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Found 75 papers, 3 papers with code
Super Non-singular Decompositions of Polynomials and their Application to Robustly Learning Low-degree PTFs
We study the efficient learnability of low-degree polynomial threshold functions (PTFs) in the presence of a constant fraction of adversarial corruptions.
Robust Sparse Estimation for Gaussians with Optimal Error under Huber Contamination
Concretely, for Gaussian robust $k$-sparse mean estimation on $\mathbb{R}^d$ with corruption rate $\epsilon>0$, our algorithm has sample complexity $(k^2/\epsilon^2)\mathrm{polylog}(d/\epsilon)$, runs in sample polynomial time, and approximates the target mean within $\ell_2$-error $O(\epsilon)$.
Statistical Query Lower Bounds for Learning Truncated Gaussians
We study the problem of estimating the mean of an identity covariance Gaussian in the truncated setting, in the regime when the truncation set comes from a low-complexity family $\mathcal{C}$ of sets.
Agnostically Learning Multi-index Models with Queries
In contrast, algorithms that rely only on random examples inherently require $d^{\mathrm{poly}(1/\epsilon)}$ samples and runtime, even for the basic problem of agnostically learning a single ReLU or a halfspace.
Clustering Mixtures of Bounded Covariance Distributions Under Optimal Separation
Furthermore, under a variant of the "no large sub-cluster'' condition from in prior work [BKK22], we show that our algorithm outputs an accurate clustering, not just a refinement, even for general-weight mixtures.
Near-Optimal Algorithms for Gaussians with Huber Contamination: Mean Estimation and Linear Regression
We study the fundamental problems of Gaussian mean estimation and linear regression with Gaussian covariates in the presence of Huber contamination.
Testing Closeness of Multivariate Distributions via Ramsey Theory
Our main result is the first closeness tester for this problem with {\em sub-learning} sample complexity in any fixed dimension and a nearly-matching sample complexity lower bound.
Online Robust Mean Estimation
We note that if the algorithm is allowed to wait until time $T$ to report its estimate, this reduces to the well-studied problem of robust mean estimation.
New Lower Bounds for Testing Monotonicity and Log Concavity of Distributions
We develop a new technique for proving distribution testing lower bounds for properties defined by inequalities involving the bin probabilities of the distribution in question.
Efficiently Learning One-Hidden-Layer ReLU Networks via Schur Polynomials
Our main result is an efficient algorithm for this learning task with sample and computational complexity $(dk/\epsilon)^{O(k)}$, where $\epsilon>0$ is the target accuracy.
Information-Computation Tradeoffs for Learning Margin Halfspaces with Random Classification Noise
Our main result is a lower bound for Statistical Query (SQ) algorithms and low-degree polynomial tests suggesting that the quadratic dependence on $1/\epsilon$ in the sample complexity is inherent for computationally efficient algorithms.
SQ Lower Bounds for Learning Bounded Covariance GMMs
In the special case where the separation is on the order of $k^{1/2}$, we additionally obtain fine-grained SQ lower bounds with the correct exponent.
Nearly-Linear Time and Streaming Algorithms for Outlier-Robust PCA
Our main contribution is to develop a nearly-linear time algorithm for robust PCA with near-optimal error guarantees.
Do PAC-Learners Learn the Marginal Distribution?
We study a foundational variant of Valiant and Vapnik and Chervonenkis' Probably Approximately Correct (PAC)-Learning in which the adversary is restricted to a known family of marginal distributions $\mathscr{P}$.
Near-Optimal Cryptographic Hardness of Agnostically Learning Halfspaces and ReLU Regression under Gaussian Marginals
We study the task of agnostically learning halfspaces under the Gaussian distribution.
A Nearly Tight Bound for Fitting an Ellipsoid to Gaussian Random Points
We prove that for $c>0$ a sufficiently small universal constant that a random set of $c d^2/\log^4(d)$ independent Gaussian random points in $\mathbb{R}^d$ lie on a common ellipsoid with high probability.
A Strongly Polynomial Algorithm for Approximate Forster Transforms and its Application to Halfspace Learning
By leveraging our strongly polynomial Forster algorithm, we obtain the first strongly polynomial time algorithm for {\em distribution-free} PAC learning of halfspaces.
