ALT The evidence, Z = p(x |ℳ), of a data-generating model ℳ is a powerful tool in Bayesian frameworks: it enables model inference, model misspecification tests, and goodness-of-fit assessment as an alternative to standard discrepancy statistics, e.g. χ². While versatile, the evidence is notoriously difficult to compute, even when the likelihood is available. In both likelihood-based and simulation-based inference, estimating Z provides an immediate solution to high-dimensional goodness-of-fit or model misspecification tests: using Z(x) as the test statistic induces a minimum-volume level-α test. Such tests are essential to assess model fidelity, which is necessary to trust simulation-based parameter inference. Existing learned approaches largely target Bayes factors, which only measure evidence ratios. Instead, we provide accurate, fast, and amortized estimates of the absolute evidence. We present ENFR: Evidence Networks with Flow-matched References. Building on the Evidence Network framew

ALT Many of the most important datasets in the physical sciences live natively on the sphere, and these data often carry no preferred position or orientation. Using methods that do not reflect this symmetry leads to far higher simulation requirements, which can be catastrophic for scientific analyses (forcing reliance on cheap, imperfect models that bias results). We present a novel spherical implementation of a gauge equivariant mesh architecture that is specifically designed for the Hierarchical Equal Area isoLatitude Pixelization (healpix) of the sphere. The SO(3)-equivariant convolution is built from steerable filters, parallel transport and spin-weighted features. The architecture supplies the exact symmetry, so the network does not spend training data learning it. The convolution operations are constructed to give provably exact equivariance, given a band-limit. Our full CNNs can achieve SO(3)-invariance error at per-cent-level and lower, which is controlled through well-understood h

ALT Rènyi differential privacy, based on Rènyi divergence, has become a notion of choice for privacy guarantees. However, outside the Gaussian setting, Rènyi divergence can be difficult to compute. In particular, Rènyi differential privacy guarantees that exploit task-specific information, as in Rènyi Pufferfish privacy, can be challenging. Here we use kernel Stein discrepancies based on escort distributions to provide certificates for Rènyi differential privacy. By designing the kernel to focus on the relevant privacy directions, we also derive task-aware privacy certificates. We provide exact identities for Gaussian and Laplace mechanisms and bounds for exponential mechanisms, with separate task-aware guarantees. A real data set example illustrates how the protected comparison set and the released subspace jointly determine the resulting certificates.

ALT Primal-dual methods for constrained Markov decision processes are analyzed under a two-timescale assumption, with non-stationarity entering through two channels. In the additive channel, where the constraint level oscillates, the dual oscillation saturates at a floor set by the moving optimum and is amplified linearly by the timescale ratio. In the parametric channel, where the constraint sensitivity oscillates, a coherent modulation carries no resonant forcing at any depth: the dual update's stiffness modulation cancels the primal update's damping modulation. Instability instead comes from a mismatch between the constraint estimate used by the primal update and the measurement used by the dual update, producing a parametric resonance at twice the pair's natural frequency. The tolerated mismatch grows with the square root of the forcing bandwidth and collapses with the square root of the dual-to-primal step-size ratio. These results hold on the tangent linear-quadratic reduction around

ALT Understanding how stochastic gradient descent learns internal representations in wide neural networks trained on rich target functions is a central problem in machine learning. We study this question in two-layer neural networks learning target functions represented by networks of the same architecture, with arbitrary nonlinearity, and with network width and input dimension growing proportionally. In this regime, standard approaches for describing the learning dynamics require tracking a diverging number of overlaps, which measure correlations between learned and target features, leaving no obvious description in terms of a few order parameters. Here, we develop a mesoscopic description of learning by characterizing the distribution of overlaps. In the early stages of learning, we find that a Gaussian description of overlap statistics reduces the learning dynamics to a few collective variables that track learning of the target function's linear and quadratic components. We then estimat

