We investigate the role of feature superposition in the emergence of power-law training dynamics using a teacher-student framework. We first derive an analytic theory for training without superposition, establishing that the power-law training exponent depends on both the input data statistics and channel importance. Remarkably, we discover that a superposition bottleneck induces a transition to a universal power-law exponent of $\sim 1$, independent of data and channel statistics. This one over time training with superposition represents an up to tenfold acceleration compared to the purely sequential learning that takes place in the absence of superposition. Our finding that superposition leads to rapid training with a data-independent power law exponent may have important implications for a wide range of neural networks that employ superposition, including production-scale large language models.

Scientific machine learning (SciML) increasingly requires models that capture multimodal conditional uncertainty arising from ill-posed inverse problems, multistability, and chaotic dynamics. While recent work has favored highly expressive implicit generative models such as diffusion and flow-based methods, these approaches are often data-hungry, computationally costly, and misaligned with the structured solution spaces frequently found in scientific problems. We demonstrate that Mixture Density Networks (MDNs) provide a principled yet largely overlooked alternative for multimodal uncertainty quantification in SciML. As explicit parametric density estimators, MDNs impose an inductive bias tailored to low-dimensional, multimodal physics, enabling direct global allocation of probability mass across distinct solution branches. This structure delivers strong data efficiency, allowing reliable recovery of separated modes in regimes where scientific data is scarce. We formalize these insights through a unified probabilistic framework contrasting explicit and implicit distribution networks, and demonstrate empirically that MDNs achieve superior generalization, interpretability, and sample efficiency across a range of inverse, multistable, and chaotic scientific regression tasks.

Multivariate time series (MTS) imputation is often compromised by mismatch between observed and true data distributions -- a bias exacerbated by non-stationarity and systematic missingness. Standard methods that minimize reconstruction error or encourage distributional alignment risk overfitting these biased observations. We propose the Distributionally Robust Regularized Imputer Objective (DRIO), which jointly minimizes reconstruction error and the divergence between the imputer and a worst-case distribution within a Wasserstein ambiguity set. We derive a tractable dual formulation that reduces infinite-dimensional optimization over measures to adversarial search over sample trajectories, and propose an adversarial learning algorithm compatible with flexible deep learning backbones. Comprehensive experiments on diverse real-world datasets show DRIO consistently improves imputation under both missing-completely-at-random and missing-not-at-random settings, reaching Pareto-optimal trade-offs between reconstruction accuracy and distributional alignment.

A common approach to perform PCA on probability measures is to embed them into a Hilbert space where standard functional PCA techniques apply. While convergence rates for estimating the embedding of a single measure from $m$ samples are well understood, the literature has not addressed the setting involving multiple measures. In this paper, we study PCA in a double asymptotic regime where $n$ probability measures are observed, each through $m$ samples. We derive convergence rates of the form $n^{-1/2} + m^{-α}$ for the empirical covariance operator and the PCA excess risk, where $α>0$ depends on the chosen embedding. This characterizes the relationship between the number $n$ of measures and the number $m$ of samples per measure, revealing a sparse (small $m$) to dense (large $m$) transition in the convergence behavior. Moreover, we prove that the dense-regime rate is minimax optimal for the empirical covariance error. Our numerical experiments validate these theoretical rates and demonstrate that appropriate subsampling preserves PCA accuracy while reducing computational cost.

Backdoor and data poisoning attacks can achieve high attack success while evading existing spectral and optimisation based defences. We show that this behaviour is not incidental, but arises from a fundamental geometric mechanism in input space. Using kernel ridge regression as an exact model of wide neural networks, we prove that clustered dirty label poisons induce a rank one spike in the input Hessian whose magnitude scales quadratically with attack efficacy. Crucially, for nonlinear kernels we identify a near clone regime in which poison efficacy remains order one while the induced input curvature vanishes, making the attack provably spectrally undetectable. We further show that input gradient regularisation contracts poison aligned Fisher and Hessian eigenmodes under gradient flow, yielding an explicit and unavoidable safety efficacy trade off by reducing data fitting capacity. For exponential kernels, this defence admits a precise interpretation as an anisotropic high pass filter that increases the effective length scale and suppresses near clone poisons. Extensive experiments on linear models and deep convolutional networks across MNIST and CIFAR 10 and CIFAR 100 validate the theory, demonstrating consistent lags between attack success and spectral visibility, and showing that regularisation and data augmentation jointly suppress poisoning. Our results establish when backdoors are inherently invisible, and provide the first end to end characterisation of poisoning, detectability, and defence through input space curvature.

