Papers

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.

We study the spectral convergence of compact, self-adjoint operators on a separable Hilbert space, and derive the first-order asymptotic expansions for their eigenvalues, eigenvectors and eigenprojections, along with remainder bounds expressed in terms of weighted perturbation quantities. Our analysis focuses on eigenvalues indexed by a general subset, with minimal restrictions on their selection. In particular, when the eigenvalues of interest are clustered, i.e., the inner spectral gaps are small, we provide different types of expansions for the eigenvalues, with remainder bounds that are robust to the inner spectral gaps; this contrasts with the existing literature, which mainly focuses on isolated eigenvalues. The usefulness of the provided expansions is illustrated by an application to kernel Gram matrices, deriving concentration inequalities as well as weak convergence results, which, in contrast to existing literature, primarily rely on assumptions on the kernel that are easy to check.

Prediction sets can wrap around any ML model to cover unknown test outcomes with a guaranteed probability. Yet, it remains unclear how to use them optimally for downstream decision-making. Here, we propose a decision-theoretic framework that seeks to minimize the expected loss (risk) against a worst-case distribution consistent with the prediction set's coverage guarantee. We first characterize the minimax optimal policy for a fixed prediction set, showing that it balances the worst-case loss inside the set with a penalty for potential losses outside the set. Building on this, we derive the optimal prediction set construction that minimizes the resulting robust risk subject to a coverage constraint. Finally, we introduce Risk-Optimal Conformal Prediction (ROCP), a practical algorithm that targets these risk-minimizing sets while maintaining finite-sample distribution-free marginal coverage. Empirical evaluations on medical diagnosis and safety-critical decision-making tasks demonstrate that ROCP reduces critical mistakes compared to baselines, particularly when out-of-set errors are costly.

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.

Traditional population datasets are largely static and therefore unable to capture the strong temporal dynamics of human presence driven by daily mobility. Recent smartphone-based mobility data offer unprecedented spatiotemporal coverage, yet translating these opportunistic observations into accurate population estimates remains challenging due to incomplete sensing, spatially heterogeneous device penetration, and unstable observation processes. We propose a Stable-Attendance Anchor Calibration (SAAC) framework to reconstruct hourly population presence at the Census block group level across the United States. SAAC formulates population estimation as a balance-based population accounting problem, combining residential population with time-varying inbound and outbound mobility inferred from device-event observations. To address observation bias and identifiability limitations, the framework leverages locations with highly regular attendance as calibration anchors, using high schools in this study. These anchors enable estimation of observation scaling factors that correct for under-recorded mobility events. By integrating anchor-based calibration with an explicit sampling model, SAAC enables consistent conversion from observed device events to population presence at fine temporal resolution. The inferred population patterns are consistent with established empirical findings in prior mobility and urban population studies. SAAC provides a generalizable framework for transforming large-scale, biased digital trace data into interpretable dynamic population products, with implications for urban science, public health, and human mobility research. The hourly population estimates can be accessed at: https://gladcolor.github.io/hourly_population.

Network segmentation is a popular security practice for limiting lateral movement, yet practitioners lack a metric to measure how segmented a network actually is. We define segmentedness as the fraction of potential node-pair communications disallowed by policy – equivalently, the complement of graph edge density – and show it to be the first statistically principled scalar metric for this purpose. Then, we derive a normalized estimator for segmentedness and evaluate its uncertainty using confidence intervals. For a 95% confidence interval with a margin-of-error of $\pm 0.1$, we show that a minimum of $M=97$ sampled node pairs is sufficient. This result is independent of the total number of nodes in the network, provided that node pairs are sampled uniformly at random. We evaluate the estimator through Monte Carlo simulations on Erdős–Rényi, stochastic block models, and real-world enterprise network datasets, demonstrating accurate estimation. Finally, we discuss applications of the estimator, such as baseline tracking, zero trust assessment, and merger integration.

