# Automatic differentiation of nonsmooth iterative algorithms

arXiv: https://arxiv.org/abs/2206.00457 · HTML: https://mlsanity.com/paper/2206.00457/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 59 references that are in the map.
- First posted on arXiv 31 May 2022.

## Builds on (most cited first, 21 of 21)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Meta-Learning with Implicit Gradients](https://mlsanity.com/paper/1909.04630/) (2019)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Optimizing Millions of Hyperparameters by Implicit Differentiation](https://mlsanity.com/paper/1911.02590/) (2019)
- [Hyperparameter optimization with approximate gradient](https://mlsanity.com/paper/1602.02355/) (2016)
- [Efficient and Modular Implicit Differentiation](https://mlsanity.com/paper/2105.15183/) (2021)
- [Implicit Deep Learning](https://mlsanity.com/paper/1908.06315/) (2019)
- [Gradient Step Denoiser for convergent Plug-and-Play](https://mlsanity.com/paper/2110.03220/) (2021)
- [Conservative set valued fields, automatic differentiation, stochastic gradient method and deep learning](https://mlsanity.com/paper/1909.10300/) (2019)
- [Global convergence of the Heavy-ball method for convex optimization](https://mlsanity.com/paper/1412.7457/) (2014)
- [Adaptive piecewise polynomial estimation via trend filtering](https://mlsanity.com/paper/1304.2986/) (2013)
- [A mathematical model for automatic differentiation in machine learning](https://mlsanity.com/paper/2006.02080/) (2020)
- [Nonsmooth Implicit Differentiation for Machine Learning and Optimization](https://mlsanity.com/paper/2106.04350/) (2021)
- [Implicit differentiation of Lasso-type models for hyperparameter optimization](https://mlsanity.com/paper/2002.08943/) (2020)
- [Super-efficiency of automatic differentiation for functions defined as a minimum](https://mlsanity.com/paper/2002.03722/) (2020)
- [On Correctness of Automatic Differentiation for Non-Differentiable Functions](https://mlsanity.com/paper/2006.06903/) (2020)
- [Automatic Differentiation of Some First-Order Methods in Parametric Optimization](https://mlsanity.com/paper/1910.05696/) (2019)
- [Techniques for Gradient Based Bilevel Optimization with Nonsmooth Lower Level Problems](https://mlsanity.com/paper/1602.07080/) (2016)
- [Provably Correct Automatic Subdifferentiation for Qualified Programs](https://mlsanity.com/paper/1809.08530/) (2018)
- [Path differentiability of ODE flows](https://mlsanity.com/paper/2201.03819/) (2022)

## Built on by (most cited first, 25 of 25)

- [On Penalty-based Bilevel Gradient Descent Method](https://mlsanity.com/paper/2302.05185/) (2023)
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- [Synergies between Disentanglement and Sparsity: Generalization and Identifiability in Multi-Task Learning](https://mlsanity.com/paper/2211.14666/) (2022)
- [Principled Penalty-based Methods for Bilevel Reinforcement Learning and RLHF](https://mlsanity.com/paper/2402.06886/) (2024)
- [One-step differentiation of iterative algorithms](https://mlsanity.com/paper/2305.13768/) (2023)
- [Functional Bilevel Optimization for Machine Learning](https://mlsanity.com/paper/2403.20233/) (2024)
- [The derivatives of Sinkhorn-Knopp converge](https://mlsanity.com/paper/2207.12717/) (2022)
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- [Deep Equilibrium models for Poisson Imaging Inverse problems via Mirror Descent](https://mlsanity.com/paper/2507.11461/) (2025)
- [Differentiating Nonsmooth Solutions to Parametric Monotone Inclusion Problems](https://mlsanity.com/paper/2212.07844/) (2022)
- [Fixed-Point Automatic Differentiation of Forward--Backward Splitting Algorithms for Partly Smooth Functions](https://mlsanity.com/paper/2208.03107/) (2022)
- [An adaptively inexact first-order method for bilevel optimization with application to hyperparameter learning](https://mlsanity.com/paper/2308.10098/) (2023)
- [Nonsmooth Implicit Differentiation: Deterministic and Stochastic Convergence Rates](https://mlsanity.com/paper/2403.11687/) (2024)
- [An Adaptively Inexact Method for Bilevel Learning Using Primal-Dual Style Differentiation](https://mlsanity.com/paper/2412.06436/) (2024)
- [Auto-Differentiation of Relational Computations for Very Large Scale Machine Learning](https://mlsanity.com/paper/2306.00088/) (2023)
- [Enhancing Hypergradients Estimation: A Study of Preconditioning and Reparameterization](https://mlsanity.com/paper/2402.16748/) (2024)
- [Convergence of Decentralized Stochastic Subgradient-based Methods for Nonsmooth Nonconvex functions](https://mlsanity.com/paper/2403.11565/) (2024)
- [Automatic Differentiation of Optimization Algorithms with Time-Varying Updates](https://mlsanity.com/paper/2410.15923/) (2024)
- [DynaLay: An Introspective Approach to Dynamic Layer Selection for Deep Networks](https://mlsanity.com/paper/2312.12781/) (2023)
- [Derivatives of Stochastic Gradient Descent in parametric optimization](https://mlsanity.com/paper/2405.15894/) (2024)
- [Change of Thought: Adaptive Test-Time Computation](https://mlsanity.com/paper/2507.13569/) (2025)
- [A Single-Loop First-Order Algorithm for Linearly Constrained Bilevel Optimization](https://mlsanity.com/paper/2510.24710/) (2025)
- [Bilevel Learning via Inexact Stochastic Gradient Descent](https://mlsanity.com/paper/2511.06774/) (2025)
- [Understanding the Curse of Unrolling](https://mlsanity.com/paper/2602.19733/) (2026)

## Abstract (arXiv)

Differentiation along algorithms, i.e., piggyback propagation of derivatives, is now routinely used to differentiate iterative solvers in differentiable programming. Asymptotics is well understood for many smooth problems but the nondifferentiable case is hardly considered. Is there a limiting object for nonsmooth piggyback automatic differentiation (AD)? Does it have any variational meaning and can it be used effectively in machine learning? Is there a connection with classical derivative? All these questions are addressed under appropriate nonexpansivity conditions in the framework of conservative derivatives which has proved useful in understanding nonsmooth AD. For nonsmooth piggyback iterations, we characterize the attractor set of nonsmooth piggyback iterations as a set-valued fixed point which remains in the conservative framework. This has various consequences and in particular almost everywhere convergence of classical derivatives. Our results are illustrated on parametric convex optimization problems with forward-backward, Douglas-Rachford and Alternating Direction of Multiplier algorithms as well as the Heavy-Ball method.