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Escaping Local Minima: A Finite-Time Markov Chain Analysis of Constant-Temperature Simulated Annealing

Hansini Ramachandran, Bhaskar Krishnamachari

TL;DR

This work addresses the finite-time behavior of constant-temperature Simulated Annealing on simplified 1-D energy landscapes. It develops a discrete-state Markov-chain framework to derive exact mean escape times from a single basin and extends the analysis to a two-basin landscape, linking these times to geometry and temperature through closed-form expressions. A key finding is that the discrete escape-time predictions map to continuous SA with a radius-scaling factor that converges to $\sqrt{3}$ in relevant regimes, enabling accurate finite-time characterizations and guiding a two-temperature switching strategy. Collectively, these results provide a predictive, analytically tractable foundation for understanding and tuning SA; they also suggest pathways to extend to higher dimensions and more complex basin networks while reducing empirical tuning.

Abstract

Simulated Annealing (SA) is a widely used stochastic optimization algorithm, yet much of its theoretical understanding is limited to asymptotic convergence guarantees or general spectral bounds. In this paper, we develop a finite-time analytical framework for constant-temperature SA by studying a piecewise linear cost function that permits exact characterization. We model SA as a discrete-state Markov chain and first derive a closed-form expression for the expected time to escape a single linear basin in a one-dimensional landscape. We show that this expression also accurately predicts the behavior of continuous-state searches up to a constant scaling factor, which we analyze empirically and explain via variance matching, demonstrating convergence to a factor of sqrt(3) in certain regimes. We then extend the analysis to a two-basin landscape containing a local and a global optimum, obtaining exact expressions for the expected time to reach the global optimum starting from the local optimum, as a function of basin geometry, neighborhood radius, and temperature. Finally, we demonstrate how the predicted basin escape time can be used to guide the design of a simple two-temperature switching strategy.

Escaping Local Minima: A Finite-Time Markov Chain Analysis of Constant-Temperature Simulated Annealing

TL;DR

This work addresses the finite-time behavior of constant-temperature Simulated Annealing on simplified 1-D energy landscapes. It develops a discrete-state Markov-chain framework to derive exact mean escape times from a single basin and extends the analysis to a two-basin landscape, linking these times to geometry and temperature through closed-form expressions. A key finding is that the discrete escape-time predictions map to continuous SA with a radius-scaling factor that converges to in relevant regimes, enabling accurate finite-time characterizations and guiding a two-temperature switching strategy. Collectively, these results provide a predictive, analytically tractable foundation for understanding and tuning SA; they also suggest pathways to extend to higher dimensions and more complex basin networks while reducing empirical tuning.

Abstract

Simulated Annealing (SA) is a widely used stochastic optimization algorithm, yet much of its theoretical understanding is limited to asymptotic convergence guarantees or general spectral bounds. In this paper, we develop a finite-time analytical framework for constant-temperature SA by studying a piecewise linear cost function that permits exact characterization. We model SA as a discrete-state Markov chain and first derive a closed-form expression for the expected time to escape a single linear basin in a one-dimensional landscape. We show that this expression also accurately predicts the behavior of continuous-state searches up to a constant scaling factor, which we analyze empirically and explain via variance matching, demonstrating convergence to a factor of sqrt(3) in certain regimes. We then extend the analysis to a two-basin landscape containing a local and a global optimum, obtaining exact expressions for the expected time to reach the global optimum starting from the local optimum, as a function of basin geometry, neighborhood radius, and temperature. Finally, we demonstrate how the predicted basin escape time can be used to guide the design of a simple two-temperature switching strategy.
Paper Structure (17 sections, 25 equations, 10 figures)

This paper contains 17 sections, 25 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Single-Basin Analysis
  4. Assumptions
  5. Markov Chain / Random Walk Modeling
  6. Derived Expression for Expected Time to Exit
  7. Fit to Continuous Case
  8. Empirical results
  9. Explanation of $\sqrt{3}$ limit
  10. Two-Basin Analysis
  11. Setup
  12. Expression
  13. Empirical fit for the Continuous Case
  14. Exploration: Determining the Impact of Temperature
  15. Temperature
  16. ...and 2 more sections

Figures (10)

  • Figure 1: Energy landscape for a linear 1-basin geometry with physical parameters labeled
  • Figure 2: Discrete Markov chain to model one linear basin with absorbing boundary states at $\pm N$.
  • Figure 3: Estimated time to absorption (continuous and discrete) from state 0 vs. $\tfrac{w\mathrm{T}}{2rd}$ ratio
  • Figure 4: Optimal $k$ (radius scaling factor) vs. $\tfrac{w\mathrm{T}}{2rd}$ ratio
  • Figure 5: Simplified representation of the two-basin Markov chain showing only the right-hand half of the state space. The omitted left-hand side consists of a reflected copy with identical transition structure. State $M$ denotes the barrier between basins, and $N$ is an absorbing state.
  • ...and 5 more figures