Papers

We study multiple change-point detection in multivariate time series whose distributions change in a piecewise constant manner. Distributional changes can manifest across different moment orders, from shifts in the mean and covariance to changes in higher-order moments. Higher-order moments capture increasingly rich distributional features but become difficult to estimate in high dimensions. Our tensor representation unifies moments of different orders within a common linear algebraic framework, enabling a new method to detect changes in moments of all orders up to a prescribed fixed order $p$. The resulting procedure accommodates temporal dependence and allows the dimension of the time series to grow with the sample size. Under suitable regularity conditions, the proposed procedure achieves a localization error rate that matches a newly developed minimax lower bound. We further derive limiting distributions under both nonvanishing and vanishing moment jumps and construct asymptotically valid confidence intervals in the vanishing-jump regime. Numerical experiments and real-data analyses demonstrate the method's effectiveness in detecting moment changes and a range of distributional shifts.

In multiclass probabilistic prediction, Utility Calibration (UC), which focuses auditing on specified utilities, has recently received attention as a way to guarantee downstream decisions while controlling computational and sample requirements. At the same time, some multiclass problems have meaningful label hierarchies that play important roles in medicine and image classification, yet how UC evaluates utility within a hierarchy remains insufficiently understood. We show that the difference between realized utility and predicted mean utility admits an exact decomposition into a sum of contributions from the internal nodes of the label tree. This decomposition shows that positive and negative contributions from different nodes can cancel, and that even when UC is small, the utility errors remaining in parts of the hierarchy need not be small. To address this problem, we propose Hierarchical Utility Calibration (HUC), which evaluates each node contribution before summation while retaining the same target utility, subgroup, and predicted-utility interval. We further provide finite-sample evaluation over all predicted-utility intervals and propose HUC-Boost, which updates only violated internal nodes, with theoretical guarantees for both.

We study an unknown-transition finite-horizon Markov decision process (MDP) with a finite collection of known reward functions $\{r^1, r^2, \ldots, r^M\}$. The goal is to output an $\epsilon$-optimal policy for every reward using online episodic interaction only. Performance is measured by the policy error $V_{0}^{*, m} - V_{0}^{\widehat\pi^{m}, m}$ where $m\in [M]$ represents the reward function and $V_{0}^{*, m}=\mathbb{E}_{s_1\sim \mu}[V_{1}^{*, m}(s_1)]$. Under this setting, we design a provably efficient algorithm to establish a minimax sample complexity bound of $$ O\left(\frac{SAH^3}{\epsilon^2}\log M \mathrm{polylog}\left(\frac{SAH\log M}{\min\left\{\epsilon, 1\right\}\delta}\right)\right)$$ episodes, with no additional burn-in cost. This matches the information-theoretic lower bound up to a factor of $ \mathrm{polylog}(SAH\log M/(\min\left\{\epsilon, 1\right\}\delta))$. Our method combines three technical ingredients. First, we adapt MVP to reward-switching learning to construct optimistic value estimates. Second, we use fresh replay samples to conservatively evaluate the candidate policies. Third, gap-based multiplicative weights updates adjust the reward-sampling distribution using the differences between these estimates, converting weighted learning progress into simultaneous guarantees for all rewards.

A central question in the prediction of stochastic processes is the extent to which past information can improve the probability of correctly predicting the next state. We introduce the Memory Prediction Excess (MPE) to address this question quantitatively. The MPE measures the average improvement in prediction accuracy obtained by using the entire observed history relative to using only the static marginal distribution, in discrete-time finite-state processes. It is defined as the difference between the expected optimal conditional prediction accuracy and the optimal static prediction accuracy. Its basic properties are examined: the MPE is always non-negative; it admits an upper bound depending on the static accuracy, attained if and only if the future is almost surely a deterministic function of the past; and degenerate cases in which the MPE vanishes are characterized. A normalized version, taking values in the unit interval, is introduced as a dimensionless measure of predictive efficiency. A lower bound is derived by comparing predictions based on histories of different lengths, showing that the expected optimal prediction accuracy is monotone with respect to the history length. The framework is extended to finite-length histories, where the finite-history MPE (FH-MPE) measures the predictive gain attainable when only the most recent observations are retained. This leads to the notion of a minimal memory length required to achieve the same predictive performance as the full history. For finite-order Markov chains, this minimal memory length is shown to be bounded by the Markov order. The MPE and its variants are formulated in terms of conditional probabilities and prediction accuracies, offering a probabilistic perspective on the predictive utility of memory that is complementary to classical information-theoretic approaches.

