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State-Dependent Temperature Control for Langevin Diffusions

Xuefeng Gao, Zuo Quan Xu, Xun Yu Zhou

arXiv:2011.07456v3math.OCcs.LG

TL;DR

Classical optimal temperature control for Langevin optimization is bang-bang and sensitive to errors. The paper instead randomizes temperature through an entropy-regularized relaxed-control formulation, deriving a state-dependent truncated exponential distribution. In a one-dimensional example, its algorithm finds the global minimizer faster than three alternatives, while high-dimensional use remains constrained by PDE-solving difficulty.

  • Problem

    Classical optimal temperature control is bang-bang and overly sensitive to errors, motivating a smoother state-dependent temperature strategy for non-convex optimization.

    Source: §2.2 · p. 5

  • Method

    The paper uses entropy-regularized stochastic relaxed control to derive a state-dependent truncated exponential distribution for sampling Langevin temperatures.

    Source: §1 · p. 3, §2.3 · p. 6

  • Results

    In the one-dimensional example, the state-dependent-noise Langevin algorithm finds the global minimizer faster than constant-temperature, power-law-decay, and replica-exchange algorithms.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The framework connects non-convex optimization to an HJB PDE and indicates that the control uses global information about the objective function.

    Source: §1 · p. 3

    The authors do not advocate high-dimensional algorithmic implementation before the curse of dimensionality for PDEs is resolved.

    Source: §1 · p. 4

Abstract

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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.

1. Introduction.

The paper develops a theoretical framework for state-dependent temperature control in Langevin diffusions, using stochastic relaxed controls and entropy regularization to smooth rigid bang-bang control. It derives a state-dependent truncated exponential control and demonstrates faster global-minimizer search than three alternatives in a one-dimensional example, while cautioning against high-dimensional implementation before PDE-solving difficulties are addressed.

1. Introduction.

  • Langevin diffusion adds Gaussian noise to gradient descent, with temperature controlling exploration; its continuous-time formulation offers tools from stochastic calculus, control, and PDEs.The temperature process can be stochastic, unlike the sequence of temperatures in the discrete algorithm.Source: §1 · p. 1
  • Existing temperature schedules are typically constant or prescribed functions of time, while state-dependent temperature can vary with the current iterate.The paper motivates higher temperatures for escaping deeper local minima and lower temperatures for staying near good solutions.Source: §1 · p. 2
  • The classical optimal control is bang-bang, switching between temperature extremes, so errors can select the wrong extreme; randomized controls and entropy regularization smooth this strategy.The paper adopts exploratory control to allow intermediate temperature values and encourage randomization.Source: §1 · pp. 2–3
  • Entropy-regularized exploration is used here to escape optimization traps, rather than to address repeated learning in unknown environments as in the cited RL formulation.The paper notes that f is assumed known.Source: §1 · p. 3
  • The optimal exploratory control is a truncated exponential distribution over the prespecified temperature range, with a state-dependent parameter determined by an elliptic PDE or, in one dimension, an ODE.The authors describe it as a continuous version of Boltzmann exploration, replacing the Q-function with the generalized Hamiltonian.Source: §1 · p. 3
  • In the one-dimensional experiment, the state-dependent-noise Langevin algorithm finds the global minimizer faster than constant-temperature, power-law-decay, and replica-exchange methods.The authors frame the result as an algorithmic implication for cases where the HJB equation can be solved easily.Source: §1 · p. 3
  • The paper’s main contribution is theoretical: it connects non-convex optimization to an HJB PDE requiring global information about f, unlike the local information used by most existing simulated-annealing algorithms.The authors do not advocate high-dimensional implementation before the PDE curse of dimensionality is resolved.Source: §1 · pp. 3–4

2. Problem Background and Formulation.

The formulation controls Langevin temperature to minimize expected discounted function values, beginning with classical controls and then relaxing them to distributions. The classical HJB yields a rigid bang-bang policy; entropy-regularized distributional controls permit randomized temperatures over a bounded range.

2.1. Non-convex optimization and Langevin algorithm.

  • The Langevin algorithm discretizes an overdamped diffusion, adding Gaussian noise to gradient updates with a temperature sequence that typically cools over time.The continuous process uses an adapted, nonnegative stochastic temperature.Source: §2.1 · p. 4
  • Performance is measured by expected discounted function values along the trajectory; the discount factor downweights later values and effectively sets the computational budget.A small cost indicates lower average function values and an iterate close to the global optimum at the budgeted stopping time.Source: §2.1 · pp. 4–5
  • The classical temperature control is constrained to an adapted process in [a, 1], with the upper bound representing a user-tuned initial temperature.The upper bound should be large enough to overcome potential barriers and avoid early trapping in a poor local minimum.Source: §2.1 · p. 5

2.2. Solving problem (2.5) classically.

  • Dynamic programming gives an HJB equation whose linear dependence on temperature yields a bang-bang policy: choose the upper or lower bound according to the sign of the value function’s curvature.The authors regard the policy as too rigid for practice because computational errors can make the extreme choice undesirable.Source: §2.2 · p. 5