Understanding how the structure of language can be learned from sentences alone is a central question in both cognitive science and machine learning. Studies of the internal representations of Large Language Models (LLMs) support their ability to parse text when predicting the next word, while representing semantic notions independently of surface form. Yet, which data statistics make these feats possible, and how much data is required, remain largely unknown. Probabilistic context-free grammars (PCFGs) provide a tractable testbed for studying these questions. However, prior work has focused either on the post-hoc characterization of the parsing-like algorithms used by trained networks; or on the learnability of PCFGs with fixed syntax, where parsing is unnecessary. Here, we (i) introduce a tunable class of PCFGs in which both the degree of ambiguity and the correlation structure across scales can be controlled; (ii) provide a learning mechanism – an inference algorithm inspired by the structure of deep convolutional networks – that links learnability and sample complexity to specific language statistics; and (iii) validate our predictions empirically across deep convolutional and transformer-based architectures. Overall, we propose a unifying framework where correlations at different scales lift local ambiguities, enabling the emergence of hierarchical representations of the data.

The rapid growth of renewable energy penetration has intensified the need for reliable probabilistic forecasts to support grid operations at aggregated (fleet or system) levels. In practice, however, system operators often lack access to fleet-level probabilistic models and instead rely on site-level forecasts produced by heterogeneous third-party providers. Constructing coherent and calibrated fleet-level probabilistic forecasts from such inputs remains challenging due to complex cross-site dependencies and aggregation-induced miscalibration. Motivated by a real-world operational challenge identified through collaboration with an industry partner, this paper presents a probabilistic aggregation and calibration framework that directly converts site-level probabilistic forecasts into reliable fleet-level forecasts. The paper introduces \method~ (Conformal Calibration and Copula Aggregation), which integrates Copula-based dependence modeling to capture cross-site correlations followed by Context-Aware Conformal Prediction (CACP) to mitigate miscalibration at the aggregated level. This combination enables dependence-aware aggregation while improving empirical coverage and maintaining sharp prediction intervals. Experiments on large-scale solar generation datasets from three system operators in the United States including MISO, ERCOT, and SPP demonstrate that the proposed \method{} approach achieves near-nominal empirical coverage and consistently lower Winkler scores than competing aggregation-based baselines across different target coverage levels. Importantly, \method{} achieves performance comparable to directly calibrated system-level forecasts, demonstrating its practical value when dedicated fleet-level forecasting models are unavailable.

The Kullback-Leibler (KL) divergence is not a proper distance metric and does not satisfy the triangle inequality, posing theoretical challenges in certain practical applications. Existing work has demonstrated that KL divergence between multivariate Gaussian distributions follows a relaxed triangle inequality. Given any three multivariate Gaussian distributions $\mathcal{N}_1, \mathcal{N}_2$, and $\mathcal{N}_3$, if $KL(\mathcal{N}_1, \mathcal{N}_2)\leq \epsilon_1$ and $KL(\mathcal{N}_2, \mathcal{N}_3)\leq \epsilon_2$, then $KL(\mathcal{N}_1, \mathcal{N}_3)< 3\epsilon_1+3\epsilon_2+2\sqrt{\epsilon_1\epsilon_2}+o(\epsilon_1)+o(\epsilon_2)$. However, the supremum of $KL(\mathcal{N}_1, \mathcal{N}_3)$ is still unknown. In this paper, we investigate the relaxed triangle inequality for the KL divergence between multivariate Gaussian distributions and give the supremum of $KL(\mathcal{N}_1, \mathcal{N}_3)$ as well as the conditions when the supremum can be attained. When $\epsilon_1$ and $\epsilon_2$ are small, the supremum is $\epsilon_1+\epsilon_2+2\sqrt{\epsilon_1\epsilon_2}+o(\epsilon_1)+o(\epsilon_2)$. Finally, we demonstrate several applications of our results in out-of-distribution detection with flow-based generative models and safe reinforcement learning.

Contaminated mixture of experts (MoE) is motivated by transfer learning methods where a pre-trained model, acting as a frozen expert, is integrated with an adapter model, functioning as a trainable expert, in order to learn a new task. Despite recent efforts to analyze the convergence behavior of parameter estimation in this model, there are still two unresolved problems in the literature. First, the contaminated MoE model has been studied solely in regression settings, while its theoretical foundation in classification settings remains absent. Second, previous works on MoE models for classification capture pointwise convergence rates for parameter estimation without any guaranty of minimax optimality. In this work, we close these gaps by performing, for the first time, the convergence analysis of a contaminated mixture of multinomial logistic experts with homogeneous and heterogeneous structures, respectively. In each regime, we characterize uniform convergence rates for estimating parameters under challenging settings where ground-truth parameters vary with the sample size. Furthermore, we also establish corresponding minimax lower bounds to ensure that these rates are minimax optimal. Notably, our theories offer an important insight into the design of contaminated MoE, that is, expert heterogeneity yields faster parameter estimation rates and, therefore, is more sample-efficient than expert homogeneity.