As black-box machine learning models become increasingly common, extracting interpretations with uncertainty quantification has become a critical challenge. One popular type of interpretation is leave-one-covariate-out (LOCO) feature importance, while prior LOCO inference methods often require data-splitting or model-refitting. A recent ensemble framework, LOCO-MP, addresses these challenges using minipatches that subsample both observations and features, but massive feature subsampling can hurt prediction in high-dimensional sparse settings. Motivated by this limitation, we consider minipatch ensembles with adaptive feature sampling guided by LOCO importance, and propose LOCO-AdaMP, which enables free LOCO inference for the resulting adaptive minipatch ensemble. We show that LOCO-AdaMP yields substantially improved predictive models while retaining asymptotically valid feature importance inference without data-splitting, despite the complex dependence between the adaptive sampling distribution and the LOCO importance statistics. Our analysis relies on a careful leave-two-out perturbation bound for the iteratively updated sampling probabilities together with the stability of LOCO scores induced by observation subsampling. Empirical results on synthetic and real datasets demonstrate advantages of LOCO-AdaMP over existing methods in predictive performance, inferential power, and stability. Overall, LOCO-AdaMP provides a flexible ensemble framework (agnostic to base models) that delivers both strong predictive performance and asymptotically valid, powerful feature importance inference for regression.

Offline reinforcement learning seeks optimal decision rules from previously collected data. In some applications, a decision can be an entire function, such as a fluence map in radiation therapy or a smooth movement trajectory in robotics. In this paper, we study the finite-sample theory for fitted Q-iteration (FQI) with functional actions in a discounted infinite-horizon setting. Three major difficulties arise in this setting: first, the absence of a Lebesgue probability density for functional actions complicates coverage descriptions; second, conventional coverage requirements can be restrictive; and third, the large functional action space makes greedy optimization in FQI challenging. To address these difficulties, we study smoothness-regularized policy search under a critic-relative coverage condition. This condition measures how well logged data distinguish relevant action-value differences without requiring an action density. Our main theorem gives finite-sample guarantees for learned-policy regret relative to the best value within a fixed smooth class of functional-action policies. The results allow trajectory lengths to be either bounded or growing and Q-functions to be fitted by either functional-input kernel ridge regression or adaptive functional neural networks. For a few examples, we can obtain polynomially decaying regret bounds in the number of logged transitions, up to logarithmic factors, with logarithmically many FQI iterations. Numerical experiments show gains of learned functional-action policies over constant-action policies and support our adoption of a critic-relative coverage condition.

Tabular foundation models achieve strong performance by conditioning on labelled examples in context, but softmax attention limits their use on large datasets. Existing linear-time alternatives, however, are mostly causal, and their potential for tabular in-context learning (ICL) remains underexplored. To address this, we (1) revisit causal training setups, (2) compare linear sequence mixers, and (3) investigate their ICL generalisation beyond the pretraining context length. First, we show that the best training setup for causal models resembles next-token prediction. Then, perhaps surprisingly, the most promising linear sequence mixer is causal: DeltaNet outperforms even non-causal linear attention. However, it degrades beyond $2$-$4\times$ the pretraining context length, and existing mitigation strategies such as bidirectionality defer the problem at best. A hidden-state oracle shows that this is not a capacity problem. Instead, our analysis points to an instability in the recurrent state, which drifts in deeper layers of causal models. Since DeltaNet's learned write rates overfit to the pretraining regime, we modulate them with a time-dependent decay schedule intervention to stabilise length generalisation. Finally, re-introducing non-causality by reading out from the final state allows us to closely match a controlled softmax attention baseline on OpenML-CC18 and TabArena.

Partial Least Squares (PLS) regression extracts a few outcome-aligned directions in a high-dimensional X and is widely used across applied science, but inference on the resulting fit is either expensive, biased and discouraged, or absent. We reduce inference to held-out OLS refits of the supervised subspace, a primitive shared by PLS, supervised PCA, and linear probes, and supply two tests using held-out correlations: a Nadeau-Bengio corrected asymptotic t-test as a fast approximation, and a permutation test with comparable power, finite-sample valid under outcome-predictor independence and iid rows. Held-out predictions are unchanged under any orthogonal rebasing of the supervised span, so an interpretable basis such as varimax inherits the joint claim but not a per-axis p-value; per-component claims come from a fixed-sequence test on the PLS extraction order. We validate on synthetic geometries, two NIR chemometric datasets, and cross-lingual word-embedding regressions; the exact test also transfers to supervised PCA and a ridge probe. The proposed tests have more power than CV-permutation-Q^2, at a fraction of its cost. A pre-run check on n and the spectrum of X says when the approximation is safe. We release a Rust library with Python, R, and Julia bindings, plus a Python text pipeline.

