Papers

Causal inference has traditionally centered on scalar outcomes: whether a patient recovers, how much a worker earns, or how many visits a website receives. Modern studies increasingly ask causal questions about outcomes with richer form, such as clinical notes, open-ended survey responses, and images. A hospital may want to know how an AI documentation tool changes the notes physicians write, or how a nurse training program alters what patients say in survey responses. For such outcomes, the usual average treatment effect is ill-defined: one cannot meaningfully subtract one text or image from another. To this end, we propose a causal query for unstructured outcomes. The key idea is to learn what features of the outcome are most causally affected by the treatment, which we call the maximally contrasting feature (MCF). To estimate the MCF, we learn a feature-scoring function that maps each outcome to a scalar and exposes the sharpest contrast between treated and control potential outcomes. We develop identification conditions and estimation algorithms for this query, and extend it to heterogeneous effects by allowing the feature-scoring function to depend on observed covariates. We also handle settings where both the treatment and the outcome are unstructured. Empirical studies on text and images show that the algorithm recovers salient aspects of an outcome changed by a treatment.

Unit excitation (UE) is a common assumption in stochastic saddle avoidance: the stochastic error must have a uniformly positive component along every direction, in expectation. This condition gives a direct way to rule out convergence to strict saddles, but it also oversimplifies the actual noise structure, and does not match many stochastic optimization regimes. In overparameterized or interpolation models, the noise may vanish near stationarity. In finite-sum problems, the stochastic gradient noise may lie in a low-dimensional, data-dependent subspace. In these (common) scenarios, UE is naturally not satisfied. In this paper, we prove an abstract almost sure avoidance theorem for stochastic recursions without UE. The theorem replaces UE-type requirements by verifiable pathwise conditions. In applications, these conditions follow, e.g., from local smoothness and finite-moment assumptions under standard i.i.d. sampling, or from the finite-sum structure under without-replacement sampling. Since the stochastically sampled maps generally do not share a fixed point, the celebrated center-stable manifold argument used in deterministic analyses is not directly applicable. Instead, we use a path-dependent change of variables together with a pathwise Lyapunov–Perron-based proof strategy. As applications, we obtain strict saddle avoidance for stochastic mirror descent (including SGD) and for random reshuffling. For nonsmooth composite objectives, we prove avoidance results for a proximal-type stochastic gradient method. Combining these insights with suitable iterate convergence guarantees, this allows establishing convergence to local minimizers of the original objective function.

The standard check for contamination in LLM backtests is simple: compare scores before and after the training cutoff. We show this check is uninformative. Four flagship models fail it on questions they cannot have memorized: every scored question resolved after their cutoffs. The reason is structural. Models legitimately know more about times near their cutoff, so recency mimics leakage, and we prove no passive backtest can separate the two from genuine skill. Measurement, not just detection, requires information from outside the backtest. We supply it in two forms. A known cutoff identifies leakage at the boundary; a matched clean control identifies it globally and yields a leakage-adjusted score. We also derive where leakage hides: it concentrates on outcomes that surprised the crowd and were well covered in training, and partial memorization is disproportionately rewarded. We validate the estimators against ground truth by planting leakage in twin models, where they recover the injected dose and return null on clean questions. Deployed on frontier models, they detect one cutoff-localized signature and, at the audit's power floor, clear five models whose apparent advantages were recency alone. Backtests need not be discarded; they need one defensible reference.

Causal inference increasingly extends beyond classical causal effects defined by deterministic treatment assignments, such as the average treatment effect, to stochastic intervention effects that can weaken positivity requirements and offer greater policy relevance. Nonparametric Bayesian models are attractive for estimating these effects due to their flexibility and inherent uncertainty propagation, but this posterior uncertainty need not be well calibrated for the causal effect of interest. We develop a simple post-processing correction that can be applied to posterior samples without changing the prior or fitting algorithm. We prove that, for a broad class of stochastic interventions, the corrected posterior yields asymptotically efficient inference and credible intervals with asymptotically valid frequentist coverage; formally, it satisfies a semiparametric Bernstein-von Mises theorem. The theory covers interventions specified independently of the observed treatment process, as well as interventions that modify it, including incremental propensity score interventions and a new power-tilt intervention. A central contribution is new theory for SoftBART, including conditions under which this flexible tree-based Bayesian model supports calibrated Bayesian inference for stochastic intervention effects. In simulations, the correction reduces bias and improves coverage relative to the uncorrected Bayesian analysis while remaining competitive with frequentist alternatives. We illustrate the method by estimating how expected LDL cholesterol would change under hypothetical increases or decreases in the odds of receiving statin therapy.

