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106 records · Page 3Linked to original sources
arXiv2026

Content Exploration Beyond the Feed: Creator Supply and the Shared Corpus

Industrial recommenders give new content initial views through budgeted exploration, then use early performance to decide further delivery. On many short-video platforms, exploration is the primary way new videos reach viewers. Viewer-side tests measure consumption; the published budget objectives we review omit creator response. We analyze four experiments on a major short-video platform. An eight-month creator ablation finds production exploration raises videos posted per creator by 8.55% and creators posting at least once by 7.10% relative to a minimal floor. A budget-matched reallocation raises creator participation with no detectable short-run viewer-side change. A year-long viewer ablation finds 1.74% more video views but 2.13% less view time. A delivered view creates immediate feed value, can trigger organic take-up, and can induce creator supply. Take-up and supply replenish a shared corpus, creating two measurement limits. Viewer-side A/B tests cancel the corpus effect when both arms consume the same corpus. Giving each arm its own corpus avoids cancellation, but turnover still controls the horizon. If the corpus turns over at rate w per posting cycle, a t-cycle experiment expresses at most wt of the eventual corpus effect. More users reduce noise but do not speed turnover. Before the corpus path visibly bends, data cannot distinguish a modest fast effect from an arbitrarily large slow one, so a valid confidence interval may lack a finite upper endpoint. As predicted, the three-week co-diverted experiment cannot determine the sign of the eventual corpus effect. Within the window, it identifies the direct feed effect, and an exploratory cohort analysis detects organic lift after exploration ends. The experiments establish a positive creator response, measure the gross corpus flow visible within three weeks, and show the design and duration needed to identify total value.

cs.IR↗
arXiv2026

Sharp mean-field analysis of permutation mixtures and permutation-invariant decisions

We develop sharp bounds on the statistical distance between high-dimensional permutation mixtures and their i.i.d. counterparts. Our approach establishes a new geometric link between the spectrum of a complex channel overlap matrix and the information geometry of the channel, yielding tight dimension-independent bounds that close gaps left by previous work. Within this geometric framework, we also derive dimension-dependent bounds that uncover phase transitions in dimensionality for Gaussian and Poisson families. Applied to compound decision problems, this refined control of permutation mixtures enables sharper mean-field analyses of permutation-invariant decision rules, yielding strong non-asymptotic equivalence results between two notions of compound regret in Gaussian and Poisson models.

math.ST↗
arXiv2026

A Complete Characterization of Tensorizable $f$-divergences

Csiszar's formulation of the $f$-divergence introduced a vast family of functionals for quantifying dissimilarity between probability distributions. However, many applications in statistics and information theory rely only on a few $f$-divergences, such as the Kullback-Leibler divergence, the $χ^2$-divergence, and the squared Hellinger distance. These divergences are especially useful because they admit simple compositional formulas under product measures, a property sometimes referred to as tensorization. In this work, we refine a formalism of tensorization previously introduced in the literature. Then, we show that any possible tensorization formula has a multi-affine form characterized by a single parameter, and identify all tensorizable $f$-divergences under our adopted notion of tensorization.

cs.IT↗
arXiv2026

Generalization Error Curves for Analytic Spectral Algorithms under Power-law Decay

The generalization error curve of certain kernel regression method aims at determining the exact order of generalization error with various source condition, noise level and choice of the regularization parameter rather than the minimax rate. In this work, under mild assumptions, we rigorously provide a full characterization of the generalization error curves of the kernel gradient descent method (and a large class of analytic spectral algorithms) in kernel regression. Consequently, we could sharpen the near inconsistency of kernel interpolation and clarify the saturation effects of kernel regression algorithms with higher qualification, etc. Thanks to the neural tangent kernel theory, these results greatly improve our understanding of the generalization behavior of training the wide neural networks. A novel technical contribution, the analytic functional argument, might be of independent interest.

cs.LG↗
arXiv2026

Bernstein--von Mises theorems for Bayesian probabilistic numerics

We study probabilistic numerical methods for solving nonlinear PDEs from a Bayesian nonparametric perspective. Given noisy evaluations at random collocation points, we place a truncated Gaussian series prior on the unknown solution and establish contraction at the minimax nonparametric rate, up to a logarithmic factor. Our main results give Gaussian approximations of the posterior in positive-order Sobolev spaces and, under suitable conditions, in the uniform topology. This contrasts with classical ill-posed inverse problems, where Bernstein--von Mises theorems typically require substantially weaker topologies. Here, the observation operator is differential rather than smoothing, and inversion of its linearisation gains regularity, making these strong-topology results possible. The posterior may be centred at either the posterior mean or the posterior mode. We further prove that the Gaussian Laplace approximation is asymptotically equivalent to the true posterior at a $\sqrt{N}$-scale.