Mixture-of-Experts (MoE) layers are central to frontier language models, and recent architectures push toward more and smaller experts. In this regime, the standard linear router becomes a bottleneck: with $M$ experts and hidden dimension $h$, its per-token cost $\Theta(Mh)$ dominates the MoE layer once $M$ is large. We introduce MoRE (Mixture of Rank-reduced-routed Experts), which factorizes the router weight matrix at rank $r$ and reduces the routing cost to $O((h + M)r)$. We prove that rank logarithmic in $M$ suffices for routing expressivity when the number of active experts is fixed, and is necessary up to precision factors. We also prove that logarithmic rank preserves load balance in a Gaussian memorization model, and training on a synthetic phonebook task shows that low rank does not hurt memorization. At matched active FLOPs, the factorization allows a factor of $\Theta(h/r)$ more experts. To realize this gain in wall-clock time, we design a fused Triton kernel at inference that avoids expensive memory operations on HBM. Empirically, MoRE improves memorization on the phonebook task and performance on knowledge-intensive Q&A benchmarks after pretraining, while matching reasoning ability. Code available at https://github.com/Matheart/MoRE_code.

Deep forecasters predict from a fixed-length lookback window, and lengthening it gives diminishing returns at a growing cost. Retrieval augmentation instead shows the model how similar past situations continued. Retrieving a whole past window gives every variate the continuation of the same past moment. In multivariate series, however, the best past match differs from variate to variate. We present GNA (Granular Neighbor Assembly), a retrieval layer for forecasting backbones that assembles neighbors at two granularities: whole past windows, which keep the variates coherent, and per-variate neighbors, in which each variate takes its future from its own best-matching past. A learned gate decides, per forecast step and variate, how much to trust these futures against a persistence forecast, next to the backbone's own forecast. Candidates come from an embedding trained to predict each window's future, and retrieval is strictly causal: a past window is used only once its future has been observed. With the same lookback for every model and the same retrieval constants for all datasets, GNA improves two Transformer backbones in 85 of 96 dataset-horizon settings, gives the lowest MSE on 8 of 12 standard benchmarks and beats its backbone in every seed on 10 of them. Both granularities are needed, and neighbors of mismatched queries are worse than none. Retrieval helps most where the lookback says least: the gate shifts trust to retrieved futures further ahead. Where it fails, on hourly non-stationary series at long horizons, the loss is consistent with a drifting level of the retrieved futures.

Recent work has established that power-law spectral conditions on data enable tight convergence bounds for deterministic gradient descent, resolving the conflict between classical exponential bounds and observed power-law learning curves. In this work, we extend this result to the stochastic regime of high-dimensional machine learning. We provide two main contributions: (1) We generalize the power-law spectral theory to Stochastic Gradient Descent (SGD), showing that the same spectral exponents govern stochastic dynamics; (2) For the fundamental case of isotropic Gaussian data, we provide a precise analysis of data shuffling, deriving exact constants that prove Single Shuffle is strictly superior to Flip-Flop and IID sampling. Our results bridge the gap between abstract spectral theory and practical stochastic training choices, offering a unified picture of how data geometry drives optimization speed.

Reliable evaluation of large language models (LLMs) is essential for their development and deployment, yet is often costly, risky, and difficult to perform safely online. We study off-policy evaluation for LLMs, where limited human-labeled data from a behavior model are used to evaluate a newer target LLM. This setting is challenging because labels are scarce, behavior–target distribution shift is common, and response likelihoods are often unavailable for black-box LLMs. We propose the Optimal Transport-based Robust Off-Policy Evaluation (OTROPE), a likelihood-free evaluation that performs distributional correction in a semantic space via optimal transport to align labeled behavior-policy samples with unlabeled target-policy samples. OTROPE combines corrected human-labeled residuals with proxy predictors, yielding a doubly robust-style evaluation without behavior-policy modeling or density-ratio estimation. We theoretically characterize why baseline evaluators fail under LLM distribution shift, and establish consistency and convergence rates for OTROPE when either the reweighted behavior distribution or the proxy predictor converges. Experiments on synthetic and real LLM evaluation tasks show that OTROPE consistently outperforms baselines while enabling ensembles of weaker LLM evaluators to approach and sometimes surpass stronger evaluators. Code is available at https://github.com/LinerXiang/OTROPE.

