The Fisher Information Matrix (FIM) is the canonical local measure of the curvature of a statistical model's log-likelihood surface, and its dominant eigenvalue λmax quantifies the worst-case sensitivity of the model's output distribution to infinitesimal parameter perturbation [1, 2]. The spectral properties of the FIM of neural networks have been studied directly in the random matrix theory literature. Pennington and Worah [4] derive the limiting spectral density of the FIM of a single-hidden-layer network in the high-dimensional asymptotic regime, building on the broader programme of analysing neural network Hessian and kernel spectra via random matrix methods [5, 6], with subsequent work extending these techniques to deeper architectures and non-asymptotic regimes [7, 8]. These results characterize the typical (bulk and edge) spectral behaviour of the FIM for a fixed network and a random or structured input ensemble. This paper studies a complementary question, posed as a perturbation problem rather than an asymptotic-spectrum problem: how does the dominant eigenvalue of a fixed, evaluated empirical FIM change under two specific structured perturbations of the underlying distribution? The first perturbation is a change in the conditioning input away from a reference (in-distribution) ensemble. The second is a structured additive perturbation of the model's own parameters by finite-precision quantization noise -- a perturbation of independent mathematical interest, since it falls outside the i.i.d.-input asymptotic regime treated in the random matrix literature cited above, and instead concerns a fixed network whose parameters, not its input distribution, are perturbed by a noise process with a specific, analytically tractable structure (Definition 4.1). To our knowledge, this parameterperturbation question for the FIM's dominant eigenvalue, under either source of departure, has not been previously formalized.

We study PAC learning in tabular discounted Markov decision processes with exogenous i.i.d. contexts, with discount factor $γ$, finite state space $\mathcal X$, action space $\mathcal A$, and context space $\mathcal Z$. At each time step, a context is drawn independently from an unknown distribution $μ$ and revealed before the agent acts. This context may affect both rewards and transitions, while remaining uncontrolled by the agent. Depending on the regime, the learner has access either to a sampling oracle for $μ$, to a sampling oracle for the transition kernel conditioned on state-context-action tuples, or to both. Oracles can be accessed before and during policy execution. The sample complexity is measured by a couple $(n,m)$, where $n$ is the number of calls to the sampling oracles before execution and $m$ is the number of calls to the sampling oracles during execution. When rewards and transitions are known and only the context distribution $μ$ is sampled, we give a variance-reduced algorithm that solves policy evaluation (PE), best-value estimation (BVE), and best-policy extraction (BPE) with $\left(\widetilde O\left(1/((1-γ)^3\varepsilon^2)\right), 0 \right) $ sample complexity. The rate is independent of $|\mathcal Z|$ and minimax optimal up to logarithmic factors. As a corollary, we also obtain tight rates in the case of one-step perfect look-ahead, improving upon the existing guarantees. In the fully unknown regime, where both $μ$ and P must be learned, we show that PE remains $|\mathcal Z|$-free, with matching upper and lower bounds $\bigl(\widetilde O(|\mathcal X|/((1-γ)^3\varepsilon^2)),\, \widetilde O(1/((1-γ)^2\varepsilon^2))\bigr)$.

Diffusion LLMs have emerged as a promising alternative to conventional autoregressive LLMs, offering substantial potential for improving runtime efficiency. However, existing diffusion models fail to provably enforce user-specified formal constraints, such as regular expressions, which makes them unreliable for tasks that require structured outputs, such as fixed-schema JSON generation. Unlike autoregressive models, which generate tokens sequentially, diffusion LLMs predict a block of tokens in parallel. This parallelism makes traditional constrained decoding algorithms, designed to enforce constraints with sequential token prediction, ineffective at preserving the true output distribution. To address this limitation, we propose DINGO, a dynamic programming-based constrained decoding strategy that is both efficient and provably distribution-preserving. DINGO enables sampling of output strings with the highest probability under the model's predicted distribution while strictly adhering to any user-specified regular expression. On standard symbolic math and JSON generation benchmarks, DINGO achieves up to a 68%points of improvement over unconstrained inference. The code is available at DINGO.

Structure-agnostic (SA) models introduced by Balakrishnan et al. (2026) aim to reflect the general lack of knowledge of structural assumptions on data-generating laws such as smoothness or sparsity in practice. Roughly speaking, SA models restrict the observed-data generating law to be in some rn-neighborhood of (black-box machine learning) estimates, treated as given and fixed, where rn encodes the convergence rates of the estimates to the truth. Under SA models, Balakrishnan et al. (2026) show that the popular Double Machine Learning (DML) estimators for three functionals, the quadratic functional in the Gaussian sequence model, the quadratic density integral functional and the expected conditional covariance, are minimax. However, minimax estimators may be inadmissible. In this paper, we show that, for the first two of the three functionals, the DML estimator is asymptotically inadmissible under the SA model. In particular, we show that these two functionals fall into a class of functionals, which we refer to as the monotone bias class. For this class, we exhibit second-order (U-statistic) estimators, which asymptotically dominate DML estimators, under the SA model. These second-order estimators are empirical higher-order influence function (HOIF) estimators introduced in Liu et al. (2017). Furthermore, the empirical HOIF estimator, like the DML estimator, is minimax for the third functional (the expected conditional covariance), although neither asymptotically dominates the other.

