Related papers: Query lower bounds for log-concave sampling
Discretized Langevin diffusions are efficient Monte Carlo methods for sampling from high dimensional target densities that are log-Lipschitz-smooth and (strongly) log-concave. In particular, the Euclidean Langevin Monte Carlo sampling…
We consider the problem of identifying the parameters of an unknown mixture of two arbitrary $d$-dimensional gaussians from a sequence of independent random samples. Our main results are upper and lower bounds giving a computationally…
Given a database, a common problem is to find the pairs or $k$-tuples of items that frequently co-occur. One specific problem is to create a small space "sketch" of the data that records which $k$-tuples appear in more than an $\epsilon$…
A key task in Bayesian machine learning is sampling from distributions that are only specified up to a partition function (i.e., constant of proportionality). One prevalent example of this is sampling posteriors in parametric distributions,…
Let $\mathscr{F}_{n,d}$ be the class of all functions $f:\{-1,1\}^n\to[-1,1]$ on the $n$-dimensional discrete hypercube of degree at most $d$. In the first part of this paper, we prove that any (deterministic or randomized) algorithm which…
Polar slice sampling, a Markov chain construction for approximate sampling, performs, under suitable assumptions on the target and initial distribution, provably independent of the state space dimension. We extend the aforementioned result…
Even in low dimensions, sampling from multi-modal distributions is challenging. We provide the first sampling algorithm for a broad class of distributions -- including all Gaussian mixtures -- with a query complexity that is polynomial in…
The Gibbs sampler (a.k.a. Glauber dynamics and heat-bath algorithm) is a popular Markov Chain Monte Carlo algorithm which iteratively samples from the conditional distributions of a probability measure $\pi$ of interest. Under the…
Motivated by applications in deep learning, where the global Lipschitz continuity condition is often not satisfied, we examine the problem of sampling from distributions with super-linearly growing log-gradients. We propose a novel tamed…
We give lower bounds on the amount of memory required by one-pass streaming algorithms for solving several natural learning problems. In a setting where examples lie in $\{0,1\}^d$ and the optimal classifier can be encoded using $\kappa$…
This paper studies the complexity for finding approximate stationary points of nonconvex-strongly-concave (NC-SC) smooth minimax problems, in both general and averaged smooth finite-sum settings. We establish nontrivial lower complexity…
We consider the problem of digitalizing Euclidean segments. Specifically, we look for a constructive method to connect any two points in $\mathbb{Z}^d$. The construction must be {\em consistent} (that is, satisfy the natural extension of…
We consider the well-studied problem of learning intersections of halfspaces under the Gaussian distribution in the challenging \emph{agnostic learning} model. Recent work of Diakonikolas et al. (2021) shows that any Statistical Query (SQ)…
Analysis of Stochastic Gradient Descent (SGD) and its variants typically relies on the assumption of uniformly bounded variance, a condition that frequently fails in practical non-convex settings, such as neural network training, as well as…
Stochastic convex optimization over an $\ell_1$-bounded domain is ubiquitous in machine learning applications such as LASSO but remains poorly understood when learning with differential privacy. We show that, up to logarithmic factors the…
We consider the problem of efficiently learning mixtures of a large number of spherical Gaussians, when the components of the mixture are well separated. In the most basic form of this problem, we are given samples from a uniform mixture of…
We give an $O(\log^2 n)$-query algorithm for finding a Tarski fixed point over the $4$-dimensional lattice $[n]^4$, matching the $\Omega(\log^2 n)$ lower bound of [EPRY20]. Additionally, our algorithm yields an ${O(\log^{\lceil…
We prove structure theorems for measures on the discrete cube and on Gaussian space, which provide sufficient conditions for mean-field behavior. These conditions rely on a new notion of complexity for such measures, namely the…
In this paper we show that the family P_d of probability distributions on R^d with log-concave densities satisfies a strong continuity condition. In particular, it turns out that weak convergence within this family entails (i) convergence…
Differential entropy and log determinant of the covariance matrix of a multivariate Gaussian distribution have many applications in coding, communications, signal processing and statistical inference. In this paper we consider in the high…