ALT Adaptive experiments use bandit algorithms to favor interventions with higher expected rewards. However, decision-makers also require reliable inference on current treatment effects: differences between intervention and reference-arm mean rewards at the present time, rather than historical averages. These objectives conflict: reward maximization favors better interventions, whereas current-effect inference requires continued exploration of potentially suboptimal interventions as outcomes change over time. We characterize this tradeoff for K-arm bandits with Hölder-smooth mean rewards of exponent β∈(0,1] and constant L. Requiring high-probability simultaneous coverage of all current effects by confidence intervals of width at most w, we establish matching minimax bounds, up to logarithmic factors, on expected dynamic regret. Over horizon T, our algorithm TRACE (Tracking Rewards and Current Effects) achieves regret of order Õᵦ(√KT+Kᵝ/⁽²ᵝ⁺¹)L¹/⁽²ᵝ⁺¹)T⁽ᵝ⁺¹/(2β+1))+Kw⁻²(1+L¹/ᵝw⁻¹/ᵝ)), wher

ALT We study gradient flow in a two-layer linear network in the high-dimensional regime where the input, inner and output dimensions diverge proportionally. Under whitened inputs and isotropic imbalance, we combine a closed-form solution of the dynamics with prior random matrix theory results to derive a high-dimensional limit for the dynamics. Our method yields exact asymptotic spectral distributions for the weight matrices at all times and characterizes the evolution of the learned subspace. In the small-initialization limit, we identify a silent alignment phenomenon: the singular directions of the network align on an order-one timescale, while the singular values and empirical loss remain at their initial values. Visible learning occurs later, on a logarithmic timescale, as singular values progressively escape zero and the loss decreases. This phase is characterized by a saddle-to-saddle dynamics, which becomes a continuous process of spectral learning in the high-dimensional limit.

ALT Principal Component Analysis (PCA) is a fundamental tool for statistical dimension reduction, yet no canonical extension to distribution-valued data has emerged within the framework of optimal transport. While Wasserstein barycenters and Wasserstein least-squares regression provide canonical liftings of Euclidean averaging and linear regression in the Wasserstein space, an analogous principle for PCA has remained unexplored. In this paper, we introduce Lifted PCA, a canonical extension of Euclidean PCA to Wasserstein space, obtained through the framework of lifting functionals in the space of probability measures by convex extension. Starting from the classical variational formulations of PCA on the Stiefel and Grassmann manifolds, we derive two equivalent minimization problems over probability measures supported on these manifolds that correspond to Wasserstein barycenters with generalized costs. We establish an exact analogue of the equivalence between the Stiefel and Grassmann formu

ALT In this paper, we study quantile distributional policy evaluation, where the return distribution under a given policy is approximated by the quantile Bellman fixed point \bmη_m. We investigate the statistical accuracy of estimating \bmη_m across quantile resolutions under Wasserstein distances. Assuming access to a generative model, we construct a quantile fixed point estimator \bmη_mⁿ from an empirical Markov decision process and establish non-asymptotic error bounds under Wasserstein distances. When reward densities are uniformly bounded away from zero, we obtain a root-n high-probability bound simultaneously over all resolutions m and all Wasserstein orders 1≤ p≤∞. When reward densities vanish at the support endpoints, we show that continuity and endpoint monotonicity suffice to preserve the same resolution-independent rate for 1≤ p≤2. For p>2, we establish a root-n bound with coefficient m¹/2-1/p, together with a slower, resolution-independent n^-1/p bound for every finite p>2. We

ALT Residual-based causal discovery assumes that each variable is generated from its parents and an independent noise source. It estimates this noise and tests its independence from candidate causes. For identifiable functions such as additive-noise models (ANMs) and the broader class of location-scale noise models (LSNMs), this principle identifies the causal direction, as only the causal model admits this independence. Existing LSNM methods rely on parametric assumptions to obtain standardized residuals, risking model misspecification. In contrast, the RESIT-framework allows for nonparametric estimation of residuals to iteratively identify sink nodes, but was proposed for ANMs. To close this gap, we propose Loscar, an extension of RESIT to LSNMs that estimates the conditional mean and scale via quantile regression, enabling pretrained tabular foundation models as plug-in estimators. We compare it with a feasible generalized least squares (FGLS) variant, which is consistent given suitable

