[Paper Review] An Adaptive Sampling-based Progressive Hedging Algorithm for Stochastic Programming

Di Zhang, Zhang, Yihang+1 more|arXiv (Cornell University)|Jul 30, 2024

Risk and Portfolio OptimizationDecision Sciences3 citations

TL;DR

This paper proposes an adaptive sampling-based progressive hedging algorithm (AS-PHA) that overcomes key limitations of traditional progressive hedging in stochastic programming by dynamically sampling scenario subproblems, using stochastic conjugate subgradients and line search for dual updates, and eliminating reliance on explicit probability distributions. The method achieves faster convergence and improved scalability, with experimental results showing superior data efficiency and robustness compared to classic PHA.

ABSTRACT

The progressive hedging algorithm (PHA) is a cornerstone among algorithms for large-scale stochastic programming problems. However, its traditional implementation is hindered by some limitations, including the requirement to solve all scenario subproblems in each iteration, reliance on an explicit probability distribution, and a convergence process that is highly sensitive to the choice of certain penalty parameters. This paper introduces a sampling-based PHA which aims to overcome these limitations. Our approach employs a dynamic selection process for the number of scenario subproblems solved per iteration. It incorporates adaptive sequential sampling for determining sample sizes, a stochastic conjugate subgradient method for direction finding, and a line-search technique to update the dual variables. Experimental results demonstrate that this novel algorithm not only addresses the bottlenecks of the conventional PHA but also potentially surpasses its scalability, representing a substantial improvement in the field of stochastic programming.

Motivation & Objective

• To address the computational bottleneck of classic progressive hedging algorithms (PHA), which require solving all scenario subproblems in each iteration.
• To eliminate the need for explicit knowledge of the probability distribution of random variables, enabling application in simulation-based or large-scale uncertainty settings.
• To improve convergence robustness by replacing fixed penalty parameters with adaptive dual update strategies based on line search and conjugate subgradients.
• To enhance scalability and data efficiency through dynamic, adaptive sampling of scenarios rather than fixed sample sizes.
• To establish theoretical convergence guarantees under adaptive sampling and stochastic approximation principles.

Proposed method

• Employs a dynamic sampling strategy that adaptively selects the number of scenario subproblems to solve in each iteration based on optimization progress.
• Uses a stochastic conjugate subgradient method to compute search directions, improving convergence over standard subgradient approaches.
• Applies a line-search technique to update dual variables, enabling implicit and adaptive adjustment of the penalty parameter.
• Introduces a sample size adaptation mechanism that increases or decreases the number of scenarios based on subgradient and duality gap estimates.
• Replaces the deterministic subproblem resolution of classic PHA with a sequential, adaptive sampling process that maintains convergence under uncertainty.
• Theoretical analysis establishes convergence under adaptive sampling, with dual objective values forming a submartingale and iterates converging to an ε-optimal solution.

Experimental results

Research questions

• RQ1 Can a progressive hedging algorithm be made scalable and efficient by dynamically sampling only a subset of scenarios per iteration, rather than all?
• RQ2 How can the convergence of PHA be stabilized and improved when the underlying probability distribution is unknown or intractable?
• RQ3 Can adaptive sampling combined with stochastic conjugate subgradients and line search outperform fixed-penalty, full-scenario PHA in terms of convergence speed and data efficiency?
• RQ4 What theoretical guarantees can be established for a sampling-based PHA under adaptive scenario selection and stochastic approximation?
• RQ5 Does the proposed method maintain or improve solution quality while significantly reducing computational burden in large-scale stochastic programs?

Key findings

• The proposed adaptive sampling-based PHA achieves faster convergence than classic PHA on benchmark problems, particularly for problems requiring many scenarios to converge.
• The dual objective values of the sampling-based PHA exhibit a submartingale property, indicating consistent improvement over iterations, unlike classic PHA.
• Experimental results show the algorithm is more data-efficient and scalable, with reduced computational overhead due to dynamic scenario sampling.
• Theoretical analysis confirms that the algorithm converges to an ε-optimal solution, with |f(x*) - f(x̄k)| = O(ε) under appropriate assumptions.
• The method eliminates the need for explicit probability distributions and avoids sensitivity to penalty parameter tuning, enhancing robustness in real-world applications.
• The combination of stochastic conjugate subgradients and line search enables stable and efficient dual variable updates, improving convergence behavior.