We develop finite-sample certificates for local population-risk increments \(Pδ_v=R(θ_0+v)-R(θ_0)\), \(v\in\mathcal D\). The primitive object is an expected-valid upper endpoint \(\widehat{\mathsf U}_{\mathcal D}\) satisfying \(\mathbb E\sup_{v\in\mathcal D} \{Pδ_v-\widehat{\mathsf U}_{\mathcal D}(v)\}\le0\). This uniform criterion certifies any measurable update selected from the same sample and allows penalties to depend on empirical geometry. The main construction is a cross-fitted ridge calibration for linear feature classes. A pilot fold learns the ridge metric, the complementary fold calibrates the squared mean error in that metric, and complete split averaging recovers the full empirical covariance in the directional quadratic form \(\widehat q_{X,λ}\). The optimized diagnostic scale is \(\{\widehat q_{X,λ}(h) \widehat r_{X,n_{\rm p},λ}^{\rm cf}/n\}^{1/2}\), and the calibrated trace factor \(\widehat r_{X,n_{\rm p},λ}^{\rm cf}\) is compared with the ordinary ridge effective dimension \(\widehat r_{X,λ}\). For nonsmooth losses, an exact fixed-mask decomposition \(δ_v=J_v^0+R_v^\circ+C_v\) separates frozen Taylor fluctuations, good-path remainders, and interface crossings. Applying the linear and composite certificates componentwise yields endpoints for same-sample expected local search and concentrated release rules.

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.

The online evaluation of machine learning models is typically conducted through A/B experiments. Sequential statistical tests are valuable tools for analysing these experiments, as they enable researchers to stop data collection early without increasing the risk of false discoveries. However, existing sequential tests either limit the number of interim analyses or suffer from low statistical power. In this paper, we introduce a novel sequential test designed for the continuous monitoring of A/B experiments. We validate our method using semi-synthetic simulations and demonstrate that it outperforms current state-of-the-art sequential testing approaches. Our method is derived using a new technique that "inverts" a bound on the probability of threshold crossing, based on a classical maximal inequality.

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 $\Delta_{ij}$ between text $t_i$ and video $v_j$, redistributing gradients to relieve optimization tension and absorb noise. We derive $\Delta_{ij}$ 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 $\Delta$ 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.

Stochastic-gradient Langevin algorithms often use tamed denominators to stabilize non-globally Lipschitz drifts. This paper shows that when the denominator depends on the same stochastic-gradient realization as the numerator, the taming step changes the stochastic oracle itself and can create a stationary bias even if the original stochastic gradient is unbiased. We propose a structure-preserving framework for designing tamed denominators. It fixes the denominator before the oracle noise is sampled and uses localized deterministic envelopes to avoid unnecessary taming in typical regions. These kernels keep the stabilizing effect of taming while avoiding the bias introduced by a gradient-dependent denominator. Our theory explains how the stationary error splits into the bias caused by oracle-dependent taming and the remaining error introduced by deterministic stabilization. Within this deterministic-envelope family, the analysis identifies a far-tail condition that explains the limitation of local soft envelopes and motivates a hybrid member: soft in the typical region, but protected by hard-tail control on rare excursions. Experiments confirm the predicted stationary distortions of random denominators, the bias reduction of deterministic-envelope designs, and the stabilizing effect of the hybrid construction.

We introduce Triangular-Reference Schrödinger Bridges for Time Series (TR-SBTS), a conservative extension of the SBTS framework in which the Brownian reference is replaced by an intervalwise frozen, possibly degenerate diffusion reference, triangular across a hierarchy of latent volatility levels. The construction is a single entropy projection on the augmented state space, with the variational constraint imposed jointly across time and the latent levels and unfolded hierarchically by the disintegration of relative entropy. The variational core of SBTS is preserved: the entropy minimiser is the h-transform of the reference, and on each frozen interval the optimal dynamics admit a logarithmic-gradient drift formula on the affine leaves of the active covariance directions, valid even when the frozen covariance is rank-deficient. We establish stability of the frozen approximation and convergence of the corresponding regularised kernel estimators. The construction is realised through a finite-dimensional conditioning map assembled from three complementary reductions of the past -- a block PCR summary, a reference-aware Mahalanobis kernel on past increments induced by the runtime frozen covariance cumulants, and a past-window WLS drift regressor under the same reference metric -- together with a coupled state-covariance bridge step in which each latent level produces a dynamic reference for the level above, summarised by a covariance descriptor; the construction is evaluated on numerical experiments.

Obtaining stable diffusion-based samplers in high- and infinite-dimensional settings is challenging because errors can accumulate across high-frequency coordinates and make the dynamics unstable under refinement of the finite-dimensional approximation of the underlying function-space problem. Discretization is a typical source of such errors, and preconditioning with a suitable spectral decay is one way to control their accumulation. In this paper, we study this problem for preconditioned annealed Langevin dynamics (ALD) applied to Gaussian mixtures. We first show that Euler-Maruyama (EM) discretization, by treating the stiff linear part of the annealed score with a forward Euler step, imposes a stability constraint coupling the preconditioner with the annealed covariance scale. Together with the conditions ensuring dimension-uniform control of the annealed dynamics, this constraint forces the initial smoothed law to remain uniformly close to the target across dimensions. We then consider an exponential-integrator scheme that integrates the stiff linear part of the annealed score exactly. Under explicit spectral summability conditions coupling the smoothing covariance, the component covariance spectra, and the preconditioner, we prove a dimension-uniform Kullback-Leibler (KL) bound for this scheme. This bound can be made arbitrarily small, uniformly in dimension, by allowing enough time for annealing and then refining the time mesh accordingly. Importantly, these conditions allow regimes in which the KL divergence between the target and the initial smoothed law diverges with dimension, showing that the restrictions imposed by EM are scheme-dependent rather than intrinsic to ALD.

The signature is a canonical representation of a multidimensional path over an interval. However, it treats all historical information uniformly, offering no intrinsic mechanism for contextualising the relevance of the past. To address this, we introduce the Exponentially Weighted Signature (EWS), generalising the Exponentially Fading Memory (EFM) signature from diagonal to general bounded linear operators. These operators enable cross-channel coupling at the level of temporal weighting together with richer memory dynamics including oscillatory, growth, and regime-dependent behaviour, while preserving the algebraic strengths of the classical signature. We show that the EWS is the unique solution to a linear controlled differential equation on the tensor algebra, and that it generalises both state-space models and the Laplace and Fourier transforms of the path. The group-like structure of the EWS enables efficient computation and makes the framework amenable to gradient-based learning, with the full semigroup action parametrised by and learned through its generator. We use this framework to empirically demonstrate the expressivity gap between the EWS and both the signature and EFM on two SDE-based regression tasks.

In contextual dynamic pricing, a seller sequentially prices goods based on contextual information.