along": 94, "induc": 94, "invari": 94, "matric": 94, "rsm": 94, "reveal": 94, "support": 94, "don": 94, "potenti": 94, "versu": [94, 100], "produc": 94, "conclud": 94, "reconstruct": 94, "organ": 94, "abil": 94, "construct": 94, "meaning": 94, "space": 94, "few": 94, "avail": 94, "could": 94, "ssl": 94, "simclr": 94, "wa": 94, "cope": 94, "contrast": 94, "collaps": 94, "reduc": 94, "histogram": 94, "pair": 94, "shot": 94, "benefit": 94, "short": 94, "scenario": 94, "e": 94, "when": 94, "fraction": 94, "advantag": 94, "disadvantag": 94, "type": 94, "under": 94, "variou": 94, "chealsea": 96, "finn": 96, "chelsea": 96, "grid": 97, "shortest": 97, "path": 97, "planner": [97, 100, 102], "gridworld": 97, "gridworldplann": 97, "try": 97, "harder": 97, "markov": 97, "mdp": 97, "go": 97, "solver": 97, "iter": 97, "epsilon": 97, "greedi": [97, 100], "For": [98, 100], "thinking3": 98, "amita": 99, "kapoor": 99, "futur": [99, 101], "othellogam": 100, "player": [100, 102], "compet": 100, "othello": 100, "othellonnet": 100, "valuenetwork": 100, "mse": 100, "progress": 100, "policynetwork": 100, "policybasedplay": 100, "mont": [100, 102], "carlo": [100, 102], "rollout": 100, "montecarlo": 100, "against": [100, 102], "unbeat": 100, "oppon": 100, "19": 100, "In": 101, "memori": 101, "mct": 102, "concept": 103}, "envversion": {"sphinx.domains.c": 2, "sphinx.domains.changeset": 1, "sphinx.domains.citation": 1, "sphinx.domains.cpp": 6, "sphinx.domains.index": 1, "sphinx.domains.javascript": 2, "sphinx.domains.math": 2, "sphinx.domains.python": 3, "sphinx.domains.rst": 2, "sphinx.domains.std": 2, "sphinx.ext.intersphinx": 1, "sphinx": 56}}) \ No newline at end of file diff --git a/tutorials/Bonus_DeployModels/student/Bonus_Tutorial1.html b/tutorials/Bonus_DeployModels/student/Bonus_Tutorial1.html index 7ea47578f..a4390cd32 100644 --- a/tutorials/Bonus_DeployModels/student/Bonus_Tutorial1.html +++ b/tutorials/Bonus_DeployModels/student/Bonus_Tutorial1.html @@ -48,7 +48,7 @@ const thebe_selector_output = ".output, .cell_output" - + @@ -1077,7 +1077,7 @@

Tutorial Objectives
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Section 1: IntroductionVideo 1: Deploying Neural Networks on the Web#

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We will start by building a simple web application in Flask, which we’ll keep extending throughout the tutorial. In the end, you will have a web app where you can upload an image and have it classified automatically by a neural network model.

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Section 2.1: Your First Flask App#

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Creating a minimal Flask app is very simple. You need to create a Flask object and define the handler for the root URL returning the HTML response. You need to provide the applications module or package, but we can use __name__ as a convenient shortcut.

We need one small trick because the app will be running in a notebook. If you just run the app, it will be accessible at http://127.0.0.1:5000. The problem is that this is a local address to the server where the notebook is running, so you can’t access it. This is where ngrok helps - it creates a tunnel from the notebook server to the outside world. Make sure you use the ngrok URL when testing your app.

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Section 2.2: Using Jinja2 Templates#

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The default template engine used by Flask is Jinja2. Jinja2 offers features that help you write clean and reusable templates such as inheritance, humanizing, and formatting data (there’s an extension for this), dividing components into sub-modules, etc.

In this section, we are going to add Jinja2 templates to the app. WIth Jinja2 you can use variables and control flow commands, like ifs and loops in your HTML code. Then you can pass data from your Python code to the template when it is rendered.

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Section 2.3: Apply the MVVM Design Pattern#

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Design patterns provide a way of writing reusable, adaptable, and extendable code. Design patterns are not libraries, but rather a set of best practices to follow when designing your software.

Model View View-Model (MVVM) is a powerful design pattern commonly used in web applications (and other GUI applications).

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Section 2.4: Creating a REST API#

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REST (Representational State Transfer) is a set of rules according to which APIs are designed to enable your service to interact with other services. If HTML pages are interfaces designed for humans, you can think about REST APIs as interfaces made for computers.

A common way to implement a REST API is for your application to respond to certain requests by returning a JSON string containing the required data.

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Section 3: Vue.js#

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We already talked about the MVVM pattern and implemented it using a back end framework - Flask. Applying the same pattern on the front end can also be beneficial when creating dynamic applications.

Vue.js is a great front end library that implements the MVVM design pattern. It is widely used for creating user interfaces and single-page applications.

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Section 4: Model Presentation#

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Now (finally) we have all the tools we need to deploy our neural network! We are going to use a pre-trained DenseNet mode. In the first step, we are going to create an API entry point that accepts an image as input and classifies it. After that, we will create a dynamic UI for easier interaction.

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Section 4.1: Image Classification API#

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First, we need to load a pre-trained DenseNet trained on ImageNet. You can use torchvision.models to quickly get a pre-trained model for many popular neural network architectures.

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Section 4.2: Create a Dynamic Application#

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We will now create a Flask app that receives an image at /predict and passes it through the model. We will also implement an interactive UI to upload the image and call the API.

The UI consists of a file upload field, a classify button, and an image displaying the uploaded file.

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Section 5: Deploy a Flask app on Heroku#

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Now you are going to deploy your application as a real web server outside of the notebook. We are going to use Heroku for this. Heroku is a PaaS (Platform-as-a-Service) that offers pre-configured environments so you can deploy an application easily and quickly. They also offer a free tier which is enough for deploying simple apps.

But first, you need to test your application locally.

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Section 5.1: Preparing Your Environment#

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You need to do all the steps from here on on your own machine and not in the notebook. You need to make sure that you have Python 3 installed and some code editor (for example VS Code). You will also be using the terminal a lot in this section.

First, you need to prepare your Python environment and install all required dependencies. You should first create an empty folder where you will store your application and do the following steps.

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Section 5.2: Create Your Application#

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You are now ready to create the files needed for your application. For now, you need just 2 files.

app.py

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Section 5.4: Preparing for Deployment on Heroku#

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Before we can deploy on Heroku there are a couple of things we need to prepare.

Create Procfile

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Section 5.5: Deploying on Heroku#

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You are now finally ready to deploy to Heroku! There are just a couple of steps needed.

1. Create a Heroku account

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Summary#

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In this tutorial you learned the basics of some modern tools for creating dynamic web applications and REST APIs. You also learned how you can deploy your neural network model as a web app.

You can now build on top of that and create more sophisticated and awesome applications and make them available to millions of people!

diff --git a/tutorials/W1D1_BasicsAndPytorch/student/W1D1_Tutorial1.html b/tutorials/W1D1_BasicsAndPytorch/student/W1D1_Tutorial1.html index 50961d6f3..d223c48e0 100644 --- a/tutorials/W1D1_BasicsAndPytorch/student/W1D1_Tutorial1.html +++ b/tutorials/W1D1_BasicsAndPytorch/student/W1D1_Tutorial1.html @@ -50,7 +50,7 @@ - + @@ -1309,7 +1309,7 @@

Tutorial Objectives
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Section 1: Welcome to Neuromatch Deep learning courseVideo 1: Welcome and History#

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This will be an intensive 3 week adventure. We will all learn Deep Learning (DL) in a group. Groups need standards. Read our Code of Conduct.

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Video 2: Why DL is cool#

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Discuss with your pod: What do you hope to get out of this course? [in about 100 words]

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Section 2.1: Creating Tensors#

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There are various ways of creating tensors, and when doing any real deep learning project, we will usually have to do so.

Construct tensors directly:

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Tensor a: tensor([[0.9728, 0.2797, 0.9038]])
-Tensor b: tensor([[ 1.3048, -2.7761,  0.0065, -1.7133],
-        [-0.8749,  0.0401, -0.0865, -0.8609],
-        [ 0.5365,  0.4012, -0.8520,  0.9674]])
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Tensor a: tensor([[0.5906, 0.6871, 0.6338]])
+Tensor b: tensor([[-0.1422,  0.4803,  0.0736,  0.2737],
+        [-0.5090,  0.9543,  0.2056, -0.1748],
+        [ 0.2209,  0.6177, -2.1440,  0.6247]])
 Tensor c: tensor([[0., 0., 0.]])
-Tensor d: tensor([[0.7411, 0.6957, 0.3873]])
+Tensor d: tensor([[0.5401, 0.5953, 0.9354]])
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Section 2.2: Operations in PyTorch#

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Tensor-Tensor operations

We can perform operations on tensors using methods under torch..

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Mean values of the rows of x tensor([0.2522, 0.6170, 0.5267])
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Coding Exercise 2.2 : Simple tensor operations#

Below are two expressions involving operations on matrices.

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-(1)#\[\begin{equation} +
+(1)#\[\begin{equation} \textbf{A} = \begin{bmatrix}2 &4 \\5 & 7 \end{bmatrix} @@ -2042,8 +2044,8 @@

Coding Exercise 2.2 : Simple tensor operations -(2)#\[\begin{equation} +
+(2)#\[\begin{equation} b = \begin{bmatrix} 3 \\ 5 \\ 7 \end{bmatrix} \cdot @@ -2156,7 +2158,7 @@

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Section 2.3 Manipulating Tensors in Pytorch#

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Indexing

Just as in numpy, elements in a tensor can be accessed by index. As in any numpy array, the first element has index 0 and ranges are specified to include the first to last_element-1. We can access elements according to their relative position to the end of the list by using negative indices. Indexing is also referred to as slicing.

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Coding Exercise 2.3: Manipulating Tensors\(A\) and \(B\) and returns the column sum of A multiplied by the sum of all the elmements of \(B\), i.e., a scalar, e.g.,

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+(3)#\[\begin{equation} \text{If } A = \begin{bmatrix} 1 & 1 \\ @@ -2451,8 +2453,8 @@

Coding Exercise 2.3: Manipulating Tensors\(C\) and returns a 2D tensor consisting of a flattened \(C\) with the index of each element appended to this tensor in the row dimension, e.g.,

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+(4)#\[\begin{equation} \text{If } C = \begin{bmatrix} 2 & 3 \\ @@ -2469,8 +2471,8 @@

Coding Exercise 2.3: Manipulating Tensors\(D\) and \(E\). If the dimensions allow it, this function returns the elementwise sum of \(D\)-shaped \(E\), and \(D\); else this function returns a 1D tensor that is the concatenation of the two tensors, e.g.,

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+(5)#\[\begin{equation} \text{If } D = \begin{bmatrix} 1 & -1 \\ @@ -2487,8 +2489,8 @@

Coding Exercise 2.3: Manipulating Tensors -(6)#\[\begin{equation} +
+(6)#\[\begin{equation} \text{If } D = \begin{bmatrix} 1 & -1 \\ @@ -2632,7 +2634,7 @@

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Section 2.4: GPUs#

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By default, when we create a tensor it will not live on the GPU!

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tensor([3, 5, 7])
-tensor([ 9, 11, 13])
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tensor([ 9, 11, 13])

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Discuss!

Try and reduce the dimensions of the tensors and increase the iterations. You can get to a point where the cpu only function is faster than the GPU function. Why might this be?

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Section 2.5: Datasets and Dataloaders#

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When training neural network models you will be working with large amounts of data. Fortunately, PyTorch offers some great tools that help you organize and manipulate your data samples.

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Video 8: Train and Test#

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Training and Test Datasets

When loading a dataset, you can specify if you want to load the training or the test samples using the train argument. We can load the training and test datasets separately. For simplicity, today we will not use both datasets separately, but this topic will be adressed in the next days.

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Dataloader

Another important concept is the Dataloader. It is a wrapper around the Dataset that splits it into minibatches (important for training the neural network) and makes the data iterable. The shuffle argument is used to shuffle the order of the samples across the minibatches.

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Video 9: Data Augmentation - Transformations
Batch size: torch.Size([64, 3, 32, 32])
-../../../_images/5517ff75b0df3683c1d00c1d0cc0a9ba636f10e7d27ecfc0f92189d829f2c556.png +../../../_images/adbbb7d4b9eca41b61d38981d89b534767b785b4466f76eb683aef9080dad326.png

Transformations

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Section 3: Neural Networks#

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Generate sample data -../../../_images/a2829e1935e2bad62518a69ef55ee33889fe6ecc6762ac0b1dcb7fed5c9472dd.png +../../../_images/522d287bd530127087194d55ae3f4e772f3315f39b6f902327e01565029fd1c7.png

Prepare Data for PyTorch

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Section 3.2: Create a Simple Neural Network#

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For this example we want to have a simple neural network consisting of 3 layers:

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Section 3.3: Train Your Neural Network#

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Now it is time to train your network on your dataset. Don’t worry if you don’t fully understand everything yet - we will cover training in much more details in the next days. For now, the goal is just to see your network in action!

