This paper derives the complete theory of covariance regret functional for decision making, providing insights on steepest-descent directions and boundary-optimal solutions.
Provides a complete theoretical derivation of the covariance regret functional and its implications.
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Applications
- →portfolio tilting
- →LLM-based allocation strategies
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Abstract
More Like ThisBuilding 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 Gâteaux 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 $Σ_c$, 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.