Papers

Exhaustive subgroup treatment effect plots are constructed by displaying all subgroup treatment effects of interest against subgroup sample size, providing a useful overview of the observed treatment effect heterogeneity in a clinical trial. As in any exploratory subgroup analysis, however, the observed estimates suffer from small sample sizes and multiplicity issues. To facilitate more interpretable exploratory assessments, this paper introduces a computationally efficient method to generate homogeneity regions within exhaustive subgroup treatment effect plots. Using the Doubly Robust (DR) learner, pseudo-outcomes are used to estimate subgroup effects and derive reference distributions, quantifying how surprising observed heterogeneity is under a homogeneous effects model. Explicit formulas are derived for the homogeneity region and different methods for calculation of the critical values are compared. The method is illustrated with a cardiovascular trial and evaluated via simulation, showing well-calibrated inference and improved performance over standard approaches using simple differences of observed group means.

In this paper, we study dynamic regret in unconstrained online convex optimization (OCO) with movement costs. Specifically, we generalize the standard setting by allowing the movement cost coefficients $\lambda_t$ to vary arbitrarily over time. Our main contribution is a novel algorithm that establishes the first comparator-adaptive dynamic regret bound for this setting, guaranteeing $\widetilde{\mathcal{O}}(\sqrt{(M^2+MP_T)(T+\sum_t \lambda_t)})$ regret, where $P_T$ is the path length of the comparator sequence over $T$ rounds and $M$ is the maximal comparator norm. Our result recovers the optimal adaptive rates for both static and dynamic regret in OCO as the special case where $\lambda_t=0$ for all rounds. To demonstrate the versatility of our results, we consider two applications: OCO with delayed feedback and OCO with time-varying memory. We show that both problems can be translated into time-varying movement costs, establishing a novel reduction specifically for the delayed feedback setting that is of independent interest. A crucial observation is that the first-order dependence on movement costs in our regret bound plays a key role in enabling optimal comparator-adaptive dynamic regret guarantees in both settings.

Bayesian optimal experimental design (BOED) seeks to maximize the expected information gain (EIG) of experiments. This requires a likelihood estimate, which in many settings is intractable. Simulation-based inference (SBI) provides powerful tools for this regime. However, existing work explicitly connecting SBI and BOED is restricted to a single contrastive EIG bound. We show that the EIG admits multiple formulations which can directly leverage modern SBI density estimators, encompassing neural posterior, likelihood, and ratio estimation. Building on this perspective, we define a novel EIG estimator using neural likelihood estimation. Further, we identify optimization as a key bottleneck of gradient based EIG maximization and show that a simple multi-start parallel gradient ascent procedure can substantially improve reliability and performance. With these innovations, our SBI-based BOED methods are able to match or outperform by up to $22\%$ existing state-of-the-art approaches across standard BOED benchmarks.

Recovering a unique causal graph from observational data is an ill-posed problem because multiple generating mechanisms can lead to the same observational distribution. This problem becomes solvable only by exploiting specific structural or distributional assumptions. While recent work has separately utilized time-series dynamics or multi-environment heterogeneity to constrain this problem, we integrate both as complementary sources of heterogeneity. This integration yields unified necessary identifiability conditions and enables a rigorous analysis of the statistical limits of recovery under thin versus heavy-tailed noise. In particular, temporal structure is shown to effectively substitute for missing environmental diversity, possibly achieving identifiability even under insufficient heterogeneity. Extending this analysis to heavy-tailed (Student's t) distributions, we demonstrate that while geometric identifiability conditions remain invariant, the sample complexity diverges significantly from the Gaussian baseline. Explicit information-theoretic bounds quantify this cost of robustness, establishing the fundamental limits of covariance-based causal graph recovery methods in realistic non-stationary systems. This work shifts the focus from whether causal structure is identifiable to whether it is statistically recoverable in practice.