You will usually implement the train method directly when implementing your class NaiveNet. Here, we will implement it as a function outside of the class in order to have it in a separate cell.

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Helper function to plot the decision boundary
Epoch 14000 loss is 0.2128836214542389
-../../../_images/5adc4e702a00e5378fb581feb37714785469a4a637c9024550a29ebc765a8315.png +../../../_images/b66c28d1e7df9a9bb7418cf1fd3dcad9402533eba8bfc3b56b9f52bb5d7012a2.png

Plot the loss during training

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Helper function to plot the decision boundary
Text(0, 0.5, 'Loss')
-../../../_images/4633ec72b53c4f88006a88008cb2fa937738c206757d2c6c59ee0e0e48e762f5.png +../../../_images/58d3ce2721c5d4f2d4bef02242e9b514a75acef3b3fe60aa340c14e62d074fe8.png

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Visualize the training process#

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Video 14: XOR Widget#

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Exclusive OR (XOR) logical operation gives a true (1) output when the number of true inputs is odd. That is, a true output result if one, and only one, of the inputs to the gate is true. If both inputs are false (0) or both are true or false output results. Mathematically speaking, XOR represents the inequality function, i.e., the output is true if the inputs are not alike; otherwise, the output is false.

In case of two inputs (\(X\) and \(Y\)) the following truth table is applied:

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+(7)#\[\begin{matrix} X & Y & \text{XOR}\\ \hline 0 & 0 & 0\\ @@ -3968,15 +3972,15 @@

Interactive Demo 3.3: Solving XOR\(f(x)\) is ReLU is:

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+(8)#\[\begin{equation} y = f(x_1)+f(x_2)-f(x_1+x_2) \end{equation}\]

Try to set the weights and biases to implement this function after you played enough :)

Play with the parameters to solve XOR

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Do you think we can solve the discrete XOR (only 4 possibilities) with only 2 hidden units?

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Section 4: Ethics And Course Info#

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Video 16: Be a group#

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Video 17: Syllabus#

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Define Visualization using ALtair
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Edit the AUTHOR
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Install and import feedback gadget#

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diff --git a/tutorials/W1D2_LinearDeepLearning/student/W1D2_Tutorial1.html b/tutorials/W1D2_LinearDeepLearning/student/W1D2_Tutorial1.html index 17778fc6d..eac37b68c 100644 --- a/tutorials/W1D2_LinearDeepLearning/student/W1D2_Tutorial1.html +++ b/tutorials/W1D2_LinearDeepLearning/student/W1D2_Tutorial1.html @@ -50,7 +50,7 @@ - + @@ -1043,7 +1043,7 @@

Tutorial Objectives
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Section 0: IntroductionVideo 0: Introduction#

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Section 1.1: Gradients & Steepest Ascent#

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Before introducing the gradient descent algorithm, let’s review a very important property of gradients. The gradient of a function always points in the direction of the steepest ascent. The following exercise will help clarify this.

Analytical Exercise 1.1: Gradient vector (Optional)#

Given the following function:

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+(9)#\[\begin{equation} z = h(x, y) = \sin(x^2 + y^2) \end{equation}\]

find the gradient vector:

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+(10)#\[\begin{equation} \begin{bmatrix} \dfrac{\partial z}{\partial x} \\ \\ \dfrac{\partial z}{\partial y} \end{bmatrix} \end{equation}\]

Hint: Use the chain rule!

Chain rule: For a composite function \(F(x) = g(h(x)) \equiv (g \circ h)(x)\):

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-(11)#\[\begin{equation} +
+(11)#\[\begin{equation} F'(x) = g'(h(x)) \cdot h'(x) \end{equation}\]

or differently denoted:

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-(12)#\[\begin{equation} +
+(12)#\[\begin{equation} \frac{dF}{dx} = \frac{dg}{dh} ~ \frac{dh}{dx} \end{equation}\]
Click here for the solution

We can rewrite the function as a composite function:

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-(13)#\[\begin{equation} +
+(13)#\[\begin{equation} z = f\left( g(x,y) \right), ~~ f(u) = \sin(u), ~~ g(x, y) = x^2 + y^2 \end{equation}\]

Using the chain rule:

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-(14)#\[\begin{align} +
+(14)#\[\begin{align} \dfrac{\partial z}{\partial x} &= \dfrac{\partial f}{\partial g} \dfrac{\partial g}{\partial x} = \cos(g(x,y)) ~ (2x) = \cos(x^2 + y^2) \cdot 2x \\ \\ \dfrac{\partial z}{\partial y} &= \dfrac{\partial f}{\partial g} \dfrac{\partial g}{\partial y} = \cos(g(x,y)) ~ (2y) = \cos(x^2 + y^2) \cdot 2y \end{align}\]
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Video 2: Gradient Descent - Discussion#

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Section 1.2: Gradient Descent Algorithm#

Let \(f(\mathbf{w}): \mathbb{R}^d \rightarrow \mathbb{R}\) be a differentiable function. Gradient Descent is an iterative algorithm for minimizing the function \(f\), starting with an initial value for variables \(\mathbf{w}\), taking steps of size \(\eta\) (learning rate) in the direction of the negative gradient at the current point to update the variables \(\mathbf{w}\).

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-(15)#\[\begin{equation} +
+(15)#\[\begin{equation} \mathbf{w}^{(t+1)} = \mathbf{w}^{(t)} - \eta \nabla f \left( \mathbf{w}^{(t)} \right) \end{equation}\]

where \(\eta > 0\) and \(\nabla f (\mathbf{w})= \left( \frac{\partial f(\mathbf{w})}{\partial w_1}, ..., \frac{\partial f(\mathbf{w})}{\partial w_d} \right)\). Since negative gradients always point locally in the direction of steepest descent, the algorithm makes small steps at each point towards the minimum.

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Section 1.2: Gradient Descent Algorithm

Hence, all we need is to calculate the gradient of the loss function with respect to the learnable parameters (i.e., weights):

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-(16)#\[\begin{equation} +
+(16)#\[\begin{equation} \dfrac{\partial Loss}{\partial \mathbf{w}} = \left[ \dfrac{\partial Loss}{\partial w_1}, \dfrac{\partial Loss}{\partial w_2} , \dots, \dfrac{\partial Loss}{\partial w_d} \right]^{\top} \end{equation}\]
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Section 1.3: Computational Graphs and Backprop#

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Exercise 1.2 is an example of how overwhelming the derivation of gradients can get, as the number of variables and nested functions increases. This function is still extraordinarily simple compared to the loss functions of modern neural networks. So how can we (as well as PyTorch and similar frameworks) approach such beasts?

Let’s look at the function again:

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-(17)#\[\begin{equation} +
+(17)#\[\begin{equation} f(x, y, z) = \tanh \left(\ln \left[1 + z \frac{2x}{sin(y)} \right] \right) \end{equation}\]

We can build a so-called computational graph (shown below) to break the original function into smaller and more approachable expressions.

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Submit your feedback\(f\), and work our way against the arrows while calculating the gradient of each expression as we go. This is called the backward pass, from which the backpropagation of errors algorithm gets its name.

Computation Graph full

By breaking the computation into simple operations on intermediate variables, we can use the chain rule to calculate any gradient:

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-(18)#\[\begin{equation} +
+(18)#\[\begin{equation} \dfrac{\partial f}{\partial x} = \dfrac{\partial f}{\partial e}~\dfrac{\partial e}{\partial d}~\dfrac{\partial d}{\partial c}~\dfrac{\partial c}{\partial a}~\dfrac{\partial a}{\partial x} = \left( 1-\tanh^2(e) \right) \cdot \frac{1}{d+1}\cdot z \cdot \frac{1}{b} \cdot 2 \end{equation}\]

Conveniently, the values for \(e\), \(b\), and \(d\) are available to us from when we did the forward pass through the graph. That is, the partial derivatives have simple expressions in terms of the intermediate variables \(a,b,c,d,e\) that we calculated and stored during the forward pass.

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Analytical Exercise 1.3: Chain Rule (Optional)\(\dfrac{\partial f}{\partial y}\) using the computational graph and chain rule.

Click here for the solution -
-(19)#\[\begin{equation} +
+(19)#\[\begin{equation} \dfrac{\partial f}{\partial y} = \dfrac{\partial f}{\partial e}~\dfrac{\partial e}{\partial d}~\dfrac{\partial d}{\partial c}~\dfrac{\partial c}{\partial b}~\dfrac{\partial b}{\partial y} = \left( 1-\tanh^2(e) \right) \cdot \frac{1}{d+1}\cdot z \cdot \frac{-a}{b^2} \cdot \cos(y) \end{equation}\]

For more: Calculus on Computational Graphs: Backpropagation

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Section 2: PyTorch AutoGrad#

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Deep learning frameworks such as PyTorch, JAX, and TensorFlow come with a very efficient and sophisticated set of algorithms, commonly known as Automatic Differentiation. AutoGrad is PyTorch’s automatic differentiation engine. Here we start by covering the essentials of AutoGrad, and you will learn more in the coming days.

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Section 2.2: Backward Propagation -
Gradient function = <AddBackward0 object at 0x7fb5727ae700>
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Gradient function = <AddBackward0 object at 0x7f8a7bb7ba90>
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Section 3: PyTorch’s Neural Net module (nn module#

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PyTorch provides us with ready-to-use neural network building blocks, such as layers (e.g., linear, recurrent, etc.), different activation and loss functions, and much more, packed in the torch.nn module. If we build a neural network using torch.nn layers, the weights and biases are already in requires_grad mode and will be registered as model parameters.

For training, we need three things:

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Generate the sample dataset
Random seed 2021 has been set.
-../../../_images/4c1ed1868ba96066a2e7d7703df52a32f0302c2ace25d15ed08ee848fea75ec2.png +../../../_images/83f933dbc6d50564dbbf98f58f6d5322fb90d93971785b0d778011045bf4d158.png

Let’s define a very wide (512 neurons) neural net with one hidden layer and nn.Tanh() activation function.

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Summary#

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diff --git a/tutorials/W1D2_LinearDeepLearning/student/W1D2_Tutorial2.html b/tutorials/W1D2_LinearDeepLearning/student/W1D2_Tutorial2.html index bc0f456e2..4c449756b 100644 --- a/tutorials/W1D2_LinearDeepLearning/student/W1D2_Tutorial2.html +++ b/tutorials/W1D2_LinearDeepLearning/student/W1D2_Tutorial2.html @@ -50,7 +50,7 @@ - + @@ -1064,7 +1064,7 @@

Tutorial Objectives
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Section 1: A Shallow Narrow Linear Neural NetworkVideo 1: Shallow Narrow Linear Net#

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Coding Exercise 1.1: Implement simple narrow LNN (Optional)forward pass for our model from scratch without using PyTorch.

Also, although our model gets a single input feature and outputs a single prediction, we could calculate the loss and perform training for multiple samples at once. This is the common practice for neural networks, since computers are incredibly fast doing matrix (or tensor) operations on batches of data, rather than processing samples one at a time through for loops. Therefore, for the loss function, please implement the mean squared error (MSE), and adjust your analytical gradients accordingly when implementing the dloss_dw function.

Finally, complete the train function for the gradient descent algorithm:

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-(20)#\[\begin{equation} +
+(20)#\[\begin{equation} \mathbf{w}^{(t+1)} = \mathbf{w}^{(t)} - \eta \nabla loss (\mathbf{w}^{(t)}) \end{equation}\]
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Section 1.2: Learning landscapes#

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As you may have already asked yourself, we can analytically find \(w_1\) and \(w_2\) without using gradient descent:

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-(21)#\[\begin{equation} +
+(21)#\[\begin{equation} w_1 \cdot w_2 = \dfrac{y}{x} \end{equation}\]

In fact, we can plot the gradients, the loss function and all the possible solutions in one figure. In this example, we use the \(y = 1x\) mapping:

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Submit your feedback -../../../_images/16bacb1df7ed9615ca2b3a5b1b2d017eea911c94d95e697331ef9f174cda5ad0.png +../../../_images/b2c72bebca4f83cb4f4eaf6f8abd90f6e9200fbfb1d2b4c301ce441b41803f8a.png

Here, we also visualize the loss landscape in a 3-D plot, with two training trajectories for different initial conditions. @@ -2295,7 +2295,7 @@

Submit your feedback -../../../_images/4035348be18abfca2094f2231f8015e24075adf0e363ced40e8af2346bef57a4.png +../../../_images/a929da8da1b2ef2bcf7555e64b02623d058e275714fc2799073034aa8b006012.png

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Submit your feedback#

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Section 2.1: The effect of depth#

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Why might depth be useful? What makes a network or learning system “deep”? The reality is that shallow neural nets are often incapable of learning complex functions due to data limitations. On the other hand, depth seems like magic. Depth can change the functions a network can represent, the way a network learns, and how a network generalizes to unseen data.

