arXiv subjects

Daniel M. Kane

Publications and source records attributed to Daniel M. Kane.

At least 73 records · Page 4Linked to original sources

arXiv2022

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. We develop novel and simple optimization formulations for these problems such that any approximate stationary point of the associated optimization problem yields a near-optimal solution for the underlying robust estimation task. As a corollary, we obtain that any first-order method that efficiently converges to stationarity yields an efficient algorithm for these tasks. The obtained algorithms are simple, practical, and succeed under broader distributional assumptions compared to prior work.

cs.LG↗

arXiv2022

Gaussian Mean Testing Made Simple

We study the following fundamental hypothesis testing problem, which we term Gaussian mean testing. Given i.i.d. samples from a distribution $p$ on $\mathbb{R}^d$, the task is to distinguish, with high probability, between the following cases: (i) $p$ is the standard Gaussian distribution, $\mathcal{N}(0,I_d)$, and (ii) $p$ is a Gaussian $\mathcal{N}(μ,Σ)$ for some unknown covariance $Σ$ and mean $μ\in \mathbb{R}^d$ satisfying $\|μ\|_2 \geq ε$. Recent work gave an algorithm for this testing problem with the optimal sample complexity of $Θ(\sqrt{d}/ε^2)$. Both the previous algorithm and its analysis are quite complicated. Here we give an extremely simple algorithm for Gaussian mean testing with a one-page analysis. Our algorithm is sample optimal and runs in sample linear time.

math.ST↗

arXiv2022

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. Specifically, for a known activation function $f: \mathbb{R} \to \mathbb{R}$, the learner is given access to labeled examples $(\mathbf{x}, y) \in \mathbb{R}^d \times \mathbb{R}$, where the marginal distribution of $\mathbf{x}$ is arbitrary and the corresponding label $y$ is a Massart corruption of $f(\langle \mathbf{w}, \mathbf{x} \rangle)$. The goal of the learner is to output a hypothesis $h: \mathbb{R}^d \to \mathbb{R}$ with small squared loss. For a range of activation functions, including ReLUs, we establish super-polynomial Statistical Query (SQ) lower bounds for this learning problem. In more detail, we prove that no efficient SQ algorithm can approximate the optimal error within any constant factor. Our main technical contribution is a novel SQ-hard construction for learning $\{ \pm 1\}$-weight Massart halfspaces on the Boolean hypercube that is interesting on its own right.

cs.LG↗

arXiv2022

Quantum Money from Quaternion Algebras

We propose a new idea for public key quantum money. In the abstract sense, our bills are encoded as a joint eigenstate of a fixed system of commuting unitary operators. We perform some basic analysis of this black box system and show that it is resistant to black box attacks. In order to instantiate this protocol, one needs to find a cryptographically complicated system of computable, commuting, unitary operators. To fill this need, we propose using Brandt operators acting on the Brandt modules associated to certain quaternion algebras. We explain why we believe this instantiation is likely to be secure.

quant-ph↗

arXiv2022

Cryptographic Hardness of Learning Halfspaces with Massart Noise

We study the complexity of PAC learning halfspaces in the presence of Massart noise. In this problem, we are given i.i.d. labeled examples $(\mathbf{x}, y) \in \mathbb{R}^N \times \{ \pm 1\}$, where the distribution of $\mathbf{x}$ is arbitrary and the label $y$ is a Massart corruption of $f(\mathbf{x})$, for an unknown halfspace $f: \mathbb{R}^N \to \{ \pm 1\}$, with flipping probability $η(\mathbf{x}) \leq η< 1/2$. The goal of the learner is to compute a hypothesis with small 0-1 error. Our main result is the first computational hardness result for this learning problem. Specifically, assuming the (widely believed) subexponential-time hardness of the Learning with Errors (LWE) problem, we show that no polynomial-time Massart halfspace learner can achieve error better than $Ω(η)$, even if the optimal 0-1 error is small, namely $\mathrm{OPT} = 2^{-\log^{c} (N)}$ for any universal constant $c \in (0, 1)$. Prior work had provided qualitatively similar evidence of hardness in the Statistical Query model. Our computational hardness result essentially resolves the polynomial PAC learnability of Massart halfspaces, by showing that known efficient learning algorithms for the problem are nearly best possible.