Prediction-based methods, including Large Language Models (LLMs) and other machine learning techniques, are often used to construct measures of political phenomena that are difficult to quantify directly, such as policy positions in manifestos or emotions expressed on social media. In many applications, these prediction-generated measures are used as explanatory variables in regression models, even though they are measured with error. This leads to biased estimates. In this paper, we propose a simple solution to these biases: instrumental variables constructed from multiple measures created on independent splits of the original data. This approach is theoretically valid, easy to implement, and does not require new data. Through simulations, we show that this approach recovers estimates close to the true values, even in relatively small samples, while the standard approach can produce substantial bias in practice. We illustrate the method by revisiting two applications: whether gendered speech affects legislative outcomes in the German Parliament, and whether political risk influences poverty alleviation programs in China.

We develop a class of general diffusion-based stochastic particle optimisation methods for loss functions with intractable gradients. Specifically, we consider problems in which the loss gradient is an integral with respect to a parameter-dependent distribution, a structure that includes training generative models, fine-tuning, and learning latent-variable models. We introduce mean-field dynamics and its interacting-particle approximations, which contain several existing algorithms as special cases and provide a route to constructing new methods. Under well-posedness and joint contractivity assumptions, we prove exponential convergence and show that the continuous-time particle system admits a non-asymptotic error bound. We illustrate our framework by developing higher-order Langevin and momentum optimisers and evaluating them on maximum marginal-likelihood estimation and energy-based-model training.

Reinforcement learning is a subfield of machine learning that studies how an agent interacts with an environment in order to extract as large a reward as possible. A standard approach to study such interaction is through Markov Decision Processes (MDPs) and the task of choosing an optimal policy — a function that tells the agent which action to take. In this work, we study two types of MDPs — finite-horizon and infinite-horizon discounted — and propose new quantum algorithms for computing approximate optimal policies. Our quantum algorithms are based on a new combination of standard value iteration and quantum subroutines like quantum mean estimation and quantum maximum finding, overall enhanced with techniques from sample-optimal classical algorithms. Our resulting query complexities improve upon previous works, thus approaching already established quantum lower bounds.

Score-based diffusion models are typically formulated using continuous-time stochastic differential equations and measure-theoretic stochastic calculus. In this paper, we develop a hyperfinite formulation of score-based generative modeling within the framework of Nonstandard Analysis. Starting from an internal diffusion process on a hyperfinite grid, we derive the associated infinitesimal generator and establish its correspondence with the classical Fokker–Planck equation. We then obtain a hyperfinite backward-mean identity that yields the reverse-time drift and provides a constructive derivation of the reverse-time SDE. Building on these results, we show that minimization of an internal score-matching objective recovers the score function required by the reverse-time dynamics, thereby connecting score estimation with generative sampling directly at the hyperfinite level. Under suitable assumptions, we further derive a hyperfinite Girsanov formula and establish a relationship between likelihood optimization and Fisher-divergence objectives. Finally, we analyze the second-order consistency of the hyperfinite dynamics and show that the leading correction term depends explicitly on the fourth moment of the increment distribution, with the Gaussian value $\kappa=3$ eliminating the leading dispersion contribution. Taken together, these results provide a unified hyperfinite framework for diffusion-based generative modeling–while laying foundations for further extensions–that links discrete grid dynamics, reverse-time diffusion, score matching, and likelihood-based formulations within a common nonstandard setting.

