arXiv subjects

Nikos Zarifis

Publications and source records attributed to Nikos Zarifis.

29 records · Page 2Linked to original sources

arXiv2023

Efficient Testable Learning of Halfspaces with Adversarial Label Noise

We give the first polynomial-time algorithm for the testable learning of halfspaces in the presence of adversarial label noise under the Gaussian distribution. In the recently introduced testable learning model, one is required to produce a tester-learner such that if the data passes the tester, then one can trust the output of the robust learner on the data. Our tester-learner runs in time $\poly(d/\eps)$ and outputs a halfspace with misclassification error $O(\opt)+\eps$, where $\opt$ is the 0-1 error of the best fitting halfspace. At a technical level, our algorithm employs an iterative soft localization technique enhanced with appropriate testers to ensure that the data distribution is sufficiently similar to a Gaussian.

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arXiv2022

Learning a Single Neuron with Adversarial Label Noise via Gradient Descent

We study the fundamental problem of learning a single neuron, i.e., a function of the form $\mathbf{x}\mapstoσ(\mathbf{w}\cdot\mathbf{x})$ for monotone activations $σ:\mathbb{R}\mapsto\mathbb{R}$, with respect to the $L_2^2$-loss in the presence of adversarial label noise. Specifically, we are given labeled examples from a distribution $D$ on $(\mathbf{x}, y)\in\mathbb{R}^d \times \mathbb{R}$ such that there exists $\mathbf{w}^\ast\in\mathbb{R}^d$ achieving $F(\mathbf{w}^\ast)=ε$, where $F(\mathbf{w})=\mathbf{E}_{(\mathbf{x},y)\sim D}[(σ(\mathbf{w}\cdot \mathbf{x})-y)^2]$. The goal of the learner is to output a hypothesis vector $\mathbf{w}$ such that $F(\mathbb{w})=C\, ε$ with high probability, where $C>1$ is a universal constant. As our main contribution, we give efficient constant-factor approximate learners for a broad class of distributions (including log-concave distributions) and activation functions. Concretely, for the class of isotropic log-concave distributions, we obtain the following important corollaries: For the logistic activation, we obtain the first polynomial-time constant factor approximation (even under the Gaussian distribution). Our algorithm has sample complexity $\widetilde{O}(d/ε)$, which is tight within polylogarithmic factors. For the ReLU activation, we give an efficient algorithm with sample complexity $\tilde{O}(d\, \polylog(1/ε))$. Prior to our work, the best known constant-factor approximate learner had sample complexity $\tildeΩ(d/ε)$. In both of these settings, our algorithms are simple, performing gradient-descent on the (regularized) $L_2^2$-loss. The correctness of our algorithms relies on novel structural results that we establish, showing that (essentially all) stationary points of the underlying non-convex loss are approximately optimal.

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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.

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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.

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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.

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arXiv2020

A Polynomial Time Algorithm for Learning Halfspaces with Tsybakov Noise

We study the problem of PAC learning homogeneous halfspaces in the presence of Tsybakov noise. In the Tsybakov noise model, the label of every sample is independently flipped with an adversarially controlled probability that can be arbitrarily close to $1/2$ for a fraction of the samples. {\em We give the first polynomial-time algorithm for this fundamental learning problem.} Our algorithm learns the true halfspace within any desired accuracy $ε$ and succeeds under a broad family of well-behaved distributions including log-concave distributions. Prior to our work, the only previous algorithm for this problem required quasi-polynomial runtime in $1/ε$. Our algorithm employs a recently developed reduction \cite{DKTZ20b} from learning to certifying the non-optimality of a candidate halfspace. This prior work developed a quasi-polynomial time certificate algorithm based on polynomial regression. {\em The main technical contribution of the current paper is the first polynomial-time certificate algorithm.} Starting from a non-trivial warm-start, our algorithm performs a novel "win-win" iterative process which, at each step, either finds a valid certificate or improves the angle between the current halfspace and the true one. Our warm-start algorithm for isotropic log-concave distributions involves a number of analytic tools that may be of broader interest. These include a new efficient method for reweighting the distribution in order to recenter it and a novel characterization of the spectrum of the degree-$2$ Chow parameters.

