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Iteration Complexity and Finite-Time Efficiency of Adaptive Sampling Trust-Region Methods for Stochastic Derivative-Free Optimization

Yunsoo Ha, Sara Shashaani

TL;DR

Two key refinements to reduce the dependence of ASTRO-DF on the problem dimension and boost its performance in finite time are presented, namely: local models with diagonal Hessians constructed on interpolation points based on a coordinate basis and direct search using the interpolation points whenever possible.

Abstract

Adaptive sampling with interpolation-based trust regions or ASTRO-DF is a successful algorithm for stochastic derivative-free optimization with an easy-to-understand-and-implement concept that guarantees almost sure convergence to a first-order critical point. To reduce its dependence on the problem dimension, we present local models with diagonal Hessians constructed on interpolation points based on a coordinate basis. We also leverage the interpolation points in a direct search manner whenever possible to boost ASTRO-DF's performance in a finite time. We prove that the algorithm has a canonical iteration complexity of $\mathcal{O}(ε^{-2})$ almost surely, which is the first guarantee of its kind without placing assumptions on the quality of function estimates or model quality or independence between them. Numerical experimentation reveals the computational advantage of ASTRO-DF with coordinate direct search due to saving and better steps in the early iterations of the search.

Iteration Complexity and Finite-Time Efficiency of Adaptive Sampling Trust-Region Methods for Stochastic Derivative-Free Optimization

TL;DR

Two key refinements to reduce the dependence of ASTRO-DF on the problem dimension and boost its performance in finite time are presented, namely: local models with diagonal Hessians constructed on interpolation points based on a coordinate basis and direct search using the interpolation points whenever possible.

Abstract

Adaptive sampling with interpolation-based trust regions or ASTRO-DF is a successful algorithm for stochastic derivative-free optimization with an easy-to-understand-and-implement concept that guarantees almost sure convergence to a first-order critical point. To reduce its dependence on the problem dimension, we present local models with diagonal Hessians constructed on interpolation points based on a coordinate basis. We also leverage the interpolation points in a direct search manner whenever possible to boost ASTRO-DF's performance in a finite time. We prove that the algorithm has a canonical iteration complexity of almost surely, which is the first guarantee of its kind without placing assumptions on the quality of function estimates or model quality or independence between them. Numerical experimentation reveals the computational advantage of ASTRO-DF with coordinate direct search due to saving and better steps in the early iterations of the search.
Paper Structure (26 sections, 11 theorems, 46 equations, 5 figures, 1 algorithm)

This paper contains 26 sections, 11 theorems, 46 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. List of Contributions
  3. Preliminaries
  4. Literature Review
  5. uniformly bounded $|E(\bm{x})|$:
  6. one-sided sub-exponential $E(\bm{x})$:
  7. zero-mean $E(\bm{x})$ with a finite variance:
  8. zero-mean sub-exponential $E(\bm{x})$ with bounded $v$-th moment for $v\leq 8$:
  9. bounded variance of $E(\bm{x})$:
  10. Notation
  11. Definitions
  12. ASTRO-DF with Coordinate Direct Search
  13. Model Construction
  14. Updating next incumbent
  15. Simulation budget
  16. ...and 11 more sections

Key Result

Theorem 4.1

(Theorem 2.7 by ha2023siamopt) Let Assumptions assum:error and assum:lambda hold and define $|\bar{E}_k|:=N_k^{-1}\sum_{j=1}^{N_k}E_j$. Then for a given $c>0$, we have that $|\bar{E}_k|\leq c \Delta_k^2$ for sufficiently large $k$ almost surely. In other words, $\mathbb{P}\{|\bar{E}_k|\geq c\Delta_k

Figures (5)

  • Figure 1: Fraction of 60 problems from SimOpt librarysimoptgithub solved to $0.1$-optimality with 95% confidence intervals from 20 runs of each algorithm shows a clear advantage in finite-time performance of ASTRO-DF with coordiate direct search.
  • Figure 2: Slower rate of trust-region radius decay in the early iterations of the search as a result of using direct search allows for larger step size and hence better progress per iteration early on. The mean and 95% confidence interval of trajectory of $\Delta_k$ exhibits this behavior for two problems.
  • Figure 3: Better progress of ASTRO-DF in the early iterations with direct search is evident in the mean function estimates and 95% confidence intervals on both problems.
  • Figure 4: The total budget spent per iteration is less due to savings from using direct search, as tested in SAN with 95% confidence intervals from 20 macroreplications.
  • Figure 5: Mean value and 95% confidence interval of objective function trajectory per spent budget for both problems exhibits significant finite-time improvement with direct search.

Theorems & Definitions (30)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 4.1
  • Remark 1
  • Lemma 4.2
  • Remark 2
  • Theorem 4.3
  • ...and 20 more