We study distributed testing of $\mathrm{Ber}(\alpha)$ versus $\mathrm{Ber}(\beta)$ in the broadcast, or shared-blackboard, model. For protocols with constant advantage, we characterise up to universal constant factors the information complexity under either hypothesis for every pair $\beta<\alpha$. The characterisation shows that the two information costs can be quite different and identifies three parameter regimes, with optimal protocols based respectively on clean samples, a noisy binary symmetric channel, and an asymmetric $Z$-channel. The lower bounds rely on a novel mixed Hellinger–Jensen–Shannon inequality that may be of independent interest. We also characterise the constant-advantage information complexity of testing arbitrary discrete distributions via an optimisation problem over channels, and show that binary-output channels suffice. We obtain bounds for bounded likelihood-ratio distributions, and give general upper bounds in terms of $\chi^2$ divergence. As applications, we recover the broadcast-model set-disjointness lower bound, and derive stronger lower bounds in the multi-pass streaming setting for some problems considered in prior work.

Multimodal supervised learning seeks to leverage multiple heterogeneous data sources to improve predictive performance. A central challenge is determining the fusion granularity across modalities: over-integration may amplify noise while under-integration fails to exploit cross-modal dependence. Existing approaches rely on pre-specified fusion architectures, from early to late fusion, that may not adapt to the underlying dependence structure among modalities. We propose DAIF, a data adaptive intermediate fusion framework that combines random matrix theory and non-parametric dependence measures to learn fusion structure directly from data. We operate under a Bayesian multimodal factor model where the prior on the latent factors determines the cross-modal dependence. Our method clusters modalities based on estimated intermodal dependence, then performs clusterwise empirical Bayes estimation of the priors. These estimated priors are used to construct denoisers within an approximate message passing (AMP) framework, yielding denoised low-dimensional features that borrow strength across related modalities while preserving modality-specific signal. The resulting embeddings are used for downstream supervised prediction. We evaluate the framework through simulations under varying dependence structures and signal regimes, comparing against several benchmark methods, and demonstrate its practical utility on two multimodal datasets, namely a trimodal TEA-seq dataset (Swanson et al., 2021) and TCGA-BRCA dataset (Goldman et al., 2020). In the first example, we predict the expression level of a T-cell differentiation marker protein and in the second case we analyze patient survival prediction based on multimodal information. Our method competes with or outperforms the state-of-the-art techniques in both prediction problems, demonstrating its versatility across diverse supervised learning tasks.

Comparing stochastically trained models requires estimating both a performance difference and its uncertainty from repeated runs. We study whether training logs from those same runs can make such comparisons more precise. Because training-log covariates are produced during training rather than measured before it, we use arm-specific covariate adjustment: each model is adjusted only with statistics from its own runs, and the raw mean difference remains the reported effect. In a vision study spanning three architectures and three datasets, simple adjustments based on early training logs often reduce uncertainty in model comparisons. The main limitation is covariate selection. Broadly searching the log pool for the most correlated statistic often adds more noise than it removes, even when useful statistics exist in hindsight. Training logs therefore appear useful for more precise model comparisons, but only when the adjustment avoids large selection noise.

Stochastic learning objectives are typically written as expectations over abstract random variables. Actual training, however, uses concrete random inputs that enter both the realized loss and its gradient. Structure in these inputs that is accessible to the learning system can therefore be learned and exploited. Much prior structured-noise work asks how noise should be distributed or designed; we instead ask what structure in the concrete realized randomness becomes exploitable by the learner. We develop this general view and analyze its mechanism in diffusion models: in noise prediction, clean data and realized noise jointly form the noisy input, so the model can improve prediction by learning clean-data regularities or exploiting structure introduced through the noise, and the two routes can interact. Using pseudorandom streams as controlled, reproducible instances of structured noise, we provide mechanistic evidence on MNIST and CIFAR-10: random-role ablations localize the dominant effect to diffusion noise; in a diffusion probe, structured-noise training can reduce prediction loss below the IID reference, but replacing the test noise with IID reverses this advantage; and shuffling the same values largely removes the source-dependent loss reduction. This learned dependence can also affect generation. The same view offers a unified interpretation of data-dependent noise assignment, noise-based backdoors, and temporally correlated noise in video diffusion: although these methods introduce different structures, all alter what the model can exploit through noise and its interaction with clean-data learning. Our results indicate that diffusion noise is not merely a passive stochastic perturbation, but a learnable—and therefore potentially designable—input dimension.