ALT In this paper, we develop a unified framework connecting proper scoring rules and Stein's method for the estimation of unnormalized models. We construct a class of proper scoring rules that encompasses score matching, ratio matching, pseudolikelihood, and noise contrastive estimation. We show that these proper scoring rules naturally induce Stein operators. This connection yields a new construction principle for Stein operators, and enables us to study estimators based on these scoring rules through the geometry associated with the induced Stein operators. Using this geometric perspective, we provide a unified characterization of the asymptotic efficiency of the resulting estimators.

ALT We are the first to give a certified rule for whether unseen data belong in a pretrained model's context or its weights, built on an exact law for their information value. That value, the code length saved on held-out queries, equals in expectation the bits stored about the query, plus a prior repair, minus a decoding loss, for any model, architecture, dataset and task. The law's ceilings certify the choice from one measurement, and an anytime-valid test makes it from held-out queries, with error controlled whenever it stops. Every certificate the ceilings issue on a trained transformer, TabPFN, an occupancy network and a stereo network is correct. The test picks TabPFN over XGBoost and a trained surrogate over an in-context learner on airfoil pressure, calls the better of GPT-2's cache and fine-tune in 61,948 of 64,000 query streams over 32 comparisons, and errs in at most 3.3% of null streams. The law also diagnoses when learning in context helps and when it does not. Deleting part o

ALT Barycenter models, based on stochastic and deterministic optimal transport bridges between probability measures, have proven to be useful for supervised and unsupervised learning for (probability) measure-valued data. In this paper, we utilize the directional asymmetry and dual drift-diffusion control arising in the recently proposed Schrödinger-Bass Bridge (SBB) between two measures, and propose SBB-encoder, SBB-decoder, SBB-symmetric, and SBB-Markovian barycenter models. For each of these novel models, our first theoretical result establishes the existence of the SBB-barycenter under mild assumptions. Our second theoretical result characterizes the decoder SBB-barycenter of Gaussians where we show that for Gaussian input measures, the decoder formulation admits a Gaussian minimizer, reducing barycenter computation to a finite-dimensional optimization and yielding a compact algorithm. For general input measures, we derive a hybrid Wasserstein gradient particle descent scheme for all f

ALT Reward guidance methods sample an exponential tilt of a base density using a diffusion model trained with only samples from the base distribution. Existing methods use what we call external guidance: additive terms in the drift or multiplicative terms on the transition probabilities. We propose internal guidance which leaves the output of the base denoiser unmodified and instead changes the point at which it is evaluated. For discrete distributions embedded in the hypercube {-1,1}ᵈ, the denoiser of any linearly tilted density is exactly the base denoiser evaluated at a shifted and rescaled input. In practice the base denoiser is a neural network trained on noised samples of the base distribution. The required evaluation points for sampling the tilted density would be out-of-distribution for basic noising of the base distribution. We modify the training algorithm with random diagonal rescalings of the data, so that the denoiser network learns both in- and out-of-distribution evaluations

ALT In-context reinforcement learning (ICRL) exploits the impressive in-context learning ability of large language models to zero-shot generalize to new, unseen tasks at inference time without any parameter updates. Among many approaches, the Decision-Pretrained Transformer (DPT) is particularly appealing for its theoretical grounding: under idealized assumptions such as a sufficiently expressive model and infinite pretraining data, DPT can perform Bayesian posterior sampling. Whether this characterization survives in practice remains open. Therefore, we ask: is ICRL, specifically DPT and our proposed DPT variants, a Bayesian posterior sampler in practice? To approach this question, we establish two necessary conditions that a Bayesian posterior sampler must satisfy: sufficient-statistic invariance and the martingale property. We then construct statistical hypothesis tests to statistically assess if practically-trained ICRL models satisfy these conditions. Applying these tests in Gaussian,

