An adaptive importance sampling algorithm for risk-averse optimization

TL;DR

This paper addresses risk-averse optimization by minimizing $CVaR_\beta[f(z,\boldsymbol{\xi})]$ and proposes an adaptive importance sampling algorithm that jointly tunes the sampling distribution and the sample size using a reduced-order model (ROM) of $f(z,\cdot)$. The ROM-based biasing densities oversample the current risk region, reducing gradient variance and enabling smaller per-iteration samples while preserving the linear convergence rate. A convergence analysis for the alternating optimization is provided, and numerical experiments on PDE-constrained problems demonstrate substantial computational savings. The work situates itself among adaptive sampling and ROM-based CVaR evaluation, offering a practical, scalable framework for risk-averse stochastic optimization.

Abstract

Adaptive sampling algorithms are modern and efficient methods that dynamically adjust the sample size throughout the optimization process. However, they may encounter difficulties in risk-averse settings, particularly due to the challenge of accurately sampling from the tails of the underlying distribution of random inputs. This often leads to a much faster growth of the sample size compared to risk-neutral problems. In this work, we propose a novel adaptive sampling algorithm that adapts both the sample size and the sampling distribution at each iteration. The biasing distributions are constructed on the fly, leveraging a reduced-order model of the objective function to be minimized, and are designed to oversample a so-called risk region. As a result, a reduction of the variance of the gradients is achieved, which permits to use fewer samples per iteration compared to a standard algorithm, while still preserving the asymptotic convergence rate. Our focus is on the minimization of the Conditional Value-at-Risk (CVaR), and we establish the convergence of the proposed computational framework. Numerical experiments confirm the substantial computational savings achieved by our approach.

Figures (4)

• Figure 1: Graphical representation of the map $\boldsymbol\xi\rightarrow f(0,\boldsymbol\xi)$ for $\kappa_1$ and $\kappa_2$.
• Figure 2: Top row: growth of the sample size $M_k$ with iterations. Bottom row: estimated empirical variance of the gradients along the iterations. Left panels refer to $\kappa_1$, the right panels to $\kappa_2$. The continuous lines refer to Alg \ref{['alg:adaptive_sampling']} while the dashed-dotted ones to Alg. \ref{['alg:adaptive_sampling_rom']}.
• Figure 3: Decay of the relative error with respect to the number of iterations (top) and number of PDEs solved (bottom). The left column refers to $\kappa_1$, while the right one to $\kappa_2$. The continuous line corresponds to Alg. \ref{['alg:adaptive_sampling']}, the dashed-dotted lines to Alg. \ref{['alg:adaptive_sampling_rom']}. The circle marker highlights the last iteration of Alg. \ref{['alg:adaptive_sampling_rom']}.
• Figure 4: Comparison of the computational times in seconds required by Alg. \ref{['alg:adaptive_sampling']} and Alg. \ref{['alg:adaptive_sampling_rom']} to reach approximately the same relative error. Left panel refers to $\kappa_1$, while the right one to $\kappa_2$.

Theorems & Definitions (3)

• Proposition 1
• Lemma 2
• Theorem 3