# Data-Driven Mirror Descent with Input-Convex Neural Networks

arXiv: https://arxiv.org/abs/2206.06733 · HTML: https://mlsanity.com/paper/2206.06733/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jun 2022.

## Built on by (most cited first, 8 of 8)

- [Learning to Warm-Start Fixed-Point Optimization Algorithms](https://mlsanity.com/paper/2309.07835/) (2023)
- [FSNet: Feasibility-Seeking Neural Network for Constrained Optimization with Guarantees](https://mlsanity.com/paper/2506.00362/) (2025)
- [Neural Approximate Mirror Maps for Constrained Diffusion Models](https://mlsanity.com/paper/2406.12816/) (2024)
- [Data-Driven Performance Guarantees for Classical and Learned Optimizers](https://mlsanity.com/paper/2404.13831/) (2024)
- [Finding Regularized Competitive Equilibria of Heterogeneous Agent Macroeconomic Models with Reinforcement Learning](https://mlsanity.com/paper/2303.04833/) (2023)
- [Unsupervised approaches based on optimal transport and convex analysis for inverse problems in imaging](https://mlsanity.com/paper/2311.08972/) (2023)
- [Blessing of Dimensionality for Approximating Sobolev Classes on Manifolds](https://mlsanity.com/paper/2408.06996/) (2024)
- [Majorization-Minimization Networks for Inverse Problems: An Application to EEG Imaging](https://mlsanity.com/paper/2602.03855/) (2026)

## Abstract (arXiv)

Learning-to-optimize is an emerging framework that seeks to speed up the solution of certain optimization problems by leveraging training data. Learned optimization solvers have been shown to outperform classical optimization algorithms in terms of convergence speed, especially for convex problems. Many existing data-driven optimization methods are based on parameterizing the update step and learning the optimal parameters (typically scalars) from the available data. We propose a novel functional parameterization approach for learned convex optimization solvers based on the classical mirror descent (MD) algorithm. Specifically, we seek to learn the optimal Bregman distance in MD by modeling the underlying convex function using an input-convex neural network (ICNN). The parameters of the ICNN are learned by minimizing the target objective function evaluated at the MD iterate after a predetermined number of iterations. The inverse of the mirror map is modeled approximately using another neural network, as the exact inverse is intractable to compute. We derive convergence rate bounds for the proposed learned mirror descent (LMD) approach with an approximate inverse mirror map and perform extensive numerical evaluation on various convex problems such as image inpainting, denoising, learning a two-class support vector machine (SVM) classifier and a multi-class linear classifier on fixed features.