Sampling from unnormalised discrete distributions is a fundamental problem across various domains. While Markov chain Monte Carlo offers a principled approach, it often suffers from slow mixing and poor convergence. In this paper, we propose Discrete Neural Flow Samplers (DNFS), a trainable and efficient framework for discrete sampling. DNFS learns the rate matrix of a continuous-time Markov chain such that the resulting dynamics satisfy the Kolmogorov equation. As this objective involves the intractable partition function, we then employ control variates to reduce the variance of its Monte Carlo estimation, leading to a coordinate descent learning algorithm. To further facilitate computational efficiency, we propose locally equivaraint Transformer, a novel parameterisation of the rate matrix that significantly improves training efficiency while preserving powerful network expressiveness. Empirically, we demonstrate the efficacy of DNFS in a wide range of applications, including sampling from unnormalised distributions, training discrete energy-based models, and solving combinatorial optimisation problems.

In this paper, we present a learning algorithm that achieves asymptotically optimal regret for Markov decision processes in average reward under a communicating assumption. That is, given a communicating Markov decision process $M$, our algorithm has regret $K(M) \log(T) + \mathrm{o}(\log(T))$ where $T$ is the number of learning steps and $K(M)$ is the best possible constant. This algorithm works by explicitly tracking the constant $K(M)$ to learn optimally, then balances the trade-off between exploration (playing sub-optimally to gain information), co-exploration (playing optimally to gain information) and exploitation (playing optimally to score maximally). We further show that the function $K(M)$ is discontinuous, which is a consequence challenge for our approach. To that end, we describe a regularization mechanism to estimate $K(M)$ with arbitrary precision from empirical data.

The Restricted Boltzmann Machine (RBM) is one of the simplest generative neural networks capable of learning input distributions. Despite its simplicity, the analysis of its performance in learning from the training data is only well understood in cases that essentially reduce to singular value decomposition of the data. Here, we consider the limit of a large dimension of the input space and a constant number of hidden units. In this limit, we simplify the standard RBM training objective into a form that is equivalent to the multi-index model with non-separable regularization. This opens a path to analyze training of the RBM using methods that are established for multi-index models, such as Approximate Message Passing (AMP) and its state evolution, and the analysis of Gradient Descent (GD) via the dynamical mean-field theory. We then give rigorous asymptotics of the training dynamics of RBM on data generated by the spiked covariance model as a prototype of a structure suitable for unsupervised learning. We show in particular that RBM reaches the optimal computational weak recovery threshold, aligning with the BBP transition, in the spiked covariance model.

Mixture of Experts (MoE) models constitute a widely utilized class of ensemble learning approaches in statistics and machine learning, known for their flexibility and computational efficiency. They have become integral components in numerous state-of-the-art deep neural network architectures, particularly for analyzing heterogeneous data across diverse domains. Despite their practical success, the theoretical understanding of model selection, especially concerning the optimal number of mixture components or experts, remains limited and poses significant challenges. These challenges primarily stem from the inclusion of covariates in both the Gaussian gating functions and expert networks, which introduces intrinsic interactions governed by partial differential equations with respect to their parameters. In this paper, we revisit the concept of dendrograms of mixing measures and introduce a novel extension to Gaussian-gated Gaussian MoE models that enables consistent estimation of the true number of mixture components and achieves the pointwise optimal convergence rate for parameter estimation in overfitted scenarios. Notably, this approach circumvents the need to train and compare a range of models with varying numbers of components, thereby alleviating the computational burden, particularly in high-dimensional or deep neural network settings. Experimental results on synthetic data demonstrate the effectiveness of the proposed method in accurately recovering the number of experts. It outperforms common criteria such as the Akaike information criterion, the Bayesian information criterion, and the integrated completed likelihood, while achieving optimal convergence rates for parameter estimation and accurately approximating the regression function.

Bisimulation metrics are powerful tools for measuring similarities between stochastic processes, and specifically Markov chains. Recent advances have uncovered that bisimulation metrics are, in fact, optimal-transport distances, which has enabled the development of fast algorithms for computing such metrics with provable accuracy and runtime guarantees. However, these recent methods, as well as all previously known methods, assume full knowledge of the transition dynamics. This is often an impractical assumption in most real-world scenarios, where typically only sample trajectories are available. In this work, we propose a stochastic optimization method that addresses this limitation and estimates bisimulation metrics based on sample access, without requiring explicit transition models. Our approach is derived from a new linear programming (LP) formulation of bisimulation metrics, which we solve using a stochastic primal-dual optimization method. We provide theoretical guarantees on the sample complexity of the algorithm and validate its effectiveness through a series of empirical evaluations.

Many real-world decision-making problems face the off-dynamics challenge: the agent learns a policy in a source domain and deploys it in a target domain with different state transitions. The distributionally robust Markov decision process (DRMDP) addresses this challenge by finding a robust policy that performs well under the worst-case environment within a pre-specified uncertainty set of transition dynamics. Its effectiveness heavily hinges on the proper design of these uncertainty sets, based on prior knowledge of the dynamics. In this work, we propose a novel linear mixture DRMDP framework, where the nominal dynamics is assumed to be a linear mixture model. In contrast with existing uncertainty sets directly defined as a ball centered around the nominal kernel, linear mixture DRMDPs define the uncertainty sets based on a ball around the mixture weighting parameter. We show that this new framework provides a more refined representation of uncertainties compared to conventional models based on $(s,a)$-rectangularity and $d$-rectangularity, when prior knowledge about the mixture model is present. We propose a meta algorithm for robust policy learning in linear mixture DRMDPs with general $f$-divergence defined uncertainty sets, and analyze its sample complexities under three divergence metrics instantiations: total variation, Kullback-Leibler, and $χ^2$ divergences. These results establish the statistical learnability of linear mixture DRMDPs, laying the theoretical foundation for future research on this new setting.