Outlier-Robust Sparse Mean Estimation for Heavy-Tailed Distributions
We study the fundamental task of outlier-robust mean estimation for heavy-tailed distributions in the presence of sparsity.
Gaussian Mean Testing Made Simple
Here we give an extremely simple algorithm for Gaussian mean testing with a one-page analysis.
SQ Lower Bounds for Learning Single Neurons with Massart Noise
We study the problem of PAC learning a single neuron in the presence of Massart noise.
Cryptographic Hardness of Learning Halfspaces with Massart Noise
We study the complexity of PAC learning halfspaces in the presence of Massart noise.
Near-Optimal Bounds for Testing Histogram Distributions
We investigate the problem of testing whether a discrete probability distribution over an ordered domain is a histogram on a specified number of bins.
List-Decodable Sparse Mean Estimation via Difference-of-Pairs Filtering
We study the problem of list-decodable sparse mean estimation.
Optimal SQ Lower Bounds for Robustly Learning Discrete Product Distributions and Ising Models
We establish optimal Statistical Query (SQ) lower bounds for robustly learning certain families of discrete high-dimensional distributions.
Robust Sparse Mean Estimation via Sum of Squares
In this work, we develop the first efficient algorithms for robust sparse mean estimation without a priori knowledge of the covariance.
Streaming Algorithms for High-Dimensional Robust Statistics
In this work, we develop the first efficient streaming algorithms for high-dimensional robust statistics with near-optimal memory requirements (up to logarithmic factors).
Non-Gaussian Component Analysis via Lattice Basis Reduction
Non-Gaussian Component Analysis (NGCA) is the following distribution learning problem: Given i. i. d.
Realizable Learning is All You Need
The equivalence of realizable and agnostic learnability is a fundamental phenomenon in learning theory.
Outlier-Robust Sparse Estimation via Non-Convex Optimization
We explore the connection between outlier-robust high-dimensional statistics and non-convex optimization in the presence of sparsity constraints, with a focus on the fundamental tasks of robust sparse mean estimation and robust sparse PCA.
Learning General Halfspaces with General Massart Noise under the Gaussian Distribution
We study the general problem and establish the following: For $\eta <1/2$, we give a learning algorithm for general halfspaces with sample and computational complexity $d^{O_{\eta}(\log(1/\gamma))}\mathrm{poly}(1/\epsilon)$, where $\gamma =\max\{\epsilon, \min\{\mathbf{Pr}[f(\mathbf{x}) = 1], \mathbf{Pr}[f(\mathbf{x}) = -1]\} \}$ is the bias of the target halfspace $f$.
Forster Decomposition and Learning Halfspaces with Noise
A Forster transform is an operation that turns a distribution into one with good anti-concentration properties.
Statistical Query Lower Bounds for List-Decodable Linear Regression
We study the problem of list-decodable linear regression, where an adversary can corrupt a majority of the examples.
Clustering Mixture Models in Almost-Linear Time via List-Decodable Mean Estimation
We leverage this result, together with additional techniques, to obtain the first almost-linear time algorithms for clustering mixtures of $k$ separated well-behaved distributions, nearly-matching the statistical guarantees of spectral methods.
Agnostic Proper Learning of Halfspaces under Gaussian Marginals
We study the problem of agnostically learning halfspaces under the Gaussian distribution.
The Optimality of Polynomial Regression for Agnostic Learning under Gaussian Marginals
We study the problem of agnostic learning under the Gaussian distribution.
Outlier-Robust Learning of Ising Models Under Dobrushin's Condition
We study the problem of learning Ising models satisfying Dobrushin's condition in the outlier-robust setting where a constant fraction of the samples are adversarially corrupted.
The Sample Complexity of Robust Covariance Testing
This lower bound is best possible, as $O(d^2)$ samples suffice to even robustly {\em learn} the covariance.
Near-Optimal Statistical Query Hardness of Learning Halfspaces with Massart Noise
The best known $\mathrm{poly}(d, 1/\epsilon)$-time algorithms for this problem achieve error of $\eta+\epsilon$, which can be far from the optimal bound of $\mathrm{OPT}+\epsilon$, where $\mathrm{OPT} = \mathbf{E}_{x \sim D_x} [\eta(x)]$.
Small Covers for Near-Zero Sets of Polynomials and Learning Latent Variable Models
Our result is constructive yielding an algorithm to compute such an $\epsilon$-cover that runs in time $\mathrm{poly}(M)$.