ALT Most convergence guarantees for iterative stochastic algorithms are stated at a fixed iteration count and do not justify stopping at a data-dependent time. Recent work has constructed anytime-valid envelopes that do, but deterministic envelopes are typically too conservative for practical stopping, and existing data-adaptive ones are derived case-by-case, each coupling a specific algorithm to a specific noise model and boundary construction. This paper provides a general recipe for constructing such anytime-valid data-adaptive boundaries for a large class of iterative algorithms. Performance measures of many of these algorithms satisfy a one-step recursion that reduces the entire error analysis to a single general requirement: time-uniform concentration for predictably weighted sums of noise. We formalize this through a strongly adapted sub-ψ condition, and derive off-the-shelf boundary constructions via continuous and discrete mixtures, stitching, and product betting martingales. We s

ALT We introduce a conformal prediction framework for regression with Grassmann-valued responses. The proposed method constructs prediction regions in the form of metric balls directly on the Grassmann manifold, using basis-invariant geometric distances as non-conformity measures and scaling the corresponding non-conformity scores through a learned prediction difficulty estimator. This yields locally adaptive, input-dependent prediction regions that respect the underlying manifold geometry, while retaining the finite-sample marginal coverage guarantee of split conformal prediction under exchangeability. We evaluate several Grassmannian distance metrics and compare their prediction-region efficiency on synthetic benchmarks and an airfoil inverse shape-design problem. The results highlight the advantages of geometry-aware non-conformity scores, while local adaptation improves empirical coverage uniformity across prediction difficulty levels and, in strongly heteroscedastic settings, can subs

ALT We consider the problem of online linear classification, where a stream of unlabeled samples is available, and annotation is costly. We investigate the optimal query policy, that prescribes whether and when a sample should be labeled and used for a training step, or instead discarded, so as to minimize a given objective. We focus on the realizable case, under Gaussian design, in the high-dimensional regime where data dimension and number of unlabeled samples grow proportionally. We consider two objectives: 1) the generalization error under a labeling budget, and 2) a cumulative cost function where mistakes have unit cost and queries cost Δ. We first show that the myopic policies that greedily minimize at all times the increment in the objective correspond to either small or intermediate-margin policies, that prescribe the selection of samples whose margins fall within certain intervals that shrink with time. Within this class, we numerically ascertain the optimal schedules for the sele

ALT Often, eg in healthcare and predictive maintenance, what matters is which of several competing causes will strike first, and with what probability: survival analysis with competing risks. Corresponding deep learning methods are developed as bundles: a new architecture together with a new training loss, tested on one to five datasets against a handful of baselines. Thus, a state-of-the-art result cannot be attributed to the architecture, the loss, or the evaluation protocol. The coupling is not accidental: with one incidence curve per cause, a network's head is tied to the losses expressible on it. We vary the two axes independently, three losses, two of them strictly proper, on four backbones, across two modalities that tease the axes apart: 20 tabular and 3 pathology-image datasets. One loss wins in both: modelling cumulative incidence directly under censoring weighting. It brings twice as much as benefits a good choice of backbone on competing-risks data, and no more than it on singl

ALT Conformal test martingales (CTMs) are e-processes for sequentially testing exchangeability via p-values produced by a conformity function A. However, any distribution shift that preserves the push-forward P∘ A⁻¹ leaves the p-value distribution invariant, rendering it invisible to every CTM based on A. We formalise this blind spot through the A-equivalence class [P]_A and show that, for any target distribution T, there is a unique closest element Q∗∈[P]_A in KL divergence—the A-cryptic I-projection—with explicit density q∗(z) = t(z) p_S(A(z))/t_S(A(z)) and a Pythagorean decomposition into detectable and cryptic components. When A is the Neyman-Pearson likelihood ratio, the cryptic component vanishes, connecting our result to optimal e-value design. We demonstrate the theory on high-dimensional image data by empirically constructing this I-projection. By conditionally resampling EMNIST letters to identically match the conformity scores of MNIST digits, we demonstrate that a distribution