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arXiv2020

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. In the former problem, given labeled examples $(\mathbf{x}, y)$ from an unknown distribution on $\mathbb{R}^d \times \{ \pm 1\}$, whose marginal distribution on $\mathbf{x}$ is the standard Gaussian and the labels $y$ can be arbitrary, the goal is to output a hypothesis with 0-1 loss $\mathrm{OPT}+ε$, where $\mathrm{OPT}$ is the 0-1 loss of the best-fitting halfspace. In the latter problem, given labeled examples $(\mathbf{x}, y)$ from an unknown distribution on $\mathbb{R}^d \times \mathbb{R}$, whose marginal distribution on $\mathbf{x}$ is the standard Gaussian and the labels $y$ can be arbitrary, the goal is to output a hypothesis with square loss $\mathrm{OPT}+ε$, where $\mathrm{OPT}$ is the square loss of the best-fitting ReLU. We prove Statistical Query (SQ) lower bounds of $d^{\mathrm{poly}(1/ε)}$ for both of these problems. Our SQ lower bounds provide strong evidence that current upper bounds for these tasks are essentially best possible.

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arXiv2020

Algorithms and SQ Lower Bounds for PAC Learning One-Hidden-Layer ReLU Networks

We study the problem of PAC learning one-hidden-layer ReLU networks with $k$ hidden units on $\mathbb{R}^d$ under Gaussian marginals in the presence of additive label noise. For the case of positive coefficients, we give the first polynomial-time algorithm for this learning problem for $k$ up to $\tilde{O}(\sqrt{\log d})$. Previously, no polynomial time algorithm was known, even for $k=3$. This answers an open question posed by~\cite{Kliv17}. Importantly, our algorithm does not require any assumptions about the rank of the weight matrix and its complexity is independent of its condition number. On the negative side, for the more general task of PAC learning one-hidden-layer ReLU networks with arbitrary real coefficients, we prove a Statistical Query lower bound of $d^{Ω(k)}$. Thus, we provide a separation between the two classes in terms of efficient learnability. Our upper and lower bounds are general, extending to broader families of activation functions.

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arXiv2020

Learning Halfspaces with Tsybakov Noise

We study the efficient PAC learnability of halfspaces in the presence of Tsybakov noise. In the Tsybakov noise model, each label is independently flipped with some probability which is controlled by an adversary. This noise model significantly generalizes the Massart noise model, by allowing the flipping probabilities to be arbitrarily close to $1/2$ for a fraction of the samples. Our main result is the first non-trivial PAC learning algorithm for this problem under a broad family of structured distributions -- satisfying certain concentration and (anti-)anti-concentration properties -- including log-concave distributions. Specifically, we given an algorithm that achieves misclassification error $ε$ with respect to the true halfspace, with quasi-polynomial runtime dependence in $1/\epsilin$. The only previous upper bound for this problem -- even for the special case of log-concave distributions -- was doubly exponential in $1/ε$ (and follows via the naive reduction to agnostic learning). Our approach relies on a novel computationally efficient procedure to certify whether a candidate solution is near-optimal, based on semi-definite programming. We use this certificate procedure as a black-box and turn it into an efficient learning algorithm by searching over the space of halfspaces via online convex optimization.

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arXiv2020

Non-Convex SGD Learns Halfspaces with Adversarial Label Noise

We study the problem of agnostically learning homogeneous halfspaces in the distribution-specific PAC model. For a broad family of structured distributions, including log-concave distributions, we show that non-convex SGD efficiently converges to a solution with misclassification error $O(\opt)+\eps$, where $\opt$ is the misclassification error of the best-fitting halfspace. In sharp contrast, we show that optimizing any convex surrogate inherently leads to misclassification error of $ω(\opt)$, even under Gaussian marginals.

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arXiv2020

Learning Halfspaces with Massart Noise Under Structured Distributions

We study the problem of learning halfspaces with Massart noise in the distribution-specific PAC model. We give the first computationally efficient algorithm for this problem with respect to a broad family of distributions, including log-concave distributions. This resolves an open question posed in a number of prior works. Our approach is extremely simple: We identify a smooth {\em non-convex} surrogate loss with the property that any approximate stationary point of this loss defines a halfspace that is close to the target halfspace. Given this structural result, we can use SGD to solve the underlying learning problem.

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