We investigate how machine learning models acquire the ability to compose music and how musical information is internally represented within such models. We develop a composition algorithm based on a restricted Boltzmann machine (RBM), a simple generative model capable of producing musical pieces of arbitrary length. We convert musical scores into piano-roll image representations and train the RBM in an unsupervised manner. We confirm that the trained RBM can generate new musical pieces; however, by analyzing the model's responses and internal structure, we find that the learned information is not stored in a form directly interpretable by humans. This study contributes to a better understanding of how machine learning models capable of music composition may internally represent musical structure and highlights issues related to the interpretability of generative models in creative tasks.

Large Language Models (LLMs) have demonstrated strong performance across a wide range of tasks, yet they still struggle with complex mathematical reasoning, a challenge fundamentally rooted in deep structural dependencies. To address this challenge, we propose \textbf{CA}usal \textbf{MA}thematician (\textbf{CAMA}), a two-stage causal framework that equips LLMs with explicit, reusable mathematical structure. In the learning stage, CAMA first constructs the \textbf{M}athematical \textbf{C}ausal \textbf{G}raph (\textbf{MCG}), a high-level representation of solution strategies, by combining LLM priors with causal discovery algorithms applied to a corpus of question-solution pairs. The resulting MCG encodes essential knowledge points and their causal dependencies. To better align the graph with downstream reasoning tasks, CAMA further refines the MCG through iterative feedback derived from a selected subset of the question-solution pairs. In the reasoning stage, given a new question, CAMA dynamically extracts a task-relevant subgraph from the MCG, conditioned on both the question content and the LLM's intermediate reasoning trace. This subgraph, which encodes the most pertinent knowledge points and their causal dependencies, is then injected back into the LLM to guide its reasoning process. Empirical results on real-world datasets show that CAMA significantly improves LLM performance on challenging mathematical problems. Furthermore, our experiments demonstrate that structured guidance consistently outperforms unstructured alternatives, and that incorporating asymmetric causal relationships yields greater improvements than using symmetric associations alone.

Markov Chain Monte Carlo (MCMC) is a flexible approach to approximate sampling from intractable probability distributions, with a rich theoretical foundation and comprising a wealth of exemplar algorithms. While the qualitative correctness of MCMC algorithms is often easy to ensure, their practical efficiency is contingent on the `target' distribution being reasonably well-behaved. In this work, we concern ourself with the scenario in which this good behaviour is called into question, reviewing an emerging line of work on `robust' MCMC algorithms which can perform acceptably even in the face of certain pathologies. We focus on two particular pathologies which, while simple, can already have dramatic effects on standard `local' algorithms. The first is roughness, whereby the target distribution varies so rapidly that the numerical stability of the algorithm is tenuous. The second is flatness, whereby the landscape of the target distribution is instead so barren and uninformative that one becomes lost in uninteresting parts of the state space. In each case, we formulate the pathology in concrete terms, review a range of proposed algorithmic remedies to the pathology, and outline promising directions for future research.

V ariational inference (VI) is a cornerstone of modern Bayesian learning, enabling approximate inference in complex models that would otherwise be intractable. However, its formulation depends on expectations and divergences defined through high-dimensional integrals, often rendering analytical treatment impossible and necessitating heavy reliance on approximate learning and inference techniques. Possibility theory, an imprecise probability framework, allows to directly model epistemic uncertainty instead of leveraging subjective probabilities. While this framework provides robustness and interpretability under sparse or imprecise information, adapting VI to the possibilistic setting requires rethinking core concepts such as entropy and divergence, which presuppose additivity. In this work, we develop a principled formulation of possibilistic variational inference and apply it to a special class of exponential-family functions, highlighting parallels with their probabilistic counterparts and revealing the distinctive mathematical structures of possibility theory.

We consider the problem of estimating Ising models over $n$ variables in Total Variation (TV) distance, given $l$ independent samples from the model. While the statistical complexity of the problem is well-understood [DMR20], identifying computationally and statistically efficient algorithms has been challenging. In particular, remarkable progress has occurred in several settings, such as when the underlying graph is a tree [DP21, BGPV21], when the entries of the interaction matrix follow a Gaussian distribution [GM24, CK24], or when the bulk of its eigenvalues lie in a small interval [AJK+24, KLV24], but no unified framework for polynomial-time estimation in TV exists so far. Our main contribution is a unified analysis of the Maximum Pseudo-Likelihood Estimator (MPLE) for two general classes of Ising models. The first class includes models that have bounded operator norm and satisfy the Modified Log-Sobolev Inequality (MLSI), a functional inequality that was introduced to study the convergence of the associated Glauber dynamics to stationarity. In the second class of models, the interaction matrix has bounded infinity norm (or bounded width), which is the most common assumption in the literature for structure learning of Ising models. We show how our general results for these classes yield polynomial-time algorithms and optimal or near-optimal sample complexity guarantees in a variety of settings. Our proofs employ a variety of tools from tensorization inequalities to measure decompositions and concentration bounds.