Robustly Learning Mixtures of $k$ Arbitrary Gaussians
We give a polynomial-time algorithm for the problem of robustly estimating a mixture of $k$ arbitrary Gaussians in $\mathbb{R}^d$, for any fixed $k$, in the presence of a constant fraction of arbitrary corruptions.
List-Decodable Mean Estimation in Nearly-PCA Time
Our algorithm runs in time $\widetilde{O}(ndk)$ for all $k = O(\sqrt{d}) \cup \Omega(d)$, where $n$ is the size of the dataset.
A Polynomial Time Algorithm for Learning Halfspaces with Tsybakov Noise
{\em We give the first polynomial-time algorithm for this fundamental learning problem.}
Optimal Testing of Discrete Distributions with High Probability
To illustrate the generality of our methods, we give optimal algorithms for testing collections of distributions and testing closeness with unequal sized samples.
Outlier Robust Mean Estimation with Subgaussian Rates via Stability
We study the problem of outlier robust high-dimensional mean estimation under a finite covariance assumption, and more broadly under finite low-degree moment assumptions.
The Complexity of Adversarially Robust Proper Learning of Halfspaces with Agnostic Noise
We study the computational complexity of adversarially robust proper learning of halfspaces in the distribution-independent agnostic PAC model, with a focus on $L_p$ perturbations.
Robust Learning of Mixtures of Gaussians
We resolve one of the major outstanding problems in robust statistics.
Near-Optimal SQ Lower Bounds for Agnostically Learning Halfspaces and ReLUs under Gaussian Marginals
We study the fundamental problems of agnostically learning halfspaces and ReLUs under Gaussian marginals.
Algorithms and SQ Lower Bounds for PAC Learning One-Hidden-Layer ReLU Networks
For the case of positive coefficients, we give the first polynomial-time algorithm for this learning problem for $k$ up to $\tilde{O}(\sqrt{\log d})$.
List-Decodable Mean Estimation via Iterative Multi-Filtering
We study the problem of {\em list-decodable mean estimation} for bounded covariance distributions.
Point Location and Active Learning: Learning Halfspaces Almost Optimally
Given a finite set $X \subset \mathbb{R}^d$ and a binary linear classifier $c: \mathbb{R}^d \to \{0, 1\}$, how many queries of the form $c(x)$ are required to learn the label of every point in $X$?
Recent Advances in Algorithmic High-Dimensional Robust Statistics
Learning in the presence of outliers is a fundamental problem in statistics.
Nearly Tight Bounds for Robust Proper Learning of Halfspaces with a Margin
We study the problem of {\em properly} learning large margin halfspaces in the agnostic PAC model.
The Power of Comparisons for Actively Learning Linear Classifiers
While previous results show that active learning performs no better than its supervised alternative for important concept classes such as linear separators, we show that by adding weak distributional assumptions and allowing comparison queries, active learning requires exponentially fewer samples.
Communication and Memory Efficient Testing of Discrete Distributions
We study distribution testing with communication and memory constraints in the following computational models: (1) The {\em one-pass streaming model} where the goal is to minimize the sample complexity of the protocol subject to a memory constraint, and (2) A {\em distributed model} where the data samples reside at multiple machines and the goal is to minimize the communication cost of the protocol.
Learning Ising Models with Independent Failures
We give the first efficient algorithm for learning the structure of an Ising model that tolerates independent failures; that is, each entry of the observed sample is missing with some unknown probability p. Our algorithm matches the essentially optimal runtime and sample complexity bounds of recent work for learning Ising models due to Klivans and Meka (2017).
Degree-$d$ Chow Parameters Robustly Determine Degree-$d$ PTFs (and Algorithmic Applications)
Our robust identifiability result gives the following algorithmic applications: First, we show that Boolean degree-$d$ PTFs can be efficiently approximately reconstructed from approximations to their degree-$d$ Chow parameters.
Testing Identity of Multidimensional Histograms
We give the first identity tester for this problem with {\em sub-learning} sample complexity in any fixed dimension and a nearly-matching sample complexity lower bound.
Sever: A Robust Meta-Algorithm for Stochastic Optimization
In high dimensions, most machine learning methods are brittle to even a small fraction of structured outliers.