cs.LG↗

arXiv2022

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. One of the most common tools for the succinct approximation of data, $k$-histograms over $[n]$, are probability distributions that are piecewise constant over a set of $k$ intervals. The histogram testing problem is the following: Given samples from an unknown distribution $\mathbf{p}$ on $[n]$, we want to distinguish between the cases that $\mathbf{p}$ is a $k$-histogram versus $\varepsilon$-far from any $k$-histogram, in total variation distance. Our main result is a sample near-optimal and computationally efficient algorithm for this testing problem, and a nearly-matching (within logarithmic factors) sample complexity lower bound. Specifically, we show that the histogram testing problem has sample complexity $\widetilde Θ(\sqrt{nk} / \varepsilon + k / \varepsilon^2 + \sqrt{n} / \varepsilon^2)$.

cs.DS↗

arXiv2022

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. In particular, we show that no efficient SQ algorithm with access to an $ε$-corrupted binary product distribution can learn its mean within $\ell_2$-error $o(ε\sqrt{\log(1/ε)})$. Similarly, we show that no efficient SQ algorithm with access to an $ε$-corrupted ferromagnetic high-temperature Ising model can learn the model to total variation distance $o(ε\log(1/ε))$. Our SQ lower bounds match the error guarantees of known algorithms for these problems, providing evidence that current upper bounds for these tasks are best possible. At the technical level, we develop a generic SQ lower bound for discrete high-dimensional distributions starting from low dimensional moment matching constructions that we believe will find other applications. Additionally, we introduce new ideas to analyze these moment-matching constructions for discrete univariate distributions.

cs.DS↗

arXiv2021

Non-Gaussian Component Analysis via Lattice Basis Reduction

Non-Gaussian Component Analysis (NGCA) is the following distribution learning problem: Given i.i.d. samples from a distribution on $\mathbb{R}^d$ that is non-gaussian in a hidden direction $v$ and an independent standard Gaussian in the orthogonal directions, the goal is to approximate the hidden direction $v$. Prior work \cite{DKS17-sq} provided formal evidence for the existence of an information-computation tradeoff for NGCA under appropriate moment-matching conditions on the univariate non-gaussian distribution $A$. The latter result does not apply when the distribution $A$ is discrete. A natural question is whether information-computation tradeoffs persist in this setting. In this paper, we answer this question in the negative by obtaining a sample and computationally efficient algorithm for NGCA in the regime that $A$ is discrete or nearly discrete, in a well-defined technical sense. The key tool leveraged in our algorithm is the LLL method \cite{LLL82} for lattice basis reduction.

cs.DS↗

arXiv2021

Clustering Mixture Models in Almost-Linear Time via List-Decodable Mean Estimation

We study the problem of list-decodable mean estimation, where an adversary can corrupt a majority of the dataset. Specifically, we are given a set $T$ of $n$ points in $\mathbb{R}^d$ and a parameter $0< α<\frac 1 2$ such that an $α$-fraction of the points in $T$ are i.i.d. samples from a well-behaved distribution $\mathcal{D}$ and the remaining $(1-α)$-fraction are arbitrary. The goal is to output a small list of vectors, at least one of which is close to the mean of $\mathcal{D}$. We develop new algorithms for list-decodable mean estimation, achieving nearly-optimal statistical guarantees, with running time $O(n^{1 + ε_0} d)$, for any fixed $ε_0 > 0$. All prior algorithms for this problem had additional polynomial factors in $\frac 1 α$. 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. Prior clustering algorithms inherently relied on an application of $k$-PCA, thereby incurring runtimes of $Ω(n d k)$. This marks the first runtime improvement for this basic statistical problem in nearly two decades. The starting point of our approach is a novel and simpler near-linear time robust mean estimation algorithm in the $α\to 1$ regime, based on a one-shot matrix multiplicative weights-inspired potential decrease. We crucially leverage this new algorithmic framework in the context of the iterative multi-filtering technique of Diakonikolas et al. '18, '20, providing a method to simultaneously cluster and downsample points using one-dimensional projections -- thus, bypassing the $k$-PCA subroutines required by prior algorithms.