So let’s look at the challenges that depth poses in training a neural network. Imagine a single input, single output linear network with 50 hidden layers and only one neuron per layer (i.e. a narrow deep neural network). The output of the network is easy to calculate:

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Interactive Demo 2.1: Depth widget - +

Video 5: Effect of Depth - Discussion#

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Section 2.2: Choosing a learning rate#

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Interactive Demo 2.2: Learning rate widget - +

Video 7: Learning Rate - Discussion#

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Section 2.3: Depth vs Learning Rate#

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Interactive Demo 2.3: Depth and Learning Rate - +

Submit your feedback#

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Video 9: Depth and Learning Rate - Discussion#

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Section 2.4: Why initialization is important#

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We’ve seen, even in the simplest of cases, that depth can slow learning. Why? From the chain rule, gradients are multiplied by the current weight at each layer, so the product can vanish or explode. Therefore, weight initialization is a fundamentally important hyperparameter.

Although in practice initial values for learnable parameters are often sampled from different \(\mathcal{Uniform}\) or \(\mathcal{Normal}\) probability distribution, here we use a single value for all the parameters.

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Submit your feedback -../../../_images/3cd532bdc2aa84dba045f16ff44310580f469353eef0e1144af89a8bd28c1647.png +../../../_images/d50b0bb5be799a5938cc05673941c07c15ae6d4203f35bc93551e6831ce2318c.png

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Submit your feedback#

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Summary#

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Hyperparameter interaction - +

Submit your feedback#

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diff --git a/tutorials/W1D2_LinearDeepLearning/student/W1D2_Tutorial3.html b/tutorials/W1D2_LinearDeepLearning/student/W1D2_Tutorial3.html index 6c9cfaf2c..5a12f8e97 100644 --- a/tutorials/W1D2_LinearDeepLearning/student/W1D2_Tutorial3.html +++ b/tutorials/W1D2_LinearDeepLearning/student/W1D2_Tutorial3.html @@ -50,7 +50,7 @@ - + @@ -1108,7 +1108,7 @@

Tutorial Objectives
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Section 0: Prelude

Coding Exercise 0: Re-initialization (Optional)#

Complete the function ex_initializer_, such that the weights are sampled from the following distribution:

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-(22)#\[\begin{equation} +
+(22)#\[\begin{equation} \mathcal{N}\left(\mu=0, ~~\sigma=\gamma \sqrt{\dfrac{1}{n_{in} + n_{out}}} \right) \end{equation}\]

where \(\gamma\) is the initialization scale, \(n_{in}\) and \(n_{out}\) are respectively input and output dimensions of the layer. the Underscore (“_”) in ex_initializer_ and other functions, denotes “in-place” operation.

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Section 1: Deep Linear Neural Nets#

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So far, depth just seems to slow down the learning. And we know that a single nonlinear hidden layer (given enough number of neurons and infinite training samples) has the potential to approximate any function. So it seems fair to ask: What is depth good for?

One reason can be that shallow nonlinear neural networks hardly meet their true potential in practice. In the contrast, deep neural nets are often surprisingly powerful in learning complex functions without sacrificing generalization. A core intuition behind deep learning is that deep nets derive their power through learning internal representations. How does this work? To address representation learning, we have to go beyond the 1D chain.

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Run to generate and visualize training samples from tree -../../../_images/2e10eb2fa8a8838373723213e24c9f75dc2ab57c7be16c39d0e86b84746e6dc1.png +../../../_images/1d278a6559a883fb1f41a77e46e43a5afbf4aa7f10e77be7f987643785f08e1c.png 
---------------------------------------------------------------
 Input Dimension: 8
 Output Dimension: 10000
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Make sure you execute this cell to train the network and plot
-../../../_images/3c6665d686802f0f4bf3cf60d8cab3be4c09762b585030cdf6311a15b4370ef1.png
+../../../_images/e1383b959d43a9732ddffe7257cf6666e5e98ca6eb420487f5b6507bd1e6c0ff.png

Think!

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Make sure you execute this cell to enable the widget!
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Section 2: Singular Value Decomposition (SVD)#

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In this section, we intend to study the learning (training) dynamics we just saw. First, we should know that a linear neural network is performing sequential matrix multiplications, which can be simplified to:

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-(23)#\[\begin{align} +
+(23)#\[\begin{align} \mathbf{y} &= \mathbf{W}_{L}~\mathbf{W}_{L-1}~\dots~\mathbf{W}_{1} ~ \mathbf{x} \\ &= \left(\prod_{i=1}^{L}{\mathbf{W}_{i}}\right) ~ \mathbf{x} \\ &= \mathbf{W}_{tot} ~ \mathbf{x} @@ -2398,8 +2398,8 @@

Submit your feedback\(A\) (yes, ANY) can be decomposed (factorized) to 3 matrices:

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-(24)#\[\begin{equation} +
+(24)#\[\begin{equation} \mathbf{A} = \mathbf{U} \mathbf{Σ} \mathbf{V}^{\top} \end{equation}\]

where \(U\) is an orthogonal matrix, \(\Sigma\) is a diagonal matrix, and \(V\) is again an orthogonal matrix. The diagonal elements of \(\Sigma\) are called singular values.

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Make sure you execute this cell to train the network and plot#

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Make sure you execute this cell to train the network and plot -../../../_images/ab691438ce7197a801c04a5a0512e67400c7328fdb5a2324a1b97d67fb9dbb7a.png +../../../_images/0425646389429b5ee5be251a9729a599897cc73935acbe8201f0d6bba99ac16b.png

Think!

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Video 3: SVD - Discussion#

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Section 3: Representational Similarity Analysis (RSA)Video 4: RSA#

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The previous section ended with an interesting remark. SVD helped to break our deep “wide” linear neural net into 8 deep “narrow” linear neural nets.

The first narrow net (highest singular value) converges fastest, while the last four narrow nets, converge almost simultaneously and have the smallest singular values. Can it be that the narrow net with larger mode is learning the difference between “living things” and “objects”, while another narrow net with smaller mode is learning the difference between Fish and Birds? how could we check this hypothesis?

Representational Similarity Analysis (RSA) is an approach that could help us understand the internal representation of our network. The main idea is that the activity of hidden units (neurons) in the network must be similar when the network is presented with similar input. For our dataset (hierarchically structured data), we expect the activity of neurons in the hidden layer to be more similar for Tuna and Canary, and less similar for Tuna and Oak.

For similarity measure, we can use the good old dot (scalar) product, which is also called cosine similarity. For calculating the dot product between multiple vectors (which would be our case), we can simply use matrix multiplication. Therefore the Representational Similarity Matrix for multiple-input (batch) activity could be calculated as follow:

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-(25)#\[\begin{equation} +
+(25)#\[\begin{equation} RSM = \mathbf{H} \mathbf{H}^{\top} \end{equation}\]

where \(\mathbf{H} = \mathbf{X} \mathbf{W_1}\) is the activity of hidden neurons for a given batch \(\mathbf{X}\).

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Make sure you execute this cell to enable widgets
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Let’s take a moment to analyze this more. A deep neural net is learning the representations, rather than a naive mapping (look-up table). This is thought to be the reason for deep neural nets supreme generalization and transfer learning ability. Unsurprisingly, neural nets with no hidden layer are incapable of representation learning, even with extremely small initialization.

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Video 5: RSA - Discussion#

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Section 4: Illusory Correlations#

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Let’s recall the training loss curves. There was often a long plateau (where the weights are stuck at a saddle point), followed by a sudden drop. For very deep complex neural nets, such plateaus can take hours of training, and we are often tempted to stop the training, because we believe it is “as good as it gets”! Another side effect of “immature interruption” of training is the network finding (learning) illusory correlations.

To better understand this, let’s do the next demonstration and exercise.

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Demonstration: Illusory Correlations -../../../_images/f64a1c67037ca360d4520ed87514eab2d9b77fd6d7e1f1842165a09a711452f4.png +../../../_images/3a63aa830d1503b84a7d83fb936f1171a97b791503c43847651d484a54daee9e.png

You can see the new feature shown in the last column of the plot above.

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Make sure you execute this cell to train the network and plot -../../../_images/dbccbca66516cacb8a41ced26bf0cd378c90041c9e9cbf543116fb377c8a8557.png +../../../_images/575be5eb288326275c799b4750a001e8347933129cbc93aa448a89aca03882eb.png

It seems that the network starts by learning an “illusory correlation” that sharks have bones, and in later epochs, as it learns deeper representations, it can see (learn) beyond the illusory correlation. This is important to remember that we never presented the network with any data saying that sharks have bones.

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Exercise 4: Illusory Correlations -../../../_images/569fc8f69cf7812bf8f3a04c31fc90eb497c9de16f18b9bd9de5ab65b360fc37.png +../../../_images/b3c2bb3bab70cb65e6dbcbdbd1e021ecd70d29be61ff143fb1e69a93c731189a.png
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Make sure you execute this cell to train the network and plot -../../../_images/afac7309a8b92ea40082ce82f15d62fee1ea84fde5d0e3ce3c1fb65c82d75890.png +../../../_images/a31ba78cc972eadf46b9cf6e28951a6ff3572d2d93b9f21732bbaf097acd39b8.png

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Video 7: Illusory Correlations - Discussion#

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Summary#

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Bonus#

Video 9: Linear Regression#

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Section 5.1: Linear Regression\(\mathbf{x} \in \mathbb{R}^M\), where \(M\) denotes the number of independent variables, while the dependent variables are collected in vector \(\mathbf{y} \in \mathbb{R}^N\), where \(N\) denotes the number of dependent variables. And the mapping between them is represented by the weight matrix \(\mathbf{W} \in \mathbb{R}^{N \times M}\) and a bias vector \(\mathbf{b} \in \mathbb{R}^{N}\) (generalizing to affine mappings).

The multivariate regression model can be written as:

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-(26)#\[\begin{equation} +
+(26)#\[\begin{equation} \mathbf{y} = \mathbf{W} ~ \mathbf{x} + \mathbf{b} \end{equation}\]

or it can be written in matrix format as:

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-(27)#\[\begin{equation} +
+(27)#\[\begin{equation} \begin{bmatrix} y_{1} \\ y_{2} \\ \vdots \\ y_{N} \\ \end{bmatrix} = \begin{bmatrix} w_{1,1} & w_{1,2} & \dots & w_{1,M} \\ w_{2,1} & w_{2,2} & \dots & w_{2,M} \\ \vdots & \ddots & \ddots & \vdots \\ w_{N,1} & w_{N,2} & \dots & w_{N,M} \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{M} \\ \end{bmatrix} + \begin{bmatrix} b_{1} \\ b_{2} \\ \vdots \\b_{N} \\ \end{bmatrix} \end{equation}\]

Section 5.2: Vectorized regression#

Linear regression can be simply extended to multi-samples (\(D\)) input-output mapping, which we can collect in a matrix \(\mathbf{X} \in \mathbb{R}^{M \times D}\), sometimes called the design matrix. The sample dimension also shows up in the output matrix \(\mathbf{Y} \in \mathbb{R}^{N \times D}\). Thus, linear regression takes the following form:

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-(28)#\[\begin{equation} +
+(28)#\[\begin{equation} \mathbf{Y} = \mathbf{W} ~ \mathbf{X} + \mathbf{b} \end{equation}\]

where matrix \(\mathbf{W} \in \mathbb{R}^{N \times M}\) and the vector \(\mathbf{b} \in \mathbb{R}^{N}\) (broadcasted over sample dimension) are the desired parameters to find.

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Section 5.2: Vectorized regression#

Linear regression is a relatively simple optimization problem. Unlike most other models that we will see in this course, linear regression for mean squared loss can be solved analytically.

For \(D\) samples (batch size), \(\mathbf{X} \in \mathbb{R}^{M \times D}\), and \(\mathbf{Y} \in \mathbb{R}^{N \times D}\), the goal of linear regression is to find \(\mathbf{W} \in \mathbb{R}^{N \times M}\) such that:

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-(29)#\[\begin{equation} +
+(29)#\[\begin{equation} \mathbf{Y} = \mathbf{W} ~ \mathbf{X} \end{equation}\]

Given the Squared Error loss function, we have:

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-(30)#\[\begin{equation} +
+(30)#\[\begin{equation} Loss(\mathbf{W}) = ||\mathbf{Y} - \mathbf{W} ~ \mathbf{X}||^2 \end{equation}\]

So, using matrix notation, the optimization problem is given by:

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-(31)#\[\begin{align} +
+(31)#\[\begin{align} \mathbf{W^{*}} &= \underset{\mathbf{W}}{\mathrm{argmin}} \left( Loss (\mathbf{W}) \right) \\ &= \underset{\mathbf{W}}{\mathrm{argmin}} \left( ||\mathbf{Y} - \mathbf{W} ~ \mathbf{X}||^2 \right) \\ &= \underset{\mathbf{W}}{\mathrm{argmin}} \left( \left( \mathbf{Y} - \mathbf{W} ~ \mathbf{X}\right)^{\top} \left( \mathbf{Y} - \mathbf{W} ~ \mathbf{X}\right) \right) \end{align}\]

To solve the minimization problem, we can simply set the derivative of the loss with respect to \(\mathbf{W}\) to zero.

