Paper: arXiv 2604.00178

Authors: Giovanni Amici, Sara Shashaani, Pranav Jain

Abstract

There is emerging evidence that trust-region (TR) algorithms are very effective at solving derivative-free nonconvex stochastic optimization problems in which the objective function is a Monte Carlo (MC) estimate. A recent strand of methodologies adaptively adjusts the sample size of the MC estimates by keeping the estimation error below a measure of stationarity induced from the TR radius. In this work we explore stratified adaptive sampling strategies to equip the TR framework with accurate estimates of the objective function, thus optimizing the required number of MC samples to reach a given ε-accuracy of the solution. We prove a reduced sample complexity, confirm a superior efficiency via numerical tests and applications, and explore inexpensive implementations in high dimension.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical framework with theoretical proofs regarding sample complexity reduction in stochastic trust-region optimization. Empirical validity is established via numerical simulations, though further real-world empirical financial backtests would strengthen the findings.

Research Flowchart

  flowchart TD
    A[Research Goal: Optimize MC Sample Complexity in Derivative-Free Stochastic TR Optimization] --> B[Input: Objective Function as Monte Carlo Estimates]
    B --> C[Methodology: Stratified Adaptive Sampling Framework]
    C --> D[Adaptively Adjust Sample Size via TR Radius & Error Bound]
    D --> E[Inexpensive High-Dimensional Implementation]
    E --> F[Outcomes: Reduced Sample Complexity & Higher Efficiency for ε-Accuracy]

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