Causal queries are often only partially identifiable from observational data, and experiments that could tighten the resulting bounds are typically costly. We study the problem of selecting, prior to observing experimental outcomes, a cost-constrained subset of experiments that maximally tightens bounds on a target query. We formalize this as the max-potency problem, where epistemic potency measures the worst-case reduction in bound width guaranteed by an experiment, and show that this problem is NP-hard via a reduction from 0-1 knapsack. Building on the polynomial-programming framework of Duarte et al. (2023), we give a general procedure for evaluating epistemic potency in discrete settings. To control the super-exponential search space, we introduce two graphical pruning criteria that depend only on the causal graph and the query: a novel path-interception rule that exploits district structure to certify zero potency in linear time, and an identifiability check based on the ID algorithm. On Erdos-Renyi random graphs and 11 bnlearn benchmark networks, the two criteria together prune 50-88% of candidate experiments on average without solving a single polynomial program. For the general subset search, we show that ID-pruned experiments are combinatorially inert, yielding a super-exponential reduction in the number of subsets evaluated. We close with an end-to-end demonstration on observational NHANES data, selecting optimal experiments for estimating the effect of physical activity on diabetes.

A major bottleneck in characterizing the failure modes of generative AI systems is the cost and time of annotation and evaluation. Consequently, adaptive testing paradigms have gained popularity, where one opportunistically decides which cases and how many to annotate based on past results. While this framework is highly practical, its extreme flexibility makes it difficult to draw statistically rigorous conclusions, as it violates classical assumptions: the number of observations is typically limited (often 10 to 50 cases) and decisions regarding sampling and stopping are made in the midst of data collection rather than based a pre-specified rule. To characterize what statistical inferences can be drawn from highly adaptive audits, we introduce a hypothesis testing framework from two 'dueling' perspectives: (i) the model's null that asserts there is no failure mode with performance below a target threshold versus (ii) the auditor's null that asserts they have a sampling strategy that will uncover a failure mode. Leveraging Safe Anytime-Valid Inference (SAVI), we formalize the auditor as conducting 'testing by betting', which translates into simultaneous e-processes for testing the dueling null hypotheses. Furthermore, if the auditor is sufficiently powerful, we prove that these two hypotheses are asymptotically inverses of each other, in that passage of a stringent audit does in fact certify the AI system as being globally robust. Empirically, we demonstrate that our proposed testing procedures maintain anytime-valid type-I error control, outperform pre-specified testing methods, and can reach statistically rigorous conclusions sometimes with as few as 20 observations.

Evaluation of large language models (LLMs) is increasingly critical, yet standard benchmarking methods rely on average accuracy, overlooking both the inherent stochasticity of LLM outputs and the heterogeneity of benchmark items. Item Response Theory (IRT) offers a principled framework for modeling latent model abilities and item characteristics, but conventional methods are computationally expensive and numerically unstable, limiting large-scale implementations. To address these challenges, we propose an interpretable and scalable framework for LLM evaluation based on the majorization-minimization principle. Our approach reformulates the problem as a sequence of constrained matrix factorization subproblems, enabling stable and efficient parameter estimation with theoretical guarantees for identifiability and convergence. Experiments on synthetic and real-world datasets, including MATH-500 and six Open LLM Leaderboard benchmarks, demonstrate that our method achieves superior scalability and interpretability. It delivers orders-of-magnitude speedups over competing methods while maintaining comparable or even higher estimation accuracy. Our results align with established scaling laws and offer insights into item difficulty and discrimination, informing more principled benchmark design.

We show that, in a precise sense, a broad class of feedforward neural networks learn (have finite sample complexity) in the PAC model: every fixed finite feedforward architecture whose layers are definable in an o-minimal structure has finite sample complexity in the agnostic PAC setting, even with unbounded parameters. This covers standard fixed-size MLPs, CNNs, GNNs, and transformers with fixed sequence length, together with the operations and layers typically used in such architectures, including linear projections, residual connections, attention mechanisms, pooling layers, normalization layers, and admissible positional encodings. Hence, distribution-free learnability for modern non-recurrent architectures is not an exceptional property of particular activations or architecture-specific VC arguments, but a consequence of tame feedforward computation. Our results reposition finite-sample PAC learnability as a baseline rather than a differentiator: they shift the focus of architectural comparison toward inductive biases, symmetries and geometric priors, scalability, and optimization behaviour.

Constructing valid and informative conformal prediction regions for multi-dimensional outputs remains a fundamental challenge. While conformal prediction provides finite-sample, distribution-free coverage guarantees, its practical performance critically depends on the choice of nonconformity score. Existing approaches often rely on restrictive geometric assumptions or require explicit likelihood evaluation and invertible transformations, limiting their applicability in complex generative settings. In this work, we introduce TRACE (TRansport Alignment Conformal Estimation), a conformal prediction framework that defines nonconformity through transport alignment in diffusion and flow matching models. Rather than evaluating likelihoods, we measure how well a candidate output aligns with the learned generative dynamics by averaging denoising or velocity-matching errors along stochastic transport trajectories. The resulting transport-based scores are scalar-valued and can be calibrated using split conformal prediction, yielding valid marginal coverage under exchangeability. We further analyze the statistical properties of the proposed scores and their sensitivity to computational budget. Experiments on synthetic and real datasets demonstrate valid coverage and show that the resulting regions adapt naturally to multimodal and non-convex conditional distributions.

