Simultaneous Learning and Optimization via Misspecified Saddle Point Problems

2025-10-06

AI Summary

This paper studies a class of misspecified saddle-point problems with unknown parameters that must be learned concurrently from data—relaxing the strong assumption in prior work that parameters are either known or pre-estimated. We propose a learning-aware accelerated primal-dual algorithm that tightly integrates parameter estimation with optimization: it explicitly models parameter dynamics via momentum-based updates and introduces online parameter estimation coupled with backward tracking, enabling adaptive step sizes and natural extension to multi-solution settings. Theoretically, we establish an $O(log K / K)$ convergence rate; under specific structural assumptions, this improves to $O(1/sqrt{K})$. Empirical evaluation on misspecified portfolio optimization demonstrates significant performance gains over state-of-the-art baselines, validating the framework’s flexibility, practical efficacy, and theoretical advantages.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling for targeted and personalized online advertising

Abstract

We study a class of misspecified saddle point (SP) problems, where the optimization objective depends on an unknown parameter that must be learned concurrently from data. Unlike existing studies that assume parameters are fully known or pre-estimated, our framework integrates optimization and learning into a unified formulation, enabling a more flexible problem class. To address this setting, we propose two algorithms based on the accelerated primal-dual (APD) by Hamedani & Aybat 2021. In particular, we first analyze the naive extension of the APD method by directly substituting the evolving parameter estimates into the primal-dual updates; then, we design a new learning-aware variant of the APD method that explicitly accounts for parameter dynamics by adjusting the momentum updates. Both methods achieve a provable convergence rate of $mathcal{O}(log K / K)$, while the learning-aware approach attains a tighter $mathcal{O}(1)$ constant and further benefits from an adaptive step-size selection enabled by a backtracking strategy. Furthermore, we extend the framework to problems where the learning problem admits multiple optimal solutions, showing that our modified algorithm for a structured setting achieves an $mathcal{O}(1/sqrt{K})$ rate. To demonstrate practical impact, we evaluate our methods on a misspecified portfolio optimization problem and show superior empirical performance compared to state-of-the-art algorithms.

Problem

• Simultaneously learning unknown parameters while optimizing misspecified saddle point problems
• Integrating optimization and learning into a unified framework formulation
• Developing accelerated primal-dual algorithms with provable convergence rates

Innovation

• Integrates optimization and learning into unified formulation
• Proposes accelerated primal-dual algorithms with parameter dynamics
• Uses backtracking strategy for adaptive step-size selection