The smoothness of the transformer architecture has been extensively studied in the context of generalization, training stability, and adversarial robustness. However, its role in transfer learning remains poorly understood. In this paper, we analyze the ability of vision transformer components to adapt their outputs to changes in inputs, or, in other words, their plasticity. Defined as an average rate of change, it captures the sensitivity to input perturbation; in particular, a high plasticity implies a low smoothness. Our theoretical analysis and extensive experiments – over $1,000$ finetuning runs on large-scale vision transformers – showcase that this perspective provides principled guidance in choosing the components to prioritize during adaptation. A key takeaway for practitioners is that the high plasticity of the attention modules and feedforward layers consistently leads to better finetuning performance. Our findings depart from the prevailing assumption that smoothness is desirable, offering a novel perspective on transformers' functional properties. The code is available at https://github.com/ambroiseodt/vit-plasticity.

Hybrid modeling, the combination of machine learning models and scientific mathematical models, enables flexible and robust data-driven prediction with partial interpretability. However, the unknown parameters of the scientific model cannot necessarily be estimated properly, since the flexibility of the machine learning model might make the scientific model part effectively ignored in prediction. We may avoid it by applying some regularization, but the formulation of such regularizers typically depends on model architectures and domain knowledge. In this paper, we propose an architecture-agnostic method to learn hybrid models while properly estimating the scientific parameters. The idea is to use the flatness of loss minima to achieve model simplicity, based upon the Occam's razor principle. We employ the idea of sharpness-aware minimization and adapt it to the hybrid modeling setting. Numerical experiments demonstrate the effectiveness of the SAM-based hybrid model learning for scientific parameter estimation.

We study graph matching between two correlated networks in the almost fully seeded regime, where all but a vanishing fraction of vertex correspondences are revealed. Concretely, we consider the correlated stochastic block model and assume that $n^{1-\alpha}$ vertices remain unrevealed for some $\alpha \in (0,1)$, while the remaining $n - n^{1-\alpha}$ vertices are provided as seed correspondences. Our goal is to determine when the true permutation can be recovered efficiently as the proportion of unrevealed vertices vanishes. We prove that exact recovery of the remaining correspondences is achievable in polynomial time whenever $\lambda s^{2} > 1 - \alpha$, where $\lambda = (a+b)/2$ is the SBM density parameter and $s$ denotes the edge retention parameter. This condition smoothly interpolates between the fully seeded setting and the classical unseeded threshold $\lambda s^{2} > 1$ for matching in correlated Erdős-Rényi graphs. Our analysis applies to both a simple neighborhood-overlap rule and a bistochastic relaxation followed by projection, establishing matching achievability in the almost fully seeded regime without requiring spectral methods or message passing. On the converse side, we show that below the same threshold, exact recovery is information-theoretically impossible with high probability. Thus, to our knowledge, we obtain the first tight statistical and computational characterization of graph matching when only a vanishing fraction of vertices remain unrevealed. Our results complement recent progress in semi-supervised community detection by demonstrating that revealing all but $n^{1-\alpha}$ correspondences similarly lowers the information threshold for graph matching.

We construct an asymptotic prediction interval for the population-wise error rate (PWER), which is a multiple type I error criterion for clinical trials with overlapping patient populations. The PWER is the probability that a randomly selected patient will receive an ineffective treatment. It must usually be estimated due to unknown population strata sizes, such that only an estimate can be controlled at the given significance level. We apply the delta method to find a prediction interval for the resulting true PWER, we demonstrate by simulations that the interval has the required coverage probability, and illustrate the approach with real data examples.

