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 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.