Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach (1610.03425v3)

Abstract: We study statistical inference and distributionally robust solution methods for stochastic optimization problems, focusing on confidence intervals for optimal values and solutions that achieve exact coverage asymptotically. We develop a generalized empirical likelihood framework---based on distributional uncertainty sets constructed from nonparametric $f$-divergence balls---for Hadamard differentiable functionals, and in particular, stochastic optimization problems. As consequences of this theory, we provide a principled method for choosing the size of distributional uncertainty regions to provide one- and two-sided confidence intervals that achieve exact coverage. We also give an asymptotic expansion for our distributionally robust formulation, showing how robustification regularizes problems by their variance. Finally, we show that optimizers of the distributionally robust formulations we study enjoy (essentially) the same consistency properties as those in classical sample average approximations. Our general approach applies to quickly mixing stationary sequences, including geometrically ergodic Harris recurrent Markov chains.

• The paper introduces a generalized empirical likelihood framework to derive asymptotically exact confidence intervals for robust optimization.
• It demonstrates that variance regularization is achieved by robust estimators, systematically penalizing uncertainty in stochastic optimization.
• The authors show that the consistent solutions extend to dependent data scenarios, broadening the approach's practical impact.

Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach

The paper "Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach" by Duchi, Glynn, and Namkoong explores advanced statistical inference techniques for stochastic optimization problems, particularly emphasizing the derivation of distributionally robust solutions. The research presents a framework for deriving confidence intervals that reliably cover optimal values, leveraging a generalized empirical likelihood structure.

Key Contributions

1. Generalized Empirical Likelihood Framework: The authors introduce a framework based around $f$-divergence balls, which forms the crux for constructing confidence intervals in stochastic optimization. This model enables the derivation of asymptotically exact confidence intervals for optimal values by formulating distributional uncertainty sets that encapsulate the underlying data distribution within a specified divergence distance. The authors show how this method couches an optimization problem into a statistically robust format, providing avenues for computing both one- and two-sided confidence intervals that are exact asymptotically.
2. Variance Regularization Insight: A novel aspect of this work is how robust estimators can regularize variance. The authors demonstrate that robustification through empirical likelihood systematically accounts for variance, thereby regularizing the resultant problem. The asymptotic expansion of the robust problem makes explicit how variance is mitigated as a systematic penalty, which parallels conventional regularization techniques in machine learning.
3. Consistency of Optimizers: The paper comprehensively addresses the consistency of solutions derived from their robust optimization formulation. The authors establish that the solutions are consistent and converge toward true population values under minor restrictions, akin to those in traditional sample average approximation (SAA) methods. This consistency extends even to dependent data streams, including Markov chains, making this approach adaptable to various real-world applications where data correlation exists.
4. Empirical and Asymptotic Results: By applying Hadamard differentiability, the authors guarantee smoothness for their functional mappings, thus allowing the use of powerful empirical process theory results. The paper includes rigorous proofs that derive the asymptotic properties necessary for confidence intervals—specifically, the generalized likelihood's asymptotic behaviors—paralleling classical empirical likelihood in simplicity but expanding in utility and application to stochastic optimization.

Implications and Future Directions

The implications of this work are significant for both the theoretical underpinnings of robust statistics and for practical applications where decision-making involves uncertainty. The adaptive nature of choosing $f$-divergence aligns with robust statistics' goals—improving decision quality under distributional uncertainty. This can have sizeable impacts on fields like finance and supply chain management, where decisions must optimize performance under probabilistic constraints.

Future exploration could pivot towards refining computational methods for high-dimensional settings, where computing $\chi^2$-divergence balls becomes intractable. The effectiveness of different divergence measures compared to $f$-divergences deserves examination. Investigating optimal trade-offs in variance penalization under finite samples might yield additional practical insights, improving robustness’s role in automated decision systems.

This paper establishes a structured method for understanding stochastic optimization problems through a statistical inference lens, offering a robust optimization technique infused with the precision of empirical likelihood. This represents a substantial theoretical leap with direct implications for mathematical optimization, computational efficiency, and statistical robustness.