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-(32)#\[\begin{equation} +
+(32)#\[\begin{equation} \dfrac{\partial Loss}{\partial \mathbf{W}} = 0 \end{equation}\]

Assuming that \(\mathbf{X}\mathbf{X}^{\top}\) is full-rank, and thus it is invertible, we can write:

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-(33)#\[\begin{equation} +
+(33)#\[\begin{equation} \mathbf{W}^{\mathbf{*}} = \mathbf{Y} \mathbf{X}^{\top} \left( \mathbf{X} \mathbf{X}^{\top} \right) ^{-1} \end{equation}\]

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Demonstration: Linear Regression vs. DLNN -../../../_images/7dab5cd492e47eabc79f18cfc91295b3a6fba687cd26f117f1d712277ef030da.png +../../../_images/378be3d26490c74e544d93d36b6495079af1f963fbab130eeb7397e22381bc71.png
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Demonstration: Linear Regression vs. DLNN#

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diff --git a/tutorials/W1D3_MultiLayerPerceptrons/student/W1D3_Tutorial1.html b/tutorials/W1D3_MultiLayerPerceptrons/student/W1D3_Tutorial1.html index 2f9b77b56..0d6b76edd 100644 --- a/tutorials/W1D3_MultiLayerPerceptrons/student/W1D3_Tutorial1.html +++ b/tutorials/W1D3_MultiLayerPerceptrons/student/W1D3_Tutorial1.html @@ -50,7 +50,7 @@ - + @@ -1040,7 +1040,7 @@

Tutorial objectives
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Section 0: Introduction to MLPs#

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Section 1: The Need for MLPs#

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Coding Exercise 1: Function approximation with ReLU

These are the points we will use to learn how to approximate the function. We have 10 training data points so we will have 9 ReLUs (we don’t need a ReLU for the last data point as we don’t have anything to the right of it to model).

We first need to figure out the bias term for each ReLU and compute the activation of each ReLU where:

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-(34)#\[\begin{equation} +
+(34)#\[\begin{equation} y(x) = \text{max}(0, x+b) \end{equation}\]

We then need to figure out the correct weights on each ReLU so the linear combination approximates the desired function.

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Section 2: MLPs in PyTorch#

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In the previous segment, we implemented a function to approximate any smooth function using MLPs. We saw that using Lipschitz continuity; We can prove that our approximation is mathematically correct. MLPs are fascinating, but before we get into the details on designing them, let’s familiarize ourselves with some basic terminology of MLPs - layer, neuron, depth, width, weight, bias, and activation function. Armed with these ideas, we can now design an MLP given its input, hidden layers, and output size.

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Coding Exercise 2: Implement a general-purpose MLP in PytorchLeaky ReLU) in all hidden layers

Leaky ReLU is described by the following mathematical formula:

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-(35)#\[\begin{align} +
+(35)#\[\begin{align} \text{LeakyReLU}(x) &= \text{max}(0,x) + \text{negative_slope} \cdot \text{min}(0, x) \\ &= \left\{ @@ -1664,7 +1664,7 @@

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Section 2.1: Classification with MLPs#

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The main loss function we could use out of the box for multi-class classification for N samples and C number of classes is:

To get CrossEntropyLoss of a sample \(i\), we could first calculate \(-\log(\text{softmax}(x))\) and then take the element corresponding to \(\text {labels}_i\) as the loss. However, due to numerical stability, we implement this more stable equivalent form,

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-(36)#\[\begin{equation} +
+(36)#\[\begin{equation} \operatorname{loss}(x_i, \text {labels}_i)=-\log \left(\frac{\exp (x[\text {labels}_i])}{\sum_{j} \exp (x[j])}\right)=-x_i[\text {labels}_i]+\log \left(\sum_{j=1}^C \exp (x_i[j])\right) \end{equation}\]

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Coding Exercise 2.1: Implement Batch Cross Entropy Loss

A batch of labels with shape (N, ) that ranges from 0 to C-1

Returns the average loss \(L\) calculated according to:

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-(37)#\[\begin{align} +
+(37)#\[\begin{align} \text{loss}(x_i, \text {labels}_i) &= -x_i[\text {labels}_i]+\log \left(\sum_{j=1}^C \exp (x_i[j])\right) \\ L &= \frac{1}{N} \sum_{i=1}^{N}{\text{loss}(x_i, \text {labels}_i)} \end{align}\]
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+ @@ -1802,8 +1802,8 @@

Submit your feedback#

Before we could start optimizing these loss functions, we need a dataset!

Let’s turn this fancy-looking equation into a classification dataset

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-(38)#\[\begin{equation} +
+(38)#\[\begin{equation} \begin{array}{c} X_{k}(t)=t\left(\begin{array}{c} \sin \left[\frac{2 \pi}{K}\left(2 t+k-1\right)\right]+\mathcal{N}\left(0, \sigma\right) \\ @@ -1864,7 +1864,7 @@

Section 2.2: Spiral Classification Dataset
Random seed 2021 has been set.

-../../../_images/338911a3d51a8d2332a6edb77e91f79d860a36bea2c7a8ca6c0e1c78c16ea3aa.png +../../../_images/85460b363e5ef3fea7e7fa140a35f2f2f1cfe37ab393709587f3156a10f3cd53.png
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Section 2.3: Training and Evaluation#

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And we need to make a Pytorch data loader out of it. Data loading in PyTorch can be separated in 2 parts:

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Bonus: Neuron Physiology and Motivation to Deep Learning#

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Leaky Integrate-and-fire (LIF) neuronal model#

The basic idea of LIF neuron was proposed in 1907 by Louis Édouard Lapicque, long before we understood the electrophysiology of a neuron (see a translation of Lapicque’s paper ). More details of the model can be found in the book Theoretical neuroscience by Peter Dayan and Laurence F. Abbott.

The model dynamics is defined with the following formula,

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-(39)#\[\begin{equation} +
+(39)#\[\begin{equation} \frac{d V_m}{d t}=\left\{\begin{array}{cc} \frac{1}{C_m}\left(-\frac{V_m}{R_m} + I \right) & t>t_{rest} \\ 0 & \text { otherwise } \end{array}\right. \end{equation}\]

Note that \(V_{m}\), \(C_{m}\), and \(R_{m}\) are the membrane voltage, capacitance, and resitance of the neuron, respectively, so the \(-\frac{V_{m}}{R_{m}}\) denotes the leakage current. When \(I\) is sufficiently strong such that \(V_{m}\) reaches a certain threshold value \(V_{\rm th}\), it momentarily spikes and then \(V_{m}\) is reset to \(V_{\rm reset}< V_{\rm th}\), and voltage stays at \(V_{\rm reset}\) for \(\tau_{\rm ref}\) ms, mimicking the refractoriness of the neuron during an action potential (note that \(V_{\rm reset}\) and \(\tau_{\rm ref}\) is assumed to be zero in the lecture):

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-(40)#\[\begin{eqnarray} +
+(40)#\[\begin{eqnarray} V_{m}(t)=V_{\rm reset} \text{ for } t\in(t_{\text{sp}}, t_{\text{sp}} + \tau_{\text{ref}}] \end{eqnarray}\]

where \(t_{\rm sp}\) is the spike time when \(V_{m}(t)\) just exceeded \(V_{\rm th}\).

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Leaky Integrate-and-fire (LIF) neuronal model#

In the cell below is given a function for LIF neuron model with it’s arguments described.

Note that we will use Euler’s method to make a numerical approximation to a derivative. Hence we will use the following implementation of the model dynamics,

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-(41)#\[\begin{equation} +
+(41)#\[\begin{equation} V_m^{[n]}=\left\{\begin{array}{cc} V_m^{[n-1]} + \frac{1}{C_m}\left(-\frac{V_m^{[n-1]}}{R_m}+I \right) \Delta t & t>t_{r e s t} \\ 0 & \text { otherwise } @@ -2408,7 +2408,7 @@

Simulating an LIF Neuron -../../../_images/fdf6936a17ef4f3d30a069cae1cc47564e133fba51034ae49fd322bf2d5e90b1.png +../../../_images/5b7bee7b59e463ff8e6f7eb25a97d7e743499c4f94b358ed72743336ebe7dfcb.png

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#
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Submit your feedback - + diff --git a/tutorials/W1D3_MultiLayerPerceptrons/student/W1D3_Tutorial2.html b/tutorials/W1D3_MultiLayerPerceptrons/student/W1D3_Tutorial2.html index a9ee86bc7..a638278d1 100644 --- a/tutorials/W1D3_MultiLayerPerceptrons/student/W1D3_Tutorial2.html +++ b/tutorials/W1D3_MultiLayerPerceptrons/student/W1D3_Tutorial2.html @@ -50,7 +50,7 @@ - + @@ -989,7 +989,7 @@

Tutorial Objectives
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Section 1: Wider vs deeper networks#

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Submit your feedback#

Let’s use the same spiral dataset generated before with two features. And then add more polynomial features (which makes the first layer wider). And finally, train a single linear layer. We could use the same MLP network with no hidden layers (though it would not be called an MLP anymore!).

Note that we will add polynomial terms upto \(P=50\) which means that for every \(x_1^n x_2^m\) term, \(n+m\leq P\). Now it’s fun math exercise to prove why the total number of polynomial features upto \(P\) becomes:

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-(42)#\[\begin{equation} +
+(42)#\[\begin{equation} \text{# of terms} = \frac{(P+1)(P+2)}{2} \end{equation}\]

Also, we don’t need the polynomial term with degree zero (which is the constatnt term) since nn.Linear layers have bias terms. Therefore we will have one fewer polynomial feature.

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Section 1.1: Where Wide Fails
Random seed 2021 has been set.

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Accuracy on the 3200 training samples: 69.88
+
Accuracy on the 3200 training samples: 69.88
 Accuracy on the 800 testing samples: 72.62
 

 

-../../../_images/3570e35a9991f01a0b6f3327abff86351f1c8b78d2024e75f701407ae49d30d5.png
+../../../_images/2ae68461c5418d5a111a8068f89f0042745e74bc03add970013477b9155bf7bd.png
 
Number of features: 1325
 

 

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Section 2: Deeper MLPsVideo 2: Case study#

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Coding Exercise 2: Dataloader on a real-world dataset
-../../../_images/6864477d9e8849db0385b53a73ba8cbce3690f54f52e3239e340749f14d4243c.png +../../../_images/effbe87412081f6f6827cf3b5e342e950298e3905b0ac4550c91d643e4c9191e.png
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Section 3: Ethical aspects#

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Summary#

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Bonus: The need for good initialization#

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Xavier initialization#

Let us look at the scale distribution of an output (e.g., a hidden variable) \(o_i\) for some fully-connected layer without nonlinearities. With \(n_{in}\) inputs (\(x_j\)) and their associated weights \(w_{ij}\) for this layer. Then an output is given by,

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-(43)#\[\begin{equation} +
+(43)#\[\begin{equation} o_{i} = \sum_{j=1}^{n_\mathrm{in}} w_{ij} x_j \end{equation}\]

The weights \(w_{ij}\) are all drawn independently from the same distribution. Furthermore, let us assume that this distribution has zero mean and variance \(\sigma^2\). Note that this does not mean that the distribution has to be Gaussian, just that the mean and variance need to exist. For now, let us assume that the inputs to the layer \(x_j\) also have zero mean and variance \(\gamma^2\) and that they are independent of \(w_{ij}\) and independent of each other. In this case, we can compute the mean and variance of \(o_i\) as follows:

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-(44)#\[\begin{align} +
+(44)#\[\begin{align} E[o_i] &= \sum_{j=1}^{n_\mathrm{in}} E[w_{ij} x_j] \\ \\ &= \sum_{j=1}^{n_\mathrm{in}} E[w_{ij}] E[x_j] = 0, \\ \\ \\ \mathrm{Var}[o_i] &= E[o_i^2] - (E[o_i])^2 \\ \\ @@ -2265,20 +2265,20 @@

Xavier initialization

One way to keep the variance fixed is to set \(n_{in}\sigma^2=1\) . Now consider backpropagation. There we face a similar problem, albeit with gradients being propagated from the layers closer to the output. Using the same reasoning as for forward propagation, we see that the gradients’ variance can blow up unless \(n_{out}\sigma^2=1\) , where \(n_{out}\) is the number of outputs of this layer. This leaves us in a dilemma: we cannot possibly satisfy both conditions simultaneously. Instead, we simply try to satisfy:

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-(45)#\[\begin{align} +
+(45)#\[\begin{align} \frac{1}{2} (n_\mathrm{in} + n_\mathrm{out}) \sigma^2 = 1 \text{ or equivalently } \sigma = \sqrt{\frac{2}{n_\mathrm{in} + n_\mathrm{out}}} \end{align}\]

This is the reasoning underlying the now-standard and practically beneficial Xavier initialization, named after the first author of its creators Glorot and Bengio, 2010. Typically, the Xavier initialization samples weights from a Gaussian distribution with zero mean and variance \(\sigma^2=\frac{2}{(n_{in}+n_{out})}\),

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-(46)#\[\begin{equation} +
+(46)#\[\begin{equation} w_{ij} \sim \mathcal{N} \left (\mu=0, \sigma=\sqrt{\frac{2}{(n_{in}+n_{out})}} \right) \end{equation}\]

We can also adapt Xavier’s intuition to choose the variance when sampling weights from a uniform distribution. Note that the uniform distribution \(\mathcal{U}(−a,a)\) has variance \(\frac{a^2}{3}\). Plugging this into our condition on \(\sigma^2\) yields the suggestion to initialize according to

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-(47)#\[\begin{equation} +
+(47)#\[\begin{equation} w_{ij} \sim \mathcal{U} \left(-\sqrt{\frac{6}{n_\mathrm{in} + n_\mathrm{out}}}, \sqrt{\frac{6}{n_\mathrm{in} + n_\mathrm{out}}}\right) \end{equation}\]

This explanation is mainly taken from here.

