# Consistent Approximations in Composite Optimization

arXiv: https://arxiv.org/abs/2201.05250 · HTML: https://mlsanity.com/paper/2201.05250/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 63 references that are in the map.
- First posted on arXiv 13 Jan 2022.

## Builds on (most cited first, 12 of 12)

- [Understanding Black-box Predictions via Influence Functions](https://mlsanity.com/paper/1703.04730/) (2017)
- [Influence Functions in Deep Learning Are Fragile](https://mlsanity.com/paper/2006.14651/) (2020)
- [Stochastic model-based minimization of weakly convex functions](https://mlsanity.com/paper/1803.06523/) (2018)
- [Error bounds, quadratic growth, and linear convergence of proximal methods](https://mlsanity.com/paper/1602.06661/) (2016)
- [Efficiency of minimizing compositions of convex functions and smooth maps](https://mlsanity.com/paper/1605.00125/) (2016)
- [Stochastic Methods for Composite and Weakly Convex Optimization Problems](https://mlsanity.com/paper/1703.08570/) (2017)
- [Global Guarantees for Enforcing Deep Generative Priors by Empirical Risk](https://mlsanity.com/paper/1705.07576/) (2017)
- [Graphical Convergence of Subgradients in Nonconvex Optimization and Learning](https://mlsanity.com/paper/1810.07590/) (2018)
- [A Provably Convergent Scheme for Compressive Sensing under Random Generative Priors](https://mlsanity.com/paper/1812.04176/) (2018)
- [Composite Difference-Max Programs for Modern Statistical Estimation Problems](https://mlsanity.com/paper/1803.00205/) (2018)
- [Stability and Error Analysis for Optimization and Generalized Equations](https://mlsanity.com/paper/1903.08754/) (2019)
- [A study of convex convex-composite functions via infimal convolution with applications](https://mlsanity.com/paper/1907.08318/) (2019)

## Abstract (arXiv)

Approximations of optimization problems arise in computational procedures and sensitivity analysis. The resulting effect on solutions can be significant, with even small approximations of components of a problem translating into large errors in the solutions. We specify conditions under which approximations are well behaved in the sense of minimizers, stationary points, and level-sets and this leads to a framework of consistent approximations. The framework is developed for a broad class of composite problems, which are neither convex nor smooth. We demonstrate the framework using examples from stochastic optimization, neural-network based machine learning, distributionally robust optimization, penalty and augmented Lagrangian methods, interior-point methods, homotopy methods, smoothing methods, extended nonlinear programming, difference-of-convex programming, and multi-objective optimization. An enhanced proximal method illustrates the algorithmic possibilities. A quantitative analysis supplements the development by furnishing rates of convergence.