arXiv subjects

Jeyong Lee

Publications and source records attributed to Jeyong Lee.

4 recordsLinked to original sources

arXiv2026

Bayesian online learning in the one-pass regime: Frequentist validity and uncertainty quantification

Bayesian online learning provides a coherent framework for sequential inference. However, its theoretical understanding remains limited, particularly in the one-pass setting. Existing theoretical guarantees typically require the mini-batch sample size to diverge, a condition that fails in the one-pass regime. In this paper, we propose a new Bayesian online learning algorithm tailored to the one-pass setting, which incorporates a warm-start phase to ensure stable sequential updates. For this algorithm, we show that the sequentially updated posterior attains the optimal convergence rate. Building on this, we establish an online analogue of the Bernstein-von Mises theorem, which guarantees valid uncertainty quantification without diverging mini-batch sample sizes. Our analysis is based on a novel theoretical framework that differs fundamentally from existing approaches in the online learning literature. Numerical experiments on generalized linear models show that the proposed method matches the performance of the batch estimator while outperforming existing online procedures.

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arXiv2026

Online Bernstein-von Mises theorem

Online learning is an inferential paradigm in which parameters are updated incrementally from sequentially available data, in contrast to batch learning, where the entire dataset is processed at once. In this paper, we assume that mini-batches from the full dataset become available sequentially. The Bayesian framework, which updates beliefs about unknown parameters after observing each mini-batch, is naturally suited for online learning. At each step, we update the posterior distribution using the current prior and new observations, with the updated posterior serving as the prior for the next step. However, this recursive Bayesian updating is rarely computationally tractable unless the model and prior are conjugate. When the model is regular, the updated posterior can be approximated by a normal distribution, as justified by the Bernstein-von Mises theorem. We adopt a variational approximation at each step and investigate the frequentist properties of the final posterior obtained through this sequential procedure. Under mild assumptions, we show that the accumulated approximation error becomes negligible once the mini-batch size exceeds a threshold depending on the parameter dimension. As a result, the sequentially updated posterior is asymptotically indistinguishable from the full posterior.

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arXiv2025

Advances in Bayesian model selection consistency for high-dimensional generalized linear models

Uncovering genuine relationships between a response variable of interest and a large collection of covariates is a fundamental and practically important problem. In the context of Gaussian linear models, both the Bayesian and non-Bayesian literature is well-developed and there are no substantial differences in the model selection consistency results available from the two schools. For the more challenging generalized linear models (GLMs), however, Bayesian model selection consistency results are lacking in several ways. In this paper, we construct a Bayesian posterior distribution using an appropriate data-dependent prior and develop its asymptotic concentration properties using new theoretical techniques. In particular, we leverage Spokoiny's powerful non-asymptotic theory to obtain sharp quadratic approximations of the GLM's log-likelihood function, which leads to tight bounds on the errors associated with the model-specific maximum likelihood estimators and the Laplace approximation of our Bayesian marginal likelihood. In turn, these improved bounds lead to significantly stronger, near-optimal Bayesian model selection consistency results, e.g., far weaker beta-min conditions, compared to those available in the existing literature. In particular, our results are applicable to the Poisson regression model, in which the score function is not sub-Gaussian.

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arXiv2025

Rates of convergence for nonparametric estimation of singular distributions using generative adversarial networks

It is common in nonparametric estimation problems to impose a certain low-dimensional structure on the unknown parameter to avoid the curse of dimensionality. This paper considers a nonparametric distribution estimation problem with a structural assumption under which the target distribution is allowed to be singular with respect to the Lebesgue measure. In particular, we investigate the use of generative adversarial networks (GANs) for estimating the unknown distribution and obtain a convergence rate with respect to the $L^1$-Wasserstein metric. The convergence rate depends only on the underlying structure and noise level. More interestingly, under the same structural assumption, the convergence rate of GAN is strictly faster than the known rate of VAE in the literature. We also obtain a lower bound for the minimax optimal rate, which is conjectured to be sharp at least in some special cases. Although our upper and lower bounds for the minimax optimal rate do not match, the difference is not significant.

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