We study optimal learning rate (LR) schedules under the functional scaling law (FSL) framework (Li et al., 2025), which decomposes training dynamics into signal learning and noise forgetting. In power-law kernel regression, these two components are governed by a source exponent $s>0$ and a capacity exponent $q>1$, respectively, with smaller $s$ corresponding to harder tasks. For a fixed training horizon $N$, we characterize the schedules that minimize the final-step loss under a stability constraint and reveal a sharp phase transition. In the easy-task regime $s>1-1/q$, the optimal schedule follows power decay from the beginning of training; in the hard-task regime $s<1-1/q$, it becomes warmup-stable-decay (WSD)-like (Hu et al., 2024), staying at the largest admissible LR for most of training before a final decay. In both regimes, the decay exponent is $2q-1$: task difficulty determines when to decay, while model capacity determines how to decay. Beyond the exact optimum, we study fractional schedules, whose shape is defined over relative training progress. We show that precise tuning of the decay shape is often unnecessary: a broad class of profiles attains the optimal convergence rate, while overly slow terminal decay leads to schedule-induced capacity saturation. Finally, for one-pass SGD in kernel regression, FSL-motivated power-decay schedules achieve optimal last-iterate rates. Experiments support the theoretical predictions and the task-dependent transition between early and delayed decay.

We study the problem of learning robust classifiers where the classifier will receive a perturbed input. Unlike robust PAC learning studied in prior work, here the clean data and its label are also adversarially chosen. We formulate this setting as an online learning problem and consider both the realizable and agnostic learnability of hypothesis classes. We define a new dimension of classes and show it controls the mistake bounds in the realizable setting and the regret bounds in the agnostic setting. In contrast to the dimension that characterizes learnability in the PAC setting, our dimension is rather simple and resembles the Littlestone dimension. We generalize our dimension to multiclass hypothesis classes and prove similar results in the realizable case. Finally, we study the case where the learner does not know the set of allowed perturbations for each point and only has some prior on them.

Multicalibration gradient boosting has recently emerged as a scalable method that empirically produces approximately multicalibrated predictors and has been deployed at web scale. Despite this empirical success, its convergence properties are not well understood. In this paper, we provide computational guarantees for multicalibration gradient boosting algorithms. We show that the magnitude of successive prediction updates decays at $O(1/\sqrt{T})$, which implies the same convergence rate bound for the empirical multicalibration error over rounds. Under additional smoothness assumptions on the weak learners, this rate improves to linear convergence. We further establish convergence for adaptive variants. Experiments on real-world datasets support our theory and clarify the regimes in which the method achieves fast convergence.

We consider parametric inference for an ergodic and stationary diffusion process, when the data are high-frequency observations of the integral of the diffusion process. Such data are obtained via certain measurement devices, or if positions are recorded and speed is modelled by a diffusion. In finance, realized volatility or variations thereof can be used to construct observations of the latent integrated volatility process. Specifically, we assume that the integrated process is observed at equidistant, deterministic time points and consider the high-frequency/infinite horizon asymptotic scenario, where the number of observations, the sampling frequency and the time of the last observation all go to infinity. Subject to mild standard regularity conditions on the diffusion model, we prove the asymptotic existence and uniqueness of a consistent estimator for useful and tractable classes of prediction-based estimating functions. Asymptotic normality of the estimator is obtained under an additional assumption on the rates. The proofs are based on the useful Euler-Ito expansions of transformations of diffusions and integrated diffusions, which we study in some detail.

Missing data imputation, where a model is trained on observed data to estimate unobserved values, is a fundamental problem in machine learning. In this paper, we rigorously formulate imputation model learning as a mean-squared error risk minimisation problem. We show that when the probability of missingness depends on the data, many state-of-the-art methods fail to account for the resulting distribution shift between the observed data used for training and the full data distribution used for evaluation. Consequently, these approaches do not minimise mean-squared error on the full data distribution. Instead, we propose a novel imputation algorithm designed to learn an imputation model from the observed data while explicitly accounting for this distribution shift. Simulation studies show consistent improvements over otherwise identical uncorrected baselines, with average reductions of 3% in RMSE and 7% in Wasserstein distance.