cs.DS↗

arXiv2021

Near-Optimal Statistical Query Hardness of Learning Halfspaces with Massart Noise

We study the problem of PAC learning halfspaces with Massart noise. Given labeled samples $(x, y)$ from a distribution $D$ on $\mathbb{R}^{d} \times \{ \pm 1\}$ such that the marginal $D_x$ on the examples is arbitrary and the label $y$ of example $x$ is generated from the target halfspace corrupted by a Massart adversary with flipping probability $η(x) \leq η\leq 1/2$, the goal is to compute a hypothesis with small misclassification error. The best known $\mathrm{poly}(d, 1/ε)$-time algorithms for this problem achieve error of $η+ε$, which can be far from the optimal bound of $\mathrm{OPT}+ε$, where $\mathrm{OPT} = \mathbf{E}_{x \sim D_x} [η(x)]$. While it is known that achieving $\mathrm{OPT}+o(1)$ error requires super-polynomial time in the Statistical Query model, a large gap remains between known upper and lower bounds. In this work, we essentially characterize the efficient learnability of Massart halfspaces in the Statistical Query (SQ) model. Specifically, we show that no efficient SQ algorithm for learning Massart halfspaces on $\mathbb{R}^d$ can achieve error better than $Ω(η)$, even if $\mathrm{OPT} = 2^{-\log^{c} (d)}$, for any universal constant $c \in (0, 1)$. Furthermore, when the noise upper bound $η$ is close to $1/2$, our error lower bound becomes $η- o_η(1)$, where the $o_η(1)$ term goes to $0$ when $η$ approaches $1/2$. Our results provide strong evidence that known learning algorithms for Massart halfspaces are nearly best possible, thereby resolving a longstanding open problem in learning theory.

cs.LG↗

arXiv2021

Learning General Halfspaces with General Massart Noise under the Gaussian Distribution

We study the problem of PAC learning halfspaces on $\mathbb{R}^d$ with Massart noise under the Gaussian distribution. In the Massart model, an adversary is allowed to flip the label of each point $\mathbf{x}$ with unknown probability $η(\mathbf{x}) \leq η$, for some parameter $η\in [0,1/2]$. The goal is to find a hypothesis with misclassification error of $\mathrm{OPT} + ε$, where $\mathrm{OPT}$ is the error of the target halfspace. This problem had been previously studied under two assumptions: (i) the target halfspace is homogeneous (i.e., the separating hyperplane goes through the origin), and (ii) the parameter $η$ is strictly smaller than $1/2$. Prior to this work, no nontrivial bounds were known when either of these assumptions is removed. We study the general problem and establish the following: For $η<1/2$, we give a learning algorithm for general halfspaces with sample and computational complexity $d^{O_η(\log(1/γ))}\mathrm{poly}(1/ε)$, where $γ=\max\{ε, \min\{\mathbf{Pr}[f(\mathbf{x}) = 1], \mathbf{Pr}[f(\mathbf{x}) = -1]\} \}$ is the bias of the target halfspace $f$. Prior efficient algorithms could only handle the special case of $γ= 1/2$. Interestingly, we establish a qualitatively matching lower bound of $d^{Ω(\log(1/γ))}$ on the complexity of any Statistical Query (SQ) algorithm. For $η= 1/2$, we give a learning algorithm for general halfspaces with sample and computational complexity $O_ε(1) d^{O(\log(1/ε))}$. This result is new even for the subclass of homogeneous halfspaces; prior algorithms for homogeneous Massart halfspaces provide vacuous guarantees for $η=1/2$. We complement our upper bound with a nearly-matching SQ lower bound of $d^{Ω(\log(1/ε))}$, which holds even for the special case of homogeneous halfspaces.