Coverage analysis is essential for validating the safety of autonomous driving systems, yet existing approaches typically assess coverage factors individually or in limited combinations, struggling to capture the complex interactions inherent in traffic scenes. This paper proposes a graph-based framework for coverage analysis that represents traffic scenes as hierarchical graphs, combining map topology with actor relationships. The framework introduces a two-phase graph construction algorithm that systematically captures spatial relationships between traffic participants, including leading, following, neighboring, and opposing configurations. Two complementary coverage analysis methods are presented. First, a sub-graph isomorphism approach matches traffic scenes against a set of manually defined archetype graphs representing common driving scenarios. Second, a graph embedding approach utilizes Graph Isomorphism Networks with Edge features (GINE) trained via self-supervised contrastive learning to project traffic scenes into a vector space, enabling similarity-based coverage assessment. The framework is validated on both real-world data from the Argoverse 2.0 dataset and synthetic data from the CARLA simulator. The subgraph isomorphism method is used to calculate node coverage percentages using predefined archetypes, while the embedding approach reveals meaningful structure in the latent space suitable for clustering and anomaly detection. The proposed approach offers significant advantages over traditional methods by scaling efficiently to diverse traffic scenarios without requiring scenario-specific handling, and by naturally accommodating varying numbers of actors in a scene.

We consider the efficient inference of finite dimensional parameters arising in the context of inverse problems. Our setup is the observation of a transformation of an unknown infinite dimensional signal $f$ corrupted by statistical noise, with the transformation $K_\theta$ being linear but unknown up to a scalar $\theta$. We adopt a Bayesian approach and put a prior on the pair $(\theta,f)$ and prove a Bernstein-von Mises theorem for the marginal posterior of $\theta$ under regularity conditions on the operators $K_\theta$ and on the prior. We apply our results to the recovery of location parameters in semi-blind deconvolution problems and to the recovery of attenuation constants in X-ray tomography.

To address spatial boundary effects in climate networks, two surrogate-based correction methods, (1) subtraction and (2) division, have been widely applied in the literature. In the subtraction method, an original network measure is adjusted by subtracting the expected value obtained from a surrogate ensemble, whereas in the division method, it is normalized by dividing by this expected value. However, to the best of our knowledge, no prior study has assessed whether these two correction approaches yield statistically different results. In this study, we constructed complex networks of extreme precipitation and temperature events (EPEs and ETEs) across the CONUS for both summer (June-August, JJA) and winter (December-February, DJF) seasons. We computed key network metrics degree centrality (DC), clustering coefficient (CC), mean geographic distance (MGD), and betweenness centrality (BC) and applied both correction methods. Although the corrected spatial patterns generally appeared visually similar, statistical analyses revealed that the network measures derived from the subtraction and division methods were significantly different at the 95 percent confidence level. Across the CONUS, network hubs of EPEs were primarily concentrated in the northwestern United States during summer and shifted toward the east during winter, reflecting seasonal differences in the dominant atmospheric drivers. In contrast, the ETE networks showed strong spatial coherence and pronounced regional teleconnections in both seasons, with higher connectivity and longer synchronization distances in winter, consistent with large-scale circulation patterns such as the Pacific-North American and North Atlantic Oscillation modes. Our results indicated that the network metrics CC and MGD were more sensitive to the correction methods than the DC and BC, particularly in the EPE networks.

We consider the heat equation with absorption in a bounded domain of $\mathbb{R}^d$, where both the scalar diffusivity and the absorption function are unknown. We investigate a Bayesian approach for recovering the diffusivity from a noisy observation of the solution to the PDE over the domain. Given a Gaussian process prior on the absorption function, we derive a Bernstein-von Mises theorem for the marginal posterior distribution of the diffusivity under assumptions on the prior and on smoothness properties of the absorption.

