Uniform Generalization Bound on Time and Inverse Temperature for Gradient Descent Algorithm and its Application to Analysis of Simulated Annealing

TL;DR

This paper establishes a novel uniform generalization bound for SGLD that is independent of time and inverse temperature, and applies it to analyze the effectiveness of simulated annealing in non-convex optimization.

Contribution

It introduces a Rademacher complexity-based generalization bound that removes dependence on time and inverse temperature, advancing theoretical understanding of SGLD and simulated annealing.

Findings

Generalization bound independent of time and inverse temperature

Evaluation of simulated annealing effectiveness in non-convex settings

Sample size and time evaluations with specific convergence rates

Abstract

In this paper, we propose a novel uniform generalization bound on the time and inverse temperature for stochastic gradient Langevin dynamics (SGLD) in a non-convex setting. While previous works derive their generalization bounds by uniform stability, we use Rademacher complexity to make our generalization bound independent of the time and inverse temperature. Using Rademacher complexity, we can reduce the problem to derive a generalization bound on the whole space to that on a bounded region and therefore can remove the effect of the time and inverse temperature from our generalization bound. As an application of our generalization bound, an evaluation on the effectiveness of the simulated annealing in a non-convex setting is also described. For the sample size $n$ and time $s$, we derive evaluations with orders $\sqrt{n^{-1} \log (n+1)}$ and $|(\log)^4(s)|^{-1}$, respectively. Here, $(\log)^4$ denotes the $4$ times composition of the logarithmic function.