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Xavier initializationInitialization with transfer function#

Let’s derive the optimal gain for LeakyReLU following similar steps.

LeakyReLU is described mathematically:

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-(48)#\[\begin{equation} +
+(48)#\[\begin{equation} f(x)=\left\{ \begin{array}{ll} \alpha \cdot x & \text { for } x<0 \\ @@ -2298,15 +2298,15 @@

Initialization with transfer function\(\alpha\) controls the angle of the negative slope.

Considering a single layer with this activation function gives,

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-(49)#\[\begin{align} +
+(49)#\[\begin{align} o_{i} &= \sum_{j=1}^{n_\mathrm{in}} w_{ij} x_j\\ z_{i} &= f\left( o_{i} \right) \end{align}\]

where \(z_i\) denotes the activation of node \(i\).

The expectation of the output is still zero, i.e., \(\mathbb{E}[f(o_i)=0]\), but the variance changes, and assuming that the probability \(P(x < 0) = 0.5\), we have that:

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-(50)#\[\begin{align} +
+(50)#\[\begin{align} \mathrm{Var}[f(o_i)] &= \mathbb{E}[f(o_i)^2] - \left( \mathbb{E}[f(o_i)] \right)^{2} \\ \\ &= \frac{\mathrm{Var}[o_i] + \alpha^2 \mathrm{Var}[o_i]}{2} \\ \\ &= \frac{1+\alpha^2}{2}n_\mathrm{in} \sigma^2 \gamma^2 @@ -2314,8 +2314,8 @@

Initialization with transfer function\(\gamma\) is the variance of the distribution of the inputs \(x_j\) and \(\sigma\) is the variance of the distribution of weights \(w_{ij}\), as before.

Therefore, following the rest of derivation as before,


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-(51)#\[\begin{equation} +
+(51)#\[\begin{equation} \sigma = gain\sqrt{\frac{2}{n_\mathrm{in} + n_\mathrm{out}}}, \, \text{where} \,\, gain = \sqrt{\frac{2}{1+\alpha^2}} \end{equation}\]

As we can see from the derived formula of \(\sigma\), the transfer function we choose is related with the variance of the distribution of the weights. As the negative slope of the LeakyReLU \(\alpha\) becomes larger, the \(gain\) becomes smaller and thus, the distribution of the weights is narrower. On the other hand, as \(\alpha\) becomes smaller and smaller, the distribution of the weights is wider. Recall that, we initialize our weights, for example, by sampling from a normal distribution with zero mean and variance \(\sigma^2\).

diff --git a/tutorials/W1D5_Optimization/student/W1D5_Tutorial1.html b/tutorials/W1D5_Optimization/student/W1D5_Tutorial1.html index 13668f763..81b71c569 100644 --- a/tutorials/W1D5_Optimization/student/W1D5_Tutorial1.html +++ b/tutorials/W1D5_Optimization/student/W1D5_Tutorial1.html @@ -50,7 +50,7 @@ - + @@ -1246,7 +1246,7 @@

Tutorial Objectives
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Section 1. IntroductionVideo 1: Introduction#

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Section 2: Case study: successfully training an MLP for image classification

Video 2: Case Study - MLP Classification#

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Run me! -../../../_images/94521ffc12e0f43997e97e204432b3e669bdfe84ffb02c1c42e216b27772bc27.png +../../../_images/b8061ee4e4d4e3abd502af240b0102a0ebb5490a872dfc818001e9a245bee996.png

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Section 2.4: Interpretability -../../../_images/53200be50c429ba186794203173fbe578ab9800d98547ee4ad2ef3bb6d9e015a.png +../../../_images/a89102cec942cb125061c328e1d0676155805a2f0051d2961c1236f4cb4e0053.png @@ -1912,7 +1912,7 @@

Section 3: High dimensional search#

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Section 4: Poor conditioning#

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We illustrate this issue in a 2-dimensional setting. We freeze all but two parameters of the network: one of them is an element of the weight matrix (filter) for class 0, while the other is the bias for class 7. These results in an optimization with two decision variables.

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Coding Exercise 4: Implement momentum#

In this exercise you will implement the momentum update given by:

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-(52)#\[\begin{equation} +
+(52)#\[\begin{equation} w_{t+1} = w_t - \eta \nabla J(w_t) + \beta (w_t - w_{t-1}) \end{equation}\]

It is convenient to re-express this update rule in terms of a recursion. For that, we define ‘velocity’ as the quantity:

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-(53)#\[\begin{equation} +
+(53)#\[\begin{equation} v_{t-1} := w_{t} - w_{t-1} \end{equation}\]

which leads to the two-step update rule:

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-(54)#\[\begin{equation} +
+(54)#\[\begin{equation} v_t = - \eta \nabla J(w_t) + \beta (\underbrace{w_t - w_{t-1}}_{v_{t-1}}) \end{equation}\]
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-(55)#\[\begin{equation} +
+(55)#\[\begin{equation} w_{t+1} \leftarrow w_t + v_{t} \end{equation}\]

Pay attention to the positive sign of the update in the last equation, given the definition of \(v_t\), above.

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Interactive Demo 4: Momentum vs. GD
Random seed 2021 has been set.

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Submit your feedback#

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Section 5: Non-convexity#

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Take a couple of minutes to play with a more complex 3D visualization of the loss landscape of a neural network on a non-convex problem. Visit https://losslandscape.com/explorer.

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    Interactive Demo 5: Overparameterization to the rescue!
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    Submit your feedback#

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Section 6: Full gradients are expensive#

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Interactive Demo 6.1: Cost of computation - +

Submit your feedback#

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Interactive Demo 6.2: Compare different minibatch sizes - +

Remarks: SGD works! We have an algorithm that can be applied (with due precautions) to learn datasets of arbitrary size.

However, note the difference in the vertical scale across the plots above. When using a larger minibatch, we can perform fewer parameter updates as the forward and backward passes are more expensive.

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Section 7: Adaptive methods#

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Coding Exercise 7: Implement RMSprop#

In this exercise you will implement the update of the RMSprop optimizer:

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-(56)#\[\begin{align} +
+(56)#\[\begin{align} v_{t} &= \alpha v_{t-1} + (1 - \alpha) \nabla J(w_t)^2 \\ \\ w_{t+1} &= w_t - \eta \frac{\nabla J(w_t)}{\sqrt{v_t + \epsilon}} \end{align}\]
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Interactive Demo 7: Compare optimizers - +

Submit your feedback#

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Remarks: Note that RMSprop allows us to use a ‘per-dimension’ learning rate without having to tune one learning rate for each dimension ourselves. The method uses information collected about the variance of the gradients throughout training to adapt the step size for each of the parameters automatically. The savings in tuning efforts of RMSprop over SGD or ‘plain’ momentum are undisputed on this task.

Moreover, adaptive optimization methods are currently a highly active research domain, with many related algorithms like Adam, AMSgrad, Adagrad being used in practical application and theoretically investigated.

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Section 8: Ethical concerns#

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Bonus: Putting it all together#

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Download parameters of the benchmark model
WARNING: For this notebook to perform best, if possible, in the menu under `Runtime` -> `Change runtime type.`  select `GPU`
+
---------------------------------------------------------------------------
+EOFError                                  Traceback (most recent call last)
+Cell In[65], line 15
+     13   benchmark_state_dict = torch.load(fname)
+     14 else:
+---> 15   benchmark_state_dict = torch.load(fname, map_location=torch.device('cpu'))
+
+File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/torch/serialization.py:1040, in load(f, map_location, pickle_module, weights_only, mmap, **pickle_load_args)
+   1038     except RuntimeError as e:
+   1039         raise pickle.UnpicklingError(UNSAFE_MESSAGE + str(e)) from None
+-> 1040 return _legacy_load(opened_file, map_location, pickle_module, **pickle_load_args)
+
+File /opt/hostedtoolcache/Python/3.9.19/x64/lib/python3.9/site-packages/torch/serialization.py:1258, in _legacy_load(f, map_location, pickle_module, **pickle_load_args)
+   1252 if not hasattr(f, 'readinto') and (3, 8, 0) <= sys.version_info < (3, 8, 2):
+   1253     raise RuntimeError(
+   1254         "torch.load does not work with file-like objects that do not implement readinto on Python 3.8.0 and 3.8.1. "
+   1255         f"Received object of type \"{type(f)}\". Please update to Python 3.8.2 or newer to restore this "
+   1256         "functionality.")
+-> 1258 magic_number = pickle_module.load(f, **pickle_load_args)
+   1259 if magic_number != MAGIC_NUMBER:
+   1260     raise RuntimeError("Invalid magic number; corrupt file?")
+
+EOFError: Ran out of input
+
+
@@ -3688,6 +3713,7 @@

Download parameters of the benchmark model @@ -3823,35 +3849,7 @@

Exercise Bonus: Train your own model -
Random seed 2021 has been set.
-
-
-
[VALID] Epoch 1 - Batch 200 - Loss: 2.235 - Acc: 38.193%
-[TRAIN] Epoch 1 - Batch 200 - Loss: 2.274 - Acc: 30.930%
-
-
-
[VALID] Epoch 1 - Batch 400 - Loss: 2.068 - Acc: 52.979%
-[TRAIN] Epoch 1 - Batch 400 - Loss: 2.166 - Acc: 44.289%
-
-
-
[VALID] Epoch 1 - Batch 600 - Loss: 1.789 - Acc: 56.807%
-[TRAIN] Epoch 1 - Batch 600 - Loss: 1.935 - Acc: 55.352%
-
-
-
[VALID] Epoch 2 - Batch 200 - Loss: 1.266 - Acc: 70.420%
-[TRAIN] Epoch 2 - Batch 200 - Loss: 1.381 - Acc: 66.203%
-
-
-
[VALID] Epoch 2 - Batch 400 - Loss: 1.066 - Acc: 77.568%
-[TRAIN] Epoch 2 - Batch 400 - Loss: 1.166 - Acc: 74.414%
-
-
-
[VALID] Epoch 2 - Batch 600 - Loss: 0.930 - Acc: 80.361%
-[TRAIN] Epoch 2 - Batch 600 - Loss: 0.989 - Acc: 79.312%
-
-
-

+
@@ -3876,7 +3874,7 @@

Exercise Bonus: Train your own model -../../../_images/fcb2cefa233355c15605817e37826c6ed3e69137306d11d3bf0e009a8f290296.png +../../../_images/526bbfc5df5c554dcedb08f917749ac49a6112047439f59c84109ab3df812acb.png

@@ -3895,7 +3893,7 @@

Submit your feedback - +

@@ -3919,7 +3917,7 @@

Submit your feedback - +
@@ -3942,21 +3940,7 @@

Evaluation -
Your model...
-
-
-
Train Loss 0.826 / Test Loss 0.810
-Train Accuracy 82.171% / Test Accuracy 83.252%
 
-Benchmark model
-
-
-
Train Loss 0.011 / Test Loss 0.025
-Train Accuracy 99.784% / Test Accuracy 99.316%
-
-
-

diff --git a/tutorials/W2D1_Regularization/student/W2D1_Tutorial1.html b/tutorials/W2D1_Regularization/student/W2D1_Tutorial1.html index 4cd58a7e8..54da6d358 100644 --- a/tutorials/W2D1_Regularization/student/W2D1_Tutorial1.html +++ b/tutorials/W2D1_Regularization/student/W2D1_Tutorial1.html @@ -50,7 +50,7 @@ - + @@ -1005,7 +1005,7 @@

Tutorial Objectives
-
+
@@ -1615,7 +1615,7 @@

Section 1: Regularization is Shrinkage#

-
+
@@ -1634,7 +1634,7 @@

Submit your feedback
-
+

A key idea of neural nets is that they use models that are “too complex” - complex enough to fit all the noise in the data. One then needs to “regularize” them to make the models fit complex enough, but not too complex. The more complex the model, the better it fits the training data, but if it is too complex, it generalizes less well; it memorizes the training data but is less accurate on future test data.