This paper introduces a rigorous framework for defining generative diffusion models in infinite dimensions via Doob's h-transform. Rather than relying on time reversal of a noising process, a reference diffusion is forced towards the target distribution by an exponential change of measure. Compared to existing methodology, this approach readily generalises to the infinite-dimensional setting, hence offering greater flexibility in the diffusion model. The construction is derived rigorously under verifiable conditions, and bounds with respect to the target measure are established. We show that the forced process under the changed measure can be approximated by minimising a score-matching objective and validate our method on both synthetic and real data.

Standard chain-of-thought reasoning generates a solution in a single forward pass, committing irrevocably to each token and lacking a mechanism to recover from early errors. We introduce Inference-Time Rethinking, a generative framework that enables iterative self-correction by decoupling declarative latent thought vectors from procedural generation. We factorize reasoning into a continuous latent thought vector (what to reason about) and a decoder that verbalizes the trace conditioned on this vector (how to reason). Beyond serving as a declarative buffer, latent thought vectors compress the reasoning structure into a continuous representation that abstracts away surface-level token variability, making gradient-based optimization over reasoning strategies well-posed. Our prior model maps unstructured noise to a learned manifold of valid reasoning patterns, and at test time we employ a Gibbs-style procedure that alternates between generating a candidate trace and optimizing the latent vector to better explain that trace, effectively navigating the latent manifold to refine the reasoning strategy. Training a 0.2B-parameter model from scratch on GSM8K, our method with 30 rethinking iterations surpasses baselines with 10 to 15 times more parameters, including a 3B counterpart. This result demonstrates that effective mathematical reasoning can emerge from sophisticated inference-time computation rather than solely from massive parameter counts.

This article addresses online variational estimation in parametric state-space models. We propose a new procedure for efficiently computing the evidence lower bound and its gradient in a streaming-data setting, where observations arrive sequentially. The algorithm allows for the simultaneous training of the model parameters and the distribution of the latent states given the observations. It is based on i.i.d. Monte Carlo sampling, coupled with a well-chosen deep architecture, enabling both computational efficiency and flexibility. The performance of the method is illustrated on both synthetic data and real-world air-quality data. The proposed approach is theoretically motivated by the existence of an asymptotic contrast function and the ergodicity of the underlying Markov chain, and applies more generally to the computation of additive expectations under posterior distributions in state-space models.

This paper provides sufficient conditions over the sequence of samples and parameters of an adaptive Markov Chain Monte Carlo (MCMC) algorithm to ensure ergodicity with respect to a target distribution that can have unbounded support. These conditions aim to make more easily usable the conditions of Containment and Diminishing Adaptation from Roberts and Rosenthal [2007] formulated over the transition kernels, without needing, as was done in other works, an artificial assumption of the compactness over both sample and parameter spaces. The paper shows that the condition of compactness can be relaxed to a more realistic bound in probability over the sequence of both samples and parameters.

Graph Neural Networks (GNNs) have established themselves as the leading models for learning on graph-structured data, generally categorized into spatial and spectral approaches. Central to these architectures is the Graph Shift Operator (GSO), a matrix representation of the graph structure used to filter node signals. However, selecting the optimal GSO, whether fixed or learnable, remains largely empirical. In this paper, we introduce a novel alignment gain metric that quantifies the geometric distortion between the input signal and label subspaces. Crucially, our theoretical analysis connects this alignment directly to generalization bounds via a spectral proxy for the Lipschitz constant. This yields a principled, computation-efficient criterion to rank and select the optimal GSO for any prediction task prior to training, eliminating the need for extensive search.