Closed-form recourse moves a rejected user along the unit gradient $\hat g$ of the classifier score $f$ by the promised distance $d_p=|f(x)|/\|\nabla f(x)\|$, at which the linearized score reaches zero. We ask when this one-shot step succeeds and what additional model queries change. To leading order the step ends on the favorable side exactly when the path curvature $\kappa=\hat g^\top\nabla^2 f(x)\,\hat g$ is nonnegative. Across 80 shallow models, the fraction of rejected users whose step ends there and the fraction with $\kappa\ge0$ correlate at $r=0.985$, although on Fashion-MNIST the first falls below the second by 8.2 points on average. No rule that uses only the score value and gradient can be valid for every score with path curvature bounded by $K$ without overshooting some by order $Kd_p^2/\|\nabla f(x)\|$. When the curvature is also Lipschitz and the step is short, one evaluation of $f$ at the promised point attains the minimax rate among deterministic one-query rules that know the curvature bound and its Lipschitz constant, and split-conformal calibration makes such a rule reach the first crossing or abstain with probability at least $1-\delta$. Training with an asymmetric curvature penalty lets 99-100% of paths cross within the promised step on undershoot-prone shallow data, at about 4-22 times the overshoot of symmetric penalties (Fashion-MNIST, COMPAS). Because $\kappa$ and $d_p$ depend on how the score is scaled, part of this gain can be a longer promised step, and at matched validity a smaller audit of briefly trained models finds no uniform advantage over tuned inflation. Where a per-user line search along the ray is affordable, it is exact to grid resolution and preferable.

Does representation learning stop when the training loss stops improving? We study this question for matrix Muon, whose polar-normalised updates have a step length set by the gradient's rank rather than its norm. Near the edge of stability, full-batch Muon on teacher-student problems enters approximately period-2 loss oscillations that persist for thousands of steps: the cycle-mean loss stays flat or rises, yet the weights keep moving and the learned features continue to align with the teacher subspace. For linear teacher-student learning toys, we derive explicit cycle and alignment formulas and conditional plateau and decay bounds. For a population mean-field ReLU model, we prove that, under stated dimension, initialisation and small-head conditions, the leading eigenspace of the average gradient outer product (AGOP) recovers the teacher subspace exactly during a loss plateau, before the loss later drops. In all 33 ReLU, GELU and SiLU teacher configurations we study, direction-only alignment metrics show the student AGOP aligned with, or still aligning to, the teacher subspace during the period-2 oscillations; projected head refitting on selected configurations shows that the learned directions are useful for prediction, and further measurements distinguish AGOP alignment from weight-mass concentration. In deep residual ReLU students, freezing the downstream layers while the first layer trains with full-batch exact polar updates recreates a nearly flat cycle-mean loss with improving input-AGOP alignment; freezing and unfreezing switch between this plateau and loss decrease, and the effect is sensitive to momentum and to the choice of orthogonaliser.

Next-latent prediction fits a map from the current embedding to the next one. LeNEPA carries this objective to time series, replacing the stop-gradient of next-embedding prediction with the isotropy penalty of LeJEPA. A world model is a transition kernel that can be rolled out. The one-step regression identifies a conditional mean, and a mean is a kernel only in special cases. For a linear-Gaussian Markov latent, the mean transition and the innovation covariance are fixed by the one-step problem, and the open-loop squared error at horizon $K$ equals the trace of the sum of the pushed-forward innovation covariances. That error grows with $K$ after the one-step fit is exact. If the conditional mean is nonlinear, composing it is not the multi-step conditional mean. If the observation is a non-injective function of a Markov state, a memoryless one-step map does not determine future observations, while a short window can. An isotropy penalty is a function of the embedding marginal, so its partial derivative in the transition weights is zero. On a scalar autoregression with coefficient $0.9$, the one-step mean squared error is $0.998$ and the $16$-step open-loop error is $5.10$. On a hidden rotation, an eight-step window reaches $16$-step error $0.056$, while the current scalar alone reaches $0.778$. Raising the isotropy weight from $0.1$ to $10$ leaves eight-step latent error inside $[0.78,0.85]$ on three seeds.