math.ST↗
arXiv2026

Branch Scaling Manifests as Implicit Architectural Regularization for Improving Generalization in Overparameterized ResNets

Scaling factors in residual branches have emerged as a prevalent method for boosting neural network performance, especially in normalization-free architectures. While prior work has primarily examined scaling effects from an optimization perspective, this paper investigates their role in residual architectures through the lens of generalization theory. Specifically, we establish that wide residual networks (ResNets) with constant scaling factors become asymptotically unlearnable as depth increases. In contrast, when the scaling factor exhibits rapid depth-wise decay combined with early stopping, over-parameterized ResNets achieve minimax-optimal generalization rates. To establish this, we demonstrate that the generalization capability of wide ResNets can be approximated by kernel regression associated with the Neural Tangent Kernel (NTK). Our theoretical findings are validated through experiments on synthetic data and real-world classification tasks, including MNIST and CIFAR-100.

cs.LG↗
arXiv2026

Geometric Optics Approximation Sampling: A Reflector-Induced Transport Map Framework

In this paper, we propose Geometric Optics Approximation Sampling (GOAS), a reflector-induced transport-map framework for sampling from target measures. Once a reflecting surface is constructed, the associated transport map is explicitly determined by the physical law of reflection. As a concrete realization, we develop a supporting-hyperellipsoid construction that requires only a discrete approximation of the target measure and does not require gradient information of the target density. The formulation accommodates both density-based and sample-based target representations. A softmin smoothing technique is introduced to obtain a smooth approximate transport map from this piecewise hyperellipsoidal construction. We establish well-posedness and stability of the reflector-induced push-forward measure and derive quantitative error estimates in the maximum mean discrepancy metric, and convergence of continuous statistical observables, including fixed-order moments. Numerical experiments on an analytically tractable example, strongly non-Gaussian targets, sample-based target approximations, and Bayesian inverse problems demonstrate the accuracy and flexibility of GOAS.

math.NA↗
arXiv2026

Categorical algebra of conditional probability

In the field of categorical probability, one uses concepts and techniques from category theory, such as monads and monoidal categories, to study the structures of probability and statistics. In this paper, we connect some ideas from categorical algebra, namely weakly cartesian functors and natural transformations, to the idea of conditioning in probability theory, using Markov categories and probability monads. First of all, we show that under some conditions, the monad associated to a Markov category with conditionals has a weakly cartesian functor and weakly cartesian multiplication. In particular, we show that this is the case for the Giry monad on standard Borel spaces. We then connect this theory to existing results on statistical experiments. We show that for deterministic statistical experiments, the so-called standard measure construction (which can be seen as a generalization of the ``hypernormalizations'' introduced by Jacobs) satisfies a universal property, allowing an equivalent definition which does not rely on the existence of conditionals.

math.CT↗
arXiv2026

Sub-Gaussian Concentration and Entropic Normality of the Maximum Likelihood Estimator

It is well known that, under standard regularity conditions, the maximum likelihood estimator (MLE) satisfies a central limit theorem and converges in distribution to a Gaussian random variable as the sample size grows. This paper strengthens this classical result by developing several stronger forms of asymptotic normality for the normalized MLE. With additional assumptions on the score, we first establish sub-Gaussian tail bounds and convergence of all moments for the normalized estimation error. We then prove an entropic central limit theorem for a smoothed version of the estimator, showing convergence in relative entropy to the limiting Gaussian law. When the Fisher information of the normalized estimate is bounded, or its density has bounded first derivative, we further show that the smoothing can be removed, yielding entropic normality of the MLE itself. The proofs develop auxiliary tools that may be of independent interest, including exponential consistency bounds, high-moment estimates, and entropy-control arguments for the estimator.

cs.IT↗
arXiv2026

Windowed thinning and query complexity for the bouncy particle and Zigzag samplers

Let $μ(d x)\propto e^{-U(x)} d x$ on $\R^d$, where $U$ is $m$-strongly convex and $L$-smooth, and denote by $κ=L/m$ the condition number. We consider windowed thinning, an exact simulation method for the bouncy particle sampler and the coordinate Zigzag process. The method divides a trajectory into deterministic windows and uses a gradient evaluation at the beginning of each window to construct a tractable local envelope for the event rate. Combining this construction with quantitative mixing estimates and finite-time bounds on the expected numbers of bounces and flips yields query complexity guarantees from a Gaussian cold start. For total-variation error $\varepsilon$, the expected query counts are $O(κ^{1/2}d\,(d\logκ+\log\frac1\varepsilon))$ gradient queries for the bouncy particle sampler and $O(κd^{1/4}(d\logκ+\log\frac1\varepsilon))$ full-gradient equivalents for Zigzag, where $d$ coordinate-partial queries count as one equivalent.