Adversarial online linear optimization (OLO) is essentially about making performance tradeoffs with respect to the unknown difficulty of the adversary. In the setting of one-dimensional fixed-time OLO on a bounded domain, it has been observed since Cover (1966) that achievable tradeoffs are governed by probabilistic inequalities, and these descriptive results can be converted into algorithms via dynamic programming, which, however, is not computationally efficient. We address this limitation by showing that Stein's method, a classical framework underlying the proofs of probabilistic limit theorems, can be operationalized as computationally efficient OLO algorithms. The associated regret and total loss upper bounds are "additively sharp", meaning that they surpass the conventional big-O optimality and match normal-approximation-based lower bounds by additive lower order terms. Our construction is inspired by the remarkably clean proof of a Wasserstein martingale central limit theorem (CLT) due to Röllin (2018). Several concrete benefits can be obtained from this general technique. First, with the same computational complexity, the proposed algorithm improves upon the total loss upper bounds of online gradient descent (OGD) and multiplicative weight update (MWU). Second, our algorithm can realize a continuum of optimal two-point tradeoffs between the total loss and the maximum regret over comparators, improving upon prior works in parameter-free online learning. Third, by allowing the adversary to randomize on an unbounded support, we achieve sharp in-expectation performance guarantees for OLO with noisy feedback.

The Sliced Wasserstein Kernel (SWK) for persistence diagrams was introduced in (Carrière et al. 2017) as a powerful tool to implicitly embed persistence diagrams in a Hilbert space with reasonable distortion. This kernel is built on the intuition that the Figalli-Gigli distance-that is the partial matching distance routinely used to compare persistence diagrams-resembles the Wasserstein distance used in the optimal transport literature, and that the later could be sliced to define a positive definite kernel on the space of persistence diagrams. This efficient construction nonetheless relies on ad-hoc tweaks on the Wasserstein distance to account for the peculiar geometry of the space of persistence diagrams. In this work, we propose to revisit this idea by directly using the Figalli-Gigli distance instead of the Wasserstein one as the building block of our kernel. On the theoretical side, our sliced Figalli-Gigli kernel (SFGK) shares most of the important properties of the SWK of Carrière et al., including distortion results on the induced embedding and its ease of computation, while being more faithful to the natural geometry of persistence diagrams. In particular, it can be directly used to handle infinite persistence diagrams and persistence measures. On the numerical side, we show that the SFGK performs as well as the SWK on benchmark applications.

We present new auditors to assess Differential Privacy (DP) of an algorithm based on output samples. Such empirical auditors are common to check for algorithmic correctness and implementation bugs. Most existing auditors are batch-based or targeted toward the traditional notion of $(\varepsilon,\delta)$-DP; typically both. In this work, we shift the focus to the highly expressive privacy concept of $f$-DP, in which the entire privacy behavior is captured by a single tradeoff curve. Our auditors detect violations across the full privacy spectrum with statistical significance guarantees, which are supported by theory and simulations. Most importantly, and in contrast to prior work, our auditors do not require a user-specified sample size as an input. Rather, they adaptively determine a near-optimal number of samples needed to reach a decision, thereby avoiding the excessively large sample sizes common in many auditing studies. This reduction in sampling cost becomes especially beneficial for expensive training procedures such as DP-SGD. Our method supports both whitebox and blackbox settings and can also be executed in one-run frameworks.

The Stein class used in method of moments parameter estimation has two functions whichneedtobespecified, forwhichthereisnopersuasiveargumentsforanyparticular choice. We show that by setting one to be the derivative of the density score function with respect to the parameter leads to a generalized score matching estimator with a choice of weight function. However, choosing a suitable weight function for generalized score matching is not straightforward. We show the weight function is equivalent to a transform of the data and using a score estimator, with an optimal transform being to a normal sample, using for example Box-Cox. We compare our proposal with an alternative means by which to handle the weight function, which is to use a generalized method of moment estimator.

