Repository: Freie Universität Berlin, Math Department

Stochastic gradient descent and fast relaxation to thermodynamic equilibrium: a stochastic control approach

Breiten, T. and Hartmann, Carsten and Neureither, Lara and Sharma, Upanshu (2021) Stochastic gradient descent and fast relaxation to thermodynamic equilibrium: a stochastic control approach. Journal of Mathematical Physics, 62 (12). pp. 1-19.

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ABSTRACT We study the convergence to equilibrium of an underdamped Langevin equation that is controlled by a linear feedback force. Specifically, we are interested in sampling the possibly multimodal invariant probability distribution of a Langevin system at small noise (or low temperature), for which the dynamics can easily get trapped inside metastable subsets of the phase space. We follow Chen et al. [J. Math. Phys. 56, 113302 (2015)] and consider a Langevin equation that is simulated at a high temperature, with the control playing the role of a friction that balances the additional noise so as to restore the original invariant measure at a lower temperature. We discuss different limits as the temperature ratio goes to infinity and prove convergence to a limit dynamics. It turns out that, depending on whether the lower (“target”) or the higher (“simulation”) temperature is fixed, the controlled dynamics converges either to the overdamped Langevin equation or to a deterministic gradient flow. This implies that (a) the ergodic limit and the large temperature separation limit do not commute in general and that (b) it is not possible to accelerate the speed of convergence to the ergodic limit by making the temperature separation larger and larger. We discuss the implications of these observations from the perspective of stochastic optimization algorithms and enhanced sampling schemes in molecular dynamics.

Item Type:Article
Subjects:Mathematical and Computer Sciences > Mathematics > Applied Mathematics
Divisions:Department of Mathematics and Computer Science > Institute of Mathematics
ID Code:2803
Deposited By: Monika Drueck
Deposited On:14 Mar 2022 14:51
Last Modified:14 Mar 2022 14:51

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