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 $\hatπ(c) = Ac+b$, 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.
We show that net demand for liquidity by algo strategies is identifiable from its trade and price history alone, with no knowledge of its signal or optimization problem. An exact multi-period regret decomposition implies that the sign of this statistic classifies a linear strateg…
We propose the first deep learning algorithm, the Certainty Equivalent Learning (CEL) algorithm, for solving high-dimensional discrete-time dynamic programming problems with recursive utility. Dynamic programming with recursive utility is numerically challenging because the recur…
A specialist tolerates blind spots that a generalist does not. Usually this is treated as a cost to be minimized. We treat it as a design variable: a deficiency can be kept because it pays and removed on demand in the rare situation where it would be fatal, by routing to a compen…
We propose one-step and two-step methods for policy learning with retrieval-augmented generation (RAG). We formulate RAG-based action selection under the potential outcome framework. In the two-step method, vector search retrieves action-specific neighboring evidence in an embedd…
Automating theoretical research is constrained not only by the generation of candidate results, but also by their reliable evaluation. A common approach is to close the research loop with a large language model (LLM) reviewer. However, such reviewers remain empirically unreliable…
We study horizon-free regret minimization for finite-horizon time-homogeneous tabular Markov decision processes with $S$ states, $A$ actions, horizon $H$, and per-trajectory total reward bounded by $1$. We propose a new algorithm and prove a regret upper bound \[\tilde O(\sqrt{SA…