Optimizing Regret

Irene Aldridge

Papers from arXiv.org

Abstract: Building on the identity that expected regret equals the covariance between costs and decisions, this paper develops the complete derivative theory of the covariance regret functional. We derive the G\^ateaux derivative, showing that the universal steepest-descent direction is the contrarian policy $-(c-\bar{c})$, while ascent yields momentum. For linear policies $\hat\pi(c) = Ac+b$, the gradient is the cost covariance matrix $\Sigma_c$, 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, with applications to portfolio tilting and LLM-based allocation strategies.

Date: 2026-07
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