Training machine learning and statistical models often involves optimizing a data-driven risk criterion.

Flow models are a cornerstone of modern machine learning. They are generative models that progressively transform probability distributions according to learned dynamics. Specifically, they learn a continuous-time Markov process that efficiently maps samples from a simple source distribution into samples from a complex target distribution. We show that these models are naturally related to the Schrödinger equation, for an unusual Hamiltonian on continuous variables. Moreover, we prove that the dynamics generated by this Hamiltonian can be efficiently simulated on a quantum computer. Together, these results give a quantum algorithm for preparing coherent encodings (a.k.a., qsamples) for a vast family of probability distributions--namely, those expressible by flow models--by reducing the task to an existing classical learning problem, plus Hamiltonian simulation. For statistical problems defined by flow models, such as mean estimation and property testing, this enables the use of quantum algorithms tailored to qsamples, which may offer advantages over classical algorithms based only on samples from a flow model. More broadly, these results reveal a close connection between state-of-the-art machine learning models, such as flow matching and diffusion models, and one of the main expected capabilities of quantum computers: simulating quantum dynamics.

Normalizing flows with a Gaussian base provide a computationally efficient way to approximate posterior distributions in Bayesian inference, but they often struggle to capture complex posteriors with multimodality and heavy tails. We propose a stick-breaking mixture base with component-wise tail adaptation (StiCTAF) for posterior approximation. The method first learns a flexible mixture base to mitigate the mode-seeking bias of reverse KL divergence through a weighted average of component-wise ELBOs. It then estimates local tail indices of unnormalized densities and finally refines each mixture component using a shared backbone combined with component-specific tail transforms calibrated by the estimated indices. This design enables accurate mode coverage and anisotropic tail modeling while retaining exact density evaluation and stable optimization. Experiments on synthetic posteriors demonstrate improved tail recovery and better coverage of multiple modes compared to benchmark models. We also present a real-data analysis illustrating the practical benefits of our approach for posterior inference.

Adaptive testing and sequential estimation problems have recently gained substantial attention due to their foundational role in modern artificial intelligence and interactive systems. Prominent applications include online preference learning, where systems dynamically adapt to user feedback to refine personalized recommendations, and reinforcement learning from human feedback (RLHF), which aims to align AI agents with human values by adaptively querying users. In these contexts, the main focus is to efficiently extract maximal information from human responses, which are inherently stochastic and limited in quantity. Among various types of such problems, this work particularly considers a fundamental yet illustrative case involving stochastic binary responses. Here, a decision-maker sequentially selects questions of varying difficulty from a continuous pool to pose to a candidate and aims to efficiently estimate the candidate's ability (represented by an unknown continuous parameter) by utilizing the binary feedback (e.g., correct/incorrect) collected, which depends probabilistically on the candidate's ability and the question's difficulty. This setup is arguably the simplest scenario that captures the essence of continuous parameter estimation under uncertainty, making it an ideal benchmark for developing fundamental theoretical insights and practical algorithms. Variants of this fundamental adaptive estimation problem have been studied in several communities.

We propose to perform mean-field variational inference (MFVI) in a rotated coordinate system that reduces correlations between variables. The rotation is determined by principal component analysis (PCA) of a cross-covariance matrix involving the target's score function. Compared with standard MFVI along the original axes, MFVI in this rotated system often yields substantially more accurate approximations with negligible additional cost. MFVI in a rotated coordinate system defines a rotation and a coordinatewise map that together move the target closer to Gaussian. Iterating this procedure yields a sequence of transformations that progressively transforms the target toward Gaussian. The resulting algorithm provides a computationally efficient way to construct flow-like transport maps: it requires only MFVI subproblems, avoids large-scale optimization, and yields transformations that are easy to invert and evaluate. In Bayesian inference tasks, we demonstrate that the proposed method achieves higher accuracy than standard MFVI, while maintaining much lower computational cost than conventional normalizing flows.

