# Nonsmooth Implicit Differentiation: Deterministic and Stochastic Convergence Rates

arXiv: https://arxiv.org/abs/2403.11687 · HTML: https://mlsanity.com/paper/2403.11687/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 41 references that are in the map.
- First posted on arXiv 18 Mar 2024.

## Builds on (most cited first, 29 of 29)

- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Meta-Learning with Implicit Gradients](https://mlsanity.com/paper/1909.04630/) (2019)
- [Meta-Learning with Differentiable Convex Optimization](https://mlsanity.com/paper/1904.03758/) (2019)
- [Bilevel Programming for Hyperparameter Optimization and Meta-Learning](https://mlsanity.com/paper/1806.04910/) (2018)
- [Meta-learning with differentiable closed-form solvers](https://mlsanity.com/paper/1805.08136/) (2018)
- [Gradient-based Hyperparameter Optimization through Reversible Learning](https://mlsanity.com/paper/1502.03492/) (2015)
- [Optimizing Millions of Hyperparameters by Implicit Differentiation](https://mlsanity.com/paper/1911.02590/) (2019)
- [Approximation Methods for Bilevel Programming](https://mlsanity.com/paper/1802.02246/) (2018)
- [Hyperparameter optimization with approximate gradient](https://mlsanity.com/paper/1602.02355/) (2016)
- [Bilevel Optimization: Convergence Analysis and Enhanced Design](https://mlsanity.com/paper/2010.07962/) (2020)
- [Efficient and Modular Implicit Differentiation](https://mlsanity.com/paper/2105.15183/) (2021)
- [Towards Poisoning of Deep Learning Algorithms with Back-gradient Optimization](https://mlsanity.com/paper/1708.08689/) (2017)
- [On the Iteration Complexity of Hypergradient Computation](https://mlsanity.com/paper/2006.16218/) (2020)
- [Is feature selection secure against training data poisoning?](https://mlsanity.com/paper/1804.07933/) (2018)
- [Conservative set valued fields, automatic differentiation, stochastic gradient method and deep learning](https://mlsanity.com/paper/1909.10300/) (2019)
- [Amortized Implicit Differentiation for Stochastic Bilevel Optimization](https://mlsanity.com/paper/2111.14580/) (2021)
- [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)
- [Automatic differentiation of nonsmooth iterative algorithms](https://mlsanity.com/paper/2206.00457/) (2022)
- [Implicit differentiation for fast hyperparameter selection in non-smooth convex learning](https://mlsanity.com/paper/2105.01637/) (2021)
- [Convergence Properties of Stochastic Hypergradients](https://mlsanity.com/paper/2011.07122/) (2020)
- [Convergence of Stochastic Proximal Gradient Algorithm](https://mlsanity.com/paper/1403.5074/) (2014)
- [Bilevel Optimization with a Lower-level Contraction: Optimal Sample Complexity without Warm-start](https://mlsanity.com/paper/2202.03397/) (2022)
- [BOML: A Modularized Bilevel Optimization Library in Python for Meta Learning](https://mlsanity.com/paper/2009.13357/) (2020)
- [Differentiating Nonsmooth Solutions to Parametric Monotone Inclusion Problems](https://mlsanity.com/paper/2212.07844/) (2022)
- [A Stochastic forward-backward splitting method for solving monotone inclusions in Hilbert spaces](https://mlsanity.com/paper/1403.7999/) (2014)

## Built on by (most cited first, 2 of 2)

- [Derivatives of Stochastic Gradient Descent in parametric optimization](https://mlsanity.com/paper/2405.15894/) (2024)
- [Bilevel Learning via Inexact Stochastic Gradient Descent](https://mlsanity.com/paper/2511.06774/) (2025)

## Abstract (arXiv)

We study the problem of efficiently computing the derivative of the fixed-point of a parametric nondifferentiable contraction map. This problem has wide applications in machine learning, including hyperparameter optimization, meta-learning and data poisoning attacks. We analyze two popular approaches: iterative differentiation (ITD) and approximate implicit differentiation (AID). A key challenge behind the nonsmooth setting is that the chain rule does not hold anymore. We build upon the work by Bolte et al. (2022), who prove linear convergence of nonsmooth ITD under a piecewise Lipschitz smooth assumption. In the deterministic case, we provide a linear rate for AID and an improved linear rate for ITD which closely match the ones for the smooth setting. We further introduce NSID, a new stochastic method to compute the implicit derivative when the contraction map is defined as the composition of an outer map and an inner map which is accessible only through a stochastic unbiased estimator. We establish rates for the convergence of NSID, encompassing the best available rates in the smooth setting. We also present illustrative experiments confirming our analysis.