The aggregation with exponential weights (AEW) estimator is not fully understood in the basic setting of model selection aggregation with squared loss. In particular, whether it is minimax-rate optimal in expectation for large enough fixed temperatures and under random design has been an open problem since its introduction, which was explicitly posed by Lecué and Mendelson (2013). In this paper, we settle this problem by showing that \emph{without} requiring a Bernstein-type assumption, the AEW indeed achieves the excess risk $T \log (M) / (n+1)$ in expectation, whenever the temperature $T$ satisfies $(L^2/T)\exp(B/T)\leq μ/2$. Here, the number of dictionary elements is $M$, the estimator has observed $n$ i.i.d. samples from any distribution, and the loss is assumed to be bounded by $B$, $L$-Lipschitz continuous and $μ$-strongly convex. For squared loss, we show that $T\geq 4 b^2$ suffices when the predictions and labels are $[0,b]$-valued. Because AEW is known to be suboptimal in expectation for temperatures below some constant, this shows that AEW has a sharp phase transition when the temperature is large enough but constant, as conjectured by Lecué and Mendelson.

Pervasive data contamination -- stemming from measurement errors, outliers, or adversarial corruption -- has motivated the development of robust statistical methods. In this context, we propose a two-stage Adversarial Contamination-resistant Iterative Hard Thresholding (AC-IHT) algorithm for high-dimensional regression with contamination. Our nonconvex algorithm achieves minimax near-optimal (up to logarithmic terms) estimation by iteratively updating the coefficient vector and the contamination vector with different thresholding scales. We further demonstrate that our AC-IHT estimator is signal-adaptive: under proper signal conditions, it adaptively attains a sharper estimation rate and more accurate support recovery. Moreover, it enjoys the strong oracle property, laying a theoretical foundation for asymptotic inference. Numerical experiments confirm its superior finite-sample performance. Finally, we discuss theoretical extensions of the proposed procedure to generalized linear models and to heavy-tailed noise settings.

Causal effect estimation from observational data is fundamental across various applications.

In this paper, we study the behavior of the Upper Confidence Bound-Variance (UCB-V) algorithm for the Multi-Armed Bandit (MAB) problems, a variant of the canonical Upper Confidence Bound (UCB) algorithm that incorporates variance estimates into its decision-making process. More precisely, we provide an asymptotic characterization of the arm-pulling rates for UCB-V, extending recent results for the canonical UCB in [21] and [23]. In an interesting contrast to the canonical UCB, our analysis reveals that the behavior of UCB-V can exhibit instability, meaning that the arm-pulling rates may not always be asymptotically deterministic. Besides the asymptotic characterization, we also provide non-asymptotic bounds for the arm-pulling rates in the high probability regime, offering insights into the regret analysis. As an application of this high probability result, we establish that UCB-V can achieve a more refined regret bound, previously unknown even for more complicate and advanced variance-aware online decision-making algorithms. A matching regret lower bound is also established, demonstrating the optimality of our result.

What property of the data distribution determines the excess risk of principal component analysis? In this paper, we provide a precise answer to this question. We establish a central limit theorem for the error of the principal subspace estimated by PCA, and derive the asymptotic distribution of its excess risk under the reconstruction loss. We obtain a non-asymptotic upper bound on the excess risk of PCA that recovers, in the large sample limit, our asymptotic characterization. Underlying our contributions is the following result: we prove that the negative block Rayleigh quotient, defined on the Grassmannian, is generalized self-concordant along geodesics emanating from its minimizer of maximum rotation less than π/4.

Mechanistic interpretability methods summarize a transformer component by a single importance score, conflating two distinct roles: a component may matter because it transports task-relevant content, or because the forward computation degrades when its contribution is removed. We introduce \emph{Interchange-Group Sobol Decomposition} (IGSD), a paired-intervention framework that compares matched activation replacement with zero ablation on the same component, estimates two Sobol-style variance indices, and uses their signed difference to separate the two roles, with intervention validity monitored by a symmetric off-manifold diagnostic $\widehat{\mathrm{ST}}>1$. In factual recall, IGSD identifies an early-layer content channel in both GPT-2 small and Qwen2.5-1.5B that standard importance methods underestimate. A controlled subject and relation donor design shows that the early channel transports relation-frame content while late attention transports subject-retrieval content, refining at head granularity to the known $\mathrm{Attn}_{L9H8}$ head. Late-layer clamping confirms that the early signal is expressed through downstream transformations rather than residual pass-through. These results show that replacement and deletion are not interchangeable controls and their divergence provides a practical statistical diagnostic for content transport in transformer components.

As sequential learning algorithms are increasingly applied to real life, ensuring data privacy while maintaining their utilities emerges as a timely question. In this context, regret minimisation in stochastic bandits under ϵ-global Differential Privacy (DP) has been widely studied. The present literature poses a significant gap between the best-known regret lower and upper bound in this setting, though they "match in order". Thus, we revisit the regret lower and upper bounds of ϵ-global DP bandits and improve both. First, we prove a tighter regret lower bound involving a novel information-theoretic quantity characterising the hardness of ϵ-global DP in stochastic bandits.

Learning dynamics is essential for model-based control and Reinforcement Learning in engineering systems, such as robotics and power systems.

We utilise a sampler originating from nonequilibrium statistical mechanics, termed here Jarzynski-adjusted Langevin algorithm (JALA), to build statistical estimation methods in latent variable models. We achieve this by leveraging Jarzynski's equality and developing algorithms based on a weighted version of the unadjusted Langevin algorithm (ULA) with recursively updated weights. Adapting this for latent variable models, we develop a sequential Monte Carlo (SMC) method that provides the maximum marginal likelihood estimate of the parameters, termed JALA-EM. Under suitable regularity assumptions on the marginal likelihood, we provide a nonasymptotic analysis of the JALA-EM scheme implemented with stochastic gradient descent and show that it provably converges to the maximum marginal likelihood estimate. We demonstrate the performance of JALA-EM on a variety of latent variable models and show that it performs comparably to existing methods in terms of accuracy and computational efficiency. Importantly, the ability to recursively estimate marginal likelihoods--an uncommon feature among scalable methods--makes our approach particularly suited for model selection, which we validate through dedicated experiments.

We revisit the problem of constructing predictive confidence sets for which we wish to obtain some type of conditional validity. We provide new arguments showing how "split conformal" methods achieve near desired coverage levels with high probability, a guarantee conditional on the validation data rather than marginal over it. In addition, we directly consider (approximate) conditional coverage, where, e.g., conditional on a covariate X belonging to some group of interest, we seek a guarantee that a predictive set covers the true outcome Y. We show that the natural method of performing quantile regression on a held-out (validation) dataset yields minimax optimal guarantees of coverage in these cases. Complementing these positive results, we also provide experimental evidence highlighting work that remains to develop computationally efficient valid predictive inference methods.