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

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 dy...

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Bibliographic Details
Published in:Journal of mathematical physics 2021-12, Vol.62 (12)
Main Authors: Breiten, Tobias, Hartmann, Carsten, Neureither, Lara, Sharma, Upanshu
Format: Article
Language:English
Subjects:
Algorithms
Convergence
Ergodic processes
Gradient flow
High temperature
Invariants
Low temperature
Microbalances
Molecular dynamics
Optimal control
Optimization
Physics
Sampling
Separation
Stochastic processes
Temperature ratio
Thermodynamic equilibrium
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Summary: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.
ISSN:0022-2488
1089-7658
DOI:10.1063/5.0051796