We study how macroscopic observational constraints restrict admissible microscopic explanatory structures when no intrinsic order or dynamics is assumed a priori. Starting from an unordered collection of measurement outcomes, we formulate inference as a constrained large deviation problem, selecting probability assignments that minimize relative entropy with respect to a reference measure determined solely by the measurement setup. We show that among all microscopic structures compatible with a given macroscopic constraint, those rendering the observation statistically most typical are selected. As an explicit illustration, we demonstrate how ordered microscopic structures can emerge purely from inference under constraint, even when the reference measure is fully permutation symmetric. Order is thus not assumed but inferred, serving here only as an illustrative example of a broader class of relational explanatory hypotheses constrained by observation.

This paper studies estimation and inference of heterogeneous social interactions featuring group fixed effects and slope heterogeneity under latent structure. Theoretically, in a binary choice model with social interactions we characterize the asymptotic bias of the slope coefficient due to the presence of high dimensional incidental parameters. Computationally, we invoke parametric bootstrap method to debias and establish asymptotic validity. Furthermore, tailored algorithms are employed to explore the latent structures and determine the number of clusters. Monte Carlo simulations confirm strong finite sample performance of our methods. In an application to students' risky behaviors, the algorithm detects two latent clusters and finds that peer effects are significant within one of the clusters, demonstrating the practical applicability in uncovering heterogeneous social interactions.

This study proposes a linear-rational joint survival mortality model based on the Wishart process. The Wishart process, which is a stochastic continuous matrix affine process, allows for a general dependency between the mortality intensities that are constructed to be positive. Using the linear-rational framework along with the Wishart process as state variable, we derive a closed-form expression for the joint survival annuity, as well as the guaranteed joint survival annuity option. Exploiting our parameterisation of the Wishart process, we explicit the distribution of the mortality intensities and their dependency. We provide the distribution (density and cumulative distribution) of the joint survival annuity. We also develop some polynomial expansions for the underlying state variable that lead to fast and accurate approximations for the guaranteed joint survival annuity option. These polynomial expansions also significantly simplify the implementation of the model. Overall, the linear-rational Wishart mortality model provides a flexible and unified framework for modelling and managing joint mortality risk.

Adaptive clinical trials rely on interim analyses, flexible stopping, and data-dependent design modifications that complicate statistical guarantees when fixed-horizon test statistics are repeatedly inspected or reused after adaptations. E-values and e-processes provide anytime-valid tests and confidence sequences that remain valid under optional stopping and optional continuation without requiring a prespecified monitoring schedule. This paper is a methodology guide for practitioners. We develop the betting-martingale construction of e-processes for two-arm randomized controlled trials, show how e-values naturally handle composite null hypotheses and support futility monitoring, and provide guidance on when e-values are appropriate, when established alternatives are preferable, and how to integrate e-value monitoring with group sequential and Bayesian adaptive workflows. A numerical study compares five monitoring rules – naive and calibrated versions of frequentist, Bayesian, and e-value approaches – in a two-arm binary-endpoint trial. Naive repeated testing and naive posterior thresholds inflate Type I error substantially under frequent interim looks. Among the valid methods, the calibrated group sequential rule achieves the highest power, the e-value rule provides robust anytime-valid control with moderate power, and the calibrated Bayesian rule is the most conservative. Extended simulations show that the power gap between group sequential and e-value methods depends on the monitoring schedule and reverses under continuous monitoring. The methodology, including futility monitoring, platform trial multiplicity control, and hybrid strategies combining e-values with established methods, is implemented in the open-source R package `evalinger` and situated within the regulatory framework of the January 2026 FDA draft guidance on Bayesian methodology.

We introduce a problem of fairly allocating indivisible goods (items) in which the agents' valuations cannot be observed directly, but instead can only be accessed via noisy queries. In the two-agent setting with Gaussian noise and bounded valuations, we derive upper and lower bounds on the required number of queries for finding an envy-free allocation in terms of the number of items, $m$, and the negative-envy of the optimal allocation, $\Delta$. In particular, when $\Delta$ is not too small (namely, $\Delta \gg m^{1/4}$), we establish that the optimal number of queries scales as $\frac{\sqrt m }{(\Delta / m)^2} = \frac{m^{2.5}}{\Delta^2}$ up to logarithmic factors. Our upper bound is based on non-adaptive queries and a simple thresholding-based allocation algorithm that runs in polynomial time, while our lower bound holds even under adaptive queries and arbitrary computation time.

