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A kernel-based stochastic approximation framework for contextual optimization

Hao Cao, Jian-Qiang Hu, Jiaqiao Hu

TL;DR

The paper develops KBSA, a kernel smoothing–based stochastic approximation framework for contextual stochastic optimization with blackbox objectives. It unifies estimation, sensitivity analysis, and optimization of a broad class of contextual measures (including conditional expectations, conditional quantiles, CoVaR, and CoES) by recasting conditioning events via kernel smoothing and solving a coupled system with two timescales. Theoretical results establish almost-sure convergence to the projected ODE equilibrium and delineate finite-time convergence rates, with an acceleration mechanism using high-order kernels to boost performance. The approach is extended to multivariate conditioning events and validated through numerical experiments on synthetic and nonlinear portfolio problems, demonstrating practical efficiency and accuracy with interpretable convergence diagnostics.

Abstract

We present a kernel-based stochastic approximation (KBSA) framework for solving contextual stochastic optimization problems with differentiable objective functions. The framework only relies on system output estimates and can be applied to address a large class of contextual measures, including conditional expectations, conditional quantiles, CoVaR, and conditional expected shortfalls.Under appropriate conditions, we show the strong convergence of KBSA and characterize its finite-time performance in terms of bounds on the mean squared errors of the sequences of iterates produced. In addition, we discuss variants of the framework, including a version based on high-order kernels for further enhancing the convergence rate of the method and an extension of KBSA for handling contextual measures involving multiple conditioning events.Simulation experiments are also carried out to illustrate the framework.

A kernel-based stochastic approximation framework for contextual optimization

TL;DR

The paper develops KBSA, a kernel smoothing–based stochastic approximation framework for contextual stochastic optimization with blackbox objectives. It unifies estimation, sensitivity analysis, and optimization of a broad class of contextual measures (including conditional expectations, conditional quantiles, CoVaR, and CoES) by recasting conditioning events via kernel smoothing and solving a coupled system with two timescales. Theoretical results establish almost-sure convergence to the projected ODE equilibrium and delineate finite-time convergence rates, with an acceleration mechanism using high-order kernels to boost performance. The approach is extended to multivariate conditioning events and validated through numerical experiments on synthetic and nonlinear portfolio problems, demonstrating practical efficiency and accuracy with interpretable convergence diagnostics.

Abstract

We present a kernel-based stochastic approximation (KBSA) framework for solving contextual stochastic optimization problems with differentiable objective functions. The framework only relies on system output estimates and can be applied to address a large class of contextual measures, including conditional expectations, conditional quantiles, CoVaR, and conditional expected shortfalls.Under appropriate conditions, we show the strong convergence of KBSA and characterize its finite-time performance in terms of bounds on the mean squared errors of the sequences of iterates produced. In addition, we discuss variants of the framework, including a version based on high-order kernels for further enhancing the convergence rate of the method and an extension of KBSA for handling contextual measures involving multiple conditioning events.Simulation experiments are also carried out to illustrate the framework.

Paper Structure

This paper contains 20 sections, 19 theorems, 41 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Blackbox Contextual Optimization
  3. Notation
  4. Problem Setting
  5. A Kernel-based Stochastic Approximation Method
  6. Convergence Analysis
  7. Preliminary Results
  8. Asymptotic Behavior of Measure and Gradient Estimates
  9. Main Convergence Theorem
  10. Rate of Convergence
  11. Convergence Rate of KBSA
  12. Accelerated KBSA
  13. Multivariate Conditioning Events
  14. Numerical Experiments
  15. Blackbox Test Functions
  16. ...and 5 more sections

Key Result

Lemma 1

If Assumptions A.q-A.q2, A.beta_uni, and A.multi_timescale_uni hold, then (a) $\sup_n\mathbb{E}[\nu_n^2]<\infty$; (b) $\sup_n|\nu_n|<\infty$ a.s.

Figures (2)

  • Figure 1: Convergence behaviors of KBSA (test functions).
  • Figure 2: Convergence behaviors of KBSA (nonlinear portfolios).

Theorems & Definitions (22)

  • Example 1: Risk-neutral decision-making
  • Example 2: Systemic risk analysis
  • Remark
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Proposition 1
  • ...and 12 more