Graph Neural Networks (GNNs) are widely used to compute representations of node pairs for downstream tasks such as link prediction. Yet, theoretical understanding of their expressive power has focused almost entirely on graph-level representations. In this work, we shift the focus to links and provide the first comprehensive study of GNN expressiveness in link representation. We introduce a unifying framework, the kϕ-kρ-mframework, that subsumes existing messagepassing link models and enables formal expressiveness comparisons. Using this framework, we derive a hierarchy of state-of-the-art methods and offer theoretical tools to analyze future architectures. To complement our analysis, we propose a synthetic evaluation protocol comprising the first benchmark specifically designed to assess link-level expressiveness. Finally, we ask: does expressiveness matter in practice? We use a graph symmetry metric that quantifies the difficulty of distinguishing links and show that while expressive models may underperform on standard benchmarks, they significantly outperform simpler ones as symmetry increases, highlighting the need for dataset-aware model selection.

Reverse-Kullback-Leibler (KL) regularization has emerged to be a predominant technique to enhance policy optimization in reinforcement learning (RL) and reinforcement learning from human feedback (RLHF), which forces the learned policy to stay close to a reference policy. While the effectiveness of KL-regularization has been empirically demonstrated in various practical scenarios, current theoretical analyses of KL-regularized RLHF still yield the same O(1/ϵ2) sample complexity as ones without KL-regularization. To understand the fundamental distinction between objectives with KL-regularization and ones without KLregularization, we are the first to theoretically demonstrate the power of KLregularization by providing a sharp analysis for KL-regularized contextual bandits and RLHF, revealing an O(1/ϵ) sample complexity when ϵ is sufficiently small. We also prove matching lower bounds for both settings. More specifically, we study how the coverage of the reference policy affects the sample complexity of KL-regularized online contextual bandits and RLHF. We show that with sufficient coverage from the reference policy, a simple two-stage mixed sampling algorithm can achieve an O(1/ϵ) sample complexity with only an additive dependence on the coverage coefficient, thus proving the benefits of online data even without explicit exploration. Our results provide a comprehensive understanding of the roles of KL-regularization and data coverage in online decision making, shedding light on the design of more efficient algorithms.

We study the sample complexity of Bayesian recovery for solving inverse problems with general prior, forward operator and noise distributions. We consider posterior sampling according to an approximate prior P, and establish sufficient conditions for stable and accurate recovery with high probability. Our main result is a non-asymptotic bound that shows that the sample complexity depends on (i) the intrinsic complexity of P, quantified by its approximate covering number, and (ii) concentration bounds for the forward operator and noise distributions. As a key application, we specialize to generative priors, where P is the pushforward of a latent distribution via a Deep Neural Network (DNN). We show that the sample complexity scales log-linearly with the latent dimension k, thus establishing the efficacy of DNN-based priors. Generalizing existing results on deterministic (i.e., non-Bayesian) recovery for the important problem of random sampling with an orthogonal matrix U, we show how the sample complexity is determined by the coherence of U with respect to the support of P. Hence, we establish that coherence plays a fundamental role in Bayesian recovery as well. Overall, our framework unifies and extends prior work, providing rigorous guarantees for the sample complexity of solving Bayesian inverse problems with arbitrary distributions.

We study the inductive biases of diffusion models with a conditioning-variable, which have seen widespread application as both text-conditioned generative image models and observationconditioned continuous control policies. We observe that when these models are queried conditionally, their generations consistently deviate from the idealized "denoising" process upon which diffusion models are formulated, inducing disagreement between popular sampling algorithms (e.g.

In this article, we present a robust $Q$-learning algorithm for discrete-time mean-field control problems under Wasserstein uncertainty in the common noise law. The algorithm combines a quantization-and-projection scheme with a Wasserstein dual reformulation on the common-noise space. We establish its convergence together with finite-time iteration bounds for both synchronous and asynchronous learning schemes. Numerical experiments on systemic risk and epidemic models compare the asynchronous implementation with an idealized Bellman iteration, illustrate the robustness-performance tradeoff under common-noise misspecification, and report the observed convergence behavior of the asynchronous $Q$-learning algorithm.

Imitation learning seeks to estimate policies reflecting the values of demonstrated behaviors. Prevalent approaches learn to match or exceed the demonstrator's performance in expectation without knowing the demonstrator's reward function. Unfortunately, this does not induce pluralistic imitators that learn to support distinct demonstrations.