Graph Neural Networks (GNNs) have become a prevailing tool for learning physical dynamics. However, they still encounter several challenges: 1) Physical laws abide by symmetry, which is a vital inductive bias accounting for model generalization and should be incorporated into the model design. Existing simulators either consider insufficient symmetry, or enforce excessive equivariance in practice when symmetry is partially broken by gravity.

Mixability has been shown to be a powerful tool to obtain algorithms with optimal regret. However, the resulting methods often suffer from high computational complexity which has reduced their practical applicability. For example, in the case of multiclass logistic regression, the aggregating forecaster (Foster et al. (2018)) achieves a regret of O(log(Bn)) whereas Online Newton Step achieves O(eBlog(n)) obtaining a double exponential gain in B (a bound on the norm of comparative functions). However, this high statistical performance is at the price of a prohibitive computational complexity O(n37). In this paper, we use quadratic surrogates to make aggregating forecasters more efficient. We show that the resulting algorithm has still high statistical performance for a large class of losses. In particular, we derive an algorithm for multi-class logistic regression with a regret bounded by O(Blog(n)) and a computational complexity of only O(n4).

Our main contribution in this work is an empirical finding that random General Value Functions (GVFs), i.e., deep action-conditional predictions--random both in what feature of observations they predict as well as in the sequence of actions the predictions are conditioned upon--form good auxiliary tasks for reinforcement learning (RL) problems. In particular, we show that random deep action-conditional predictions when used as auxiliary tasks yield state representations that produce control performance competitive with state-of-the-art hand-crafted auxiliary tasks like value prediction, pixel control, and CURL in both Atari and DeepMind Lab tasks. In another set of experiments we stop the gradients from the RL part of the network to the state representation learning part of the network and show, perhaps surprisingly, that the auxiliary tasks alone are sufficient to learn state representations good enough to outperform an end-to-end trained actor-critic baseline.

Unsupervised reinforcement learning aims at learning a generalist policy in a reward-free manner for fast adaptation to downstream tasks. Most of the existing methods propose to provide an intrinsic reward based on surprise. Maximizing or minimizing surprise drives the agent to either explore or gain control over its environment. However, both strategies rely on a strong assumption: the entropy of the environment's dynamics is either high or low. This assumption may not always hold in real-world scenarios, where the entropy of the environment's dynamics may be unknown.

We investigate the problem of minimizing the excess generalization error with respect to the best expert prediction in a finite family in the stochastic setting, under limited access to information. We assume that the learner only has access to a limited number of expert advices per training round, as well as for prediction. Assuming that the loss function is Lipschitz and strongly convex, we show that if we are allowed to see the advice of only one expert per round for T rounds in the training phase, or to use the advice of only one expert for prediction in the test phase, the worst-case excess risk is Ω(1/ T) with probability lower bounded by a constant. However, if we are allowed to see at least two actively chosen expert advices per training round and use at least two experts for prediction, the fast rate O(1/T) can be achieved. We design novel algorithms achieving this rate in this setting, and in the setting where the learner has a budget constraint on the total number of observed expert advices, and give precise instance-dependent bounds on the number of training rounds and queries needed to achieve a given generalization error precision.

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As large language models (LLMs) transition from chat interfaces to integral components of stochastic pipelines and systems approaching general intelligence, the ability to faithfully sample from specified probability distributions has become a functional requirement rather than a theoretical curiosity. We present the first large-scale, statistically powered audit of native probabilistic sampling in frontier LLMs, benchmarking 11 models across 15 distributions. To disentangle failure modes, we employ a dual-protocol design: Batch Generation, where a model produces $N{=}1000$ samples within one response, and Independent Requests, comprising $N{=}1000$ stateless calls. We observe a sharp protocol asymmetry: batch generation achieves only modest statistical validity, with a 7% median pass rate, while independent requests collapse almost entirely, with 10 of 11 models passing none of the distributions. Beyond this asymmetry, we reveal that sampling fidelity degrades monotonically with distributional complexity and aggravates as the sampling horizon $N$ increases. Finally, we demonstrate how the propagation of these failures into downstream real-world application tasks introduces systematic biases: models fail to enforce uniform answer-position constraints in Multiple Choice Question generation and systematically violate demographic targets in attribute-constrained text-to-image prompt synthesis. These findings indicate that current LLMs lack a functional internal sampler, necessitating external tools for applications requiring statistical guarantees.

Network meta-analysis (NMA) combines direct and indirect comparisons across a connected treatment network to estimate relative treatment effects. However, there is a lack of exact contribution decompositions that reproduce NMA estimates, particularly in the presence of multi-arm trials that induce within-study correlations. We address this reproducibility gap by developing a contrast-space projection formulation of NMA. Working in the space of all estimable pairwise treatment contrasts, we express the NMA estimator as an explicit linear mapping of the observed contrasts onto the consistency-constrained contrast space induced by orthogonal projection. Building on this representation, we introduce a rigorous study-based definition of direct and indirect evidence through a canonical within-study reduction that removes algebraic redundancy and yields a unique, invariant decomposition. This leads to exact covariance-aware decompositions of the NMA estimator into study-level direct and indirect contributions, with indirect evidence further resolved into path-level components. The resulting weights are directly analogous to inverse-variance weights in pairwise meta-analysis and enable, to our knowledge, the first forest-plot representation that exactly reconstructs the NMA estimator. The framework also yields projection-based diagnostic and graphical tools, including forest plots, tension plots, and path-based visualizations. Applications to empirical datasets demonstrate how the proposed approach provides a reproducible and interpretable framework for understanding evidence contributions in network meta-analysis, supporting transparent interpretation and reporting.