In Bayesian inference problems with non-Gaussian observation noise, the posterior is only as accurate as the noise density, and gradient-based samplers need that density and its gradient evaluable pointwise, whether from an explicit expression or from code, and without an inner solve. We propose Copula Active Subspaces (CAS) to represent this noise density. A componentwise rank transform isolates the noise law's dependence in its copula, and a rank-$r$ reduction keeps only the directions along which that dependence varies. These directions are the leading eigenvectors of the copula score covariance $\boldsymbol{C} := \mathrm{Cov}_{\pi_{\boldsymbol{Z}}}(\nabla\log c^{Z})$, which is what makes the reduction a copula active subspace. Because $\boldsymbol{C}$ vanishes when the coordinates are independent, these are directions of dependence, which the covariance of the data need not identify. From this construction follow a Gaussian-reference KL divergence bound with the explicit constant $\tfrac{1}{2}$, minimized over all rank-$r$ reductions by exactly this eigenspace; a diagnostic for the error the reduction leaves behind, computable from the samples alone; and, from Hermite score matching, a reduced log-density and gradient in closed form, with the truncation orders and the Stage-2 regularization constants chosen on validation samples. The reduction replaces a $d$-dimensional density estimation problem by an $r$-dimensional one. On a $d=20$ noise law and a Bayesian inference problem with that noise, CAS lowers noise KL divergence more than fivefold and posterior KL divergence more than sevenfold against Gaussian-copula, product-of-marginals, and PCA-subspace baselines, and lowers noise KL divergence by factors of about $3.5$ and $2.7$ on two further $d=20$ examples.

In spatial observational studies, treatment assignment and outcomes often exhibit spatial dependence patterns, and treatment effects may vary across space and subpopulations due to both measured and unmeasured spatially structured confounders. Accounting for spatial dependence while estimating heterogeneous treatment effects (HTEs) is a central task in spatial causal inference. Causal Bayesian additive regression tree methods are popular nonparametric methods for modeling and estimating HTEs. Despite their flexibility and uncertainty quantification, the axis-aligned split rules often adopted in these models are not suitable for modeling spatial structures. We propose a spatial structure-aware Bayesian nonparametric method, called Graph-Split Bayesian Causal Forest (GSBCF), that integrates graph-split Bayesian additive regression trees (GS-BART) with the Bayesian causal forest propensity-score regression framework for spatial heterogeneous causal inference. Spatial confounding is accommodated through graph-guided split rules in modeling decision trees of the prognostic and HTE functions. We develop an efficient informed proposal sampling algorithm for posterior computation, enabling full Bayesian inference of the spatial conditional average treatment effect function. Simulations and a real data study demonstrate substantially improved estimation accuracy and uncertainty quantification over existing causal BART methods.

Geometry does more than constrain an associative memory: curvature determines what it remembers and which states it creates. We develop intrinsic dense associative memories on Riemannian manifolds by casting memory as Epanechnikov kernel-density mode seeking. We compare geodesic and volume-corrected energies and show that curvature separates their behavior. We prove that geodesic memory always retains an isolated pattern, while corrected memory obeys a sharp Ricci-curvature threshold: positive curvature can erase memories in high dimensions, while negative curvature reinforces them. We derive geodesic capacity scalings of $q_\beta^{-1/2}$ for retaining every pattern and $q_\beta^{-1}$ for a typical one, where $q_\beta$ is the pairwise kernel-overlap probability. We show how overlap creates novel memories: designed $N$-pattern configurations realize all $2^N-1$ subset modes, but random data at the storage threshold yield only a Poisson number. We establish exact one-step recall using Riemannian mean shift. In simulations, we recover the predicted curvature transition and every designed mode. On WordNet's full noun hierarchy, we demonstrate that volume correction improves low-capacity retrieval. Together, our work shows that curvature is a design variable for associative memory, not merely a property of the data.