@@ -1642,7 +1642,7 @@

Submit your feedbackVideo 2: Regularization as Shrinkage#

-
+
@@ -1661,7 +1661,7 @@

Submit your feedback - +

One way to think about Regularization is to think in terms of the magnitude of the overall weights of the model. A model with big weights can fit more data perfectly, whereas a model with smaller weights tends to underperform on the train set but can surprisingly do very well on the test set. Having the weights too small can also be an issue as it can then underfit the model.

In these tutorials, we use the sum of the Frobenius norm of all the tensors in the model as a measure of the “size of the model”.

@@ -1670,8 +1670,8 @@

Submit your feedback#

Before we start, let’s define the Frobenius norm, sometimes also called the Euclidean norm of an \(m×n\) matrix \(A\) as the square root of the sum of the absolute squares of its elements.


-
-(57)#\[\begin{equation} +
+(57)#\[\begin{equation} ||A||_F= \sqrt{\sum_{i=1}^m\sum_{j=1}^n|a_{ij}|^2} \end{equation}\]

@@ -1743,7 +1743,7 @@

Submit your feedback -

+

Apart from calculating the weight size for an entire model, we could also determine the weight size in every layer. For this, we can modify our calculate_frobenius_norm function as shown below.

Have a look how it works!!

@@ -1797,7 +1797,7 @@

Submit your feedback +../../../_images/7f96db181f1246bf7033cc953839a1c263a55b81faf06b89209f8ed1237fcf00.png

Using the last function, calculate_frobenius_norm, we can also obtain the Frobenius norm per layer for a whole ANN model and use the plot_weigts function to visualize them.

@@ -1822,7 +1822,7 @@

Submit your feedback +../../../_images/c3d6f36997c074b0a8a6d31836859b54f631b834fd9bab539fc1c013a9b96c37.png

@@ -1836,7 +1836,7 @@

Section 2: OverfittingVideo 3: Overparameterization and Overfitting#

-
+
@@ -1855,7 +1855,7 @@

Submit your feedback - +

@@ -1889,7 +1889,7 @@

Section 2.1: Visualizing Overfitting
Random seed 2021 has been set.
-../../../_images/eaff432b30bae519b5199bab45f93f26b9bc95e7f7a3ba0f7eb32dd0cbd0cbcf.png +../../../_images/0367ec8db8ae4d65ea628497d7d5f591ff01c92e4fc01ec7dad4cccfb52e97ba.png

Let’s create an overparametrized Neural Network that can fit on the dataset that we just created and train it.

@@ -2013,7 +2013,7 @@

Section 2.1: Visualizing Overfitting
Random seed 2021 has been set.
- +

Now that we have finished training, let’s see how the model has evolved over the training process.

@@ -2119,7 +2119,7 @@

Animation (Run Me!)RuntimeError: Requested MovieWriter (ffmpeg) not available -../../../_images/4cb1e04a307d81d656f10cdd6f20e9d547bb9f89a008a8e0c786186fd442e89b.png +../../../_images/66423f415c5ed2c155e6d4c8df35ddcaef422be800f1752e6da3c3185e7cab70.png

@@ -2146,7 +2146,7 @@

Plot the train and test losses -../../../_images/d90acb9fd01dedb1a3276986f7b1eb8f5fe7d446d98eac8459a2be7e04269511.png +../../../_images/a5d3cad3009c0a9f95bd70801d179e595c8bf2e6d474bcc0b05541d7407725f3.png

@@ -2174,7 +2174,7 @@

Submit your feedback - +

Now let’s visualize the Frobenious norm of the model as we trained. You should see that the value of weights increases over the epochs.

Frobenious norm of the model

@@ -2197,7 +2197,7 @@

Submit your feedback -../../../_images/630efa1975514c4e86d9488d4675e6e15ff7695ac20e920ee98c7805fd6cbaf7.png +../../../_images/9d2db9ca12067f24eff8013482b01d974c3ad921f256ab2cec85fa57d86e789d.png

Finally, you can compare the Frobenius norm per layer in the model, before and after training.

@@ -2221,8 +2221,8 @@

Submit your feedback -../../../_images/f78468b1fd9d04097ff8de8c8037eee381278208493f358d6a14244b194bb0ba.png -../../../_images/307916fb13ee9b3502c1b5695ea462b08284f61dcdd50e464d44748b9d9aa93e.png +../../../_images/f7677587eea0099d87ca72136e5798438cc531f37a309517501b606513a5fb2d.png +../../../_images/096424eaa56ce6bb1650b00527c1964ed06db0349fcf1d81cfe3926ced2d0ebd.png @@ -2466,10 +2466,10 @@

Section 3: Memorization
Random seed 2021 has been set.
-
Time to memorize the dataset: 241.6447606086731
+
Time to memorize the dataset: 223.69637608528137
 

 

-../../../_images/02c33c12e77d12754a4f51ba67fa89bce734a4c0d766e2935d2f27d56b1acc27.png
+../../../_images/8ae7cd982c48a5de102c8d6905769be84e5acc537ae8b89cd6292cd2f0993030.png
@@ -2491,8 +2491,8 @@

Frobenius norm for AnimalNet before and after training
-../../../_images/b0de31c52ce78630640d081720a3b57ef3508edc188b3a277bf0f64307ec7f08.png -../../../_images/0fffeba293de5453c66782e5f84871228cdf184f3cdb32706f755776007d1baa.png +../../../_images/da42e2626a961a450ddd110e5ba09e05e306fde26455d97b2eb04873f11bbfc0.png +../../../_images/7115401207a7c015679b0137158af5a34f99ab1c583958840de04e8653a18543.png

@@ -2549,7 +2549,7 @@

Data Visualizer
The image belongs to : cat
-../../../_images/a1a77da928eec784639eb05c87e63483f8f041a1dc6fa10f34391b8187021844.png +../../../_images/2e0155f095b8d0008f42c3e8e6d2607c01f87f0e446fed56b38405866f541fb3.png

Now let’s train the network on the shuffled data and see if it memorizes.

@@ -2594,7 +2594,7 @@

Data Visualizer
Random seed 2021 has been set.
-../../../_images/e2bbfa7fcba2c15582af5f938bb59bfd0604fbbcd2a43c8c054346d49a7ad0b6.png +../../../_images/f39b59eb6c7c677dfbd0cccfa034f14897e0e44ce39cf11a9b5b0662b7e14383.png

Isn’t it surprising to see that the ANN was able to achieve 100% training accuracy on randomly shuffled labels? This is one of the reasons why training accuracy is not a good indicator of model performance.

@@ -2608,7 +2608,7 @@

Section 4: Early Stopping#

-
+
@@ -2627,7 +2627,7 @@

Submit your feedback - +

Now that we have established that the validation accuracy reaches the peak well before the model overfits, we want to stop the training somehow early. You should have also observed from the above plots that the train/test loss on real data is not very smooth, and hence you might guess that the choice of the epoch can play a crucial role in the validation/test accuracy.

Early stopping stops training when the validation accuracies stop increasing.

@@ -2756,7 +2756,7 @@

Submit your feedback - +

@@ -2783,7 +2783,7 @@

Submit your feedback - + @@ -2838,7 +2838,7 @@

Bonus: Train with randomized labels
Random seed 2021 has been set.
-../../../_images/95f88f7e75611a5dfb12461c69670991eb7a9f23c58d715c1ed41a11777107d3.png +../../../_images/ba1198882221690cb498e83f0ecebef51d6b091fcff67ca9438698ea18db5d67.png
@@ -2869,7 +2869,7 @@

Plotting them all together (Run Me!) -../../../_images/4c430f878a4bdde5171ac45480975bfd8f1cd968428f3debd9b04c31f22b5df5.png +../../../_images/48b2f3dcdc7f8ce5bfffc4c5841322d8b0dc4f8448f79906826ce8a8013cc772.png

@@ -2897,7 +2897,7 @@

Submit your feedback - +

Also, it is interesting to note that sometimes the model trained on slightly shuffled data does slightly better than the one trained on pure data. Shuffling some of the data is a form of regularization, i.e., one of many ways of adding noise to the training data.

diff --git a/tutorials/W2D1_Regularization/student/W2D1_Tutorial2.html b/tutorials/W2D1_Regularization/student/W2D1_Tutorial2.html index 16978d4fb..7cf9ef2a0 100644 --- a/tutorials/W2D1_Regularization/student/W2D1_Tutorial2.html +++ b/tutorials/W2D1_Regularization/student/W2D1_Tutorial2.html @@ -50,7 +50,7 @@ - + @@ -1098,7 +1098,7 @@

Tutorial Objectives
-
+
@@ -1985,7 +1985,7 @@

Section 1: L1 and L2 Regularization#

-
+
@@ -2004,12 +2004,12 @@

Submit your feedback
-
+

Some of you might have already come across L1 and L2 regularization before in other courses. L1 and L2 are the most common types of regularization. These update the general cost function by adding another term known as the regularization term.


-
-(58)#\[\begin{equation} +
+(58)#\[\begin{equation} \text{Cost function} = Loss(\text{e.g., binary cross entropy}) + \text{Regularization term} \end{equation}\]

@@ -2093,7 +2093,7 @@

Dataloaders for Regularization
Random seed 2021 has been set.

-../../../_images/b9ae3a198b7e9657cfcd1bb3e387ab53c26153ea18f0437e4d25ce65a1b1658e.png +../../../_images/b1be39be0840384dec6887d3741e6245a06ade1140d68ec90e56fa51bbb92d83.png 
Maximum Validation Accuracy reached: 51.0
@@ -2104,19 +2104,19 @@

Dataloaders for Regularization

Section 1.2: L1 Regularization#

L1 Regularization (or LASSO\(^{\ddagger}\)) uses a penalty which is the sum of the absolute value of all the weights in the Deep Learning architecture, resulting in the following loss function (\(L\) is the usual Cross-Entropy loss):

-
-(59)#\[\begin{equation} +
+(59)#\[\begin{equation} L_R = L + \lambda \sum \left| w^{(r)}_{ij} \right| \end{equation}\]

where \(r\) denotes the layer, and \(ij\) the specific weight in that layer.

At a high level, L1 Regularization is similar to L2 Regularization since it leads to smaller weights (you will see the analogy in the next subsection). It results in the following weight update equation when using Stochastic Gradient Descent:

-
-(60)#\[\begin{equation} +
+(60)#\[\begin{equation} w^{(r)}_{ij}←w^{(r)}_{ij} − \eta \cdot \lambda \cdot \text{sgn}\left(w^{(r)}_{ij}\right)−\eta \frac{\partial L}{\partial w_{ij}^{(r)}} \end{equation}\]

where \(\text{sgn}(\cdot)\) is the sign function, such that

-
-(61)#\[\begin{equation} +
+(61)#\[\begin{equation} \text{sgn}(w) = \left\{ \begin{array}{ll} @@ -2191,7 +2191,7 @@

Submit your feedback -

+

Now, let’s train a classifier that uses L1 regularization. Tune the hyperparameter lambda1 such that the validation accuracy is higher than that of the unregularized model.

@@ -2255,7 +2255,7 @@

Submit your feedback -

+

@@ -2263,19 +2263,19 @@

Submit your feedback

Section 1.3: L2 / Ridge Regularization#

L2 Regularization (or Ridge), also referred to as “Weight Decay”, is widely used. It works by adding a quadratic penalty term to the Cross-Entropy Loss Function \(L\), which results in a new Loss Function \(L_R\) given by:

-
-(62)#\[\begin{equation} +
+(62)#\[\begin{equation} L_R = L + \lambda \sum \left( w^{(r)}_{ij} \right)^2 \end{equation}\]

where, again, \(r\) superscript denotes the layer, and \(ij\) the specific weight in that layer.

To get further insight into L2 Regularization, we investigate its effect on the Gradient Descent based update equations for the weight and bias parameters. Taking the derivative on both sides of the above equation, we obtain

-
-(63)#\[\begin{equation} +
+(63)#\[\begin{equation} \frac{\partial L_R}{\partial w^{(r)}_{ij}}=\frac{\partial L}{\partial w^{(r)}_{ij}} + 2\lambda w^{(r)}_{ij} \end{equation}\]

Thus the weight update rule becomes:

-
-(64)#\[\begin{equation} +
+(64)#\[\begin{equation} w^{(r)}_{ij}←w^{(r)}_{ij}−η\frac{\partial L}{\partial w^{(r)}_{ij}}−2 \eta \lambda w^{(r)}_{ij}=(1−2 \eta \lambda)w^{(r)}_{ij} − \eta \frac{\partial L}{\partial w^{(r)}_{ij}} \end{equation}\]

where \(\eta\) is the learning rate.

@@ -2343,7 +2343,7 @@

Submit your feedback -

+

Now we’ll train a classifier that uses L2 regularization. Tune the hyperparameter lambda2 such that the validation accuracy is higher than that of the unregularized model.

@@ -2407,7 +2407,7 @@

Submit your feedback -

+

Now, let’s run a model with both L1 and L2 regularization terms.

@@ -2534,7 +2534,7 @@

Section 2: Dropout#

-
+
@@ -2553,7 +2553,7 @@

Submit your feedback - +

With Dropout, we literally drop out (zero out) some neurons during training. Throughout the training, the standard dropout zeros out some fraction (usually 50%) of the nodes in each layer, and on each iteration, before calculating the subsequent layer. Randomly selecting different subsets to drop out introduces noise into the process and reduces overfitting.


