Shuffle order can be a larger source of fine-tuning noise than a memoryless analysis predicts: fixed-clock optimizer memory makes local equal-multiset contrasts first order in the learning rate rather than second order, and the resulting order channel can be large enough for a single seed to flip a close A/B comparison. We isolate this mechanism and derive a fit-free way to size the noise it produces. For a memoryless optimizer, reordering an equal multiset has no first-order endpoint term; the leading local contrast is the $O(η^2)$ gradient bracket. Fixed-clock optimizers such as AdamW are different. Their moment buffers, preconditioner state, and de-biasing counters advance with the step index rather than with the learning-rate-scaled time $τ=ηk$, so the same gradient can receive a position-dependent endpoint weight. For any fixed finite measurement window, a lifted-state expansion gives an $O(η)$ equal-multiset contrast whenever the first-order replay coefficient is nonzero, while regular and clock-matched controls remain $O(η^2)$; a bare fixed-$β$ momentum buffer is already enough. A bitwise-deterministic replay from one warmed optimizer state isolates the mechanism, giving order-variance slopes 1.83 for AdamW, 2.00 for fixed-$β$ momentum, and 4.00 for SGD; matching the memory clock to $τ$ restores the regular exponent. For AdamW with a frozen preconditioner, the same impulse-weight kernel gives a closed-form asymptotic order-variance floor after the local potentials are measured, with no fitted coefficients. The result is local to the measurement window (independent reshuffling can average the channel across windows), but it yields order-noise error bars, positional attribution weights, and a seed-budget criterion for fine-tuning comparisons.

Bogomolny, Bohigas and Schmit (BBS) found that the spectrum of the pairwise distance matrix on N points sampled from a smooth d-dimensional manifold encodes a signature of the underlying geometry. We develop I-BBS (Inference-BBS), a coordinate-free method that identifies a low-dimensional latent sub-manifold embedded in a high-dimensional ambient distance matrix alone, without accessing an ambient high-dimensional vector space. It therefore applies even when that space is only partly observable or undefined. We model the ambient embedding by two classes of generative noise, model-based and model-free. The noise mixes the latent signal with off-manifold components, so the eigenvalues reorganise collectively and the latent geometry cannot be read off eigenvalue by eigenvalue. We recover it instead from two integer-stable signatures that survive the noise: the multiplicity of the top non-Perron multiplet, which fixes $d$, and a parameter-free law for how the multiplet positions shrink as the noise grows. On synthetic spheres $S^1$, $S^2$ and $S^3$ these integer signatures are far more stable under noise than the continuous spectral slope, and a blind test recovers both the manifold and the noise model from a single distance matrix. Applications to neural-network representations and to the dynamic training regime are developed in two companion papers.

The m-out-of-n bootstrap--proposed by Bickel et al. [1992]--approximates the distribution of a statistic by repeatedly drawing msubsamples (m n) without replacement from an original sample of size n; it is now routinely used for robust inference with heavy-tailed data, bandwidth selection, and other large-sample applications. Despite this broad applicability across econometrics, biostatistics, and machine-learning workflows, rigorous parameter-free guarantees for the soundness of the m-out-of-n bootstrap when estimating sample quantiles have remained elusive. This paper establishes such guarantees by analysing the estimator of sample quantiles obtained from m-out-of-n resampling of a dataset of length n. We first prove a central limit theorem for a fully data-driven version of the estimator that holds under a mild moment condition and involves no unknown nuisance parameters. We then show that the moment assumption is essentially tight by constructing a counter-example in which the CLT fails. Strengthening the assumptions slightly, we derive an Edgeworth expansion that delivers exact convergence rates and, as a corollary, a Berry-Esséen bound on the bootstrap approximation error. Finally, we illustrate the scope of our results by obtaining parameter-free asymptotic distributions for practical statistics, including the quantiles for random walk MH, and rewards of ergodic MDP's, thereby demonstrating the usefulness of our theory in modern estimation and learning tasks.

Recent progress in text-video retrieval has been largely driven by contrastive learning. However, existing methods often overlook the effect of the modality gap, which causes anchor representations to undergo in-place optimization (i.e., optimization tension) that limits their alignment capacity. Moreover, noisy hard negatives further distort the semantics of anchors. To address these issues, we propose GARE, a Gap-Aware Retrieval framework that introduces a learnable, pair-specific increment ij between text ti and video vj, redistributing gradients to relieve optimization tension and absorb noise. We derive ij via a multivariate first-order Taylor expansion of the InfoNCE loss under a trust-region constraint, showing that it guides updates along locally consistent descent directions. A lightweight neural module conditioned on the semantic gap couples increments across batches for structure-aware correction. Furthermore, we regularize through a variational information bottleneck with relaxed compression, enhancing stability and semantic consistency. Experiments on four benchmarks demonstrate that GARE consistently improves alignment accuracy and robustness, validating the effectiveness of gap-aware tension mitigation.

