arXiv subjects

Anzo Teh

Publications and source records attributed to Anzo Teh.

5 recordsLinked to original sources

arXiv2026

Universal priors: solving empirical Bayes via Bayesian inference and pretraining

We theoretically justify the recent empirical finding of [Teh et al., 2025] that a transformer pretrained on synthetically generated data achieves strong performance on empirical Bayes (EB) problems. We take an indirect approach to this question: rather than analyzing the model architecture or training dynamics, we ask why a pretrained Bayes estimator, trained under a prespecified training distribution, can adapt to arbitrary test distributions. Focusing on Poisson EB problems, we identify the existence of universal priors such that training under these priors yields a near-optimal regret bound of $\widetilde{O}(\frac{1}{n})$ uniformly over all test distributions. Our analysis leverages the classical phenomenon of posterior contraction in Bayesian statistics, showing that the pretrained transformer adapts to unknown test distributions precisely through posterior contraction. This perspective also explains the phenomenon of length generalization, in which the test sequence length exceeds the training length, as the model performs Bayesian inference using a generalized posterior.

stat.ML↗

arXiv2026

Function estimation in the empirical Bayes setting

We study function estimation in the empirical Bayes setting for Poisson and normal means. Specifically, given observations $X_i\sim f(\cdot; \theta_i)$ with latent parameters $\theta_i\sim \pi$, the goal is to estimate $\mathbb{E}_{\pi}[\ell(\theta)|X = x]$. This task lies between classical deconvolution (recovering the full prior $\pi$), and standard empirical Bayes mean estimation. While the minimax risk for estimating $\pi$ in the Wasserstein distance is known to decay only logarithmically, we show that estimating the corresponding posterior smooth functionals admits dramatically faster rates. In particular, for polynomial functions of degree $k$ in the Poisson model, we establish a tight total regret bound of $\Theta((\frac{\log n}{\log \log n})^{k+1})$ and $\Theta((\log n)^{2k+1})$ for bounded and subexponential priors, respectively, attainable by estimators mimicking those that achieve optimal regret for the mean estimation problem (Robbins, minimum distance, ERM). In the normal means model, we establish tight total regret bound of $\Theta((\frac{\log n}{\log \log n})^{k+1})$ for bounded priors, and bounds that match up to a polylogarithmic factor for subgaussian priors. Our analysis identifies the approximation-theoretic origin of this improvement: smooth functions can be well-approximated by low-degree polynomials, whereas Lipschitz functions have only $O(\frac{1}{k})$ degree-$k$ polynomial approximation error. The results reveal a sharp hierarchy in the difficulty of empirical Bayes problems: ranging from slow, logarithmic deconvolution to near-parametric convergence for smooth posterior functionals, and establish new connections between nonparametric empirical Bayes theory, polynomial approximation, and statistical inverse problems.

math.ST↗

arXiv2025

Solving Empirical Bayes via Transformers

This work applies modern AI tools (transformers) to solving one of the oldest statistical problems: Poisson means under empirical Bayes (Poisson-EB) setting. In Poisson-EB a high-dimensional mean vector $\theta$ (with iid coordinates sampled from an unknown prior $\pi$) is estimated on the basis of $X=\mathrm{Poisson}(\theta)$. A transformer model is pre-trained on a set of synthetically generated pairs $(X,\theta)$ and learns to do in-context learning (ICL) by adapting to unknown $\pi$. Theoretically, we show that a sufficiently wide transformer can achieve vanishing regret with respect to an oracle estimator who knows $\pi$ as dimension grows to infinity. Practically, we discover that already very small models (100k parameters) are able to outperform the best classical algorithm (non-parametric maximum likelihood, or NPMLE) both in runtime and validation loss, which we compute on out-of-distribution synthetic data as well as real-world datasets (NHL hockey, MLB baseball, BookCorpusOpen). Finally, by using linear probes, we confirm that the transformer's EB estimator appears to internally work differently from either NPMLE or Robbins' estimators.

cs.LG↗

arXiv2023

Empirical Bayes via ERM and Rademacher complexities: the Poisson model

We consider the problem of empirical Bayes estimation for (multivariate) Poisson means. Existing solutions that have been shown theoretically optimal for minimizing the regret (excess risk over the Bayesian oracle that knows the prior) have several shortcomings. For example, the classical Robbins estimator does not retain the monotonicity property of the Bayes estimator and performs poorly under moderate sample size. Estimators based on the minimum distance and non-parametric maximum likelihood (NPMLE) methods correct these issues, but are computationally expensive with complexity growing exponentially with dimension. Extending the approach of Barbehenn and Zhao (2022), in this work we construct monotone estimators based on empirical risk minimization (ERM) that retain similar theoretical guarantees and can be computed much more efficiently. Adapting the idea of offset Rademacher complexity Liang et al. (2015) to the non-standard loss and function class in empirical Bayes, we show that the shape-constrained ERM estimator attains the minimax regret within constant factors in one dimension and within logarithmic factors in multiple dimensions.

math.ST↗

arXiv2023

Comparing Poisson and Gaussian channels (extended)

Consider a pair of input distributions which after passing through a Poisson channel become $\epsilon$-close in total variation. We show that they must necessarily then be $\epsilon^{0.5+o(1)}$-close after passing through a Gaussian channel as well. In the opposite direction, we show that distributions inducing $\epsilon$-close outputs over the Gaussian channel must induce $\epsilon^{1+o(1)}$-close outputs over the Poisson. This quantifies a well-known intuition that ''smoothing'' induced by Poissonization and Gaussian convolution are similar. As an application, we improve a recent upper bound of Han-Miao-Shen'2021 for estimating mixing distribution of a Poisson mixture in Gaussian optimal transport distance from $n^{-0.1 + o(1)}$ to $n^{-0.25 + o(1)}$.

cs.IT↗