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This paper advances the theoretical framework of regret minimization by deriving complete derivative theory and linking it directly to covariance structures in decision-making. By establishing connections between gradient descent, contrarian policies, and applications like portfolio tilting or LLM-based allocation strategies, it provides a rigorous mathematical foundation for optimizing performance under uncertainty. The work bridges econometric methods with machine learning algorithms, offering new tools for analyzing sequential decisions where minimizing regret is the primary objective.
Optimizing Regret
arXiv:2607.18866v1 Announce Type: cross Abstract: 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.
Abstract: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 Gâteaux 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.
Econometrics (econ.EM); Machine Learning (cs.LG); Machine Learning (stat.ML)
arXiv:2607.18866 [econ.EM]
(or arXiv:2607.18866v1 [econ.EM] for this version)
https://doi.org/10.48550/arXiv.2607.18866 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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