This paper is concerned with one-bit mean estimation, where each independent sample is represented by a single binary message. We consider distributions on $\mathbb{R}$ with mean in $[-\lambda,\lambda]$ and absolute $k$-th central moment at most $\sigma^k$, where $k>1$ is fixed. For this class, previous work attained the optimal sample complexity for general queries using a two-stage protocol. The first stage localizes the mean. The second-stage queries are chosen after localization and refine the estimate around the decoded center. We show that this interaction can be avoided by constructing a randomized fully non-adaptive protocol that fixes all queries before observing the data and matches the optimal adaptive sample complexity. For target accuracy $\epsilon$ and confidence $1-\delta$, its sample complexity scales as \[ \log\frac{\lambda}{\sigma} + \begin{cases} (\sigma/\epsilon)^2\log(1/\delta), & k>2,\\ (\sigma/\epsilon)^2\log(\sigma/\epsilon)\log(1/\delta), & k=2,\\ (\sigma/\epsilon)^{k/(k-1)}\log(1/\delta), & 1<k<2, \end{cases} \] up to constants depending only on $k$. In the range covered by the known lower bound, this rate is minimax optimal even among fully adaptive protocols. This gives a negative answer to the COLT 2026 open problem asking whether interaction is necessary for order-optimal one-bit mean estimation with general queries \citep[Open Problem~1]{lau2026open}.

Recently, rectified flow has emerged as a fundamental framework for large-scale image generation, powering state-of-the-art systems such as FLUX.1 and Stable Diffusion 3. Despite its remarkable empirical success, the computational and statistical guarantees of iterative rectified flow have remained largely unexplored. We address this problem by studying c-rectified flow, a cost-aware class of rectified flow that projects velocity fields onto a gradient class while preserving endpoint marginals. The ordinary rectified flow can fail to recover the optimal transport coupling: in a Gaussian case study, the iteration converges to the optimal coupling if and only if the source and target covariance matrices commute. In contrast, under suitable compactness and uniform-integrability assumptions, iterative c-rectified flow always converges to the optimal transport coupling. We further establish quantitative one-step contraction and exponential convergence guarantees under projection-stability assumptions for both quadratic and strongly convex displacement costs. Finally, under a Hölder ball assumption, we develop new minimax-optimal score estimation rates and show that, when combined with iterative c-rectified flow, they yield a rate-optimal estimator of the optimal transport for the dimension \(d \ge 3\) and a nearly parametric rate for \(d=1,2\).

Two active learning algorithms for hyperspectral image (HSI) classification are proposed that combine density-aware Fermat distances with Poisson-reweighted harmonic label propagation. Our methods actively query points using an uncertainty-based acquisition function, extending Poisson ReWeighted Laplace Learning (PWLL). Our first algorithm, Fermat Active Laplace Learning (FALL), builds an affinity matrix using Fermat distances between all data points. Then, PWLL is run with a diagonal perturbation using the minimum-norm acquisition function. In contrast, Approximate FALL (A-FALL) computes Fermat distances between each data point and landmark pixels selected via farthest-point sampling and constructs the affinity matrix using landmark multidimensional scaling. After several query rounds, A-FALL selects the Fermat exponent $p$ using a leave-one-out cross-validation variant. FALL and A-FALL leverage Fermat distances and subsequent harmonic label propagation to provide a density-aware estimation of the data manifold, improving labeling accuracy. Experiments on Salinas A and Pavia show the effectiveness of FALL and the scalability of A-FALL to large HSI scenes.

With AI systems gaining more access to individuals' information, it is important to protect privacy when reporting statistical answers. Equally important is to privatize the reporting of uncertainty in such answers. To this end, we adopt a Bayesian likelihood-free framework and make simulation from the posterior private. In particular, we propose a new private instantiation of the Bayesian bootstrap using a blocking strategy. Rather than assigning idiosyncratic random weights to each individual, we randomly group individuals and assign a single weight to each group. By concealing individuals' contributions within a group, we fortify differential privacy gates. We harness amortized inference that decouples private learning from posterior sampling. A push-forward map from observation weights to posterior samples is learned privately by adding calibrated noise during training. Subsequent posterior draws require no additional privacy and computation budget. We call the resulting method the Private Generative Bayesian Bootstrap (PGBB). We establish a differential privacy guarantee, analyze convergence to the non-private blocked-bootstrap target, and quantify the discrepancy between the ordinary and blocked Bayesian-bootstrap posteriors. In addition, we derive data-free tuning of the block Dirichlet concentration parameter that restores posterior dispersion asymptotically. We also show a single fit of PGBB can support a family of loss-based decision rules simultaneously without additional privacy cost. In simulations and in applications to U.S. Census returns to schooling and U.S. natality birthweight quantiles, PGBB gives competitive private uncertainty quantification and improves over private Bayesian alternatives that require a specified data-generating model in common settings.

