We prove the following strong hardness result for learning: Given a distribution of labeled examples from the hypercube such that there exists a monomial consistent with $(1-\eps)$ of the examples, it is NP-hard to find a halfspace that is correct on $(1/2+\eps)$ of the examples, for arbitrary constants $\eps > 0$. In learning theory terms, weak agnostic learning of monomials is hard, even if one is allowed to output a hypothesis from the much bigger concept class of halfspaces. This hardness result subsumes a long line of previous results, including two recent hardness results for the proper learning of monomials and halfspaces. As an immediate corollary of our result we show that weak agnostic learning of decision lists is NP-hard.
Our techniques are quite different from previous hardness proofs for learning. We define distributions on positive and negative examples for monomials whose first few moments match. We use the invariance principle to argue that regular halfspaces (all of whose coefficients have small absolute value relative to the total $\ell_2$ norm) cannot distinguish between distributions whose first few moments match. For highly non-regular subspaces, we use a structural lemma from recent work on fooling halfspaces to argue that they are ``junta-like'' and one can zero out all but the top few coefficients without affecting the performance of the halfspace. The top few coefficients form the natural list decoding of a halfspace in the context of dictatorship tests/Label Cover reductions.
We note that unlike previous invariance principle based proofs which are only known to give Unique-Games hardness, we are able to reduce from a version of Label Cover problem that is known to be NP-hard. This has inspired follow-up work on bypassing the Unique Games conjecture in some optimal geometric inapproximability results.
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 …
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
We analyze the properties of gradient descent on convex surrogates for the zero-one loss for the agnostic learning of linear halfspaces. If $\mathsf{OPT}$ is the best classification error achieved by a halfspace, by appealing to the notion of soft margins we are able to show that gradient descent finds halfspaces with classification error $\tilde O(\mathsf{OPT}^{1/2}) + \varepsilon$ in $\mathrm{poly}(d,1/\varepsilon)$ time and sample complexity for a broad class of distributions that includes log-concave isotropic distributions as a subclass. Along the way we answer a question recently posed b…
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Massachusetts Institute of Technology（麻省理工学院）；Google Research（谷歌研究院）；Carnegie Mellon University（卡内基梅隆大学）
We investigate approximation guarantees provided by logistic regression for the fundamental problem of agnostic learning of homogeneous halfspaces. Previously, for a certain broad class of "well-behaved" distributions on the examples, Diakonikolas et al. (2020) proved an $\tildeΩ(\textrm{OPT})$ lower bound, while Frei et al. (2021) proved an $\tilde{O}(\sqrt{\textrm{OPT}})$ upper bound, where $\textrm{OPT}$ denotes the best zero-one/misclassification risk of a homogeneous halfspace. In this paper, we close this gap by constructing a well-behaved distribution such that the global minimizer of t…
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.