Slice sampling is a standard Monte Carlo technique for Dirichlet process (DP)-based models, widely used in posterior simulation. However, formal assessments of the scalability of posterior slice samplers have remained largely unexplored, primarily because the computational cost of a slice-sampling iteration is random and potentially unbounded. In this work, we obtain high-probability bounds on the computational complexity of DP slice samplers. Our main results show that, uniformly across posterior cluster-growth regimes, the overhead induced by slice variables, relatively to the number of clusters supported by the posterior, is $O_{\mathbb P}(\log n)$. As a consequence, even in worst-case configurations, superlinear blow-ups in per-iteration computational cost occur with vanishing probability. Our analysis applies broadly to DP-based models without any likelihood-specific assumptions, still providing complexity guarantees for posterior sampling on arbitrary datasets. These results establish a theoretical foundation for assessing the practical scalability of slice sampling in DP-based models.

Multivariate time series imputation is often compromised by mismatch between the observed and true data distributions, a bias induced by the combined effects of time-series non-stationarity and systematic missingness. Standard methods that encourage point-wise reconstruction or direct distributional alignment may overfit these biased observations. We propose the Distributionally Robust Regularized Imputer Objective (DRIO), which jointly minimizes reconstruction error and the worst-case divergence between the imputer distribution and data distributions within a Wasserstein ambiguity set. We derive a tractable upper-bound surrogate that reduces infinite-dimensional optimization over measures to adversarial search over sample trajectories, and develop an alternating learning algorithm compatible with modern deep learning backbones. Comprehensive experiments on diverse real-world datasets show that DRIO consistently provides robust imputation and suggests improved downstream forecasting under various missingness scenarios.

We generalize the potential outcome framework to time series with an intervention by defining causal effects on stochastic processes. Interventions in dynamic systems alter not only outcome levels but also evolutionary dynamics – changing persistence and transition laws. Our framework treats potential outcomes as entire trajectories, enabling causal estimands, identification conditions, and estimators to be formulated directly on path space. The resulting Dynamic Average Treatment Effect (DATE) characterizes how causal effects evolve through time and reduces to the classical average treatment effect under one period of time. For observational data, we derive a dynamic inverse-probability weighting estimator that is unbiased under dynamic ignorability and positivity. When treated units are scarce, we show that conditional mean trajectories underlying the DATE admit a linear state-space representation, yielding a dynamic linear model implementation. Simulations demonstrate that modeling time as intrinsic to the causal mechanism exposes dynamic effects that static methods systematically misestimate. An empirical study of COVID-19 lockdowns illustrates the framework's practical value for estimating and decomposing treatment effects.

Sampling from heavy-tailed and multimodal distributions is challenging when neither the target density nor the proposal density can be evaluated, as in $\alpha$-stable Lévy-driven fractional Langevin algorithms. While the target distribution can be estimated from data via score-based or energy-based models, the $\alpha$-stable proposal density and its score are generally unavailable, rendering classical density-based Metropolis–Hastings (MH) corrections impractical. Consequently, existing fractional Langevin methods operate in an unadjusted regime and can exhibit substantial finite-time errors and poor empirical control of tail behavior. We introduce the Metropolis-Adjusted Fractional Langevin Algorithm (MAFLA), an MH-inspired, fully score-based correction mechanism. MAFLA employs designed proxies for fractional proposal score gradients under isotropic symmetric $\alpha$-stable noise and learns an acceptance function via Score Balance Matching. We empirically illustrate the strong performance of MAFLA on a series of tasks including combinatorial optimization problems where the method significantly improves finite time sampling accuracy over unadjusted fractional Langevin dynamics.

Score-based methods are powerful across machine learning, but they face a paradox: theoretically path-independent, yet practically path-dependent. We resolve this by proving that practical training objectives differ from the ideal, ground-truth objective by a crucial, overlooked term: the path variance of the score function. We propose the MVP (**M**imum **V**ariance **P**ath) Principle to minimize this path variance. Our key contribution is deriving a closed-form expression for the variance, making optimization tractable. By parameterizing the path with a flexible Kumaraswamy Mixture Model, our method learns data-adaptive, low-variance paths without heuristic manual selection. This principled optimization of the complete objective yields more accurate and stable estimators, establishing new state-of-the-art results on challenging benchmarks and providing a general framework for optimizing score-based interpolation. Our code can be found in https://github.com/Hoemr/OpenDRE.git.

