# Optimizing Regret

arXiv: https://arxiv.org/abs/2607.18866 · HTML: https://mlsanity.com/paper/2607.18866/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 14 references that are in the map.
- First posted on arXiv 21 Jul 2026.

## Builds on (most cited first, 5 of 5)

- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Smart "Predict, then Optimize"](https://mlsanity.com/paper/1710.08005/) (2017)
- [Decision-Focused Learning with Directional Gradients](https://mlsanity.com/paper/2402.03256/) (2024)
- [Regret Equals Covariance: A Closed-Form Characterization for Stochastic Optimization](https://mlsanity.com/paper/2605.14019/) (2026)

## Abstract (arXiv)

Building on the identity that expected regret equals the covariance between costs and decisions, this paper develops a derivative theory of the covariance regret functional. We derive the Gateaux derivative, showing that the universal steepest-descent direction is the contrarian policy -(c-bar c), while ascent yields momentum. For linear policies hatπ(c)=Ac+b, the gradient is the cost covariance matrix Sigmac, with a zero Hessian implying boundary-optimal solutions such as the minimum-variance portfolio. We extend to constrained optimization, sign-gradient duality between regret minimization and alpha maximization, finite-sample convergence bounds paralleling Thompson Sampling, and gradient-descent algorithms requiring only input observations.