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机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文基于成本与决策协方差发展协方差遗憾泛函导数理论，推导线性策略梯度，扩展到约束优化等，得出通用最速下降方向，还给出收敛界及算法，应用于投资组合倾斜与大语言模型分配策略。
AI 中文摘要
基于预期遗憾值等于成本与决策之间协方差的恒等式，本文发展了协方差遗憾泛函的完整导数理论。我们推导了加托导数，表明通用最速下降方向是逆势策略$-(c - \bar{c})$，上升则产生动量。对于线性策略$\hat\pi(c) = Ac + b$，梯度是成本协方差矩阵$\Sigma_c$，海森矩阵为零意味着边界最优解，如最小方差投资组合。我们扩展到约束优化、遗憾最小化与阿尔法最大化之间的符号梯度对偶性、与汤普森采样平行的有限样本收敛界，以及仅需输入观测值的梯度下降算法，并应用于投资组合倾斜和基于大语言模型的分配策略。
英文摘要
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 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.
Comments12 pages