cs.LG↗

arXiv2021

Forster Decomposition and Learning Halfspaces with Noise

A Forster transform is an operation that turns a distribution into one with good anti-concentration properties. While a Forster transform does not always exist, we show that any distribution can be efficiently decomposed as a disjoint mixture of few distributions for which a Forster transform exists and can be computed efficiently. As the main application of this result, we obtain the first polynomial-time algorithm for distribution-independent PAC learning of halfspaces in the Massart noise model with strongly polynomial sample complexity, i.e., independent of the bit complexity of the examples. Previous algorithms for this learning problem incurred sample complexity scaling polynomially with the bit complexity, even though such a dependence is not information-theoretically necessary.

cs.LG↗

arXiv2021

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. Specifically, we are given a set $T$ of labeled examples $(x, y) \in \mathbb{R}^d \times \mathbb{R}$ and a parameter $0< α<1/2$ such that an $α$-fraction of the points in $T$ are i.i.d. samples from a linear regression model with Gaussian covariates, and the remaining $(1-α)$-fraction of the points are drawn from an arbitrary noise distribution. The goal is to output a small list of hypothesis vectors such that at least one of them is close to the target regression vector. Our main result is a Statistical Query (SQ) lower bound of $d^{\mathrm{poly}(1/α)}$ for this problem. Our SQ lower bound qualitatively matches the performance of previously developed algorithms, providing evidence that current upper bounds for this task are nearly best possible.

cs.DS↗

arXiv2021

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. This resolves the main open problem in several previous works on algorithmic robust statistics, which addressed the special cases of robustly estimating (a) a single Gaussian, (b) a mixture of TV-distance separated Gaussians, and (c) a uniform mixture of two Gaussians. Our main tools are an efficient \emph{partial clustering} algorithm that relies on the sum-of-squares method, and a novel \emph{tensor decomposition} algorithm that allows errors in both Frobenius norm and low-rank terms.

cs.DS↗

arXiv2021

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. We consider a standard stability condition from the recent robust statistics literature and prove that, except with exponentially small failure probability, there exists a large fraction of the inliers satisfying this condition. As a corollary, it follows that a number of recently developed algorithms for robust mean estimation, including iterative filtering and non-convex gradient descent, give optimal error estimators with (near-)subgaussian rates. Previous analyses of these algorithms gave significantly suboptimal rates. As a corollary of our approach, we obtain the first computationally efficient algorithm with subgaussian rate for outlier-robust mean estimation in the strong contamination model under a finite covariance assumption.

math.ST↗

arXiv2021

Agnostic Proper Learning of Halfspaces under Gaussian Marginals

We study the problem of agnostically learning halfspaces under the Gaussian distribution. Our main result is the {\em first proper} learning algorithm for this problem whose sample complexity and computational complexity qualitatively match those of the best known improper agnostic learner. Building on this result, we also obtain the first proper polynomial-time approximation scheme (PTAS) for agnostically learning homogeneous halfspaces. Our techniques naturally extend to agnostically learning linear models with respect to other non-linear activations, yielding in particular the first proper agnostic algorithm for ReLU regression.

cs.LG↗

arXiv2021

The Optimality of Polynomial Regression for Agnostic Learning under Gaussian Marginals

We study the problem of agnostic learning under the Gaussian distribution. We develop a method for finding hard families of examples for a wide class of problems by using LP duality. For Boolean-valued concept classes, we show that the $L^1$-regression algorithm is essentially best possible, and therefore that the computational difficulty of agnostically learning a concept class is closely related to the polynomial degree required to approximate any function from the class in $L^1$-norm. Using this characterization along with additional analytic tools, we obtain optimal SQ lower bounds for agnostically learning linear threshold functions and the first non-trivial SQ lower bounds for polynomial threshold functions and intersections of halfspaces. We also develop an analogous theory for agnostically learning real-valued functions, and as an application prove near-optimal SQ lower bounds for agnostically learning ReLUs and sigmoids.

cs.LG↗

arXiv2021

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. Our main result is to provide the first computationally efficient robust learning algorithm for this problem with near-optimal error guarantees. Our algorithm can be seen as a special case of an algorithm for robustly learning a distribution from a general exponential family. To prove its correctness for Ising models, we establish new anti-concentration results for degree-$2$ polynomials of Ising models that may be of independent interest.

cs.LG↗