[Paper Review] State-Dependent Temperature Control for Langevin Diffusions

Xuefeng Gao, Zuo Quan Xu+1 more|arXiv (Cornell University)|Nov 15, 2020

Markov Chains and Monte Carlo Methods4 citations

TL;DR

This paper proposes a state-dependent temperature control scheme for Langevin diffusions in non-convex optimization using stochastic relaxed control and entropy regularization. By solving a Hamilton-Jacobi-Bellman (HJB) equation, it derives a state-dependent, truncated exponential distribution for sampling temperatures, enabling adaptive exploration that outperforms constant temperature, power-law, and replica exchange methods in a 1D benchmark.

ABSTRACT

We study the temperature control problem for Langevin diffusions in the context of non-convex optimization. The classical optimal control of such a problem is of the bang-bang type, which is overly sensitive to errors. A remedy is to allow the diffusions to explore other temperature values and hence smooth out the bang-bang control. We accomplish this by a stochastic relaxed control formulation incorporating randomization of the temperature control and regularizing its entropy. We derive a state-dependent, truncated exponential distribution, which can be used to sample temperatures in a Langevin algorithm, in terms of the solution to an HJB partial differential equation. We carry out a numerical experiment on a one-dimensional baseline example, in which the HJB equation can be easily solved, to compare the performance of the algorithm with three other available algorithms in search of a global optimum.

Motivation & Objective

• To address the limitations of classical bang-bang temperature control in Langevin diffusions, which is overly sensitive to errors and lacks adaptability.
• To develop an endogenous, state-dependent temperature schedule that dynamically adjusts exploration based on current position in the state space.
• To provide a theoretically grounded framework for smoothing rigid optimal control policies in non-convex optimization using stochastic relaxed control.
• To enable more efficient global optimization by reducing time spent in local minima while concentrating near global minima when appropriate.

Proposed method

• Formulates the temperature control problem as a stochastic relaxed control problem with entropy regularization to smooth out classical bang-bang controls.
• Derives a state-dependent temperature distribution using the solution to a Hamilton-Jacobi-Bellman (HJB) partial differential equation.
• Introduces a truncated exponential distribution for temperature sampling, parameterized by the HJB solution, to enable adaptive exploration.
• Uses the HJB solution to define a state-dependent noise variance, where higher temperatures are applied in regions far from global minima (e.g., near local minima or saddle points).
• Solves the HJB equation numerically in a 1D example to obtain the temperature policy, which is then embedded into a Langevin algorithm.
• Employs a relaxation technique via randomization and entropy regularization to make the control policy robust and interpretable.

Experimental results

Research questions

• RQ1 How can temperature control in Langevin diffusions be made endogenous and state-dependent to improve global optimization in non-convex landscapes?
• RQ2 What is the theoretical foundation for smoothing rigid bang-bang temperature controls using stochastic relaxed control and entropy regularization?
• RQ3 Can a state-dependent temperature policy derived from the HJB equation outperform standard methods like constant temperature, power-law decay, and replica exchange?
• RQ4 How does the structure of the HJB solution influence the shape of the state-dependent temperature function and its effectiveness in escaping local minima?

Key findings

• The proposed algorithm with state-dependent temperature sampling outperforms three baselines—constant temperature, power-law decay, and replica exchange—in locating the global minimum within 500 iterations on a 1D benchmark.
• The HJB solution $ v(x) $ grows rapidly near the global minimum at $ x = 4 $, indicating reduced need for exploration and thus lower temperature in that region.
• The second derivative $ v''(x) $ exhibits a sharp spike at $ x = -2 $, which translates into a prominent kink in the temperature function, reflecting increased exploration needs near a local minimum.
• The state-dependent temperature function $ h^2/2 $ is close to zero for $ x > 3 $, indicating minimal noise (low temperature) near the global minimum, while remaining high (up to 500) elsewhere to enable escape from traps.
• The algorithm demonstrates intelligent behavior: it uses high temperature to escape local minima and saddle points, and low temperature to refine solutions near the global minimum.
• Despite higher computational cost due to solving a nonlinear ODE, the method provides a theoretically grounded, interpretable, and adaptive temperature control mechanism for Langevin diffusions.