# Differentiating Through a Cone Program

arXiv: https://arxiv.org/abs/1904.09043 · HTML: https://mlsanity.com/paper/1904.09043/

## Key facts

- Cited by 73 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Apr 2019.

## Built on by (most cited first, 30 of 73)

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## Abstract (arXiv)

We consider the problem of efficiently computing the derivative of the solution map of a convex cone program, when it exists. We do this by implicitly differentiating the residual map for its homogeneous self-dual embedding, and solving the linear systems of equations required using an iterative method. This allows us to efficiently compute the derivative operator, and its adjoint, evaluated at a vector. These correspond to computing an approximate new solution, given a perturbation to the cone program coefficients (i.e., perturbation analysis), and to computing the gradient of a function of the solution with respect to the coefficients. Our method scales to large problems, with numbers of coefficients in the millions. We present an open-source Python implementation of our method that solves a cone program and returns the derivative and its adjoint as abstract linear maps; our implementation can be easily integrated into software systems for automatic differentiation.