Many inference-time tasks for pretrained discrete diffusion models and diffusion language models reduce to drawing samples from a tilted version of the pretrained distribution. Feynman-Kac sequential Monte Carlo (SMC) makes this correction exact in principle, but its prescribed weights routinely degenerate when the proposal dynamics are misaligned with the tilt, capping the practical gains from additional particles. We introduce FluxLite, a lightweight, training-free proposal-control framework for discrete diffusion. On the sparse directed graph of pretrained reverse rates, any sparse jump-rate perturbation can be exactly compensated by a $q_t$-weighted graph-divergence term in the Feynman-Kac potential; the target path is therefore preserved while the residual reweighting variance becomes a local convex objective. We instantiate this principle as two practical samplers: a one-hop local reallocation rule (HEU) and a small nonnegative quadratic program over pretrained-rate bases (D-VCG). We further prove population stability under the standard score-entropy training loss, identifying a tilted-path coverage factor that governs robustness to score error, together with finite-particle convergence for a fixed controlled Feynman-Kac recursion. Empirically, FluxLite improves over standard Feynman-Kac SMC baselines by up to two orders of magnitude in terminal KL on an analytically tractable finite-state CTMC benchmark, and reduces row-correlation MSE on 2D Ising sampling by 5-7x in geometric mean and up to 55x at peak.

Artificial intelligence models deployed as clinical decision support tools often suffer substantial performance degradation over time and across institutions due to covariate shift, concept drift, and cross-system heterogeneity. Fully retraining complex models is frequently infeasible, particularly in EHR settings where labeled outcome data are scarce and regulatory constraints limit model modification. We propose GUARD (Guided and Uncertainty-Aware Robustness to Domain shift), a unified statistical framework for updating an existing deployed model using limited labeled target data and multiple heterogeneous source populations while guarding against future distributional shifts. GUARD formulates robust multi-source transfer learning as a semi-supervised, adversarial optimization problem anchored at the current model, and employs double machine learning with cross-fitting to obtain debiased, efficient estimates that leverage abundant unlabeled target covariates. It then constructs an uncertainty-aware, guided group DRO estimator that combines source and target information while accounting for sampling variability in the source models. Our framework jointly provides principled robustness to domain shift, efficient semi-supervised estimation, and valid statistical inference for multi-source transfer of clinical prediction models. We demonstrate the utility of GUARD via extensive simulation experiments and a real world application to predicting future disease activity in rheumatoid arthritis from EHR data spanning more than a decade. GUARD recalibrates a pretrained model with only a small number of labeled records per year and remains accurate over multi-year horizons where target-only and standard transfer estimators degrade sharply.

We study the minimum number of hidden neurons required for arbitrary-accuracy approximation of multivariate Hölder-continuous functions on $[0,1]^d$ and the associated encoding complexity. For $d\geq 2$, we construct a fixed, explicitly defined activation function for which a closed-form network with two hidden layers of widths $d$ and $1$ achieves arbitrary accuracy in the uniform norm. We prove that $d+1$ is the exact minimum total number of hidden neurons among standard feedforward networks with locally integrable activations and affine outputs. We further give a simpler construction using a single elementary activation that combines the floor and exponential functions. This construction requires three hidden layers of widths $d$, $1$, and $2$, only two neurons above the minimum. If a skip connection is allowed, widths $d$, $1$, and $1$ suffice. These constructions use explicit grid addressing and integer encoding of quantized function values. For a bounded $\alpha$-Hölder class, they require $O(\varepsilon^{-d/\alpha}\log(1/\varepsilon))$ bits, matching the metric-entropy lower bound up to a logarithmic factor.

We study two-arm contextual bandits with arm-specific single indices and a shared unknown monotone link. Monotonicity makes the optimal action depend only on the contrast between the index directions, hence arm-specific reward functions need not be estimated. We introduce Natural Boundary Learning (NBL), a greedy procedure that uses a sequential Stein contrast to learn the optimal boundary directly, without estimating the reward functions or the common link. We characterize the local Riemannian dynamics of NBL through a decision stability coefficient balancing arm separation, link geometry, and the context distribution. We show that this stability is connected to the elicitation geometry of the underlying convex potential. Under local decision stability, NBL contracts toward the optimal boundary and achieves $O(\log n)$ expected regret. Numerical experiments illustrate the predicted stability regimes and compare NBL with a parametric greedy benchmark under link misspecification.