Feature learning strength (FLS), i.e., the inverse of the effective output scaling of a model, plays a critical role in shaping the optimization dynamics of neural nets. While its impact has been extensively studied under the asymptotic regimes – both in training time and FLS – existing theory offers limited insight into how FLS affects generalization in practical settings, such as when training is stopped upon reaching a target training risk. In this work, we investigate the impact of FLS on generalization in deep networks under such practical conditions. Through empirical studies, we first uncover the emergence of an $\textit{optimal FLS}$ – neither too small nor too large – that yields substantial generalization gains. This finding runs counter to the prevailing intuition that stronger feature learning universally improves generalization. To explain this phenomenon, we develop a theoretical analysis of gradient flow dynamics in two-layer ReLU nets trained with logistic loss, where FLS is controlled via initialization scale. Our main theoretical result establishes the existence of an optimal FLS arising from a trade-off between two competing effects: An excessively large FLS induces an $\textit{over-alignment}$ phenomenon that degrades generalization, while an overly small FLS leads to $\textit{over-fitting}$.

Motivated by recent work on benign overfitting in overparameterized machine learning, we study the generalization behavior of functions in Sobolev spaces $W^{k, p}(\mathbb{R}^d)$ that perfectly fit a noisy training data set. Under assumptions of label noise and sufficient regularity in the data distribution, we show that approximately norm-minimizing interpolators, which are canonical solutions selected by smoothness bias, exhibit harmful overfitting: even as the training sample size $n \to \infty$, the generalization error remains bounded below by a positive constant with high probability. Our results hold for arbitrary values of $p \in [1, \infty)$, in contrast to prior results studying the Hilbert space case ($p = 2$) using kernel methods. Our proof uses a geometric argument which identifies harmful neighborhoods of the training data using Sobolev inequalities.

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.

Accurate Hessian spectra of foundation models have remained out of reach, leading most prior work to rely on small models or strong structural approximations. We show that faithful spectral analysis of the true Hessian is tractable at frontier scale. Using shard-local finite-difference Hessian vector products compatible with Fully Sharded Data Parallelism, we perform stochastic Lanczos quadrature on open-source language models with up to 100B parameters, producing the first large-scale spectral density estimates beyond the sub-10B regime. We characterize the numerical behavior of this pipeline, including finite-difference bias, floating-point noise amplification, and their effect on Krylov stability in fp32 and bf16, and derive practical operating regimes that are validated empirically. We further provide end-to-end runtime and memory scaling laws, showing that full-operator spectral probing incurs only a modest constant-factor overhead over first-order training. Crucially, direct access to the Hessian reveals that widely used block-diagonal curvature approximations can fail catastrophically, exhibiting order-one relative error and poor directional alignment even in mid-scale LLMs. Together, our results demonstrate that foundation-model Hessian spectra are both computable and qualitatively misrepresented by prevailing approximations, opening the door to principled curvature-based analysis at scale.

Flow-based methods have achieved significant success in various generative modeling tasks, capturing nuanced details within complex data distributions. However, few existing works have exploited this unique capability to resolve fine-grained structural details beyond generation tasks. This paper presents a flow-inspired framework for representation learning. First, we demonstrate that a rectified flow trained using independent coupling is zero everywhere at $t=0.5$ if and only if the source and target distributions are identical. We term this property the zero-flow criterion. Second, we show that this criterion can certify conditional independence, thereby extracting sufficient information from the data. Third, we translate this criterion into a tractable, simulation-free loss function that enables learning amortized Markov blankets in graphical models and latent representations in self-supervised learning tasks. Experiments on both simulated and real-world datasets demonstrate the effectiveness of our approach. The code reproducing our experiments can be found at: https://github.com/probabilityFLOW/zfe.

We detail a novel Fourier-based approach (IterativeFT) for identifying deterministic network structure in the presence of both edge pruning and Gaussian noise. This technique involves the iterative execution of forward and inverse 2D discrete Fourier transforms on a target network adjacency matrix. The denoising ability of the method is achieved via the application of a sparsification operation to both the real and frequency domain representations of the adjacency matrix with algorithm convergence achieved when the real domain sparsity pattern stabilizes. To demonstrate the effectiveness of the approach, we apply it to noisy versions of several deterministic models including Kautz, lattice, tree and bipartite networks. For contrast, we also evaluate preferential attachment networks to illustrate the behavior on stochastic graphs. We compare the performance of IterativeFT against simple real domain and frequency domain thresholding, reduced rank reconstruction and locally adaptive network sparsification. Relative to the comparison network denoising approaches, the proposed IterativeFT method provides the best overall performance for lattice and Kuatz networks with competitive performance on tree and bipartite networks. Importantly, the InterativeFT technique is effective at both filtering noisy edges and recovering true edges that are missing from the observed network.

