# Dual Optimistic Ascent (PI Control) is the Augmented Lagrangian Method in Disguise

arXiv: https://arxiv.org/abs/2509.22500 · HTML: https://mlsanity.com/paper/2509.22500/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 51 references that are in the map.
- First posted on arXiv 26 Sep 2025.

## Builds on (most cited first, 27 of 27)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Safe RLHF: Safe Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2310.12773/) (2023)
- [Responsive Safety in Reinforcement Learning by PID Lagrangian Methods](https://mlsanity.com/paper/2007.03964/) (2020)
- [Differentiable Causal Discovery from Interventional Data](https://mlsanity.com/paper/2007.01754/) (2020)
- [A Variational Inequality Perspective on Generative Adversarial Networks](https://mlsanity.com/paper/1802.10551/) (2018)
- [A Unified Analysis of Extra-gradient and Optimistic Gradient Methods for Saddle Point Problems: Proximal Point Approach](https://mlsanity.com/paper/1901.08511/) (2019)
- [Optimization with Non-Differentiable Constraints with Applications to Fairness, Recall, Churn, and Other Goals](https://mlsanity.com/paper/1809.04198/) (2018)
- [The Limit Points of (Optimistic) Gradient Descent in Min-Max Optimization](https://mlsanity.com/paper/1807.03907/) (2018)
- [Adversarial Robustness with Semi-Infinite Constrained Learning](https://mlsanity.com/paper/2110.15767/) (2021)
- [Last-Iterate Convergence of Optimistic Gradient Method for Monotone Variational Inequalities](https://mlsanity.com/paper/2205.08446/) (2022)
- [Near-optimal Local Convergence of Alternating Gradient Descent-Ascent for Minimax Optimization](https://mlsanity.com/paper/2102.09468/) (2021)
- [Controlled Sparsity via Constrained Optimization or: How I Learned to Stop Tuning Penalties and Love Constraints](https://mlsanity.com/paper/2208.04425/) (2022)
- [A Lagrangian Duality Approach to Active Learning](https://mlsanity.com/paper/2202.04108/) (2022)
- [FairALM: Augmented Lagrangian Method for Training Fair Models with Little Regret](https://mlsanity.com/paper/2004.01355/) (2020)
- [ReLOAD: Reinforcement Learning with Optimistic Ascent-Descent for Last-Iterate Convergence in Constrained MDPs](https://mlsanity.com/paper/2302.01275/) (2023)
- [Automatic Data Augmentation via Invariance-Constrained Learning](https://mlsanity.com/paper/2209.15031/) (2022)
- [Learning Constrained Optimization with Deep Augmented Lagrangian Methods](https://mlsanity.com/paper/2403.03454/) (2024)
- [On PI Controllers for Updating Lagrange Multipliers in Constrained Optimization](https://mlsanity.com/paper/2406.04558/) (2024)
- [Linear Convergence of Primal-Dual Gradient Methods and their Performance in Distributed Optimization](https://mlsanity.com/paper/1904.01196/) (2019)
- [Cooper: A Library for Constrained Optimization in Deep Learning](https://mlsanity.com/paper/2504.01212/) (2025)
- [Alignment of large language models with constrained learning](https://mlsanity.com/paper/2505.19387/) (2025)
- [Tight Analysis of Extra-gradient and Optimistic Gradient Methods For Nonconvex Minimax Problems](https://mlsanity.com/paper/2210.09382/) (2022)
- [Balancing Act: Constraining Disparate Impact in Sparse Models](https://mlsanity.com/paper/2310.20673/) (2023)
- [Feasible Learning](https://mlsanity.com/paper/2501.14912/) (2025)
- [Position: Adopt Constraints Over Fixed Penalties in Deep Learning](https://mlsanity.com/paper/2505.20628/) (2025)
- [Alpaqa: A matrix-free solver for nonlinear MPC and large-scale nonconvex optimization](https://mlsanity.com/paper/2112.02370/) (2021)
- [Neural Networks with Quantization Constraints](https://mlsanity.com/paper/2210.15623/) (2022)

## Built on by (most cited first, 1 of 1)

- [Matrix-Valued Optimism is Matrix-Valued Augmentation: Additive Hybrid Designs for Constrained Optimization](https://mlsanity.com/paper/2605.06141/) (2026)

## Abstract (arXiv)

Constrained optimization is a powerful framework for enforcing requirements on neural networks. These constrained deep learning problems are typically solved using first-order methods on their min-max Lagrangian formulation, but such approaches often suffer from oscillations and can fail to find all local solutions. While the Augmented Lagrangian method (ALM) addresses these issues, practitioners often favor dual optimistic ascent schemes (PI control) on the standard Lagrangian, which perform well empirically but lack formal guarantees. In this paper, we establish a previously unknown equivalence between these approaches: dual optimistic ascent on the Lagrangian is equivalent to gradient descent-ascent on the Augmented Lagrangian. This finding allows us to transfer the robust theoretical guarantees of the ALM to the dual optimistic setting, proving it converges linearly to all local solutions. Furthermore, the equivalence provides principled guidance for tuning the optimism hyper-parameter. Our work closes a critical gap between the empirical success of dual optimistic methods and their theoretical foundation in the single-step, first-order regime commonly used in constrained deep learning.