Human-centered assessment tasks, which are essential for systematic decision-making, rely heavily on human judgment and typically lack verifiable ground truth. Existing approaches face a dilemma: methods using only human judgments suffer from heterogeneous expertise and inconsistent rating scales, while methods using only model-generated scores must learn from imperfect proxies or incomplete features. We propose Aggregate-then-Calibrate (AtC), a two-stage framework that combines these complementary sources. Stage-1 aggregates heterogeneous comparative judgments into a consensus ranking using a rank-aggregation model that accounts for annotator reliability. Stage-2 calibrates any predictive model's scores by an isotonic projection onto the order, enforcing ordinal consistency while preserving as much of the model's quantitative information as possible. Theoretically, we show: (1) modeling annotator heterogeneity yields strictly more efficient consensus estimation than homogeneity; (2) isotonic calibration enjoys risk bounds even when the consensus ranking is misspecified; and (3) AtC asymptotically outperforms model-only assessment. Across semi-synthetic and real-world datasets, AtC consistently improves accuracy and robustness over human-only or model-only assessments. Our results bridge judgment aggregation with model-free calibration, providing a principled recipe for human-centered assessment when ground truth is costly, scarce, or unverifiable.

We introduce a unified framework for the error analysis of generative models based on the entropy production rate of the forward-reverse diffusion process pair. For a pair of continuity equation flows, the rate admits a closed velocity form identity whose time integral decomposes the terminal Kullback–Leibler (KL) divergence into the sum of an initialization error, a score approximation error, and a time-discretization error. By analyzing the entropy production at the level of marginal distributions, rather than in path space, our framework yields a sharp convergence rate of $\mathcal{O}(h^2)$ for the Euler-Maruyama sampler, where $h$ is the step size. This improves upon the $\mathcal{O}(h)$ rates typically obtained from Girsanov's path-space analyses. Furthermore, our framework unifies the analysis of score-based SDEs, probability-flow ODEs, and stochastic interpolants by varying diffusion coefficients within a single inequality, revealing the trade-off between deterministic and stochastic sampling. Numerical experiments confirm the predicted scaling with step size and terminal time.

Given an algebraic statistical model, a challenging problem is classifying the data according to the number of positive critical points of the likelihood function. The positive critical points are the positive solutions to an algebraic system, say likelihood equations. So, identifying the number of positive critical points is a real root classification problem for the likelihood equations. A discriminant variety of a likelihood-equation system geometrically describes the data for which the number of real solutions becomes unusual. As an essential component of the discriminant variety, the nonproperness set collects the data such that the likelihood-equation system has a solution at infinity. So, the number of real solutions varies when the data passes the nonproperness set, and identifying the nonproperness set plays a crucial role in the real root classification. In this work, we develop a novel method for computing nonproperness sets of likelihood-equation systems. We prove the correctness of this method. We show experimentally that it is far more efficient than the known methods in the literature.

We analyze approximate message passing (AMP) with an independent Gaussian initialization for noiseless phase retrieval in the proportional asymptotic regime. A random initialization has overlap of order $d^{-1/2}$ with the signal, and AMP requires a growing number of iterations to attain non-vanishing overlap. Thus, its precise behavior cannot be characterized by classical fixed-time state evolution. We prove a Gaussian decomposition of the AMP trajectory and control its error over the horizons required for recovery. The resulting analysis shows that random initialization attains the weak-recovery threshold $\delta_{\rm weak}=1/2$. For $\delta\in(\delta_{\rm weak},\delta_{\rm str})$, where $\delta_{\rm str}\approx1.13$, the signal strength follows state evolution and approaches its stable finite fixed point uniformly for \(n^{1/3}/\operatorname{polylog}(n)\) iterations. For $\delta>\delta_{\rm str}$, AMP reaches any prescribed fixed recovery accuracy within $O_{\delta,\varepsilon}(\log n)$ iterations. The majority of our analysis applies more generally to generalized AMP for single-index models.

