Distributionally Robust Stochastic Optimization with Wasserstein Distance (1604.02199v3)

Abstract: Distributionally robust stochastic optimization (DRSO) is an approach to optimization under uncertainty in which, instead of assuming that there is a known true underlying probability distribution, one hedges against a chosen set of distributions. In this paper we first point out that the set of distributions should be chosen to be appropriate for the application at hand, and that some of the choices that have been popular until recently are, for many applications, not good choices. We next consider sets of distributions that are within a chosen Wasserstein distance from a nominal distribution. Such a choice of sets has two advantages: (1) The resulting distributions hedged against are more reasonable than those resulting from other popular choices of sets. (2) The problem of determining the worst-case expectation over the resulting set of distributions has desirable tractability properties. We derive a strong duality reformulation of the corresponding DRSO problem and construct approximate worst-case distributions explicitly via the first-order optimality conditions of the dual problem. Our contributions are four-fold. (i) We identify necessary and sufficient conditions for the existence of a worst-case distribution, which are naturally related to the growth rate of the objective function. (ii) We show that the worst-case distributions resulting from an appropriate Wasserstein distance have a concise structure and a clear interpretation. (iii) Using this structure, we show that data-driven DRSO problems can be approximated to any accuracy by robust optimization problems, and thereby many DRSO problems become tractable by using tools from robust optimization. (iv) Our strong duality result holds in a very general setting. As examples, we show that it can be applied to infinite-dimensional process control and intensity estimation for point processes.

• The paper presents a tractable reformulation of distributionally robust stochastic optimization using a Wasserstein-based ambiguity set that ensures strong duality.
• The methodology characterizes worst-case distributions at the boundary of the Wasserstein ball under finite objective growth conditions.
• The framework connects data-driven DRSO with robust optimization techniques, enabling practical numerical approximations for efficient decision-making under uncertainty.

A Distributionally Robust Stochastic Optimization Framework with Wasserstein Distance

The paper presents an in-depth analysis of Distributionally Robust Stochastic Optimization (DRSO) using Wasserstein distance as a key metric for constructing ambiguity sets. Wasserstein distance, a measure rooted in the concept of optimal transport, offers a refined method of considering uncertainty by including all distributions within a specified distance from a nominal distribution.

Key Contributions

1. Tractable Reformulation: The authors tackle the challenge of expressing DRSO problems in a tractable form by leveraging strong duality. For the general case, a dual reformulation is achieved where the primal optimization seeks to minimize over a set of distributions constrained by Wasserstein distance, and the dual problem becomes a one-dimensional minimization problem over Lagrange multipliers. Strong duality is demonstrated, ensuring the equivalence of these reformulations.
2. Conditions and Characterizations: The paper provides necessary and sufficient conditions for the existence of a worst-case distribution. Specifically, it characterizes worst-case distributions in terms of their belonging to the boundary of the Wasserstein ball when the growth rate of the objective function is finite, providing insights into their structure.
3. Numerical Approximations through Robust Optimization: A relationship between robust optimization and Wasserstein-based DRSO is proposed. Specifically, data-driven DRSO problems can be reformulated with robust optimization techniques, allowing practitioners to benefit from existing computational tools dedicated to robust optimization, which can approximate these problems to arbitrary accuracy.
4. Applications and Implications: The theoretical advancements are demonstrated through applications such as on/off system control and intensity estimation for non-homogeneous Poisson processes. These examples illustrate how the theoretical framework can be applied to complex decision-making problems under uncertainty.

Practical and Theoretical Implications

The implications of this research are significant: The robust characterization of the uncertainty set improves the realism and tractability of optimization under ambiguity, compared to traditional methods like moment-based or $\phi$-divergence approaches. The Wasserstein distance allows for a more nuanced understanding of uncertainty, accounting for how distributions might be close in a practical, rather than theoretical, sense.

Moreover, the paper presents a pathway for handling uncertainty in high-dimensional problems, using tools from optimal transport to offer tractable formulations and efficient computational strategies.

Future Prospects in AI

The approach delineated in this paper could be extended to broader AI applications involving data-driven models under uncertainty. For instance, in reinforcement learning, such uncertainty sets might be applicable to scenario-based policy optimization. The theoretical advancements in strong duality and the structural insights into worst-case distributions offer promising directions for extending DRSO methodologies to dynamic and interactive AI systems, where decisions are contingent not only on stochastic outcomes but also on learning model uncertainties and errors.

In summary, this paper contributes fundamentally to the understanding and application of distributionally robust optimization using the Wasserstein metric, providing a promising venue for both theoretical exploration and practical implementation in various AI processes and systems operating under uncertainty.