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.

A.1 Details of "stronger recipe" In Table 1 of our main paper, we evaluate the impact of limited model capacity [1] and small-scale dataset by comparing the results of using "previous training recipe" and our "stronger recipe". We summarize the details of "stronger recipe" and present them in Table 13. Table 13: Stronger training strategy used for distillation. "B" and "C" represent strategies for training students on ImageNet-1K and CIFAR100, respectively. A.2 Numerical results In Figure 1 of our main paper, we present a comparison of performance gaps among vanilla KD and two logits-based baselines, i.e., DKD [2] and DIST [3], on two datasets of varying scales, to demonstrate the underestimation of vanilla KD on small-scale datasets.

Machine learning (ML) algorithms can often differ in performance across domains. Understanding why their performance differs is crucial for determining what types of interventions (e.g., algorithmic or operational) are most effective at closing the performance gaps. Aggregate decompositions express the total performance gap as the gap due to a shift in the feature distribution $p(X)$ plus the gap due to a shift in the outcome's conditional distribution $p(Y|X)$. While this coarse explanation is helpful for guiding root cause analyses, it provides limited details and can only suggest coarse fixes involving all variables in an ML system. Detailed decompositions quantify the importance of each variable to each term in the aggregate decomposition, which can provide a deeper understanding and suggest more targeted interventions. Although parametric methods exist for conducting a full hierarchical decomposition of an algorithm's performance gap at the aggregate and detailed levels, current nonparametric methods only cover parts of the hierarchy; many also require knowledge of the entire causal graph. We introduce a nonparametric hierarchical framework for explaining why the performance of an ML algorithm differs across domains, without requiring causal knowledge. Furthermore, we derive debiased, computationally-efficient estimators and statistical inference procedures to construct confidence intervals for the explanations.

In the era of large language models, model merging is a promising way to combine multiple task-specific models into a single multitask model without extra training. However, two challenges remain: (a) interference between different models and (b) heterogeneous data during testing. Traditional model merging methods often show significant performance gaps compared to fine-tuned models due to these issues. Additionally, a one-size-fits-all model lacks flexibility for diverse test data, leading to performance degradation. We show that both shared and exclusive task-specific knowledge are crucial for merging performance, but directly merging exclusive knowledge hinders overall performance. In view of this, we propose Twin-Merging, a method that encompasses two principal stages: (1) modularizing knowledge into shared and exclusive components, with compression to reduce redundancy and enhance efficiency; (2) dynamically merging shared and task-specific knowledge based on the input. This approach narrows the performance gap between merged and fine-tuned models and improves adaptability to heterogeneous data. Extensive experiments on $20$ datasets for both language and vision tasks demonstrate the effectiveness of our method, showing an average improvement of $28.34\%$ in absolute normalized score for discriminative tasks and even surpassing the fine-tuned upper bound on the generative tasks.

Machine learning (ML) algorithms can often differ in performance across domains.

Proof of Theorem 4.7 We first prove a Lemma. Theorem A.2. (Theorem 1 in [36]) Let ϵ = max Theorem 4.7 In a two-player game, suppose that According to Theorem A.2, we have J ( π, µ) J ( π, α) E CQL [20] puts regularization on the learning of Q function to penalize out-of-distribution actions. The CSP algorithm is illustrated in Algorithm 1. The proxy model is trained adversarially against our agent, therefore, we set the proxy's reward function to be the negative of our agent's reward. We show experiment details of the Maze example in this section.

This leads to a new MU paradigm, termed prune first, then unlearn, which infuses a sparse model prior into the unlearning process.