Establishing almost sure convergence rates for stochastic approximation and reinforcement learning under Markovian noise is a fundamental theoretical challenge. We make progress towards this challenge for a class of stochastic approximation algorithms whose expected updates are contractive, a setting that arises in many reinforcement learning algorithms such as $Q$-learning and linear temporal difference learning. Specifically, for a power-law learning rate $O(n^{-η})$ with $η\in (1/2, 1)$, we obtain an almost sure convergence rate arbitrarily close to $o(n^{1 - 2η})$. For a harmonic learning rate $O(n^{-1})$, we obtain an almost sure convergence rate arbitrarily close to $o(n^{-1})$, which we argue is a strong result because it is close to the optimal rate $O(n^{-1}\log\log n)$ given by the law of the iterated logarithm (for a special case of i.i.d. noise). Key to our analysis is a novel Lyapunov drift construction that applies a Poisson-equation based correction for Markovian noise to the well-established Moreau-envelope smoothing for the contractive mapping.

Stochastic bandit algorithms are usually analyzed under a mean-reward criterion, yet many problems favor arms with strong upper-tail performance, which we study herein. For a fixed miscoverage level \(α\), the natural upper-tail target of arm \(j\) is the upper endpoint \(F_j^{-1}(1-α/2)\) of a central prediction interval. This target can rank arms differently from their means, creating a central mismatch with the classical bandit objective. To this end, we propose ACP-UCB1, a conformal-style policy that combines an adaptive conformal estimate of the upper endpoint with a UCB-type optimism bonus. The technical challenge is that the conformity scores used by ACP-UCB1 are recomputed from evolving empirical quantile estimates and evaluated at an adaptive level. We control this endpoint through reward-quantile concentration, a perturbation argument for recomputed score quantiles, and deterministic localization of the adaptive level. ACP-UCB1 achieves logarithmic upper-quantile regret with per-arm contribution \(O(\nicefrac{\log n}{Δ_j^{\mathrm{ACP}}})\). We also provide metric-specific regret decompositions comparing ACP-UCB1 with UCB1 and use numerical experiments to validate performance and improvement.

The classic multi-armed bandit (MAB) problem tackles the challenge of accruing maximum reward while making decisions under uncertainty. However, in applications, often the goal is to minimize cost subject to a constraint on the minimum permissible reward, an objective captured by multi-armed bandits with cost-subsidy (MAB-CS). Of interest to this paper is the setting where the quality (reward) constraint is specified relative to the unknown best reward and the cost of each arm is known. We characterize the expected sub-optimal samples required by any policy by proving instance-dependent lower bounds that offer new insight into the problem and are a strict generalization of prior bounds. Then, we propose an algorithm called Cost-Ordered Feasibility (COF) that leverages our insight and intelligently combine samples from all arms to gauge the feasibility of a cheap arm. Thereafter, we analyze COF to establish instance-dependent upper bounds on its expected cumulative cost and quality regret, i.e., relative to the cheapest feasible arm. Finally, we empirically validate the merits of COF, comparing it to baselines from the literature through extensive simulation experiments on the MovieLens and Goodreads datasets as well as representative synthetic instances. Not only does our paper develop qualitatively better theoretical regret upper bounds, but COF also convincingly demonstrates improved empirical performance.

For continuous state-action space scenarios, classical reinforcement learning (RL) theory predominantly focuses on low-rank Markov decision processes (MDPs), which provide sample-efficient guarantees at the expense of restrictive structural assumptions. Kernel smoothing model-based approaches offer a promising alternative paradigm that instead leverages the smoothness of the MDP and employs non-parametric kernel smoothing estimates of transition dynamics. This paper proposes a new kernel-smoothing model-based approach for online reinforcement learning in finite-horizon settings under Lipschitz continuity assumptions on the MDP. By incorporating a Bernstein-style exploration bonus into the kernel smoothing framework, our method achieves a regret bound which improves upon the state-of-the-art regret bound in its dependence on the horizon. The theoretical advancement relies on a delicate analysis of the synergy between Bernstein-style bonuses and kernel smoothing, where a new tight Bernstein-type concentration inequality for martingales may be of independent interest.

Understanding why trained Transformers generalize well is a fundamental problem in modern machine learning theory, and complexity-based generalization bounds provide a principled way to study this question. While existing norm-based bounds for Transformers remove the explicit polynomial dependence on the hidden dimension, they typically impose fixed norm constraints specified a priori and can exhibit unfavorable exponential dependence on depth. In this paper, we derive spectrum-adaptive post hoc generalization bounds for multi-layer Transformers. Under layerwise spectral norm control, the bounds are expressed in terms of layerwise Schatten quantities of the query-key, value, and feedforward weight matrices. Since the Schatten indices need not be fixed a priori and can instead be selected after training, separately for each matrix type and layer, the bounds adaptively trade off spectral complexity against the dimension- and depth-dependent factors according to the learned singular-value profiles. Empirical comparisons of BERT-adapted proxies for the leading complexity factors suggest that the proxies induced by our bounds grow more slowly with depth and hidden dimension than the corresponding norm-based proxies. Overall, our results provide a complexity-based perspective on how the spectral structure of trained Transformers is reflected in generalization analyses.