math.NA↗
arXiv2026

A composite generalization of Ville's martingale theorem using e-processes

We provide a composite version of Ville's theorem that an event has zero measure if and only if there exists a nonnegative martingale which explodes to infinity when that event occurs. This is a classic result connecting measure-theoretic probability to the sequence-by-sequence game-theoretic probability, recently developed by Shafer and Vovk. Our extension of Ville's result involves appropriate composite generalizations of nonnegative martingales and measure-zero events: these are respectively provided by ``e-processes'', and a new inverse capital outer measure. We then develop a novel line-crossing inequality for sums of random variables which are only required to have a finite first moment, which we use to prove a composite version of the strong law of large numbers (SLLN). This allows us to show that violation of the SLLN is an event of outer measure zero and that our e-process explodes to infinity on every such violating sequence, while this is provably not achievable with a nonnegative (super)martingale.

math.PR↗
arXiv2026

Coherent information deletion: Bayes' theorem and generalized Bayesian unlearning

Bayes' theorem admits an information-processing interpretation due to Zellner (1988): under the Shannon-information criterion, the posterior is the unique rule that processes prior and data information without information loss. We revisit these ideas, but from the perspective of information deletion. Given a posterior based on a complete dataset, what distribution should replace it when a subset of the data is removed? We define information deletion using the same information conservation principle as Zellner (1988), and show that the optimalpost-deletion distribution is exactly the leave-data-out posterior. We then extend the framework beyond likelihood-based inference from Bayes to the generalized Bayesian updating of Bissiri et al. (2016) based on loss functions. We introduce a sequential coherence requirement for deletion, under which, removing two pieces of information jointly is equivalent to removing them successively. The resulting coherent deletion rule exactly recovers the generalized Bayesian posterior based only on the retained data. Restricting these optimization problems to variational families yields corresponding formulations of variational Bayesian and generalized Bayesian unlearning.

stat.ME↗
arXiv2026

Minimax bounds for watermarked and masked recursive discrete distribution estimation

Watermarking has been proposed as a way to identify synthetic samples in estimation settings where no metadata is available to distinguish them from real samples, but its precise effects remain unexplored. In the absence of a distinguishing mechanism, it has been shown that adding synthetic samples significantly reduces the marginal efficacy of new real samples. In this work, we study the minimax loss of such recursive discrete distribution estimation in the presence of watermarks in contrast to the unassisted and oracle-assisted losses. When the fraction of real samples vanishes asymptotically, we provide a lower bound that shows that it is impossible to improve performance by adding watermarks unless the false negative rate of detection also vanishes. Additionally, we show that in most regimes, the worst-case losses of a sequence of simple deterministic estimators match the corresponding lower bounds up to constants. Finally, we propose masking, a randomization procedure that narrows the gap in the remaining regimes to a Jensen gap. We conjecture that a tighter lower bound argument can close this gap.

cs.IT↗
arXiv2026

Minimax Lower Bound for Estimating Diffusion-based Local Intrinsic Dimension

While diffusion-based methods have recently emerged as effective tools for probing the intrinsic geometry of high-dimensional data, their statistical difficulty remains largely unexplored. We study estimation of the finite-scale population functional underlying FLIPD (Kamkari et al., 2024; arXiv:2406.03537), a diffusion-based local intrinsic dimension (LID) quantity defined through the logarithmic scale derivative of a Gaussian-smoothed density. Intuitively, Gaussian smoothing turns local dimension into a scale law: near a $d$-dimensional manifold, the kernel mass grows like $σ^d$, so differentiating with respect to the noise scale reveals the intrinsic exponent. Under a regular manifold model, we show uniformly over the model class that the finite-scale field differs from the manifold dimension $d$ by at most $O(σ^2)$. We then establish a minimax lower bound of order $(nσ^d)^{-1}$ for estimating this finite-scale field from $n$ observations, for $n^{-1/(2α+d)}\lesssimσ\leσ_0$. At the smallest scale covered by our lower-bound construction, the bound becomes the nonparametric rate $n^{-2α/(2α+d)}$.

stat.ML↗
arXiv2026

Sharp Restricted Isometry Thresholds for Global Minima of Rank-Restricted Matrix LASSO

