# Simultaneous Learning and Optimization via Misspecified Saddle Point Problems

arXiv: https://arxiv.org/abs/2510.05241 · HTML: https://mlsanity.com/paper/2510.05241/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 55 references that are in the map.
- First posted on arXiv 6 Oct 2025.

## Builds on (most cited first, 15 of 15)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Data-Driven Robust Optimization](https://mlsanity.com/paper/1401.0212/) (2013)
- [Integrated Conditional Estimation-Optimization](https://mlsanity.com/paper/2110.12351/) (2021)
- [An Accelerated Linearized Alternating Direction Method of Multipliers](https://mlsanity.com/paper/1401.6607/) (2014)
- [Exploiting Problem Structure in Optimization under Uncertainty via Online Convex Optimization](https://mlsanity.com/paper/1709.02490/) (2017)
- [Adaptive Sequential Stochastic Optimization](https://mlsanity.com/paper/1610.01970/) (2016)
- [On the solution of stochastic optimization and variational problems in imperfect information regimes](https://mlsanity.com/paper/1402.1457/) (2014)
- [Asynchronous Schemes for Stochastic and Misspecified Potential Games and Nonconvex Optimization](https://mlsanity.com/paper/1711.03963/) (2017)
- [Data-Driven Minimax Optimization with Expectation Constraints](https://mlsanity.com/paper/2202.07868/) (2022)
- [A Polyhedral Approximation Framework for Convex and Robust Distributed Optimization](https://mlsanity.com/paper/1303.6092/) (2013)
- [On the resolution of misspecified convex optimization and monotone variational inequality problems](https://mlsanity.com/paper/1408.5532/) (2014)
- [On the analysis of inexact augmented Lagrangian schemes for misspecified conic convex programs](https://mlsanity.com/paper/1608.01879/) (2016)
- [Distributed Stochastic Nash Equilibrium Learning in Locally Coupled Network Games with Unknown Parameters](https://mlsanity.com/paper/2204.00100/) (2022)
- [Distributed computation of equilibria in misspecified convex stochastic Nash games](https://mlsanity.com/paper/1308.5448/) (2013)
- [On the rate analysis of inexact augmented Lagrangian schemes for convex optimization problems with misspecified constraints](https://mlsanity.com/paper/1510.00490/) (2015)

## Abstract (arXiv)

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 O(log K / K), while the learning-aware approach attains a tighter 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 O(1/√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.