Outliers can seriously distort statistical inference by inducing excessive sensitivity in the likelihood function, thereby compromising the reliability of Bayesian estimation. To address this issue, we develop a robust Bayesian estimation method for conditionally heteroscedastic time series models by extending the density power divergence (DPD) framework to the Bayesian setting. The resulting DPD-based posterior distribution, controlled by a tuning parameter, achieves a smooth balance between efficiency and robustness. We establish the asymptotic properties of the proposed estimator; specifically, the DPD-based posterior is shown to satisfy a Bernstein-von Mises type theorem, converging to a normal distribution centered at the minimum DPD estimator (MDPDE). Furthermore, the corresponding Bayes estimator, defined as the posterior mean under the DPD-based posterior (EDPE), is asymptotically equivalent to the MDPDE. Monte Carlo simulations based on GARCH(1,1) models confirm that the proposed EDPE performs well under both uncontaminated and contaminated data, maintaining robustness where the ordinary Bayes estimator becomes severely biased. An empirical application to BTC-USD returns further demonstrates the practical advantages of the proposed robust Bayesian framework for financial time series analysis.

Hazard functions play a central role in survival analysis, providing insight into the underlying risk dynamics of time-to-event data, with broad applications in medicine, epidemiology, and related fields. First-order ordinary differential equation (ODE) formulations of the hazard function have been explored as extensions beyond classical parametric models. However, such approaches typically produce monotonic hazard patterns, limiting their ability to represent oscillatory behavior, nonlinear damping, or coupled growth-decay dynamics. We propose a general statistical framework for modeling and simulating hazard functions governed by higher-order ODEs, allowing the hazard to depend on both its current level, its rate of change, and time. This formulation accommodates complex temporal risk behaviors arising in a range of applications. Building on this framework, we develop a class of nonlinear and oscillatory hazard models, each associated with an interpretable dynamical mechanism and an induced survival distribution. We also present a simulation procedure for solving a system of non-linear higher-order ODEs, with failure times generated via cumulative hazard inversion. Likelihood-based Bayesian inference under right censoring is also developed, and moment generating function analysis is used to characterize tail behavior. The proposed framework is evaluated through simulation studies and illustrated using real data, demonstrating its ability to capture temporal risk patterns not well represented by standard monotone models. In contrast to existing linear ODE-based hazard models, the proposed approach accommodates nonlinear and non-equilibrium dynamics, enabling the representation of temporal risk patterns that are not well captured by first-order or linear oscillator-based formulations.

Modern machine learning models are typically trained via multi-pass stochastic gradient descent (SGD) with small batch sizes, and understanding their dynamics in high dimensions is of great interest. However, an analytical framework for describing the high-dimensional asymptotic behavior of multi-pass SGD with small batch sizes for nonlinear models is currently missing. In this study, we address this gap by analyzing the high-dimensional dynamics of a stochastic differential equation called a stochastic gradient flow (SGF), which approximates multi-pass SGD in this regime. In the limit where the number of data samples $n$ and the dimension $d$ grow proportionally, we derive a closed system of low-dimensional and continuous-time equations and prove that it characterizes the asymptotic distribution of the SGF parameters. Our theory is based on the dynamical mean-field theory (DMFT) and is applicable to a wide range of models encompassing generalized linear models and two-layer neural networks. We further show that the resulting DMFT equations recover several existing high-dimensional descriptions of SGD dynamics as special cases, thereby providing a unifying perspective on prior frameworks such as online SGD and high-dimensional linear regression. Our proof builds on the existing DMFT technique for gradient flow and extends it to handle the stochasticity in SGF using tools from stochastic calculus.