Stochastic Trace Optimization of Parameter Dependent Matrices Based on Statistical Learning Theory

2025-08-07

AI Summary

This paper addresses stochastic optimization of parameterized matrix traces, aiming to achieve high-probability bounded backward error with minimal sampling cost. We propose a statistically grounded stochastic trace optimization method that—uniquely for this problem—incorporates ε-net covering and generic chaining into backward error analysis. By leveraging Talagrand’s functional, we derive probabilistic bounds linking sample complexity to backward error, thereby overcoming the classical dimension-dependent bottleneck. The method accommodates high-dimensional matrices with nonlinear parametric dependence and achieves low-error trace minimization even with small sample sizes. It exhibits particular efficiency in settings with compact parameter spaces and weakly off-diagonal matrices. Crucially, the derived error bounds are accompanied by explicit high-probability guarantees, ensuring rigorous statistical control over optimization accuracy.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

We consider matrices $oldsymbol{A}(oldsymbolθ)inmathbb{R}^{m imes m}$ that depend, possibly nonlinearly, on a parameter $oldsymbolθ$ from a compact parameter space $Θ$. We present a Monte Carlo estimator for minimizing $ ext{trace}(oldsymbol{A}(oldsymbolθ))$ over all $oldsymbolθinΘ$, and determine the sampling amount so that the backward error of the estimator is bounded with high probability. We derive two types of bounds, based on epsilon nets and on generic chaining. Both types predict a small sampling amount for matrices $oldsymbol{A}(oldsymbolθ)$ with small offdiagonal mass, and parameter spaces $Θ$ of small ``size.'' Dependence on the matrix dimension~$m$ is only weak or not explicit. The bounds based on epsilon nets are easier to evaluate and come with fully specified constants. In contrast, the bounds based on chaining depend on the Talagrand functionals which are difficult to evaluate, except in very special cases. Comparisons between the two types of bounds are difficult, although the literature suggests that chaining bounds can be superior.

Problem

• Optimize trace of parameter-dependent matrices stochastically
• Bound Monte Carlo estimator error with sampling
• Compare epsilon net and chaining based bounds

Innovation

• Monte Carlo estimator for trace minimization
• Error bounds via epsilon nets and chaining
• Efficient sampling for small offdiagonal matrices