We study reinforcement learning in hybrid discrete-continuous action spaces, such as settings where the discrete component selects a regime (or index) and the continuous component optimizes within it -- a structure common in robotics, control, and operations problems. Standard model-free policy gradient methods rely on score-function (SF) estimators and suffer from severe credit-assignment issues in high-dimensional settings, leading to poor gradient quality. On the other hand, differentiable simulation largely sidesteps these issues by backpropagating through a simulator, but the presence of discrete actions or non-smooth dynamics yields biased or uninformative gradients. To address this, we propose Hybrid Policy Optimization (HPO), which backpropagates through the simulator wherever smoothness permits, using a mixed gradient estimator that combines pathwise and SF gradients while maintaining unbiasedness. We also show how problems with action discontinuities can be reformulated in hybrid form, further broadening its applicability. Empirically, HPO substantially outperforms PPO on inventory control and switched linear-quadratic regulator problems, with performance gaps increasing as the continuous action dimension grows. Finally, we characterize the structure of the mixed gradient, showing that its cross term -- which captures how continuous actions influence future discrete decisions -- becomes negligible near a discrete best response, thereby enabling approximate decentralized updates of the continuous and discrete components and reducing variance near optimality.

In two-player zero-sum extensive-form games, Nash equilibrium prescribes optimal strategies against perfectly rational opponents. However, it does not guarantee rational play in parts of the game tree that can only be reached by the players making mistakes. This can be problematic when operationalizing equilibria in the real world among imperfect players. Trembling-hand refinements are a sound remedy to this issue, and are subsets of Nash equilibria that are designed to handle the possibility that any of the players may make mistakes. In this paper, we initiate the study of equilibrium refinements for settings where one of the players is perfectly rational (the "machine") and the other may make mistakes.

In this section we prove the theoretical results around the dual curriculum game and use these results to show approximation bounds for our methods, given that they have reached a Nash equilibrium (NE). The first theorem is the main result that allows us to analyze dual curriculum games. The high-level result says that the NE of a dual curriculum game are approximate NE of the base game from the perspective of any of the individual players, or from the perspective of the joint strategy. Let Bbe the maximum difference between U1t and U2t, and let (π,θ1,θ2) be a NE for G. Then (π,pθ1 + (1 p)θ2) is an approximate NE for the base game with either teacher or for a teacher optimizing their joint objective. More precisely, it is a 2Bp(1 p)-approximate NE when Ut = pU1t + (1 p)U2t, a 2B(1 p)-approximate NE when Ut = U1t, and a 2Bp-approximate NE when Ut = U2t. At a high level, this is true because, for low values of p, the best-response strategies for the individual players can be thought of as approximate-best response strategies for the joint-player, and vis-versa. Since the Nash Equilibrium consists of each of the players playing their own best response, they must be playing an approximate best response for the joint-player. We provide a formal proof below: Proof. Let B be the maximum difference between U1t and U2t, and let (π,θ1,θ2) be a Nash Equilibrium for G. Then consider pθ1 + (1 p)θ2 as a strategy in the base game for the joint player pU1t + (1 p)U2t.

A.1 Proof of Theorem 1 To prove Theorem 1, we need the help of the following Lemma See Proposition 7.1 in [3]. Now we can prove our Theorem 1. Proof. For games with only one step (normal-form games, functional-form games), there is only one fixed state. Therefore, the distribution of state-action is equivalent to the distribution of the action. A.2 Proof of Theorem 2 Let us restate our Theorem 2 Theorem 2. For a given empirical payoff matrix A RM N and the reward vector aM+1 for policy M + ||(I A>(A>))aM+1||2, (18) where (A>) is the Moore-Penrose pseudoinverse of A>, and σmin(A) is the minimum singular value of A. Proof. The last equation comes from the analytic calculation of min1>β=1 ||β (A>) aM+1||2 using Lagrangian.

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There has been a resurgence of interest in multiagent reinforcement learning (MARL), due partly to the recent success of deep neural networks. The simplest form of MARL is independent reinforcement learning (InRL), where each agent treats all of its experience as part of its (non stationary) environment. In this paper, we first observe that policies learned using InRL can overfit to the other agents' policies during training, failing to sufficiently generalize during execution. We introduce a new metric, joint-policy correlation, to quantify this effect. We describe a meta-algorithm for general MARL, based on approximate best responses to mixtures of policies generated using deep reinforcement learning, and empirical game theoretic analysis to compute meta-strategies for policy selection.

Machine Learning critically relies on the assumption that the training data is representative of the unseen instances a learner faces at test time.