Identifying truly predictive covariates while strictly controlling false discoveries remains a fundamental challenge in nonlinear, highly correlated, and low signal-to-noise regimes, where deep learning based feature selection methods are most attractive. We propose Group Regularization Importance Persistence in 2 Dimensions (GRIP2), a deep knockoff feature importance statistic that integrates first-layer feature activity over a two-dimensional regularization surface controlling both sparsity strength and sparsification geometry. To approximate this surface integral in a single training run, we introduce efficient block-stochastic sampling, which aggregates feature activity magnitudes across diverse regularization regimes along the optimization trajectory. The resulting statistics are antisymmetric by construction, ensuring finite-sample FDR control. In extensive experiments on synthetic and semi-real data, GRIP2 demonstrates improved robustness to feature correlation and noise level: in high correlation and low signal-to-noise ratio regimes where standard deep learning based feature selectors may struggle, our method retains high power and stability. Finally, on real-world HIV drug resistance data, GRIP2 recovers known resistance-associated mutations with power better than established linear baselines, confirming its reliability in practice.

A/B testing has become a gold standard for modern technological companies to conduct policy evaluation. Yet, its application to time series experiments, where policies are sequentially assigned over time, remains challenging. Existing designs suffer from two limitations: (i) they do not fully leverage the entire history for treatment allocation; (ii) they rely on strong assumptions to approximate the objective function (e.g., the mean squared error of the estimated treatment effect) for optimizing the design. We first establish an impossibility theorem showing that failure to condition on the full history leads to suboptimal designs, due to the dynamic dependencies in time series experiments. To address both limitations simultaneously, we next propose a transformer reinforcement learning (RL) approach which leverages transformers to condition allocation on the entire history and employs RL to directly optimize the MSE without relying on restrictive assumptions. Empirical evaluations on synthetic data, a publicly available dispatch simulator, and a real-world ridesharing dataset demonstrate that our proposal consistently outperforms existing designs.

Conformal Prediction (CP) provides a statistical framework for uncertainty quantification that constructs prediction sets with coverage guarantees. While CP yields uncontrolled prediction set sizes, Backward Conformal Prediction (BCP) inverts this paradigm by enforcing a predefined upper bound on set size and estimating the resulting coverage guarantee. However, the looseness induced by Markov's inequality within the BCP framework causes a significant gap between the estimated coverage bound and the empirical coverage. In this work, we introduce ST-BCP, a novel method that introduces a data-dependent transformation of nonconformity scores to narrow the coverage gap. In particular, we develop a computable transformation and prove that it outperforms the baseline identity transformation. Extensive experiments demonstrate the effectiveness of our method, reducing the average coverage gap from 4.20\% to 1.12\% on common benchmarks.

Most existing offline RL methods presume the availability of action labels within the dataset, but in many practical scenarios, actions may be missing due to privacy, storage, or sensor limitations. We formalise the setting of action-free offline-to-online RL, where agents must learn from datasets consisting solely of $(s,r,s')$ tuples and later leverage this knowledge during online interaction. To address this challenge, we propose learning state policies that recommend desirable next-state transitions rather than actions. Our contributions are twofold. First, we introduce a simple yet novel state discretisation transformation and propose Offline State-Only DecQN (\algo), a value-based algorithm designed to pre-train state policies from action-free data. \algo{} integrates the transformation to scale efficiently to high-dimensional problems while avoiding instability and overfitting associated with continuous state prediction. Second, we propose a novel mechanism for guided online learning that leverages these pre-trained state policies to accelerate the learning of online agents. Together, these components establish a scalable and practical framework for leveraging action-free datasets to accelerate online RL. Empirical results across diverse benchmarks demonstrate that our approach improves convergence speed and asymptotic performance, while analyses reveal that discretisation and regularisation are critical to its effectiveness.

Evidential Deep Learning (EDL) is a popular framework for uncertainty-aware classification that models predictive uncertainty via Dirichlet distributions parameterized by neural networks. Despite its popularity, its theoretical foundations and behavior under distributional shift remain poorly understood. In this work, we provide a principled statistical interpretation by proving that EDL training corresponds to amortized variational inference in a hierarchical Bayesian model with a tempered pseudo-likelihood. This perspective reveals a major drawback: standard EDL conflates epistemic and aleatoric uncertainty, leading to systematic overconfidence on out-of-distribution (OOD) inputs. To address this, we introduce Density-Informed Pseudo-count EDL (DIP-EDL), a new parametrization that decouples class prediction from the magnitude of uncertainty by separately estimating the conditional label distribution and the marginal covariate density. This separation preserves evidence in high-density regions while shrinking predictions toward a uniform prior for OOD data. Theoretically, we prove that DIP-EDL achieves asymptotic concentration. Empirically, we show that our method enhances interpretability and improves robustness and uncertainty calibration under distributional shift.