2.3. Relaxed control and entropy regularization.

  • The relaxed formulation replaces a single temperature control with a distribution over the allowed temperature range, from which temperatures can be sampled.The state evolves according to the relaxed-control SDE.Source: §2.2 · p. 6, §2.3 · p. 6
  • Entropy regularization rewards randomized temperature distributions, while λ sets the trade-off between optimization and randomization.The paper distinguishes λ, the higher-level regularization temperature, from β, the local Langevin noise temperature.Source: §2.3 · pp. 6–7
  • Admissible distributional controls are defined weakly, with measurability and solution conditions ensuring the performance measure is well-defined.The formulation includes the filtered probability space and Brownian motion as part of the control.Source: §2.3 · p. 7

3. Optimal Temperature Control.

Entropy-regularized stochastic control yields a state-dependent temperature distribution for Langevin diffusion, with its optimality established under stated regularity and admissibility conditions. The resulting feedback dynamics can be discretized into a state-dependent-noise Langevin algorithm.

3.1. A formal derivation.

  • The optimal temperature control is a truncated exponential distribution over the admissible range, with a state-dependent parameter derived from the HJB solution.The parameter is tr(vxx(x))/λ; it need not be positive, nor does tr(vxx(x)) need to be positive.Source: §3.1 · p. 8
  • Entropy regularization yields a continuous-time theoretical underpinning for Boltzmann exploration, using the generalized Hamiltonian to rank controls in the minimization problem.The negative sign in the exponential weighting reflects minimization rather than reward maximization.Source: §3.1 · p. 8

3.2. Admissibility and optimality.

  • The formal verification result establishes that a C2 HJB solution of polynomial growth equals the value function and induces an optimal distributional control.The stated assumptions include a temperature range [a, 1] with a > 0 and a differentiable objective with bounded gradient.Source: §3.2 · p. 9
  • The HJB solution is unique among C2 functions of polynomial growth, while the viscosity-solution route for non-C2 value functions is not pursued.The uniqueness result uses a boundary condition associated with the HJB equation.Source: §3.2 · p. 12
  • Discretizing the optimal state-process SDE yields a Langevin algorithm with state-dependent noise; in one dimension, the HJB equation is an ODE.The paper identifies Euler–Maruyama as one possible discretization.Source: §4 · p. 12

4. Algorithm and Numerical Example.

In the one-dimensional double-well experiment, the state-dependent Langevin algorithm reached the global minimum faster than three alternatives, while requiring a nonlinear ODE solve. Its temperature was mostly high away from the global minimum and close to zero near it; the authors note that higher-dimensional HJB equations may be difficult to solve numerically.

4. Algorithm and Numerical Example.

  • The experiment used an asymmetric double-well function with local minima at −3 and 4, where 4 is the unique global minimum, and initialized all algorithms at −3.The allowed temperature upper bound was 500, and the expected function value was estimated from 500 samples; the comparison ran for up to 1,000 iterations.Source: §4 · pp. 13–14
  • The state-dependent Langevin algorithm found the global minimum faster than the other three methods, while replica exchange also found it quickly but required two algorithms.The proposed method was computationally more expensive than the alternatives because it required solving a nonlinear ODE.Source: §4 · pp. 14–15, Figure 2 · p. 15
  • Temperature was close to zero for x > 3 and mostly large elsewhere, with a kink at x = −2 linked to a spike in v′′.The temperature is proportional to the algorithm's noise variance; the authors interpret the profile as using low temperature near the global minimum and high temperature to escape traps.Source: Figure 5 · p. 16, §4 · p. 16
  • The temperature schedule depends on an ODE solution, and the authors identify numerical solution of the higher-dimensional HJB PDE as potentially subject to the curse of dimensionality.The reported numerical example is one-dimensional, where the HJB equation is an ODE that can be solved efficiently.Source: §4 · p. 14, §5 · p. 17

5. Conclusion.

The conclusion-section passage acknowledges research funding and thanks colleagues for comments and experimental assistance.

5. Conclusion.

  • Xuefeng Gao acknowledged financial support from Hong Kong GRF grants No. 14201117 and No. 14201421; Zuo Quan Xu is separately credited with NSFC, Hong Kong GRF, PolyU-SDU center, and CAS AMSS-POLYU laboratory support.Source: Acknowledgments · p. 17
  • Zuo Quan Xu acknowledged support from NSFC grant No. 11971409 and Hong Kong GRF grants No. 15202817 and No. 15202421.His acknowledgments also named the PolyU-SDU Joint Research Center on Financial Mathematics and the CAS AMSS-POLYU Joint Laboratory of Applied Mathematics.Source: Acknowledgments · p. 17
  • The authors thanked Mert Gürbüzbalaban and Lingjiong Zhu for comments, and Yi Xiong for help with the experiments.Xun Yu Zhou acknowledged a start-up grant at Columbia University and the Nie Center for Intelligent Asset Management.Source: Acknowledgments · p. 17
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