ALT We study distinguishability and recovery of a sparse rank-one signal in a spiked Wigner model with independently sparse noise through a variety of algorithms and parameter regimes. The signal and noise have separate sparsity parameters, while the signal-to-noise ratio (SNR) remains of constant order as the matrix size grows. For sub-Gaussian noise and spikes whose nonzero coordinates are bounded away from zero, a leave-one-out quadratic thresholding procedure (a generalization of Ding et al., 2024) identifies the support. Sign synchronization yields exact recovery for a Rademacher spike, and spectral refinement yields strong recovery. The quadratic thresholding search parameter gives an explicit tradeoff between signal sparsity, noise sparsity, and computational cost. We also study extensions to sub-Gaussian spike priors through diagonal and quadratic thresholding. For Rademacher spikes with continuous nonzero noise, we develop an identical search algorithm that exploits unblurred sign

ALT Causal foundation models (CFMs) show promising results for treatment-effect estimation, with unit-level estimates that are competitive on standard benchmarks. Many causal analyses, however, inform decisions through effects averaged over groups of units, such as the full sample, the treated units, or a subgroup, and for such decisions a CFM is useful only if its credible intervals are calibrated. We examine these intervals and find that their coverage degrades, both in simple synthetic settings and on standard benchmarks. On ACIC 2016, for example, temperature-calibrated 95% intervals from CausalPFN fall below nominal coverage beyond groups of 30 out of 4802 units, and intervals for the full-sample average contain the true value in only 20% of datasets. We trace this problem to how these models are trained and how their intervals are formed. They output a predictive distribution for one unit at a time and, at inference time, combine the unit-level variances as if the errors on different

ALT We study uniform approximation of the Hölder ball ℋ₁ˢ([0,1]ᴰ) by fully connected ReLU networks. Except in the one-dimensional Lipschitz case, existing constructions remain separated from the information-theoretic efficiency limit. This article closes the gap for every s>0 and D≥1: networks with width O(N), depth O(L), and parameter magnitude O(1) attain uniform approximation error of order (N²L²log N)⁻ˢ/ᴰ. No relation between width and depth is required. The construction uses new encoding and decoding mechanisms that exploit higher-order finite-difference bounds for local Taylor coefficients of the target function. As a statistical consequence, empirical risk minimizers over the corresponding clipped network classes attain the minimax regression rate n⁻²ˢ/⁽²ˢ⁺ᴰ) without additional logarithmic factors. The paper also gives a detailed exposition of the key approximation techniques, covering both established and new mechanisms.

ALT The linear representation hypothesis holds that pretrained vision models encode generative factors as linearly decodable directions in feature space, and comes with a standard methodology: train a linear probe, declare a factor represented if accuracy is high. We show this test is too permissive. Asymptotically, any sufficiently rich feature map, including a random one, admits a linear readout, so high probe accuracy is consistent with two very different geometries: a true linear world model, where the latent-aligned subspace dominates the representation, or a kernel machine, where the latents lie linearly inside a much larger curved manifold. This distinction matters because the curved structure orthogonal to the probe is relevant for out-of-distribution generalization, robustness, and compositional transfer. We propose three asymmetric diagnostics that distinguish the two cases: reverse predictivity, spectral concentration, and out-of-distribution generalization. We validate them in

ALT In modern commerce, firms and platforms typically use customer features from different sources to personalize prices: some features are verified using seller records or third-party data, while others are self-reported and may be strategically manipulated by buyers seeking lower prices. We therefore introduce a novel source-aware framework for contextual dynamic pricing with strategic buyers in a high-dimensional semiparametric setting, where buyer valuations follow a sparse linear model with an unknown noise distribution. To account for differences in manipulability across feature sources while enabling effective valuation learning, we develop the Source-Aware Localized exploration and calibrated Exploitation (SALE) pricing policy, which draws on verified features to localize price exploration and combines verified and reported features to correct for strategic manipulation during exploitation. Our localized exploration design allows converting binary purchase feedback into unbiased si

