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Hyperparameters

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Eric Hermes edited this page Sep 10, 2019 · 2 revisions

Hyperparameter overview

Sella has a number of hyperparameters that can potentially have a large impact on performance. For more information, read our publication located on ChemRxiv.

from sella import Sella
...
dyn = Sella(myatoms,
            ...
            eta=1e-4,        # Finite difference step size
            update='jd0',    # Hessian diagonalization method
            gamma=0.4,       # Convergence criterion for iterative diagonalization
            delta0=1.3e-3,   # Initial trust radius
            rho_inc=1.035,   # Threshold for increasing trust radius
            rho_dec=5.0,     # Threshold for decreasing trust radius
            sigma_inc=1.15,  # Trust radius increase factor
            sigma_dec=0.65)  # Trust radius decrease factor

Iterative diagonalization

For saddle point refinement, it is necessary to identify the leftmost eigenvector of the true Hessian matrix. Sella does this using an iterative diagonalization method.

eta

eta controls the magnitude of the finite difference step that is used by the iterative diagonalization routine. By default, this value is 1e-4 Angstrom.

If eta is chosen to be too large, then third- or higher-order effects will introduce error into the finite difference curvature estimates. This may result in the iterative diagonalization failing to converge, which will increase the number of gradient evaluations needed drastically.

If eta is chosen to be too small, then the true change in the gradients may be smaller than the intrinsic error present in the gradients. This is particularly true when using an electronic structure theory method like DFT that must be solved self-consistently. When the gradients are analytical and exact, e.g. when using a molecular dynamics force field, eta can be reduced.

method

method specifies which iterative diagonalization method used by Sella. Valid values are lanczos, gd (Generalized Davidson), and jd0 (Jacobi-Davidson without inner iterations). The default is jd0. Changing this default is not advised, as jd0 is expected to have superior performance in all cases.

gamma

gamma specifies the convergence criteria for the iterative eigensolver. The approximate eigenvector (Ritz vector) is considered converged when:

equation

where r(j) is the residual vector of the j th Ritz vector and θ(j) is the corresponding Ritz value.

Smaller values of gamma will result in more iterative diagonalization steps, but will also improve the quality of the approximate Hessian, which will reduce the number of geometry steps needed to converge to the saddle point.

Trust radius

Sella uses a trust radius method to restrict step sizes.

delta0

delta0 specifies the initial trust radius in units of Angstrom per degree of freedom. For example, a system of 5 atoms with no constraints and without periodic boundary conditions (see the Constraints page) will have 3 * 5 - 6 == 9 degrees of freedom, so the initial trust radius will be 1.3e-3 * 9 == 1.17e-2 Angstrom.

rho_inc, sigma_inc, rho_dec, sigma_dec

After every geometry step, Sella evaluates the ratio of the true change in energy and the predicted change in energy:

equation

For the definition of symbols, see our paper linked at the top of this page.

When ρk is between rho_inc**-1 and rho_inc, the trust radius is set to the magnitude of the last step multiplied by sigma_inc; if this is smaller than the current trust radius, then the trust radius is not changed.

When ρk is less than rho_dec**-1 or greater than rho_dec, the trust radius is set to the magnitude of the last step multiplied by sigma_dec; if this is less than eta (see above), then the trust radius is instead set to eta.

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