Joint-embedding predictive architectures enable planning with latent world models, but accurate transition prediction alone does not ensure that the planning objective is easy to optimize. We introduce Control-Geometry Straightening (CGS), a single auxiliary loss that learns planner-friendly representations by directly straightening control geometry for sampling-efficient planning. CGS matches pairwise cosine similarities among actions to those among corresponding latent differences only using local transitions from pixel-action pairs. The loss can be applied across world-model architectures using end-to-end learned or pretrained representations. Under linear-dynamics, our theoretical analysis connects this objective to temporal straightening and more balanced terminal-cost curvature across the full planning horizon, yielding finite-budget guarantees for MPPI, local contraction results for CEM, and convergence bounds for gradient descent. Across four control environments and multiple planners, CGS improves planning with fewer sampled candidates and refinement steps, achieving success-rate gains up to 20 and 12.6 percentage points over LeWorldModel (LeWM) and its temporal-straightening variant (LeWM+TS), respectively, with sampling-based planners using 128 candidates per update. Probes, comparisons with DINO-WM architecture, and planner-side ablations clarify how latent motion organization, state dependence, and dynamical context shape planning behavior. Straightening control geometry thus makes good action sequences easier to find under limited planning budgets.

We introduce the Functional Spectral-Newton Method (FSNM) for learning the leading singular structure of a conditional expectation operator without fixing a basis or reproducing kernel Hilbert space. FSNM fits a low-rank representation of the centered joint-to-product density ratio kernel by alternating functional Newton updates. Each update reduces to a preconditioned regression, which we approximate with vector-valued regression trees in a stagewise boosting procedure. At the population level, we establish descent and an $O(1/T)$ best-iterate block-stationarity rate under a relative weak-learner accuracy condition, and show that every nondegenerate local minimum over the full centered $L^2$ spaces is a globally optimal rank-$d$ approximation. Synthetic experiments show that FSNM recovers a low-rank density ratio and its leading spectral structure, and that the same learned kernel can answer multiple conditional queries without refitting.

Diffusion models have revolutionized generative modeling for continuous data through the gradual refinement of a belief state. This iterative refinement has not yet carried over to discrete diffusion models, which discard uncertainty at intermediate steps through categorical sampling (information collapse). We propose Simplex Diffusion Models (SDMs), a framework that lifts the diffusion process to the probability simplex to represent beliefs over categories. SDMs admit probability paths with closed-form reverse transitions and can be trained with a simple cross-entropy loss. Contrary to earlier proposals such as Dirichlet Flow Matching which requires integrating an ordinary differential equation, we introduce a DDIM-like sampler with a tunable level of stochasticity. Because SDMs operate on samples on the simplex, they can carry uncertainty across denoising steps, which mitigates information collapse. On OpenWebText, SDMs are competitive with strong Discrete Diffusion baselines, achieving $17.0$ GenPPL at $5.46$ unigram entropy in 64 sampling steps, close to real validation data. Even without Self-Conditioning (SC), SDMs outperform masked and uniform diffusion (with SC or predictor-corrector sampling) on code generation (TinyGSM, $T=0.1$; $49.0\%$ vs. $45.8\%$). Distilled down to 8 steps, SDMs solve $32.1\%$ of GSM8K problems, more than distilled Discrete Diffusion models with 128 steps ($21.4\%$).

Many learning tasks map an input distribution to an output distribution. A natural way to model such an operator is to transform each input sample using a continuous function that may depend on the entire input distribution, and then take the distribution of the transformed samples. This defines a measure-dependent pushforward model and includes measure-theoretic formulations of transformers. We ask when such models can approximate arbitrary continuous operators between spaces of probability measures. We first show that universal approximation fails when atomic inputs are allowed: some continuous measure-to-measure operators that split or redistribute atomic mass cannot be approximated arbitrarily well by deterministic pushforward models. We then introduce the uniform level set condition, which requires a continuous measure-dependent scalarization whose shrinking level set neighborhoods carry uniformly vanishing mass over the input family. This condition is satisfied, in particular, by compact families of absolutely continuous measures. On every compact family satisfying this condition, we prove that any continuous measure-to-measure operator with outputs of finite $p$-th moment can be uniformly approximated, in the $p$-Wasserstein distance, by continuous measure-dependent pushforwards. Combining our theorem with existing approximation results for measure-dependent in-context maps yields universal approximation by measure-theoretic transformers. We also extend the framework to continuously-varying source measures, yielding a corresponding universality result for a class of pushforward models that are closely aligned with cross-attention architectures.

