Optimizing Regret
Irene Aldridge
econ.EM, cs.LG, stat.ML
Submitted: 2026-07-21
Comments: 12 pages
License: http://creativecommons.org/licenses/by/4.0/
The gist: 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.
Terminology
Abstract
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 G ateaux derivative, showing that the universal steepest-descent direction is the contrarian policy -(c- c), while ascent yields momentum. For linear policies pi(c)=Ac+b, the gradient is the cost covariance matrix 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.
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