List-Decodable Robust Mean Estimation and Learning Mixtures of Spherical Gaussians
We give a learning algorithm for mixtures of spherical Gaussians that succeeds under significantly weaker separation assumptions compared to prior work.
On Communication Complexity of Classification Problems
To naturally fit into the framework of learning theory, the players can send each other examples (as well as bits) where each example/bit costs one unit of communication.
Sharp Bounds for Generalized Uniformity Testing
We study the problem of generalized uniformity testing \cite{BC17} of a discrete probability distribution: Given samples from a probability distribution $p$ over an {\em unknown} discrete domain $\mathbf{\Omega}$, we want to distinguish, with probability at least $2/3$, between the case that $p$ is uniform on some {\em subset} of $\mathbf{\Omega}$ versus $\epsilon$-far, in total variation distance, from any such uniform distribution.
Learning Geometric Concepts with Nasty Noise
We give the first polynomial-time PAC learning algorithms for these concept classes with dimension-independent error guarantees in the presence of nasty noise under the Gaussian distribution.
Near-optimal linear decision trees for k-SUM and related problems
We construct near optimal linear decision trees for a variety of decision problems in combinatorics and discrete geometry.
Robustly Learning a Gaussian: Getting Optimal Error, Efficiently
We give robust estimators that achieve estimation error $O(\varepsilon)$ in the total variation distance, which is optimal up to a universal constant that is independent of the dimension.
Active classification with comparison queries
We identify a combinatorial dimension, called the \emph{inference dimension}, that captures the query complexity when each additional query is determined by $O(1)$ examples (such as comparison queries, each of which is determined by the two compared examples).
Near-Optimal Closeness Testing of Discrete Histogram Distributions
Given a set of samples from two $k$-histogram distributions $p, q$ over $[n]$, we want to distinguish (with high probability) between the cases that $p = q$ and $\|p-q\|_1 \geq \epsilon$.
Being Robust (in High Dimensions) Can Be Practical
Robust estimation is much more challenging in high dimensions than it is in one dimension: Most techniques either lead to intractable optimization problems or estimators that can tolerate only a tiny fraction of errors.
Statistical Query Lower Bounds for Robust Estimation of High-dimensional Gaussians and Gaussian Mixtures
For each of these problems, we show a {\em super-polynomial gap} between the (information-theoretic) sample complexity and the computational complexity of {\em any} Statistical Query algorithm for the problem.
Efficient Robust Proper Learning of Log-concave Distributions
We study the {\em robust proper learning} of univariate log-concave distributions (over continuous and discrete domains).
Learning Multivariate Log-concave Distributions
Prior to our work, no upper bound on the sample complexity of this learning problem was known for the case of $d>3$.
Super-Linear Gate and Super-Quadratic Wire Lower Bounds for Depth-Two and Depth-Three Threshold Circuits
$\bullet$ We give tight average-case (gate and wire) complexity results for computing PARITY with depth-two threshold circuits; the answer turns out to be the same as for depth-two majority circuits.
Properly Learning Poisson Binomial Distributions in Almost Polynomial Time
Given $\widetilde{O}(1/\epsilon^2)$ samples from an unknown PBD $\mathbf{p}$, our algorithm runs in time $(1/\epsilon)^{O(\log \log (1/\epsilon))}$, and outputs a hypothesis PBD that is $\epsilon$-close to $\mathbf{p}$ in total variation distance.
The Fourier Transform of Poisson Multinomial Distributions and its Algorithmic Applications
An $(n, k)$-Poisson Multinomial Distribution (PMD) is a random variable of the form $X = \sum_{i=1}^n X_i$, where the $X_i$'s are independent random vectors supported on the set of standard basis vectors in $\mathbb{R}^k.$ In this paper, we obtain a refined structural understanding of PMDs by analyzing their Fourier transform.
Optimal Learning via the Fourier Transform for Sums of Independent Integer Random Variables
As one of our main structural contributions, we give an efficient algorithm to construct a sparse {\em proper} $\epsilon$-cover for ${\cal S}_{n, k},$ in total variation distance.
Fast Moment Estimation in Data Streams in Optimal Space
We give a space-optimal algorithm with update time O(log^2(1/eps)loglog(1/eps)) for (1+eps)-approximating the pth frequency moment, 0 < p < 2, of a length-n vector updated in a data stream.
Data Structures and Algorithms