Machine-learning benchmarks often pair a label that aggregates a long temporal horizon with input observed through one or a few short windows. Their apparent performance ceiling may therefore be an acquisition-protocol ceiling rather than a model-capacity ceiling. We study labels of the form $\Theta_{g,T}=T^{-1}\int_0^T g\{Z(t)\}\,\mathrm{d}t$ when the latent Gaussian process contains both a stable individual trait and a correlated within-individual state. An exact protocol-conditioned Bayes-risk identity provides a common tool. First, we decompose label variance into an $O(1)$ trait component and an $O(T^{-1})$ state component, explaining why a snapshot can retain cross-sectional predictability while poorly tracking within-person change. Second, we derive task-dependent effective temporal spans: mean labels depend on the ordinary correlation time, whereas occupation-time labels depend on an entire spectrum of higher-order correlation times. Third, state-driven occupation-label variance is maximal when the stable trait lies at the threshold; window efficiency decays much more slowly away from that boundary. Under an equal segment budget, exact risks and Monte Carlo experiments show that repeated segments at one time rapidly saturate, whereas temporally dispersed observations continue to increase state explainability. The trait ceiling uses quantities available from ordinary test-retest data; only the state ceiling requires short-lag temporal calibration. The results distinguish architectural limits from protocol limits and show that the label, rather than duration or segment count alone, defines the relevant timescale.

In recent years, numerous open-source software libraries have been developed for computing sets of features from univariate time series. The type and number of features vary across these feature sets, which have been constructed with varying disciplinary perspectives on quantifying structure in time-series data. To date, the relative strengths and weaknesses of these feature sets on time-series classification problems remains largely unexplored. Here we aimed to understand the relative performance of six open-source feature sets and three baseline feature sets (based on distributional and/or basic spectral structure) across 124 univariate time-series classification problems using a normalization-based approach to problem-level benchmarking that better indexes the relative strengths and weaknesses of different algorithms compared to prior rank-based approaches. Despite their dramatic differences in size, composition, and computation time, we found that feature sets performed relatively similarly overall (85.3% of pairwise comparisons resulted in ties), with the largest feature set, tsfresh, exhibiting the strongest overall performance (29.03% wins across all pairwise comparisons against other feature sets). We also highlighted specific problems on which the specific composition of a given feature set gave it a substantial performance advantage or disadvantage, and problems where simple baselines comprised of Fourier coefficients and quantiles were sufficient to achieve strong performance. Our results demonstrate the need to consider problem-level performance when benchmarking time-series feature sets, and highlight the importance of feature make-up in driving relative classification performance.

Drifting objectives compare a target and model distribution through a vector field observed noisily at finitely many locations. We ask what distributional conclusion such a frozen measurement system warrants. For integrable antisymmetric interactions and absolutely continuous laws in a declared finite density basis, the unnormalized sampled numerator satisfies $\operatorname{vec}(V_X)=Mc$, where $c$ is an antisymmetric mismatch and $M$ is probe-dependent. This identity yields an a posteriori total-variation (TV) upper confidence bound accounting for held-out field noise, estimated-operator error, and externally validated $L^1$ residual radii around normalized density approximants in the span; a nonpositive observability margin returns the trivial TV bound and abstains. The audit recomputes this numerator from held-out samples; a normalized drift statistic requires a separate joint numerator–denominator analysis. For Gaussian-RBF interactions, a global envelope supports distribution-free and empirical-Bernstein radii without truncation, with companion bounds for the Laplace similarity in the original drifting objective. We characterize random-probe observability by a population Gram matrix, identify rank and symmetry degeneracies, and prove large-bandwidth collapse toward mean matching. Synthetic studies exercise Gaussian and Laplace numerators, separately prespecified bounded-vector and variance-adaptive radii, Monte Carlo-calibrated operators, nonzero residual radii around normalized finite-basis approximants, outward-rounded observability bounds, and designed abstention. A joint basis-size/dimension stress path extends evaluation through $m=8$. The result is a conditional diagnostic for a finite density class, or for normalized finite-basis density approximants with external residual radii, not a universal guarantee from small training drift.