Coherent estimators at different sample sizes need not describe the same population risk. We prove that dyadic estimators do so exactly when duplicating every scenario leaves the estimate unchanged, and recover the population dual set from the finite models. The risk of a Bernoulli loss determines the exact Wasserstein, Kolmogorov and total-variation moduli on a common bounded support. It also characterises uniform estimability, with an impossibility bound for arbitrary randomised estimators. An intrinsic floor on tail levels gives an explicit optimal Lipschitz constant and a uniform approximation defect; every deterministic vanishing floor preserves pointwise strong consistency. A nonstandard construction supplies compatible estimators at all finite sample sizes, explicit block-sum weights, and a formula for the worst-case mass created by infinitesimal tail levels. Finally, we derive uniform regret bounds for data-driven risk-averse optimisation and specialise them to stop-loss retention. Exact binomial calculations compare regularised and original estimators against the same population target.

Online reinforcement learning in non-episodic, finite-horizon MDPs remains underexplored and is challenged by the need to estimate returns to a fixed terminal time. Existing infinite-horizon methods, which often rely on discounted contraction, do not naturally account for this fixed-horizon structure. We introduce a modified Q-function: rather than targeting the full-horizon, we learn a K-step lookahead Q-function that truncates planning to the next K steps. To further improve sample efficiency, we introduce a thresholding mechanism: actions are selected only when their estimated K-step lookahead value exceeds a time-varying threshold. We provide an efficient tabular learning algorithm for this novel objective, proving it achieves fast finite-sample convergence: it achieves minimax optimal constant regret for $K=1$ and $\mathcal{O}(\max((K-1),C_{K-1})\sqrt{SAT\log(T)})$ regret for any $K \geq 2$. We numerically evaluate the performance of our algorithm under the objective of maximizing reward. Our implementation adaptively increases K over time, balancing lookahead depth against estimation variance. Empirical results demonstrate superior cumulative rewards over state-of-the-art tabular RL methods across synthetic MDPs and RL environments: JumpRiverswim, FrozenLake and AnyTrading. Code is provided on \href{https://github.com/jamie01713/K-Step-Lookahead}{github}.

Classifier-free guidance (CFG) is the de facto standard for conditional sampling in diffusion models, yet it often reduces sample diversity. Using tools from statistical physics, we analyze the emergence of generative distortions induced by CFG, namely the mismatch between the CFG sampling distribution and the true conditional distribution. We study this phenomenon in analytically tractable settings with exact score functions, characterizing its dependence on data dimensionality and the number of classes. For high-dimensional Gaussian mixtures, we use dynamic mean-field theory to show that distortions arise when the number of classes scales exponentially with the data dimension, whereas they vanish in the sub-exponential regime due to a dynamical phase transition. We further prove that, in the infinite-class limit, distortions remain unavoidable regardless of dimensionality because of the increasing density of classes. Finally, we show that standard CFG schedules cannot prevent variance shrinkage, and we propose a theoretically grounded guidance schedule incorporating a negative-guidance window that improves both class separability and sample diversity in real-world latent diffusion models.

In this paper, we propose a general methodology for sampling from un-normalized densities defined on Riemannian manifolds, with a particular focus on multi-modal targets that remain challenging for existing sampling methods. Inspired by the framework of diffusion models developed for generative modeling, we introduce a sampling algorithm based on the simulation of a non-equilibrium deterministic dynamics that transports an easy-to-sample noise distribution toward the target. At the marginal level, the induced density path follows a prescribed stochastic interpolant between the noise and target distributions, specifically constructed to respect the underlying Riemannian geometry. In contrast to related generative modeling approaches that rely on machine learning, our method is entirely training-free. It instead builds on iterative posterior sampling procedures using only standard Monte Carlo techniques, thereby extending recent diffusion-based sampling methodologies beyond the Euclidean setting. We complement our approach with a rigorous theoretical analysis and demonstrate its effectiveness on a range of multi-modal sampling problems, including high-dimensional and heavy-tailed examples.

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.