Zeroth-Order Methods for Nonconvex Stochastic Problems with Decision-Dependent Distributions (2412.20330v1)

Abstract: In this study, we consider an optimization problem with uncertainty dependent on decision variables, which has recently attracted attention due to its importance in machine learning and pricing applications. In this problem, the gradient of the objective function cannot be obtained explicitly because the decision-dependent distribution is unknown. Therefore, several zeroth-order methods have been proposed, which obtain noisy objective values by sampling and update the iterates. Although these existing methods have theoretical convergence for optimization problems with decision-dependent uncertainty, they require strong assumptions about the function and distribution or exhibit large variances in their gradient estimators. To overcome these issues, we propose two zeroth-order methods under mild assumptions. First, we develop a zeroth-order method with a new one-point gradient estimator including a variance reduction parameter. The proposed method updates the decision variables while adjusting the variance reduction parameter. Second, we develop a zeroth-order method with a two-point gradient estimator. There are situations where only one-point estimators can be used, but if both one-point and two-point estimators are available, it is more practical to use the two-point estimator. As theoretical results, we show the convergence of our methods to stationary points and provide the worst-case iteration and sample complexity analysis. Our simulation experiments with real data on a retail service application show that our methods output solutions with lower objective values than the conventional zeroth-order methods.

• The paper proposes two novel zeroth-order methods for nonconvex stochastic problems with decision-dependent distributions, addressing challenges where gradients are unavailable.
• The methods introduce a one-point gradient estimator with variance reduction and a flexible two-point estimator, improving optimization efficiency and convergence reliability.
• Theoretical analysis provides worst-case complexity bounds, and practical simulations demonstrate superior performance over traditional methods in real-world pricing scenarios.

Analyzing Zeroth-Order Methods for Nonconvex Stochastic Problems with Decision-Dependent Distributions

This paper explores the domain of optimization where uncertainty is contingent upon decision variables, presenting significant relevance for machine learning and pricing applications. The focus is on addressing the challenges posed by nonconvex stochastic problems where the probability distributions are linked to the decision variables, in contexts where the gradient of the objective function is not explicitly obtainable. The primary contribution is the development of two novel zeroth-order methods under mild assumptions, which offer alternative pathways in contrast to existing methods that either necessitate strong assumptions on the function and distribution or suffer from high variances in their gradient estimators.

Main Contributions

1. Zeroth-Order Methods with Variance Reduction: The authors propose a new zeroth-order method that incorporates a one-point gradient estimator with a variance reduction parameter. This novel estimator allows for iteratively updating decision variables while adjusting for variance, thus ensuring more reliable convergence to stationary points and improved optimization efficiency.
2. Two-Point Gradient Estimator: A second zeroth-order method is introduced, utilizing a two-point gradient estimator. This method provides the flexibility to switch between one-point and two-point estimators since two-point estimators tend to deliver superior performance when applicable. The paper theorizes convergence to stationary points and offers worst-case iteration and sample complexity analyses for these approaches.

Theoretical Results and Comparisons

Theoretically, the paper substantiates the convergence of the proposed methods toward stationary points. It also delivers a rigorous complexity analysis, indicating that their sample complexity is $O(d^{\frac{9}{2}^{-6})$. This analysis suggests that the proposed methods can offer an advantage, especially in scenarios where the supremum of the function values is large or unbounded—a significant advancement over traditional methods, such as those proposed by Liu et al.

Implications and Future Work

Practically, the proposed methods show promising results in simulations involving real data from a retail pricing scenario, outperforming traditional zeroth-order methods by achieving solutions with lower objective values. The improvements in sample efficiency make these methods particularly advantageous in real-world applications where sampling is expensive or logistically challenging.

Theoretically, these developments imply a broader applicability of zeroth-order methods to complex decision-dependent uncertainty problems. Future avenues for this research could explore extending the methods to accommodate dynamic environments, integrating with sophisticated variance reduction techniques, and further refining complexity to adapt to more varied problem settings. Notably, enhancing scalability through techniques like variable selection and applying these methods to other domains where decision-dependent uncertainties are prevalent present fertile grounds for future exploration.

In summary, this paper makes a substantial contribution to the understanding and application of zeroth-order methods in stochastic, nonconvex optimization problems, particularly those involving decision-dependent distributions. Through both theoretical enhancement and empirical validation, it lays foundational work that could fundamentally improve decision-making frameworks in uncertain environments.