We study the ability of language models to reason about appropriate information disclosure--a central aspect of the evolving field of agentic privacy. Whereas previous works have focused on evaluating a model's ability to align with human decisions, we examine the role of ambiguity and missing context on model performance when making information-sharing decisions. We identify context ambiguity as a crucial barrier for high performance in privacy assessments. By designing Camber, a framework for context disambiguation, we show that model-generated decision rationales can reveal ambiguities and that systematically disambiguating context based on these rationales leads to significant accuracy improvements (up to 13.3% in precision and up to 22.3% in recall) as well as reductions in prompt sensitivity. Overall, our results indicate that approaches for context disambiguation are a promising way forward to enhance agentic privacy reasoning.

We study how the choice of default prior for a common Gaussian scale affects high-dimensional shrinkage risk, highlighting the role played by high-dimensional geometry. Formally, we consider a high-dimensional setting in which the near-zero behavior of the common scale prior has first-order consequences for shrinkage risk, and show that priors that are flat on the variance and those flat on the standard deviation allocate markedly different mass near the zero-scale boundary, leading to distinct shrinkage behavior and informing principled default prior selection. Specifically, under a radial-power benchmark, we establish that the SD-flat benchmark has a one-unit asymptotic risk advantage near the origin, crosses over in the critical regime, and is second-order equivalent to the variance-flat benchmark for strong signals. Proper single global-scale hyperpriors and bounded coordinate-multiplier mixtures inherit these limits through the near-zero exponent of their SD-scale density. For heavier-tailed or sparse priors, that exponent still classifies the common global-scale component, while local-scale tails, model-size priors, or allocation priors can also affect risk.

The rapid development of Multimodal Large Language Models (MLLMs) poses increasing demands on the diversity and complexity of multimodal datasets. Yet manual annotation pipelines can no longer keep pace. Existing augmentation methods often follow fixed rules and lack verifiable control over sample diversity and reasoning complexity. To address this, we introduce Scalable COunterfactual Program Evolution (SCOPE), a framework that uses symbolic Visual Programming to guide program evolution via counterfactual reasoning. SCOPE performs the three steps of counterfactual inference: (1) Abduction, by generating verifiable programs to model reasoning associations; (2) Action, by intervening on program structure along three axes--reasoning path, visual context, and cross-instance composition; and (3) Prediction, by categorizing evolved instances by difficulty, structure, and input multiplicity. Based on this process, we build SCOPE-Train and SCOPE-Test, evolving benchmarks with expert validation. To support training, we propose MAP, a curriculum learning strategy that aligns model capacity with sample difficulty. Experiments show that SCOPEimproves reasoning performance, exposes model blind spots, and enhances visual dialog capabilities.

This paper explores the use of Hermite transform based reproducing kernel Banach space methods to construct exact or un-approximated models of feedforward neural networks of arbitrary width, depth and topology, including ResNet and Transformers networks, assuming only a feedforward topology, finite energy activations and finite (spectral-) norm weights and biases. Using this model, two straightforward but surprisingly tight bounds on Rademacher complexity are derived, precisely (1) a general bound that is width-independent and scales exponentially with depth; and (2) a width-and depth-independent bound for networks with appropriately constrained (below threshold) weights and biases.

This paper builds a hierarchy of explicit, non-asymptotic tail bounds for the supremum of the Kostlan--Shub--Smale (KSS) random field on the sphere, and applies it to two problems: Spiked Tensor PCA and the landscape of the spherical $k$-spin model. For Tensor PCA, we study the non-asymptotic statistical limits of estimating a rank-$R$ symmetric signal tensor of order~$k\ge 3$ and dimension~$d\ge 3$ from a single Gaussian observation at signal-to-noise ratio~$λ$, through the \emph{profile maximum likelihood estimator}, the MLE restricted to normalized rank-$R$ tensors of coherence at least~$κ$. Our analysis uses a single reduction: a deterministic geometric inequality (the Tube Method) and a rank-reduction step bound the estimation error by the supremum of the canonical KSS field, which the Kac--Rice formula turns into a Gaussian integral against the expected absolute characteristic polynomial of a shifted Gaussian Orthogonal Ensemble, controlled in turn by the four explicit tail bounds of our hierarchy (three from a Mehta--Fyodorov representation, one from a Ben Arous--Dembo--Guionnet large deviation). The same reduction yields two results, each with explicit constants. For estimation, a finite-$(k,d)$ error bound recovers the asymptotically optimal rate~$\sqrt{d\log k}$ of Perry, Wein and Bandeira, with explicit dependence on the rank~$R$ and the coherence~$κ$. For the landscape, a two-sided non-asymptotic bracketing of the annealed complexity of the spherical $k$-spin Hamiltonian recovers the Auffinger--Ben Arous--Černý complexity function in the high-dimensional limit.

The signature kernel [10] is a recent state-of-the-art tool for analyzing highdimensional sequential data, valued for its theoretical guarantees and strong empirical performance. In this paper, we present a novel method for efficiently computing the signature kernel of long, high-dimensional time series via adaptively truncated recursive local power series expansions. Building on the characterization of the signature kernel as the solution of a Goursat PDE [17], our approach employs tilewise Neumann-series expansions to derive rapidly converging power series approximations of the signature kernel that are locally defined on subdomains and propagated iteratively across the entire domain of the Goursat solution by exploiting the geometry of the time series. Algorithmically, this involves solving a system of interdependent Goursat PDEs via adaptively truncated local power series expansions and recursive propagation of boundary conditions along a directed graph in a topological ordering.