Discrete choice models (DCMs) are used to analyze individual decision-making in contexts such as transportation choices, political elections, and consumer preferences. DCMs play a central role in applied econometrics by enabling inference on key economic variables, such as marginal rates of substitution, rather than focusing solely on predicting choices on new unlabeled data. However, while traditional DCMs offer high interpretability and support for point and interval estimation of economic quantities, these models often underperform in predictive tasks compared to deep learning (DL) models. Despite their predictive advantages, DL models remain largely underutilized in discrete choice due to concerns about their lack of interpretability, unstable parameter estimates, and the absence of established methods for uncertainty quantification. Here, we introduce a deep learning model architecture specifically designed to integrate with approximate Bayesian inference methods, such as Stochastic Gradient Langevin Dynamics (SGLD). Our proposed model collapses to be-haviorally informed hypotheses when data is limited, mitigating overfitting and instability in underspecified settings while retaining the flexibility to capture complex nonlinear relationships when sufficient data is available. We demonstrate our approach using SGLD through a Monte Carlo simulation study, evaluating both predictive metrics--such as out-of-sample balanced accuracy--and inferential metrics--such as empirical coverage for marginal rates of substitution interval estimates. Additionally, we present results from two empirical case studies: one using revealed mode choice data in NYC, and the other based on the widely used Swiss train choice stated preference data. Introduction Discrete choice is a fundamental area of econometrics that examines how individuals make decisions among a finite set of alternatives. For example, in transportation systems, discrete choice models are often used to estimate individuals' willingness to pay for a reduction in travel time, considering factors such as cost, trip duration, level of service, and other attributes of competing transportation modes. Given that inference is fundamental in the discrete choice field, researchers often rely on transparent and interpretable statistical binary or multinomial classification models such as logistic and probit regressions, along with their more complex variations. Traditional discrete choice models (DCMs) allow for point and interval estimation of key economic quantities, including marginal rates of substitution and odds ratios.

Accurate system modeling is crucial for safe, effective control, as misidentification can lead to accumulated errors, especially under partial observability. We address this problem by formulating informative input design and model identification adaptive control (MIAC) as belief space planning problems, modeled as partially observable Markov decision processes with belief-dependent rewards ($ρ$-POMDPs). We treat system parameters as hidden state variables that must be localized while simultaneously controlling the system. We solve this problem with an adapted belief-space iterative Linear Quadratic Regulator (BiLQR). We demonstrate it on fully and partially observable tasks for cart-pole and steady aircraft flight domains. Our method outperforms baselines such as regression, filtering, and local optimal control methods, even under instantaneous disturbances to system parameters.

Using AI to create autonomous researchers has the potential to accelerate scientific discovery. A prerequisite for this vision is understanding how well an AI model can identify the underlying structure of a black-box system from its behavior. In this paper, we explore how well a large language model (LLM) learns to identify a black-box function from passively observed versus actively collected data. We investigate the reverse-engineering capabilities of LLMs across three distinct types of black-box systems, each chosen to represent different problem domains where future autonomous AI researchers may have considerable impact: Program, Formal Language, and Math Equation. Through extensive experiments, we show that LLMs fail to extract information from observations, reaching a performance plateau that falls short of the ideal of Bayesian inference. However, we demonstrate that prompting LLMs to not only observe but also intervene -- actively querying the black-box with specific inputs to observe the resulting output -- improves performance by allowing LLMs to test edge cases and refine their beliefs. By providing the intervention data from one LLM to another, we show that this improvement is partly a result of engaging in the process of generating effective interventions, paralleling results in the literature on human learning. Further analysis reveals that engaging in intervention can help LLMs escape from two common failure modes: overcomplication, where the LLM falsely assumes prior knowledge about the black-box, and overlooking, where the LLM fails to incorporate observations. These insights provide practical guidance for helping LLMs more effectively reverse-engineer black-box systems, supporting their use in making new discoveries.

We propose a Bayesian framework for training binary and spiking neural networks that achieves state-of-the-art performance without normalisation layers. Unlike commonly used surrogate gradient methods -- often heuristic and sensitive to hyperparameter choices -- our approach is grounded in a probabilistic model of noisy binary networks, enabling fully end-to-end gradient-based optimisation. We introduce importance-weighted straight-through (IW-ST) estimators, a unified class generalising straight-through and relaxation-based estimators. We characterise the bias-variance trade-off in this family and derive a bias-minimising objective implemented via an auxiliary loss. Building on this, we introduce Spiking Bayesian Neural Networks (SBNNs), a variational inference framework that uses posterior noise to train Binary and Spiking Neural Networks with IW-ST. This Bayesian approach minimises gradient bias, regularises parameters, and introduces dropout-like noise. By linking low-bias conditions, vanishing gradients, and the KL term, we enable training of deep residual networks without normalisation. Experiments on CIFAR-10, DVS Gesture, and SHD show our method matches or exceeds existing approaches without normalisation or hand-tuned gradients.