arXiv subjects

Patrik Róbert Gerber

Publications and source records attributed to Patrik Róbert Gerber.

4 recordsLinked to original sources

arXiv2025

Minimax optimal testing by classification

This paper considers an ML inspired approach to hypothesis testing known as classifier/classification-accuracy testing ($\mathsf{CAT}$). In $\mathsf{CAT}$, one first trains a classifier by feeding it labeled synthetic samples generated by the null and alternative distributions, which is then used to predict labels of the actual data samples. This method is widely used in practice when the null and alternative are only specified via simulators (as in many scientific experiments). We study goodness-of-fit, two-sample ($\mathsf{TS}$) and likelihood-free hypothesis testing ($\mathsf{LFHT}$), and show that $\mathsf{CAT}$ achieves (near-)minimax optimal sample complexity in both the dependence on the total-variation ($\mathsf{TV}$) separation $ε$ and the probability of error $δ$ in a variety of non-parametric settings, including discrete distributions, $d$-dimensional distributions with a smooth density, and the Gaussian sequence model. In particular, we close the high probability sample complexity of $\mathsf{LFHT}$ for each class. As another highlight, we recover the minimax optimal complexity of $\mathsf{TS}$ over discrete distributions, which was recently established by Diakonikolas et al. (2021). The corresponding $\mathsf{CAT}$ simply compares empirical frequencies in the first half of the data, and rejects the null when the classification accuracy on the second half is better than random.

math.ST↗

arXiv2025

Density estimation using the perceptron

We propose a new density estimation algorithm. Given $n$ i.i.d. observations from a distribution belonging to a class of densities on $\mathbb{R}^d$, our estimator outputs any density in the class whose "perceptron discrepancy" with the empirical distribution is at most $O(\sqrt{d / n})$. The perceptron discrepancy is defined as the largest difference in mass two distribution place on any halfspace. It is shown that this estimator achieves the expected total variation distance to the truth that is almost minimax optimal over the class of densities with bounded Sobolev norm and Gaussian mixtures. This suggests that the regularity of the prior distribution could be an explanation for the efficiency of the ubiquitous step in machine learning that replaces optimization over large function spaces with simpler parametric classes (such as discriminators of GANs). We also show that replacing the perceptron discrepancy with the generalized energy distance of Székely and Rizzo (2013) further improves total variation loss. The generalized energy distance between empirical distributions is easily computable and differentiable, which makes it especially useful for fitting generative models. To the best of our knowledge, it is the first "simple" distance with such properties that yields minimax optimal statistical guarantees. In addition, we shed light on the ubiquitous method of representing discrete data in domain $[k]$ via embedding vectors on a unit ball in $\mathbb{R}^d$. We show that taking $d \asymp \log (k)$ allows one to use simple linear probing to evaluate and estimate total variation distance, as well as recovering minimax optimal sample complexity for the class of discrete distributions on $[k]$.

math.ST↗

arXiv2024

Likelihood-free hypothesis testing

Consider the problem of binary hypothesis testing. Given $Z$ coming from either $\mathbb P^{\otimes m}$ or $\mathbb Q^{\otimes m}$, to decide between the two with small probability of error it is sufficient, and in many cases necessary, to have $m\asymp1/ε^2$, where $ε$ measures the separation between $\mathbb P$ and $\mathbb Q$ in total variation ($\mathsf{TV}$). Achieving this, however, requires complete knowledge of the distributions and can be done, for example, using the Neyman-Pearson test. In this paper we consider a variation of the problem which we call likelihood-free hypothesis testing, where access to $\mathbb P$ and $\mathbb Q$ is given through $n$ i.i.d. observations from each. In the case when $\mathbb P$ and $\mathbb Q$ are assumed to belong to a non-parametric family, we demonstrate the existence of a fundamental trade-off between $n$ and $m$ given by $nm\asymp n_\sf{GoF}^2(ε)$, where $n_\sf{GoF}(ε)$ is the minimax sample complexity of testing between the hypotheses $H_0:\, \mathbb P=\mathbb Q$ vs $H_1:\, \mathsf{TV}(\mathbb P,\mathbb Q)\geqε$. We show this for three families of distributions, in addition to the family of all discrete distributions for which we obtain a more complicated trade-off exhibiting an additional phase-transition. Our results demonstrate the possibility of testing without fully estimating $\mathbb P$ and $\mathbb Q$, provided $m \gg 1/ε^2$.

math.ST↗

arXiv2023

Kernel-Based Tests for Likelihood-Free Hypothesis Testing

Given $n$ observations from two balanced classes, consider the task of labeling an additional $m$ inputs that are known to all belong to \emph{one} of the two classes. Special cases of this problem are well-known: with complete knowledge of class distributions ($n=\infty$) the problem is solved optimally by the likelihood-ratio test; when $m=1$ it corresponds to binary classification; and when $m\approx n$ it is equivalent to two-sample testing. The intermediate settings occur in the field of likelihood-free inference, where labeled samples are obtained by running forward simulations and the unlabeled sample is collected experimentally. In recent work it was discovered that there is a fundamental trade-off between $m$ and $n$: increasing the data sample $m$ reduces the amount $n$ of training/simulation data needed. In this work we (a) introduce a generalization where unlabeled samples come from a mixture of the two classes -- a case often encountered in practice; (b) study the minimax sample complexity for non-parametric classes of densities under \textit{maximum mean discrepancy} (MMD) separation; and (c) investigate the empirical performance of kernels parameterized by neural networks on two tasks: detection of the Higgs boson and detection of planted DDPM generated images amidst CIFAR-10 images. For both problems we confirm the existence of the theoretically predicted asymmetric $m$ vs $n$ trade-off.

stat.ML↗