We determine the sharp restricted isometry threshold for recovery at global minima of the rank-restricted matrix LASSO. For target rank $r_{\star}$, if the rank-$k$ RIP constant satisfies $δ<δ_{\mathrm{sharp}}(k/r_{\star})$, where $δ_{\mathrm{sharp}}(t)=t/(4-t)$ for $0<t<4/3$ and $δ_{\mathrm{sharp}}(t)=\sqrt{(t-1)/t}$ for $t\ge4/3$, then every global minimizer has Frobenius error $\lesssim\sqrt{r_{\star}}λ$ for all $λ\gtrsim\|\mathcal{A}^{*}(ξ)\|_{\mathrm{op}}$ and at every search rank $r\ge r_{\star}$. The constants depend only on the RIP constant and $t=k/r_{\star}$, and in particular are independent of the search rank. When the rank restriction is inactive, the result specializes to the ordinary convex matrix LASSO. We also obtain the analogous results for sparsity-restricted vector LASSO. Conversely, we show that the threshold $δ<δ_{\mathrm{sharp}}(k/r_{\star})$ cannot be improved, due to the existence of counterexamples whose global minimizers fail to recover the ground truth.

stat.ML↗
arXiv2026

Symmetries and Causality: Causal Effect Identification Beyond IID Data

In the natural sciences, symmetries and cause-effect relationships are ubiquitous. Yet for complex machine-learning tasks, like world-modeling in reinforcement learning, they appear difficult to harness. We propose a formal description of statistical systems based on symmetries in data leaving causal mechanisms invariant. The result is an abstract, simple and general mathematical language for causal reasoning. This paper provides formal descriptions of models and queries, setting up this language, and the formal infrastructure and strategies for their mathematically rigorous identification from data within this formalism. This approach reproduces and matches standard theoretical results on IID data and transport of experimental and non-experimental data. But its main purpose is to unify and substantially extend the scope of causal reasoning, in going beyond IID data and in approaching complex causal queries not captured by do- or soft-interventions. This new perspective on causally relevant aspects of data-modeling additionally sheds new light on well-known structures like c-components or hedges but also includes aspects of missing data and is inherently well-suited for the description of transfer and robustness properties.

math.ST↗
arXiv2026

Generalized Regret Analysis of Thompson Sampling using Fractional Posteriors

Thompson sampling (TS) is one of the most popular and earliest algorithms to solve stochastic multi-armed bandit problems. We consider a variant of TS, named $α$-TS, where we use a fractional or $α$-posterior ($α\in(0,1)$) instead of the standard posterior distribution. To compute an $α$-posterior, the likelihood in the definition of the standard posterior is tempered with a factor $α$. For $α$-TS we obtain both instance-dependent $\mathcal{O}\left(\sum_{k \neq i^*} Δ_k\left(\frac{\log(T)}{C(α)Δ_k^2} + \frac{1}{2} \right)\right)$ and instance-independent $\mathcal{O}(\sqrt{KT\log K})$ frequentist regret bounds under very mild conditions on the prior and reward distributions, where $Δ_k$ is the gap between the true mean rewards of the $k^{th}$ and the best arms, and $C(α)$ is a known constant. Both the sub-Gaussian and exponential family models satisfy our general conditions on the reward distribution. Our conditions on the prior distribution can be easily satisfied by a density that is positive, continuous, and bounded. We also establish another instance-dependent regret upper bound that matches (up to constants) to that of improved UCB [Auer and Ortner, 2010]. Our regret analysis carefully adapts and combines recent theoretical developments in the non-asymptotic concentration analysis and Bernstein-von Mises type results for the $α$-posterior distribution. Moreover, our analysis does not require additional structural properties such as closed-form posteriors or conjugate priors.

stat.ML↗
arXiv2026

A computational approach to maximum likelihood thresholds for colored Gaussian graphical models

Gaussian graphical models (GGMs) are essential tools for interpretable structure learning. However, in high-dimensional, small-sample regimes, the available data is often insufficient for the maximum likelihood estimator to exist. Colored Gaussian graphical models (CGGMs) mitigate this limitation by imposing symmetry constraints through graph coloring, which reduces the required sample size. This minimal number of observations needed to guarantee that the estimator exists almost surely is defined as the maximum likelihood threshold (MLT). Here, we address the computation of the MLT for CGGMs by focusing on its geometric formulation: finding the minimum rank of a sample covariance matrix such that its projection lies almost surely within the interior of the cone of sufficient statistics. We establish a unified theoretical framework, extending results from uncolored to colored models and introducing new symbolic algorithms. Furthermore, we present a computational study integrating sampling with topological data analysis (TDA) to investigate the local geometry of the cone of sufficient statistics. Our results demonstrate the potential of TDA to overcome the computational bottlenecks of traditional symbolic algebraic methods, particularly Groebner basis computations, in analyzing the likelihood geometry of CGGMs.

stat.ML↗