Cooperative multi-agent reinforcement learning (MARL) commonly adopts centralized training with decentralized execution (CTDE), where centralized critics leverage global information to guide decentralized actors. However, centralized-decentralized mismatch (CDM) arises when the suboptimal behavior of one agent degrades others' learning. Prior approaches mitigate CDM through value decomposition, but linear decompositions allow per-agent gradients at the cost of limited expressiveness, while nonlinear decompositions improve representation but require centralized gradients, reintroducing CDM. To overcome this trade-off, we propose the multi-agent cross-entropy method (MCEM), combined with monotonic nonlinear critic decomposition (NCD). MCEM updates policies by increasing the probability of high-value joint actions, thereby excluding suboptimal behaviors. For sample efficiency, we extend off-policy learning with a modified k-step return and Retrace. Analysis and experiments demonstrate that MCEM outperforms state-of-the-art methods across both continuous and discrete action benchmarks.

In humans and other animals, category learning enhances discrimination between stimuli close to the category boundary. This phenomenon, called categorical perception, was also empirically observed in artificial neural networks trained on classification tasks. In previous modeling works based on neuroscience data, we show that this expansion/compression is a necessary outcome of efficient learning. Here we extend our theoretical framework to artificial networks. We show that minimizing the Bayes cost (mean of the cross-entropy loss) implies maximizing the mutual information between the set of categories and the neural activities prior to the decision layer. Considering structured data with an underlying feature space of small dimension, we show that maximizing the mutual information implies (i) finding an appropriate projection space, and, (ii) building a neural representation with the appropriate metric. The latter is based on a Fisher information matrix measuring the sensitivity of the neural activity to changes in the projection space. Optimal learning makes this neural Fisher information follow a category-specific Fisher information, measuring the sensitivity of the category membership. Category learning thus induces an expansion of neural space near decision boundaries. We characterize the properties of the categorical Fisher information, showing that its eigenvectors give the most discriminant directions at each point of the projection space. We find that, unexpectedly, its maxima are in general not exactly at, but near, the class boundaries. Considering toy models and the MNIST dataset, we numerically illustrate how after learning the two Fisher information matrices match, and essentially align with the category boundaries. Finally, we relate our approach to the Information Bottleneck one, and we exhibit a bias-variance decomposition of the Bayes cost, of interest on its own.

Higher-order networks, naturally described as hypergraphs, are essential for modeling real-world systems involving interactions among three or more entities. Stochastic block models offer a principled framework for characterizing mesoscale organization, yet their extension to hypergraphs involves a trade-off between expressive power and computational complexity. A recent simplification, a single-order model, mitigates this complexity by assuming a single affinity pattern governs interactions of all orders. This universal assumption, however, may overlook order-dependent structural details. Here, we propose a framework that relaxes this assumption by introducing a multi-order block structure, in which different affinity patterns govern distinct subsets of interaction orders. Our framework is based on a multi-order stochastic block model and searches for the optimal partition of the set of interaction orders that maximizes out-of-sample hyperlink prediction performance. Analyzing a diverse range of real-world networks, we find that multi-order block structures are prevalent. Accounting for them not only yields better predictive performance over the single-order model but also uncovers sharper, more interpretable mesoscale organization. Our findings reveal that order-dependent mechanisms are a key feature of the mesoscale organization of real-world higher-order networks.

Embodied cognition argues that intelligence arises from sensorimotor interaction rather than passive observation. It raises an intriguing question: do modern vision-language models (VLMs), trained largely in a disembodied manner, exhibit signs of embodied cognition? We introduce ENACT, a benchmark that casts evaluation of embodied cognition as world modeling from egocentric interaction in a visual question answering (VQA) format. Framed as a partially observable Markov decision process (POMDP) whose actions are scene graph changes, ENACT comprises two complementary sequence reordering tasks: forward world modeling (reorder shuffled observations given actions) and inverse world modeling (reorder shuffled actions given observations). While conceptually simple, solving these tasks implicitly demands capabilities central to embodied cognition-affordance recognition, action-effect reasoning, embodied awareness, and interactive, long-horizon memory from partially observable egocentric input, while avoiding low-level image synthesis that could confound the evaluation. We provide a scalable pipeline that synthesizes QA pairs from robotics simulation (BEHAVIOR) and evaluates models on 8,972 QA pairs spanning long-horizon home-scale activities. Experiments reveal a performance gap between frontier VLMs and humans that widens with interaction horizon. Models consistently perform better on the inverse task than the forward one and exhibit anthropocentric biases, including a preference for right-handed actions and degradation when camera intrinsics or viewpoints deviate from human vision. Website at https://enact-embodied-cognition.github.io/.

Existing studies on reinforcement learning (RL) for sepsis management have mostly followed an established problem setup, in which patient data are aggregated into 4-hour time steps. Although concerns have been raised regarding the coarseness of this time-step size, which might distort patient dynamics and lead to suboptimal treatment policies, the extent to which this is a problem in practice remains unexplored. In this work, we conducted empirical experiments for a controlled comparison of four time-step sizes ($Δt\!=\!1,2,4,8$ h) on this domain, following an identical offline RL pipeline. To enable a fair comparison across time-step sizes, we designed action re-mapping methods that allow for evaluation of policies on datasets with different time-step sizes, and conducted cross-$Δt$ model selections under two policy learning setups. Our goal was to quantify how time-step size influences state representation learning, behavior cloning, policy training, and off-policy evaluation. Our results show that performance trends across $Δt$ vary as learning setups change, while policies learned at finer time-step sizes ($Δt = 1$ h and $2$ h) using a static behavior policy achieve the overall best performance and stability. Our work highlights time-step size as a core design choice in offline RL for healthcare and provides evidence supporting alternatives beyond the conventional 4-hour setup.