[Paper Review] Kernel Distributionally Robust Optimization

Jia-Jie Zhu, Wittawat Jitkrittum+2 more|arXiv (Cornell University)|Jun 12, 2020

Risk and Portfolio Optimization51 references11 citations

TL;DR

This paper introduces Kernel Distributionally Robust Optimization (Kernel DRO), a method that uses reproducing kernel Hilbert spaces (RKHS) to construct flexible convex ambiguity sets for distributionally robust optimization. By proving a generalized duality theorem, it reformulates the worst-case risk minimization into a dual problem over RKHS functions, enabling robust optimization for general loss functions without requiring Lipschitz assumptions or kernel-specific constraints.

ABSTRACT

We propose kernel distributionally robust optimization (Kernel DRO) using insights from the robust optimization theory and functional analysis. Our method uses reproducing kernel Hilbert spaces (RKHS) to construct a wide range of convex ambiguity sets, which can be generalized to sets based on integral probability metrics and finite-order moment bounds. This perspective unifies multiple existing robust and stochastic optimization methods. We prove a theorem that generalizes the classical duality in the mathematical problem of moments. Enabled by this theorem, we reformulate the maximization with respect to measures in DRO into the dual program that searches for RKHS functions. Using universal RKHSs, the theorem applies to a broad class of loss functions, lifting common limitations such as polynomial losses and knowledge of the Lipschitz constant. We then establish a connection between DRO and stochastic optimization with expectation constraints. Finally, we propose practical algorithms based on both batch convex solvers and stochastic functional gradient, which apply to general optimization and machine learning tasks.

Motivation & Objective

• Address the limitations of existing robust optimization methods that require knowledge of the Lipschitz constant or are restricted to polynomial loss functions.
• Develop a general framework for distributionally robust optimization (DRO) that applies to a broad class of loss functions and data-driven ambiguity sets.
• Unify existing robust and stochastic optimization methods through the use of reproducing kernel Hilbert spaces (RKHS) and integral probability metrics (IPM).
• Establish a connection between DRO and stochastic optimization with expectation constraints, enabling scalable algorithms.
• Provide a theoretical foundation using conic duality and functional analysis to generalize classical duality results in the mathematical problem of moments.

Proposed method

• Formulate the DRO problem as a min-max optimization over probability measures within an ambiguity set defined via RKHS norms and integral probability metrics (IPM).
• Prove a generalized duality theorem (Theorem 3.1) that transforms the primal DRO problem into a dual problem over RKHS functions, enabling convex optimization over function space.
• Construct ambiguity sets using RKHS-based norms and moment bounds, including sets defined via MMD (maximum mean discrepancy) and IPM, ensuring convexity and tractability.
• Leverage universal RKHSs to ensure the duality result applies to a wide class of loss functions, including non-polynomial and non-Lipschitz ones.
• Derive a semi-infinite conic formulation of the dual problem and reformulate it into a quadratically constrained linear program (QCLP), which can be solved via semidefinite programming (SDP) using the S-lemma.
• Propose a stochastic functional gradient DRO (SFG-DRO) algorithm for scalable training on large-scale machine learning tasks, combining stochastic approximation with functional gradient updates.

Experimental results

Research questions

• RQ1 Can the duality in the mathematical problem of moments be generalized to apply to arbitrary loss functions beyond polynomial or Lipschitz-continuous ones?
• RQ2 How can RKHS-based ambiguity sets be constructed to unify various existing robust optimization methods, including those based on moment bounds and IPMs?
• RQ3 To what extent can the duality framework be extended to non-kernelized models, enabling robustness beyond kernelized learning?
• RQ4 What is the connection between distributionally robust optimization and stochastic optimization with expectation constraints, and how can it be exploited algorithmically?
• RQ5 Can scalable, practical algorithms be developed for DRO that avoid the need for explicit knowledge of the Lipschitz constant or kernel structure?

Key findings

• The generalized duality theorem (Theorem 3.1) establishes that the worst-case risk in DRO can be reformulated as a convex optimization problem over RKHS functions, lifting the need for Lipschitz continuity or polynomial loss assumptions.
• The ambiguity set defined via RKHS norms is compact and convex when the input space is compact, ensuring the existence of optimal solutions.
• By using universal RKHSs, the proposed framework applies to a broad class of loss functions, including non-polynomial and non-Lipschitz ones, overcoming a key limitation of prior DRO methods.
• The connection between DRO and stochastic optimization with expectation constraints is formally established, enabling the design of a novel stochastic functional gradient DRO (SFG-DRO) algorithm.
• The SFG-DRO algorithm scales to modern machine learning tasks by combining stochastic approximation with functional gradient updates, achieving practical efficiency.
• Theoretical analysis confirms that the duality framework generalizes classical results from the mathematical problem of moments, with complete self-contained proofs provided in the appendix.