@@ -2674,7 +2674,7 @@

Run to train the default network
Random seed 2021 has been set.
- +
@@ -2721,7 +2721,7 @@

Run to train the default network
Random seed 2021 has been set.

-
+

Now that we have finished the training, let’s see how the model has evolved over the training process.

Animation! (Run Me!)

@@ -2793,7 +2793,7 @@

Run to train the default networkRuntimeError: Requested MovieWriter (ffmpeg) not available -../../../_images/d964234cc1439e0aaf1deb690d9bad92d4f76256534fcab0ac85899a77c99c2c.png +../../../_images/b0a991aa1fd778f2f0ef3d4f2bb95a33c4e1819dc6126275b87c8b2891aa6fde.png

Plot the train and test losses with epoch

@@ -2819,7 +2819,7 @@

Run to train the default network -../../../_images/9cac0e245302cda4f4f277a43ee4c8b4b74eb2da2af25944be5022b4c6721177.png +../../../_images/d08f387ea927ba54e51cf3eac1e928d4cc5d6864c238f0e4664842ca7087a7fa.png

Plot the train and test losses with epoch

@@ -2845,7 +2845,7 @@

Run to train the default network -../../../_images/dd316b865a698b994ee3b13ad6dfe0de1aa3197969512a9600b4eebb980581aa.png +../../../_images/499b33296fd75f367398a8553dd918bb8947197f08ced98162d08fb3bc2387ad.png

Plot model weights with epoch

@@ -2870,7 +2870,7 @@

Run to train the default network -../../../_images/67a8d761ab6a11cdfddb806c3bb84ec6a68eb16014104c97ad3a53da8a969d84.png +../../../_images/5de67be11870e5173f10f30c2aa1331af6cd02956d72a7ad17e696a0ce5d2705.png

@@ -2895,7 +2895,7 @@

Submit your feedback - + @@ -3009,10 +3009,10 @@

Section 2.1: Dropout Implementation Caveats
Random seed 2021 has been set.
-
Random seed 2021 has been set.
+
Random seed 2021 has been set.
 

 

-../../../_images/47359cdf22e90dceb054e9bd8166287f7d88a2f652faf51fd3b3d90b464ec7de.png
+../../../_images/3f322b109c93aa7bfd89efd8694090ad39571d6ab42da715e35728eff66c27de.png
@@ -3035,7 +3035,7 @@

Submit your feedback - +

@@ -3049,7 +3049,7 @@

Section 3: Data Augmentation#

-
+
@@ -3068,7 +3068,7 @@

Submit your feedback - +

Data augmentation is often used to increase the number of training samples. Now we will explore the effects of data augmentation on regularization. Here regularization is achieved by adding noise into training data after every epoch.

PyTorch’s torchvision module provides a few built-in data augmentation techniques, which we can use on image datasets. Some of the techniques we most frequently use are:

@@ -3206,10 +3206,10 @@

Data Loader without Data Augmentation
Random seed 2021 has been set.
-
Random seed 2021 has been set.
+
Random seed 2021 has been set.
 

 

-../../../_images/c463454c7b069521e9e2351b0002bac6cd2b0228a2619239ac92dcabc5f64185.png
+../../../_images/a8766aa3dd02a279ca9ed688dde42f61fb55f03cbe1b2fc5f2c72f7a9f6880f0.png
@@ -3229,7 +3229,7 @@

Data Loader without Data Augmentation -../../../_images/ac41c39c0cd72dbce7b201b4a42c7dd3d712835703a37a4981d910f8078d5270.png +../../../_images/a2233e8f46644b2c4781d7a01db67e577fe810b6811c7b651aead58d939f5228.png

@@ -3253,7 +3253,7 @@

Submit your feedback - + @@ -3277,7 +3277,7 @@

Submit your feedback - + @@ -3291,7 +3291,7 @@

Section 4: Stochastic Gradient Descent#

-
+
@@ -3310,7 +3310,7 @@

Submit your feedback - +

@@ -3407,13 +3407,13 @@

Generating Data Loaders
Random seed 2021 has been set.
-
Random seed 2021 has been set.
+
Random seed 2021 has been set.
 

 

-
Random seed 2021 has been set.
+
Random seed 2021 has been set.
 

 

-

+

Plot Train and Validation accuracy (Run me)

@@ -3447,7 +3447,7 @@

Generating Data Loaders

-../../../_images/99db78bd1535fe12692e1f5329d742f857f7af1cfaeb27d07a71910ae1911136.png +../../../_images/9c8f994db318385a11b92995d28fee11addaa2d6c634d95c5fd115df3b530ed4.png

Plot parametric norms (Run me)

@@ -3472,7 +3472,7 @@

Generating Data Loaders
-../../../_images/3887af56402df0c0078b7262a0fc6f9b5635ae9181292ee8d6fc1f9554b57d75.png +../../../_images/2d59c9a8b535a107b091da62bd4a34913660601ebd751fd91c267099443fedb9.png

In the model above, we observe something different from what we expected. Why do you think this is happening?

@@ -3487,7 +3487,7 @@

Section 5: Hyperparameter Tuning#

-
+

@@ -3506,7 +3506,7 @@

Submit your feedback - +

Hyperparameter tuning is often tricky and time-consuming, and it is a vital part of training any Deep Learning model to give good generalization. There are a few techniques that we can use to guide us during the search.

@@ -3565,7 +3565,7 @@

Bonus: Adversarial Attacks#

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+
@@ -3584,7 +3584,7 @@

Submit your feedback - +

Designing perturbations to the input data to trick a machine learning model is called an “adversarial attack”. These attacks are an inevitable consequence of learning in high dimensional space using complex decision boundaries. Depending on the application, these attacks can be very dangerous.


diff --git a/tutorials/W2D2_ConvnetsAndDlThinking/student/W2D2_BonusLecture.html b/tutorials/W2D2_ConvnetsAndDlThinking/student/W2D2_BonusLecture.html index b9f7a4936..5e334ef04 100644 --- a/tutorials/W2D2_ConvnetsAndDlThinking/student/W2D2_BonusLecture.html +++ b/tutorials/W2D2_ConvnetsAndDlThinking/student/W2D2_BonusLecture.html @@ -58,7 +58,7 @@ const thebe_selector_output = ".output, .cell_output" - + @@ -1453,11 +1453,11 @@

Install and import feedback gadget#

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+
-
+

@@ -1476,7 +1476,7 @@

Submit your feedback
-
+

diff --git a/tutorials/W2D2_ConvnetsAndDlThinking/student/W2D2_Tutorial1.html b/tutorials/W2D2_ConvnetsAndDlThinking/student/W2D2_Tutorial1.html index b8175d084..7bd804ffc 100644 --- a/tutorials/W2D2_ConvnetsAndDlThinking/student/W2D2_Tutorial1.html +++ b/tutorials/W2D2_ConvnetsAndDlThinking/student/W2D2_Tutorial1.html @@ -50,7 +50,7 @@ - + @@ -1343,7 +1343,7 @@

Tutorial Objectives
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@@ -1886,7 +1886,7 @@

Section 0: Recap the Experience from Last Week#

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+
@@ -1905,7 +1905,7 @@

Submit your feedback
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+

@@ -1934,7 +1934,7 @@

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Coming Up

The rest of these lectures focus on another way to reduce parameters: weight-sharing. Weight-sharing is based on the idea that some sets of weights can be used at multiple points in a network. We will focus primarily on CNNs today, where the weight-sharing is across the 2D space of an image. This weight-sharing technique (across space) can reduce the number of parameters and increase a network’s ability to generalize. For completeness, a similar approach is the Recurrent Neural Networks (RNNs), which share parameters across time, but we will not dive into this in this tutorial.

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Section 1: Neuroscience motivation, General CNN structure#

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Section 2: Convolutions and Edge Detection#

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Before jumping into coding exercises, take a moment to look at this animation that steps through the process of convolution.

Recall from the video that convolution involves sliding the kernel across the image, taking the element-wise product, and adding those products together.

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Definitional Note

Coding Exercise 2.1: Convolution of a Simple Kernel#

At its core, convolution is just repeatedly multiplying a matrix, known as a kernel or filter, with some other, larger matrix (in our case the pixels of an image). Consider the below image and kernel:

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Think! 2.2.1: Edge Detection#

One of the simpler tasks performed by a convolutional layer is edge detection; that is, finding a place in the image where there is a large and abrupt change in color. Edge-detecting filters are usually learned by the first layers in a CNN. Observe the following simple kernel and discuss whether this will detect vertical edges (where the trace of the edge is vertical; i.e. there is a boundary between left and right), or whether it will detect horizontal edges (where the trace of the edge is horizontal; i.e., there is a boundary between top and bottom).

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Consider the image below, which has a black vertical stripe with white on the side. This is like a very zoomed-in vertical edge within an image!

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As you can see, this kernel detects vertical edges (the black stripe corresponds to a highly positive result, while the white stripe corresponds to a highly negative result. However, to display the image, all the pixels are normalized between 0=black and 1=white).

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Dataset/DataLoader Functions (Run me!) -../../../_images/3f6d3f33c0fcdc25fc5308b9f32b8a7a8eb8f1f22b32fda64c3085ee693dc64a.png +../../../_images/a0a6d8ea4435c866e16fc976e9f0b970931df02213bcbd9d015aab5767801928.png @@ -2893,7 +2893,7 @@

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Section 3.1: Multiple Filters -
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We apply the filters to the images.

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Submit your feedback -../../../_images/bcbeed3a7ea37656e384c7107d2396b9f3c947be9eeacdee26aaff7ffcc33cb3.png +../../../_images/e997693cb4498edd9c19d71a3c1b891785373cfa02e18c47b70ced7e512bf630.png

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Section 3.2: ReLU after convolutions -../../../_images/0c9ad0f380bd8fd67092f23f2f95e539674e45608d4c52ccf0c01d12a475829f.png +../../../_images/70d03ba61a7131910eebb4ce352b35f6d93a1779fc0bcaabe5e333c7dac971b3.png

Discuss with your pod how the ReLU activations help strengthen the features necessary to detect an \(X\).

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Section 3.3: PoolingVideo 4: Pooling#

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Like convolutional layers, pooling layers have fixed-shape windows (pooling windows) that are systematically applied to the input. As with filters, we can change the shape of the window and the size of the stride. And, just like with filters, every time we apply a pooling operation we produce a single output.

Pooling performs a kind of information compression that provides summary statistics for a neighborhood of the input.

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Section 4: Putting it all together#

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Interactive Demo 4.1: Number of Parameters - +

The difference in parameters is huge, and it continues to increase as the input image size increases. Larger images require that the linear layer use a matrix that can be directly multiplied with the input pixels.


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Video 6: Implement your own CNN#

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Note: We are using a softmax function here which converts a real value to a value between 0 and 1, which can be interpreted as a probability.

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Bonus 1: Write your own training loop revisited#

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Bonus 1.1: Understand the Dataset#

The dataset we are going to use for this task is called Fashion-MNIST. It consists of a training set of 60,000 examples and a test set of 10,000 examples. We further divide the test set into a validation set and a test set (8,000 and 2,000, respectively). Each example is a \(28 \times 28\) gray scale image, associated with a label from 10 classes. Following are the labels of the dataset:


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Loading Fashion-MNIST Data -../../../_images/ee2a62c7eda5e489def7f90d1dec3444d24b7a2046768887656da91b19675d09.png +../../../_images/cf8d4c96f5b4bbc479bbced72f41a91d985d373b1fcd94daf6f08f70a09a97f5.png

Take a minute with your pod and talk about which classes you think would be most confusable. How hard will it be to differentiate t-shirt/tops from shirts?

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Loading Fashion-MNIST Data#

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Load a sample dataset (EMNIST) -
Test accuracy: 93.07%
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Test accuracy: 93.07%
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Interactive Demo Bonus 2.1: Dropout exploration - +

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Submit your feedback - + diff --git a/tutorials/W2D2_ConvnetsAndDlThinking/student/W2D2_Tutorial2.html b/tutorials/W2D2_ConvnetsAndDlThinking/student/W2D2_Tutorial2.html index 3b9494977..d501ae40a 100644 --- a/tutorials/W2D2_ConvnetsAndDlThinking/student/W2D2_Tutorial2.html +++ b/tutorials/W2D2_ConvnetsAndDlThinking/student/W2D2_Tutorial2.html @@ -50,7 +50,7 @@ - + @@ -976,7 +976,7 @@

Tutorial Objectives
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Section 1: Intro to Deep Learning Thinking#

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This tutorial is a bit different from others - there will be no coding! Instead you will watch a series of vignettes about various scenarios where you want to use a neural network. This tutorial will focus on cost functions, a tutorial you will see later in the course will be similar but focused on designing architectures.

Each section below will start with a vignette where either Lyle or Konrad is trying to figure out how to set up a neural network for a specific problem. Try to think of questions you want to ask them as you watch, then pay attention to what questions Lyle and Konrad are asking. Were they what you would have asked? How do their questions help quickly clarify the situation?

