# You Shall Pass: Dealing with the Zero-Gradient Problem in Predict and Optimize for Convex Optimization

arXiv: https://arxiv.org/abs/2307.16304 · HTML: https://mlsanity.com/paper/2307.16304/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 28 references that are in the map.
- First posted on arXiv 30 Jul 2023.

## Builds on (most cited first, 14 of 14)

- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Smart "Predict, then Optimize"](https://mlsanity.com/paper/1710.08005/) (2017)
- [Hyperparameter optimization with approximate gradient](https://mlsanity.com/paper/1602.02355/) (2016)
- [Melding the Data-Decisions Pipeline: Decision-Focused Learning for Combinatorial Optimization](https://mlsanity.com/paper/1809.05504/) (2018)
- [Differentiation of Blackbox Combinatorial Solvers](https://mlsanity.com/paper/1912.02175/) (2019)
- [A Review on Bilevel Optimization: From Classical to Evolutionary Approaches and Applications](https://mlsanity.com/paper/1705.06270/) (2017)
- [MIPaaL: Mixed Integer Program as a Layer](https://mlsanity.com/paper/1907.05912/) (2019)
- [Interior Point Solving for LP-based prediction+optimisation](https://mlsanity.com/paper/2010.13943/) (2020)
- [Learning with Differentiable Perturbed Optimizers](https://mlsanity.com/paper/2002.08676/) (2020)
- [Towards Gradient-based Bilevel Optimization with Non-convex Followers and Beyond](https://mlsanity.com/paper/2110.00455/) (2021)
- [Enforcing Policy Feasibility Constraints through Differentiable Projection for Energy Optimization](https://mlsanity.com/paper/2105.08881/) (2021)
- [Automatically Learning Compact Quality-aware Surrogates for Optimization Problems](https://mlsanity.com/paper/2006.10815/) (2020)
- [Efficient Gradient Approximation Method for Constrained Bilevel Optimization](https://mlsanity.com/paper/2302.01970/) (2023)
- [Optimal Power Flow in Stand-alone DC Microgrids](https://mlsanity.com/paper/1708.05140/) (2017)

## Built on by (most cited first, 3 of 3)

- [Robust Losses for Decision-Focused Learning](https://mlsanity.com/paper/2310.04328/) (2023)
- [Sufficient Decision Proxies for Decision-Focused Learning](https://mlsanity.com/paper/2505.03953/) (2025)
- [Minimizing Surrogate Losses for Decision-Focused Learning using Differentiable Optimization](https://mlsanity.com/paper/2508.11365/) (2025)

## Abstract (arXiv)

Predict and optimize is an increasingly popular decision-making paradigm that employs machine learning to predict unknown parameters of optimization problems. Instead of minimizing the prediction error of the parameters, it trains predictive models using task performance as a loss function. The key challenge to train such models is the computation of the Jacobian of the solution of the optimization problem with respect to its parameters. For linear problems, this Jacobian is known to be zero or undefined; hence, approximations are usually employed. For non-linear convex problems, however, it is common to use the exact Jacobian. This paper demonstrates that the zero-gradient problem appears in the non-linear case as well -- the Jacobian can have a sizeable null space, thereby causing the training process to get stuck in suboptimal points. Through formal proofs, this paper shows that smoothing the feasible set resolves this problem. Combining this insight with known techniques from the literature, such as quadratic programming approximation and projection distance regularization, a novel method to approximate the Jacobian is derived. In simulation experiments, the proposed method increases the performance in the non-linear case and at least matches the existing state-of-the-art methods for linear problems.