ALT We revisit the problem of depth separation for neural networks, as studied by Eldan and Shamir, Daniely, and others. Specifically, we consider the natural setting of L² approximation of radial functions under the standard Gaussian measure in d dimensions. We show that there is a radial function in L² under the standard Gaussian measure that can be approximated to a fixed sufficiently small L² error by a depth-3 ReLU neural network of polynomial size in d but can only be approximated by depth-2 networks with subexponentially bounded weights if their size is at least exponential in d. For that, we prove a general result showing that if a radial function has substantial mass in the high frequencies of its Hermite expansion, then it cannot be well approximated by a depth-2 neural network with small width and bounded weights. All theorems, lemmas, propositions, and corollaries are verified in Lean 4.

ALT Gold's theorem shows that no learner can identify an arbitrary language in a superfinite class from positive presentations alone. We prove a positive result of a different shape. Fix an innate recursive contrast operator R mapping any string to a finite set of structurally bounded perturbations of it, and a discriminative scorer of bounded capacity trained by InfoNCE-style contrastive estimation on self-generated perturbations. Under a faithfulness condition relating operator, scorer, and target, the trained scorer achieves PAC conditional identification from positive data alone: with high probability it ranks all but an ε-fraction of positives above every out-of-language perturbation in their own orbit, with sample complexity polynomial in the capacity, the orbit size, the score bound, and 1/ε. The negatives are constructed by the learner, not supplied by an oracle, and no global normalizing constant over strings is ever computed. The misranking bound follows from the geometry of the

ALT Large retail chains must tailor assortment and pricing decisions to local customer preferences, yet even retailers with extensive sales records often have limited data for estimating demand at the store or local market level, particularly in product categories with short selling windows. Local estimation (fitting a demand model for each store on its own) can therefore be unreliable, while pooling data across stores improves estimation precision at the cost of ignoring local demand heterogeneity. We study how retailers can balance these competing effects through transfer learning under a multinomial logit (MNL) demand model. We propose a shrinkage estimator that combines store-specific demand estimates with information pooled from other stores. Unlike the linear or quadratic settings underlying much of the existing theoretical literature on information transfer in operations, the nonlinear MNL likelihood does not admit a straightforward bias-variance decomposition. By exploiting its sel

ALT In reinforcement learning, one presumes the action set to be known ahead of time, and, thus, one must know if any state-action pairs are not permissible ahead of computations. In some applications, determining this action mask is the main computational bottleneck. Isolating this problem to highlight its core properties, we study how to combine the search for an optimal policy with the learning of the mask itself, under a linear MDP model in which each attempted action reveals the exact value of a linear score determining its validity. In this framework, we first construct an algorithm based on deterministic optimistic optimization and show that it is minimax optimal, which we show is also computationally efficient. However, it is sub-optimal on non-degenerate instances, as we show. We propose an alternative, which is also computationally efficient, which plans its exploration strategy. This algorithm admits a formal interpretation as an analogue of information-directed-sampling for det

ALT Prior-data fitted networks (PFNs) have recently gained popularity as an end-to-end approach to amortized Bayesian inference. Yet, their statistical behavior for semi-parametric targets, such as means under missingness or causal effects, remains poorly understood. In this work, we develop theoretical foundations for debiased PFNs to perform inference on finite-dimensional targets in general semi-parametric settings. First, we show that, despite being Bayes-optimal with respect to the target, an idealized PFN can exhibit a posterior concentration rate that implicitly depends on the concentration rates of (latent) nuisance functions and can become slower as the prior becomes more diverse. Then, to speed up the target posterior concentration rates, we introduce two generic debiasing strategies based on efficient influence functions: (i) a label correction and (ii) a prior correction. Finally, we show that the debiased PFNs (i) & (ii) possess concentration rates that asymptotically match a