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Section 2: Cost function for neurons#

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Video 3: Spiking Neuron Predictions Set-up#

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Konrad, a neuroscientist, wants to predict what neurons in someone’s motor cortex are doing while they are riding a motorcycle.

Upon discussion with Lyle, it emerges that we have data on 12 parameters of motorcycle riding, including acceleration, angle, braking, degrees of leaning. These inputs are fairly smooth over time, the angle of the motorcycle typically does not change much in 100 ms for example.

We also have recorded data on the timing of spikes of \(N\) neurons in motor cortex. The underlying firing rate is smooth but every millisecond spikes are random and independent. This means we can assume that the number of spikes in a short interval can be modeled using a Poisson distribution with an underlying firing rate for that interval \(\lambda\).

For neuron \(i\), the probability of seeing \(k_{i}\) spikes in some interval given an underlying firing rate \(\lambda_{i}\) is:

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-(68)#\[\begin{equation} +
+(68)#\[\begin{equation} \mathcal{f(k_{i}:λ_{i})} = \mathcal{Pr(X=k_{i})} = \frac {\lambda_{i}^{k_{i}}e^{-\lambda_{i}}}{k_{i}!} \end{equation}\]

So this poisson distribution may be relevant if we want to, in a way, have a good model for the spiking of neurons.

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Think! 1: Designing a cost function to predict neural activitiesClick here for the solution

First, we will convert our spike timing data to the number of spikes per time bin for time bins of size 50 ms. This gives us \(k_{i,t}\) for every neuron \(i\) and time bin \(t\).

We are assuming a Poisson distribution for our spiking. That means that we get the probability of seeing spike count \(k_{i, t}\) given underlying firing rate \(\lambda_{i, t}\) using this equation:

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-(69)#\[\begin{equation} +
+(69)#\[\begin{equation} \mathcal{f(k_{i,t}:\lambda_{i,t})} = \mathcal{Pr}(X=k_{i,t}) = \frac {\lambda_{i,t}^{k_{i,t}}e^{-\lambda_{i,t}}}{k_{i,t}!} \end{equation}\]

That seems a pretty good thing to optimize to make our predictions as good as possible! We want a high probability of seeing the actual spike count we recorded given the neural network prediction of the underlying firing rate.

We will make this negative later so we have an equation that we want to minimize rather than maximize, so we can use all our normal tricks for minimization (instead of maximization). First though, let’s scale up to include all our neurons and time bins.

We can treat each time bin as independent because, while the underlying probability of firing changes slowly, every milisecond spiking is random and independent. From probability, we know that we can compute the probability of a set of independent events (all the spike counts) by multiplying the probabilities of each event. So the probability of seeing all of our data given the neural network predictions is all of our probabilities of \(k_{i,t}\) multiplied together:

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+(70)#\[\begin{align} \mathcal{Pr}(\text{all_data}) &= \prod_{i=1}^{N}\prod_{t=1}^\top \mathcal{Pr}(X=k_{i,t})\\ &= \prod_{i=1}^{N}\prod_{t=1}^\top \frac {\lambda_{i,t}^{k_{i,t}}e^{-\lambda_{i,t}}}{k_{i,t}!} \end{align}\]

This is also known as our likelihood!

We usually use the log likelihood instead of the likelihood when minimizing or maximizing for numerical computation reasons. W We can convert the above equation to log likelihood:

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+(71)#\[\begin{align} \text{log likelihood} &= \sum_{i=1}^N\sum_{t=1}^\top \text{log}(\mathcal{Pr}(X=k_{i,t}) \\ &= \sum_{i=1}^N\sum_{t=1}^\top k_{i,t} \text{log}(\lambda_{i,t}) - \lambda_{i,t} - \text{log}(k_{i,t}!) \end{align}\]

And last but not least, we want to make it negative so we can minimize instead of maximize:

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+(72)#\[\begin{equation} \text{negative log likelihood} = \sum_{i=1}^N\sum_{t=1}^\top - k_{i,t} \text{log}(\lambda_{i,t}) + \lambda_{i,t} + \text{log}(k_{i,t}!) \end{equation}\]
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Video 4: Spiking Neurons Wrap-up#

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Check out the papers mentioned in the above video:

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Section 3: How can an ANN know its uncertainty#

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Video 6: ANN Uncertainty Set-up#

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Lyle wants to build an artificial neural network that has a measure of its own uncertainty about it’s predictions. He wants the neural network to give a prediction/estimate and an uncertainty, or standard deviation, measurement on it.

Let’s say Lyle wants to estimate the location of an atom in a chemical molecule based on various inputs. He wants to have the estimate of the location and an estimate of the variance. We don’t train neural networks on one data point at a time though - he wants a cost function that takes in N data points (input and atom location pairings).

We think we may be able to use a Gaussian distribution to help Lyle here:

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-(73)#\[\begin{equation} +
+(73)#\[\begin{equation} g(x) = \frac{1}{\sigma\sqrt{2\pi}} \text{exp} \left( -\frac{1}{2}\frac{(x-\mu)^2}{\sigma^2} \right) \end{equation}\]

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Think! 2: Designing a cost function so we measure uncertainty\(i\), the neural network predicts the mean (\(\mu_i\)) and standard deviation (\(\sigma_i\)) of the location given the inputs. We can then compute the probability of seeing the actual recorded location (\(x_i\)) given these predictions:

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+(74)#\[\begin{equation} g(x) = \frac{1}{\sigma\sqrt{2\pi}} \text{exp}\left( -\frac{1}{2}\frac{(x_i-\mu_i)^2}{\sigma_i^2} \right) \end{equation}\]

The location of the atom is independent in each data point so we can get the overall likelihood by multiplying the probabilities for the individual data points.

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+(75)#\[\begin{equation} \text{likelihood} = \prod_{i=1}^N\frac{1}{\sigma\sqrt{2\pi}} \text{exp}\left( -\frac{1}{2}\frac{(x_i-\mu_i)^2}{\sigma_i^2} \right) \end{equation}\]

And, as before, we want to take the log of this for numerical reasons and convert to negative log likelihood:

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+(76)#\[\begin{equation} \text{negative log likelihood} = \sum_{i=1}^N \text{log} \left( \frac{1}{\sigma\sqrt{2\pi}} \text{exp}\left( -\frac{1}{2}\frac{(x_i-\mu_i)^2}{\sigma_i^2} \right) \right) \end{equation}\]

Changing the parameters of the neural network so it predicts \(\mu_i\) and \(\sigma_i\) that minimize this equation will give us (hopefully fairly accurate) predictions of the location and the network uncertainty about the location!

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Video 7: ANN Uncertainty Wrap-up#

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Check out the papers mentioned in the above video:

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Section 4: Embedding faces#

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Video 9: Embedding Faces Set-up#

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Konrad needs help recognizing faces. He wants to build a network that embeds photos of faces so that photos of the same person are nearby in the embedding space and photos of different people are far in the embedding space. We can’t just use pixel space because the pixels will be very different between a photo of someone straight on vs. from their side!

We will use a neural network to go from the pixels of each image to an embedding space. Let’s say you have a convolutional neural network with m units in the last layer. If you feed a face photo \(i\) through the CNN, the activities of the units in the last layer form an \(m\) dimensional vector \(\bar{y}_i\) - this is an embedding of that face photo in \(m\) dimensional space.

We think we might be able to incorporate Euclidean distance to help us here. The Euclidean distance between two vectors is:

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-(77)#\[\begin{equation} +
+(77)#\[\begin{equation} d(\bar{y}_i, \bar{y}_j) = \sqrt{\sum_{c=1}^m(\bar{y}_{i_c} - \bar{y}_{j_c})^2} \end{equation}\]

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Think! 3: Designing a cost function for face embedding Click here for the solution

We want the same faces to have similar embeddings. Let’s say we have one photo of Lyle \(a\) and another photo of Lyle \(p\). We want the embeddings of those photos to be very similar: we want the Euclidean distance between \(\bar{y}_a\) and \(\bar{y}_p\) (the activitys of the last layer of the CNN when photo \(a\) and \(p\) are fed through) to be small.

So one possible cost function is:

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+(78)#\[\begin{equation} \text{Cost function} = d(\bar{y}_a, \bar{y}_p) \end{equation}\]

Imagine if we just feed in pairs of the same face and minimize that though. There would be no motivation to ever have different embeddings, we would be only minimizing the distance between embeddings. If the CNN was smart, it would just have the same embedding for every single photo - then the cost function would equal 0!

This is clearly not what we want. We want to motivate the CNN to have similar embeddings only when the faces are the same. This means we need to also train it to maximize distance when the faces are different.

We could choose another two photos of different people and maximize that distance but then there’s no relation to the embeddings we’ve already established of the two photos of Lyle. Instead, we will add one more photo to the mix: a photo of Konrad \(n\). We want the distance of this photo to be far from our original photos of Lyle \(a\) and \(p\). So we want the distance between \(a\) and \(p\) to be small and the distance between \(a\) and \(n\) for example to be large:

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+(79)#\[\begin{equation} \text{Cost function} = d(\bar{y}_a, \bar{y}_p) - d(\bar{y}_a, \bar{y}_n) \end{equation}\]

We could compare \(n\) to both \(a\) and \(p\):

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+(80)#\[\begin{equation} \text{Cost function} = d(\bar{y}_a, \bar{y}_p) - d(\bar{y}_a, \bar{y}_n) - d(\bar{y}_p, \bar{y}_n) \end{equation}\]

But then the cost function is a bit unbalanced, there are two dissimiliarty terms and they might dominate (so achieving the similarity is less important). So let’s go with just including one dissimilarity term.

This is an established cost function - triplet loss! We chose the subscripts \(a\), \(p\), and \(n\) for a reason: we have an anchor image, a positive image (the same person’s face as the anchor) and a negative image (a different person’s face as the anchor). We can then sum over N data points where each data point is a set of three images:

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+(81)#\[\begin{equation} \text{Cost function} = \sum_{i=1}^N [d(\bar{y}_{a, i}, \bar{y}_{p, i}) - d(\bar{y}_{a, i}, \bar{y}_{n, i})] \end{equation}\]

There’s one little addition in triplet loss. Instead of just using the above cost function, researchers add a constant \(\alpha\) and then make the cost function 0 if it becomes negative. Why do you think they do this?

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+(82)#\[\begin{equation} \text{Cost function} = \text{max} \left( \sum_{i=1}^N \left[ d(\bar{y}_{a, i}, \bar{y}_{p, i}) - d(\bar{y}_{a, i}, \bar{y}_{n, i}) + \alpha \right], 0 \right) \end{equation}\]
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Video 10: Embedding Faces Wrap-up#

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Check out the papers mentioned in the above video:

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Section 1: Modern CNNs and Transfer Learning#

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Images are high dimensional. That is to say that image_length * image_width * image_channels is a big number, and multiplying that big number by a normal sized fully-connected layer leads to a ton of parameters to learn. Yesterday, we learned about convolutional neural networks, one way of working around high dimensionality in images and other domains.

The widget below (i.e., Interactive Demo 1) calculates the parameters required for a single convolutional or fully connected layer that operates on an image of a certain height and width.

Recall that, the number of parameters of a convolutional layer \(l\) are calculated as:

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+(83)#\[\begin{equation} \text{num_of_params}_l = \left[ \left( H \times W \times K_{l-1} \right) + 1 \right] \times K_l \end{equation}\]

where \(H\) denotes the shape of the height of the filter, \(W\) the shape of the width of the filter, and \(K_l\) denotes the number of the filters in the \(l\)-th layer. The added \(1\) is because of the bias term for each filter.

While a fully connected layer contains:

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-(84)#\[\begin{equation} +
+(84)#\[\begin{equation} \text{num_of_params}_l = \left[ \left( N_{l-1} \times N_l \right) + 1 \times N_l \right] \end{equation}\]

where \(N_l\) denotes the number of nodes in the \(l\)-th layer.

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Parameter Calculator
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Section 2: The History of Convnets#

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Section 3: Big and Deep Convnets#

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Section 3.2: What does AlexNet learn? -../../../_images/becac0473624f04086b4f6c3661bc8141ba6e6994a2cc3613887f5df90e76190.png +../../../_images/a55aa677043fbfe697ed97615dbf553411bd6fa5e5e891cc601594eb0d21e843.png
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Interactive Demo 3.2: What does AlexNet see? - +

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Section 4: Convnets After AlexNet#

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In this section we’ll be working with a state of the art CNN model called ResNet. ResNet has two particularly interesting features. First, it uses skip connections to avoid the vanishing gradient problem. Second, each block (collection of layers) in a ResNet can be treated as learning a residual function.

Mathematically, a neural network can be thought of as a series of operations that maps an input (like an image of a dog) to an output (like the label “dog”). In math-speak a mapping from an input to an output is called a function. Neural networks are a flexible way of expressing that function.

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Prepare Imagenette Data
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Section 5: Inception + ResNeXt#

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Parameter Calculator - +

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Now we want to look at the number of parameters.