Soft Actor-Critic (SAC) is widely used for entropy-regularised reinforcement learning with continuous action spaces, and practical implementations perform only a few actor steps towards an evolving target. In this work, we prove convergence guarantees when the target policy arises from policy mirror descent and compare it with the classical Gibbs target. We derive sufficient conditions for the strong convexity and smoothness of the actor objective, characterised by the curvature of the $Q$-function estimate through the Legendre differential operator, and establish an $\mathcal{O}\!\left(N^{-\frac{1}{5}}\right)$ best-iterate finite-time convergence rate up to actor and critic approximation errors. Moreover, the mirror-descent step size $\lambda$ directly controls the target drift and hence actor tracking error, whereas the analogous Gibbs bound contains a non-vanishing tracking term.

Estimating conditional statistics and learning representations of a population of conditional distributions are central problems in many data-driven applications, including uncertainty quantification and dynamical systems analysis. Conditional mean operators (CMOs), a class of linear operators between function spaces, resolve these objectives by providing access to a broad class of conditional statistics. However, existing methods typically estimate each CMO independently or constrain it to prespecified function spaces, thereby preventing the exploitation of shared structure across related distributions. In this work, we posit that related CMOs share finite-dimensional input and output function spaces, and are specialized for each task with a linear operator mapping these spaces. Based on this hypothesis, we introduce MTL-CMO, a multi-task framework that jointly learns shared function spaces and task-specific operators across multiple datasets. We further introduce T-CMO, a transfer learning method that reuses the shared spaces to estimate, in closed form, the operator of a new conditional distribution. We establish statistical guarantees quantifying the benefits of jointly learning the shared function spaces. Our experiments demonstrate that learning shared function spaces improves uncertainty quantification across a broad range of conditional distributions and, when applied to Langevin and plasma dynamics, yields compact representations of complex dynamics that retain physically meaningful information and enable parameter identification.

This paper studies convex optimization when the gradient cannot be evaluated exactly, but only approximated by a hierarchy of algorithms whose compute grows like $\delta^{-\gamma}$ in the accuracy $\delta$. When $\gamma>2$, falling into the Harder-Than-Monte-Carlo (HTMC) regime, the price of accuracy outruns the variance reduction that Monte Carlo would buy and we show that minimizing a loss function costs no more, up to a factor depending only on $\gamma$, than a single evaluation of its gradient at the accuracy the problem demands. A randomized multilevel oracle replaces the deterministic approximation of accuracy $\delta$ by an unbiased estimator of it, whose variance $\sigma^2$ becomes a second, independently priced dial: the cost of one call drops from $\delta^{-\gamma}$ to $\delta^{2-\gamma}\sigma^{-2}$. Plain inexact gradient descent driven by that oracle reaches loss $\varepsilon$ at expected compute $\Theta(\varepsilon^{-\gamma})$ in the convex case, against $\Theta(\varepsilon^{-(\gamma+1)})$ for the same method run at a fixed accuracy: randomization buys a full power of $\varepsilon$. Under $\mu$-strong convexity the exponent halves, to $\varepsilon^{-\gamma/2}$, because the iterates settle at a noise floor and the bias budget relaxes accordingly. Both bounds are independent of the step size, and hence of the smoothness constant, and we show that the cost is a functional of the underlying gradient flow rather than of any discretization of it.

Virtual simulators are widely used for studying complex physical phenomena, from particle collisions to rocket propulsion. Such "computer experiments" can be highly time-intensive, and a Bayesian surrogate model can be used for efficient emulation with reliable uncertainty quantification. To train accurate surrogates with a limited sample size $n$, recent work has explored the incorporation of monotonicity (or isotonicity) information, which can often be elicited from physical systems. In practical applications with many input variables, however, existing Bayesian isotonic models can face statistical and computational limitations, which may result in worse performance compared to models that do not incorporate isotonicity. We propose a new transformed additive isotonic model (TAIM), which aims to tame this "curse-of-dimensionality". TAIM makes use of a flexible transformed additive isotonic modeling framework, which leverages a data-estimated link transformation and a monotone basis model with spike-and-slab priors on basis weights. Prediction-wise, TAIM achieves (up to log factors) a posterior contraction rate of $O(n^{-1/3})$ when the true black-box function is in a transformed additive isotonic form with mild smoothness conditions. Such a rate does not depend on the input dimension $d$ for terms involving $n$, which softens the effect of dimensionality on posterior predictions. Computation-wise, TAIM allows for efficient posterior inference via a carefully designed Gibbs sampler, where each sampling iteration requires only linear work in $d$. We further present an extension of TAIM that can model potential deviations from transformed additivity. Numerical experiments and two applications show the effectiveness of TAIM for isotonic surrogate modeling with many input variables.