Finally, we obtain a generalization bound for meta learning withdependent episodes whose dependencyrelation ischaracterized byagraph.

The support/query episodic training strategy has been widely applied in modern meta learning algorithms. Supposing the $n$ training episodes and the test episodes are sampled independently from the same environment, previous work has derived a generalization bound of $O(1/\sqrt{n})$ for smooth non-convex functions via algorithmic stability analysis. In this paper, we provide fine-grained analysis of stability and generalization for modern meta learning algorithms by considering more general situations. Firstly, we develop matching lower and upper stability bounds for meta learning algorithms with two types of loss functions: (1) nonsmooth convex functions with $\alpha$-H{\o}lder continuous subgradients $(\alpha \in [0,1))$; (2) smooth (including convex and non-convex) functions. Our tight stability bounds show that, in the nonsmooth convex case, meta learning algorithms can be inherently less stable than in the smooth convex case.

Prototypical Networks learn a metric space in which classification can be performed by computing distances to prototype representations of each class.

Reward machines (RMs) are automata structures that encode (non-Markovian) reward functions for reinforcement learning (RL). RMs can reward any behaviour representable in regular languages and, when paired with RL algorithms that exploit RM structure, have been shown to significantly improve sample efficiency in many domains. In this work, we present pushdown reward machines (pdRMs), an extension of reward machines based on deterministic pushdown automata. pdRMs can recognise and reward temporally extended behaviours representable in deterministic context-free languages, making them more expressive than reward machines. We introduce two variants of pdRM-based policies, one which has access to the entire stack of the pdRM, and one which can only access the top $k$ symbols (for a given constant $k$) of the stack. We propose a procedure to check when the two kinds of policies (for a given environment, pdRM, and constant $k$) achieve the same optimal state values. We then provide theoretical results establishing the expressive power of pdRMs, and space complexity results for the proposed learning problems. Lastly, we propose an approach for off-policy RL algorithms that exploits counterfactual experiences with pdRMs. We conclude by providing experimental results showing how agents can be trained to perform tasks representable in deterministic context-free languages using pdRMs.

In embodied intelligence, datasets play a pivotal role, serving as both a knowledge repository and a conduit for information transfer. The two most critical attributes of a dataset are the amount of information it provides and how easily this information can be learned by models. However, the multimodal nature of embodied data makes evaluating these properties particularly challenging. Prior work has largely focused on diversity, typically counting tasks and scenes or evaluating isolated modalities, which fails to provide a comprehensive picture of dataset diversity. On the other hand, the learnability of datasets has received little attention and is usually assessed post-hoc through model training, an expensive, time-consuming process that also lacks interpretability, offering little guidance on how to improve a dataset. In this work, we address both challenges by introducing two principled, data-driven tools. First, we construct a unified multimodal representation for each data sample and, based on it, propose diversity entropy, a continuous measure that characterizes the amount of information contained in a dataset. Second, we introduce the first interpretable, data-driven algorithm to efficiently quantify dataset learnability without training, enabling researchers to assess a dataset's learnability immediately upon its release. We validate our algorithm on both simulated and real-world embodied datasets, demonstrating that it yields faithful, actionable insights that enable researchers to jointly improve diversity and learnability. We hope this work provides a foundation for designing higher-quality datasets that advance the development of embodied intelligence.

The double descent (DD) paradox, where over-parameterized models see generalization improve past the interpolation point, remains largely unexplored in the non-stationary domain of Deep Reinforcement Learning (DRL). We present preliminary evidence that DD exists in model-free DRL, investigating it systematically across varying model capacity using the Actor-Critic framework. We rely on an information-theoretic metric, Policy Entropy, to measure policy uncertainty throughout training. Preliminary results show a clear epoch-wise DD curve; the policy's entrance into the second descent region correlates with a sustained, significant reduction in Policy Entropy. This entropic decay suggests that over-parameterization acts as an implicit regularizer, guiding the policy towards robust, flatter minima in the loss landscape. These findings establish DD as a factor in DRL and provide an information-based mechanism for designing agents that are more general, transferable, and robust.

We thank all reviewers for their time and appreciate the thoughtful feedback. Below, we address the main comments. "In the example given by the author, the agent is allowed to run until it reaches a terminal state during We understand why this would be a concern, but it is actually not what we do. On the topic of terminal states, note that we have not explicitly defined any terminal states for the tasks from Figure 1. We will clarify this point further in the paper. "Their approach was marginally better than DQN on most Atari games [...] it would be nice to see some We hope that our clarification of the Figure 1 plots has increased your appreciation of low discount factors.

The synchronized flight of bird flocks, exemplified by starling murmurations, is perhaps the most striking example of collective behavior in natural systems, which fascinated scholars for quite a long time [1]. Evolutionary biologists, for instance, have long debated the advantages of living in groups [2], which should offer increased protection from predation by diluting the individual risk and 1 possibly confusing the attackers by the sheer size of the assembly. Flocking behavior involves a high degree of order in the individual directions of motion [3], and has been reproduced by minimal models of self-propelling particles (SPPs), such as Craig Reynolds Boids [4] or the celebrated Vicsek model [5] that has long captivated the attention of statistical physicists and played a pivotal role in the birth of the active matter research field. The essential ingredient of these models is the tendency of individual particles to align their direction of motion with those of their local neighbours, which is enough to promote long range order in systems with finite density (even in two spatial dimensions, due to the non-equilibrium nature of self-propelled particles) such as in toy models with periodic boundary conditions. In open systems, constituted by a finite number of individuals in an open, infinite space, purely alignment interactions are however not enough to maintain group cohesion.

Most successful applications of deep learning involve similar training and test conditions. However, tasks such as biological sequence design involve searching for sequences that improve desirable properties beyond previously known values, which requires novel hypotheses that \emph{extrapolate} beyond training data. In these settings, extrapolation may be achieved by using random search methods such as Markov chain Monte Carlo (MCMC), which, given an initial state, sample local transformations to approximate a target density that rewards states with the desired properties. However, even with a well-designed proposal, MCMC may struggle to explore large structured state spaces efficiently. Rather than relying on stochastic search, it would be desirable to have a model that greedily optimizes the properties of interest, successfully extrapolating in as few steps as possible. We propose to learn such a model from the Markov chains resulting from MCMC search. Specifically, our approach uses selected states from Markov chains as a source of training data for an autoregressive model, which is then able to efficiently generate novel sequences that extrapolate along the sequence-level properties of interest. The proposed approach is validated on three problems: protein sequence design, text sentiment control, and text anonymization. We find that the autoregressive model can extrapolate as well or better than MCMC, but with the additional benefits of scalability and significantly higher sample efficiency.