Optimization via Strategic Law of Large Numbers

TL;DR

This work develops a unifying two-armed decision framework for global optimization of continuous functions on bounded domains by establishing a Strategic Law of Large Numbers (StLLN) that equates global optimization to asymptotic maximization of $E\big[f(S_n^{\bm\theta}/n)\big]$ under strategic sampling. It introduces a PDE-guided, sign-based strategy as an asymptotically optimal reference and then proposes Strategic Monte Carlo Optimization (SMCO), a practical, gradient-sign-based method that converges to local optima and, with multiple starts, to global optima. The authors provide rigorous convergence results and rate bounds, and validate SMCO across a wide spectrum of deterministic and random test functions, including high-dimensional landscapes and ReLU-based objectives, often outperforming traditional local and global optimizers. Public code and data accompany the study, highlighting the method’s robustness, scalability, and potential for constrained extensions and further theoretical refinement.

Abstract

This paper proposes a unified framework for the global optimization of a continuous function in a bounded rectangular domain. Specifically, we show that: (1) under the optimal strategy for a two-armed decision model, the sample mean converges to a global optimizer under the Strategic Law of Large Numbers, and (2) a sign-based strategy built upon the solution of a parabolic PDE is asymptotically optimal. Motivated by this result, we propose a class of {\bf S}trategic {\bf M}onte {\bf C}arlo {\bf O}ptimization (SMCO) algorithms, which uses a simple strategy that makes coordinate-wise two-armed decisions based on the signs of the partial gradient of the original function being optimized over (without the need of solving PDEs). While this simple strategy is not generally optimal, we show that it is sufficient for our SMCO algorithm to converge to local optimizer(s) from a single starting point, and to global optimizers under a growing set of starting points. Numerical studies demonstrate the suitability of our SMCO algorithms for global optimization, and illustrate the promise of our theoretical framework and practical approach. For a wide range of test functions with challenging optimization landscapes (including ReLU neural networks with square and hinge loss), our SMCO algorithms converge to the global maximum accurately and robustly, using only a small set of starting points (at most 100 for dimensions up to 1000) and a small maximum number of iterations (200). In fact, our algorithms outperform many state-of-the-art global optimizers, as well as local algorithms augmented with the same set of starting points as ours.

Figures (6)

• Figure 1: Visualization of Test Functions
• Figure 2: (a). Classic Monte Carlo sampling from a sequence of one-armed slot machines, with arm population $\{Z_1,\cdots, Z_d\}$; (b). Strategic Monte Carlo sampling from a sequence of two-armed slot machines, with arms populations $\{X_1,\cdots, X_d\}$ and $\{Y_1,\cdots, Y_d\}$; (c) The iterative mechanism for attaining the extreme value by GD (blue arrow line) and SMCO (red arrow line) under $d=1$, where red points denote the iterative value $S_{n}^{\bm \theta^*}/n$ for SMCO given in Algorithm \ref{['alg:smco_algo']}.
• Figure 3: Visualization of the Cross-Leg Function
• Figure 4: Dynamic Binary Choice: All
• Figure 5: Dynamic Binary Choice: SMCO/-R/-BR, GenSA and DEoptim

Theorems & Definitions (20)

• Theorem 2.1: Strategic Law of Large Numbers and a Two-sided Framework for Optimization
• Theorem 2.2: Convergence Rate
• Remark 2.1
• Theorem 2.3
• Remark 2.2
• Theorem 2.4: Local Optimization under the Sign-Based Strategy
• Remark 2.3
• Remark 2.4
• Theorem 2.5
• Theorem 2.6: Convergence Rate II