Generalization bounds for neural ordinary differential equations and deep residual networks

Pierre Marion

Home/Archive/arXiv

arXiv · 2305.06648

Generalization bounds for neural ordinary differential equations and deep residual networks

Pierre Marion

View original source ↗Read source document ↗Export metadata ↓

Abstract

Neural ordinary differential equations (neural ODEs) are a popular family of continuous-depth deep learning models. In this work, we consider a large family of parameterized ODEs with continuous-in-time parameters, which include time-dependent neural ODEs. We derive a generalization bound for this class by a Lipschitz-based argument. By leveraging the analogy between neural ODEs and deep residual networks, our approach yields in particular a generalization bound for a class of deep residual networks. The bound involves the magnitude of the difference between successive weight matrices. We illustrate numerically how this quantity affects the generalization capability of neural networks.

Explore related subjects

stat.ML cs.LG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Pierre Marion. 2023-10-11. Generalization bounds for neural ordinary differential equations and deep residual networks. https://arxiv.org/abs/2305.06648

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Time-Varying Bayesian Optimization Without a Metronome

Time-Varying Bayesian Optimization (TVBO) is the go-to framework for optimizing a time-varying, expensive, noisy black-box function $f$. However, most of the asymptotic guarantees offered by TVBO algorithms rely on the assumption that observations are acquired at a constant frequency. As the GP inference complexity scales with the cube of its dataset size, this assumption is unrealistic in the long run. In this paper, we relax this assumption and derive the first upper regret bound that explicitly accounts for changes in the observations sampling frequency. Based on this analysis, we formulate practical recommendations about dataset sizes and stale data policies of TVBO algorithms. We illustrate how an algorithm (BOLT) that follows these recommendations performs better than the state-of-the-art of TVBO through experiments on synthetic and real-world problems.

stat.ML↗

arXiv2026

MultiwayPAM: Multiway Partitioning Around Medoids for LLM-as-a-Judge Score Analysis

LLM-as-a-Judge is a flexible framework for text evaluation, which allows us to obtain scores for the quality of a given text from various perspectives by changing the prompt template. Two main challenges in using LLM-as-a-Judge are computational cost of inference using a large language model (LLM), especially when evaluating a large number of instances, and inherent bias of an LLM evaluator. To address these issues and reveal the structure of score bias caused by an LLM evaluator, we propose to apply a tensor clustering method to a given LLM-as-a-Judge score tensor, whose entries are the scores for different combinations of questions, answerers, and evaluators. Specifically, we develop a new tensor clustering method MultiwayPAM, with which we can simultaneously estimate the cluster membership and the medoids for each mode of a given data tensor. By observing the medoids obtained by MultiwayPAM, we can gain knowledge about the membership of each question/answerer/evaluator cluster. We experimentally show the effectiveness of MultiwayPAM by applying it to the score tensors for two practical datasets.

stat.ML↗

arXiv2026

Learning to Fluctuate: Statistical Foundations for Causal Tabular Pretraining

Causal tabular foundation models amortize effect estimation across synthetic mechanisms, but latent-effect supervision rewards posterior shrinkage rather than encoding the repeated-sample response needed in a fixed deployment population. We introduce fluctuation-supervised pretraining (FSP): each synthetic table is labeled by its average treatment effect plus its efficient influence-function fluctuation; deployment remains a frozen forward pass. Along the path $T_{λ,P}=θ(P)+λP_nψ_P$, we prove an endpoint transition: every fixed $λ<1$ retains label ambiguity of order $(1-λ)^2/n$, whereas full fluctuation makes the Gaussian label observable and reduces optimal finite-stratum causal label-prediction risk to order $n^{-2}$. A finite-pretraining bound combines label, network, episode-sampling, and optimization errors; its sampling defect controls fixed-mechanism bias, mean squared error, variance, Gaussian approximation, and, with variance-head accuracy, studentized coverage. Complementary lower bounds separate local $n^{-1}$ ATE risk from the $\log N/M$ excess risk of generic finite-dictionary episode learning. Experiments trace the learned sampling response. Across 24 nonlinear continuous-covariate cells at trained context lengths, continuous-row FSP lowers checkpoint-mean macro RMSE by 7.0% versus S-learner and wins all 12 weak-overlap cells; validation-selected Summary FSP deploys $11.6\times$ faster per table in our warm one-thread benchmark. Under effect shift, matched Raw FSP lowers mean-checkpoint RMSE by 54.2% and teacher defect by 99.0% versus latent-effect supervision, and RMSE by 10.2% versus the released CausalPFN-S checkpoint. Known-effect semisynthesis tests coverage; two randomized-study evaluations show that lower RMSE can coexist with residual attenuation.

stat.ML↗