ALT Tabular foundation models (TFMs) now outperform gradient-boosted trees in classification and regression. Yet in survival analysis, which predicts when an event occurs from censored data, often on small clinical cohorts, we find that no existing TFM beats the half-century-old Cox model. We address this issue by introducing SurvICL, a TFM that predicts survival curves in-context. SurvICL is pretrained on purely synthetic datasets generated from a novel flexible proportional-odds prior that outperforms other prior types in our ablation study. Unlike most other survival models, SurvICL is trained with a plain quantile regression loss, which is possible because its synthetic pre-training data generator can provide uncensored event times. Additionally, we introduce the largest survival benchmark to date spanning 103 datasets. On the 63 datasets held out from all design decisions, SurvICL outperforms 20 tree-based, deep, and foundation-model baselines without any tuning: it wins 82.9% of pair

ALT Neural networks offer critical adaptability for controlling complex systems where nominal modeling proves intractable. However, when offline-trained neural networks are deployed within closed-loop control systems, their actions fundamentally alter the state distribution upon which they were trained. This paper introduces a unified taxonomy to categorize these sequential failure modes: the initial Closed-Loop Identification Error, the deployment-induced Action Error, and the continuous Tracking Error. Through discrete-time sensitivity analysis, this work formalizes how offline-learned parameters locally guarantee a trajectory divergence upon deployment, degrading performance due to local loss geometry. To mitigate this distribution shift, we propose actionable design policies centered on time-scale separated online adaptation. These mechanisms are validated through ablation studies on a nonlinear Duffing oscillator, demonstrating that bounded continuous learning successfully mitigates t

ALT In the best-arm identification problem, an algorithm adaptively selects arms to find the arm of the largest mean reward. Successive Rejects (SR) is a simple fixed-budget best-arm identification algorithm that eliminates one candidate arm at each phase. At first glance, SR seems to have limited adaptivity, since it follows predetermined phase lengths. However, for Gaussian arms with common variance, SR's worst-instance error exponent normalized by the popular complexity measure H₁(P) is optimal up to a constant factor, which partly explains why SR's performance is competitive compared with state-of-the-art adaptive sampling algorithms. In this paper, we extend the framework of SR by optimizing its predefined schedule of phase lengths. We formulate phase-schedule design as an optimization problem for the normalized error exponent. Then, we introduce a version of SR with the optimized schedule, which we call H₁-balanced Successive Rejects (H₁-SR). This schedule is optimal within the class

ALT Standard Bayesian experimental design (BED) methods assume the experimenter alone will act on the data, choosing the best action given their own beliefs and goals. In practice, data are often acted on by external parties whose procedures the experimenter cannot change and may not know, particularly when the data are released publicly. To extend BED to such settings, we introduce usage-aware experimental design, a framework in which an adaptive design policy is trained to minimise the usage risk: the experimenter's expected loss when one or more external rules act on the collected data. These rules may be non-Bayesian or black-box, and when several may apply, we either average over them or take the worst case. We show that, unlike in standard BED, more informative data can be harmful under such rules, so the usage risk cannot in general be expressed as an expected uncertainty reduction; our framework thus strictly generalises standard BED. We then use it to tackle a range of problems th

ALT Tabular foundation models based on prior-data fitted networks (PFNs) predict on a new data set in a single forward pass, but they cannot report which inputs their predictions depend on. We propose SelectPFN, a PFN that performs Bayesian variable selection and prediction jointly for sparse nonlinear regression. In one forward pass it returns posterior inclusion probabilities for every variable, and its predictions are computed from sampled subsets only, so excluded variables provably cannot affect them; we show this is exactly what the Bayesian model prescribes. The network is pretrained once on synthetic tasks from a meta-prior built around a group spike-and-slab sparse-input neural network. Column-permutation equivariance combined with random interleaving of original and knockoff columns yields knockoff statistics with exact finite-sample false discovery rate control, even when the meta-prior is wrong. We develop theory for amortized selection: a Bayes-regret bound for the amortized m