arXiv subjects

Youngtak Sohn

Publications and source records attributed to Youngtak Sohn.

21 records · Page 2Linked to original sources

arXiv2023

Upper bounds on the $2$-colorability threshold of random $d$-regular $k$-uniform hypergraphs for $k\geq 3$

For a large class of random constraint satisfaction problems (CSP), deep but non-rigorous theory from statistical physics predict the location of the sharp satisfiability transition. The works of Ding, Sly, Sun (2014, 2016) and Coja-Oghlan, Panagiotou (2014) established the satisfiability threshold for random regular $k$-NAE-SAT, random $k$-SAT, and random regular $k$-SAT for large enough $k\geq k_0$ where $k_0$ is a large non-explicit constant. Establishing the same for small values of $k\geq 3$ remains an important open problem in the study of random CSPs. In this work, we study two closely related models of random CSPs, namely the $2$-coloring on random $d$-regular $k$-uniform hypergraphs and the random $d$-regular $k$-NAE-SAT model. For every $k\geq 3$, we prove that there is an explicit $d_{\ast}(k)$ which gives a satisfiability upper bound for both of the models. Our upper bound $d_{\ast}(k)$ for $k\geq 3$ matches the prediction from statistical physics for the hypergraph $2$-coloring by Dall'Asta, Ramezanpour, Zecchina (2008), thus conjectured to be sharp. Moreover, $d_{\ast}(k)$ coincides with the satisfiability threshold of random regular $k$-NAE-SAT for large enough $k\geq k_0$ by Ding, Sly, Sun (2014).

math.CO↗

arXiv2023

The generalization error of max-margin linear classifiers: Benign overfitting and high dimensional asymptotics in the overparametrized regime

Modern machine learning classifiers often exhibit vanishing classification error on the training set. They achieve this by learning nonlinear representations of the inputs that maps the data into linearly separable classes. Motivated by these phenomena, we revisit high-dimensional maximum margin classification for linearly separable data. We consider a stylized setting in which data $(y_i,{\boldsymbol x}_i)$, $i\le n$ are i.i.d. with ${\boldsymbol x}_i\sim\mathsf{N}({\boldsymbol 0},{\boldsymbol Σ})$ a $p$-dimensional Gaussian feature vector, and $y_i \in\{+1,-1\}$ a label whose distribution depends on a linear combination of the covariates $\langle {\boldsymbol θ}_*,{\boldsymbol x}_i \rangle$. While the Gaussian model might appear extremely simplistic, universality arguments can be used to show that the results derived in this setting also apply to the output of certain nonlinear featurization maps. We consider the proportional asymptotics $n,p\to\infty$ with $p/n\to ψ$, and derive exact expressions for the limiting generalization error. We use this theory to derive two results of independent interest: $(i)$ Sufficient conditions on $({\boldsymbol Σ},{\boldsymbol θ}_*)$ for `benign overfitting' that parallel previously derived conditions in the case of linear regression; $(ii)$ An asymptotically exact expression for the generalization error when max-margin classification is used in conjunction with feature vectors produced by random one-layer neural networks.

math.ST↗

arXiv2023

Sharp thresholds in inference of planted subgraphs

A major question in the study of the Erdős--Rényi random graph is to understand the probability that it contains a given subgraph. This study originated in classical work of Erdős and Rényi (1960). More recent work studies this question both in building a general theory of sharp versus coarse transitions (Friedgut and Bourgain 1999; Hatami, 2012) and in results on the location of the transition (Kahn and Kalai, 2007; Talagrand, 2010; Frankston, Kahn, Narayanan, Park, 2019; Park and Pham, 2022). In inference problems, one often studies the optimal accuracy of inference as a function of the amount of noise. In a variety of sparse recovery problems, an ``all-or-nothing (AoN) phenomenon'' has been observed: Informally, as the amount of noise is gradually increased, at some critical threshold the inference problem undergoes a sharp jump from near-perfect recovery to near-zero accuracy (Gamarnik and Zadik, 2017; Reeves, Xu, Zadik, 2021). We can regard AoN as the natural inference analogue of the sharp threshold phenomenon in random graphs. In contrast with the general theory developed for sharp thresholds of random graph properties, the AoN phenomenon has only been studied so far in specific inference settings. In this paper we study the general problem of inferring a graph $H=H_n$ planted in an Erdős--Rényi random graph, thus naturally connecting the two lines of research mentioned above. We show that questions of AoN are closely connected to first moment thresholds, and to a generalization of the so-called Kahn--Kalai expectation threshold that scans over subgraphs of $H$ of edge density at least $q$. In a variety of settings we characterize AoN, by showing that AoN occurs if and only if this ``generalized expectation threshold'' is roughly constant in $q$. Our proofs combine techniques from random graph theory and Bayesian inference.

math.ST↗