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This paper develops a comprehensive derivative theory of the covariance regret functional, revealing that the steepest-descent direction for minimizing expected regret is the contrarian policy derived from the difference between costs and their mean. The authors demonstrate that for linear policies, the gradient corresponds to the cost covariance matrix, leading to boundary-optimal solutions like the minimum-variance portfolio. Additionally, they explore constrained optimization and establish connections between regret minimization and alpha maximization, providing finite-sample convergence bounds and practical applications in portfolio management and LLM-based allocation strategies.
The steepest-descent direction for minimizing regret reveals a contrarian policy that could transform decision-making in portfolio management and AI allocation strategies.
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.