ALT Sparse variational Gaussian processes (SVGPs) are a widely used, principled approach to scalable Gaussian process modelling. Their success stems from their theoretically interpretable structure: they provide a lower bound on the marginal likelihood of the full GP while allowing scal- able stochastic optimization. Recent work has explored modifications of the posterior conditional distribution and has shown that such extensions can improve upon this well-established framework. In particular, introducing a block-diagonal structured scal- ing matrix leads to a tighter lower bound on the GP marginal likelihood. We investigate whether selecting blocks that tighten the variational objective improves test RMSE and NLL at a fixed inducing-point budget. Since the scaling matrix contributes to the variational lower bound through an upper bound on the log-determinant of the full residual covariance matrix, our work makes the following contributions. First, we introduce a simple, scalable, geometr

ALT Learned priors are used to reconstruct regions that measurements barely reach, and to report uncertainty there. They can fail silently: the reported interval excludes the truth while every ground-truth-free diagnostic stays within its noise. We call such errors the blind set and ask whether they can be predicted before the target data are seen. For a correctly specified linear-Gaussian model averaged over the prior, the risk of a silent failure factorises exactly into non-detection and non-coverage, which does not hold for learned systems in general; we build a score that approximates this reference by linearising a VAE prior. In a prospective study on real X-ray CT objects (2DeteCT) under simulated restricted access, with predictions committed by hash before any trial, the score ranked the realised blindness of two learned inference systems (latent Laplace and ensemble Kalman inversion) with Spearman ρ=0.79 and 0.84 over 240 pre-specified errors. Simple structural predictors would als

ALT Gaussian process upper confidence bound (GP-UCB) has been widely studied, though its known regret upper bound is suboptimal. To obtain near-optimal regret upper bounds, existing studies mainly focus on non-adaptive methods, such as uncertainty sampling, whose practical effectiveness is often limited. This paper shows that, under a Gaussian noise assumption, UCB-style algorithms can achieve near-optimal regret upper bounds via data thinning, a statistical technique for creating independent datasets from a single dataset. First, we show that data thinning can yield tighter confidence intervals in the bandit setting, and that a direct extension of GP-UCB achieves a near-optimal simple regret upper bound. Furthermore, by leveraging the tighter confidence intervals and combining GP-UCB with a phased elimination algorithm, we show a near-optimal cumulative regret upper bound. Finally, we demonstrate the effectiveness of UCB-style algorithms incorporating data thinning via numerical experimen

ALT In earlier work by Jonasson, Steif and Zetterqvist (2023), fully connected deep neural networks with layers of width n, i.i.d. standard Gaussian weights, sign activations and no bias terms were shown to be annealed and quenched noise sensitive with respect to any joint probability distribution mathbbQⁿ on the input boldsymbolω and its perturbation boldsymbolη satisfying a mild positive-correlation condition and mathbbQⁿ(boldsymbolη ∈ {boldsymbolω,-boldsymbolω}) → 0. Here we add independent Gaussian biases of the natural order and show that quenched noise sensitivity is retained over a broad range of depths, indeed under the slightly weaker assumption that mathbbQⁿ(boldsymbolω = boldsymbolη) → 0. Thus, subject to the positive-correlation condition, quenched noise sensitivity holds under the minimal requirement that the perturbed input is asymptotically unlikely to coincide with the original input. At the same time, the annealed covariance remains bounded away from zero and is entirely a

ALT Offline reinforcement learning is fundamentally limited by the coverage of the available data, while collecting additional trajectories through online exploration may be costly, unsafe, or infeasible. We study whether this limitation can be overcome without directly interacting with the underlying environment. Motivated by practical settings such as collaborator-mediated simulation, where a collaborator can provide information complementary to the offline data, we introduce query-based collaborative reinforcement learning to study how to acquire this information efficiently. The learner starts with a fixed offline dataset and adaptively queries state-time pairs; the collaborator samples an action according to a fixed, possibly suboptimal policy, and the simulator returns the corresponding reward and next state. For finite-horizon tabular problems, we derive an instance-dependent information-theoretic lower bound on the additional queries required to identify an ε-optimal policy and dev