Optimizing Regret

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.