Generative models form the backbone of modern machine learning, underpinning state-of-the-art systems in text, vision, and multimodal applications. While Maximum Likelihood Estimation has traditionally served as the dominant training paradigm, recent work have highlighted its limitations, particularly in generalization and susceptibility to catastrophic forgetting compared to Reinforcement Learning techniques, such as Policy Gradient methods. However, these approaches depend on explicit reward signals, which are often unavailable in practice, leaving open the fundamental problem of how to align generative models when only high-quality datasets are accessible. In this work, we address this challenge via a Bilevel Optimization framework, where the reward function is treated as the optimization variable of an outer-level problem, while a policy gradient objective defines the inner-level. We then conduct a theoretical analysis of this optimization problem in a tractable setting and extract insights that, as we demonstrate, generalize to applications such as tabular classification and model-based reinforcement learning. We release the code at https://github.com/abenechehab/nll_to_po .

Hierarchical data with multiple observations per group is ubiquitous in empirical sciences and is often analyzed using mixed-effects regression. In such models, Bayesian inference gives an estimate of uncertainty but is analytically intractable and requires costly approximation using Markov Chain Monte Carlo (MCMC) methods. Neural posterior estimation shifts the bulk of computation from inference time to pre-training time, amortizing over simulated datasets with known ground truth targets. We propose metabeta, a transformer-based neural network model for Bayesian mixed-effects regression. Using simulated and real data, we show that it reaches stable and comparable performance to MCMC-based parameter estimation at a fraction of the usually required time.

Conditional acceptability refers to how plausible a conditional statement is perceived to be. It plays an important role in communication and reasoning, as it influences how individuals interpret implications, assess arguments, and make decisions based on hypothetical scenarios. When humans evaluate how acceptable a conditional "If A, then B" is, their judgments are influenced by two main factors: the $\textit{conditional probability}$ of $B$ given $A$, and the $\textit{semantic relevance}$ of the antecedent $A$ given the consequent $B$ (i.e., whether $A$ meaningfully supports $B$). While prior work has examined how large language models (LLMs) draw inferences about conditional statements, it remains unclear how these models judge the $\textit{acceptability}$ of such statements. To address this gap, we present a comprehensive study of LLMs' conditional acceptability judgments across different model families, sizes, and prompting strategies. Using linear mixed-effects models and ANOVA tests, we find that models are sensitive to both conditional probability and semantic relevance-though to varying degrees depending on architecture and prompting style. A comparison with human data reveals that while LLMs incorporate probabilistic and semantic cues, they do so less consistently than humans. Notably, larger models do not necessarily align more closely with human judgments.

Direct Preference Optimization (DPO) has recently emerged as a simple and effective alternative to reinforcement learning from human feedback (RLHF) for aligning large language models (LLMs) with user preferences. However, existing DPO formulations rely on a single monolithic model, which limits their expressivity in multi-task settings and their adaptability to heterogeneous or diverse preference distributions. In this work, we propose Mix- and MoE-DPO, a framework that extends DPO with both soft mixture models and mixture-of-experts (MoE) architectures, using a stochastic variational inference approach. Our method introduces a latent-variable model over expert assignments and optimizes a variational evidence lower bound (ELBO), enabling stable and efficient learning of specialized expert policies from preference data. Mix- and MoE-DPO provides three key advantages over standard DPO: (i) generalization via universal function approximation through mixtures; (ii) reward and policy specialization through expert components tailored to distinct preference modes; and (iii) contextual alignment through input-dependent soft gating that enables user-specific mixture policies. Our framework supports both shared base architectures with expert-specific policy heads and fully independent expert models, allowing flexible trade-offs between parameter efficiency and specialization. We validate our approach on a variety of model sizes and multi-preference datasets, demonstrating that Mix- and MoE-DPO offers a powerful and scalable method for preference-based LLM alignment.