We give the first efficient algorithm for learning halfspaces in the testable learning model recently defined by Rubinfeld and Vasilyan (2023). In this model, a learner certifies that the accuracy of its output hypothesis is near optimal whenever the training set passes an associated test, and training sets drawn from some target distribution -- e.g., the Gaussian -- must pass the test. This model is more challenging than distribution-specific agnostic or Massart noise models where the learner is allowed to fail arbitrarily if the distributional assumption does not hold. We consider the settin…
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 -- includ…
We study {\em online} active learning of homogeneous halfspaces in $\mathbb{R}^d$ with adversarial noise where the overall probability of a noisy label is constrained to be at most $ν$. Our main contribution is a Perceptron-like online active learning algorithm that runs in polynomial time, and under the conditions that the marginal distribution is isotropic log-concave and $ν= Ω(ε)$, where $ε\in (0, 1)$ is the target error rate, our algorithm PAC learns the underlying halfspace with near-optimal label complexity of $\tilde{O}\big(d \cdot polylog(\frac{1}ε)\big)$ and sample complexity of $\til…
Columbia University（哥伦比亚大学）；CMU（卡内基梅隆大学）；IBM Almaden（IBM阿尔马登研究中心）
Hardness results for maximum agreement problems have close connections to hardness results for proper learning in computational learning theory. In this paper we prove two hardness results for the problem of finding a low degree polynomial threshold function (PTF) which has the maximum possible agreement with a given set of labeled examples in $\R^n \times \{-1,1\}.$ We prove that for any constants $d\geq 1, \eps > 0$, {itemize} Assuming the Unique Games Conjecture, no polynomial-time algorithm can find a degree-$d$ PTF that is consistent with a $(\half + \eps)$ fraction of a given set of labe…
The increased availability of data in recent years has led several authors to ask whether it is possible to use data as a {\em computational} resource. That is, if more data is available, beyond the sample complexity limit, is it possible to use the extra examples to speed up the computation time required to perform the learning task? We give the first positive answer to this question for a {\em natural supervised learning problem} --- we consider agnostic PAC learning of halfspaces over $3$-sparse vectors in $\{-1,1,0\}^n$. This class is inefficiently learnable using $O\left(n/ε^2\right)$ exa…
University of Wisconsin, Madison（威斯康星大学麦迪逊分校）；University of California, San Diego（加利福尼亚大学圣迭戈分校）；Google Research, Mountain View（谷歌研究院（山景城））
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. We give a computationally efficient learning algorithm and a nearly matching computational hardness result for this problem. An interesting implication of our findings is that the $L_{\infty}$ perturbations case is provably computationally harder than the case $2 \leq p < \infty$.
University of Waterloo（滑铁卢大学）；National Institute of Informatics（日本国立信息学研究所）
We study distribution-free property testing and learning problems where the unknown probability distribution is a product distribution over $\mathbb{R}^d$. For many important classes of functions, such as intersections of halfspaces, polynomial threshold functions, convex sets, and $k$-alternating functions, the known algorithms either have complexity that depends on the support size of the distribution, or are proven to work only for specific examples of product distributions. We introduce a general method, which we call downsampling, that resolves these issues. Downsampling uses a notion of …
We study the task of agnostically learning halfspaces under the Gaussian distribution. Specifically, given labeled examples $(\mathbf{x},y)$ from an unknown distribution on $\mathbb{R}^n \times \{ \pm 1\}$, 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. We prove a near-optimal computational hardness result for this task, under the widely believed sub-exponential time hardness of the Learning with Errors…
Many popular learning algorithms (E.g. Regression, Fourier-Transform based algorithms, Kernel SVM and Kernel ridge regression) operate by reducing the problem to a convex optimization problem over a vector space of functions. These methods offer the currently best approach to several central problems such as learning half spaces and learning DNF's. In addition they are widely used in numerous application domains. Despite their importance, there are still very few proof techniques to show limits on the power of these algorithms. We study the performance of this approach in the problem of (agnos…
The basic problem in the PAC model of computational learning theory is to determine which hypothesis classes are efficiently learnable. There is presently a dearth of results showing hardness of learning problems. Moreover, the existing lower bounds fall short of the best known algorithms. The biggest challenge in proving complexity results is to establish hardness of {\em improper learning} (a.k.a. representation independent learning).The difficulty in proving lower bounds for improper learning is that the standard reductions from $\mathbf{NP}$-hard problems do not seem to apply in this conte…
University of Southern California（南加州大学）；University of California, San Diego（加州大学圣地亚哥分校）
We study the efficient learnability of geometric concept classes - specifically, low-degree polynomial threshold functions (PTFs) and intersections of halfspaces - when a fraction of the data is adversarially corrupted. 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. In the nasty noise model, an omniscient adversary can arbitrarily corrupt a small fraction of both the unlabeled data points and their labels. This model generalizes well-studied noise mod…
We show hardness of improperly learning halfspaces in the agnostic model, both in the distribution-independent as well as the distribution-specific setting, based on the assumption that worst-case lattice problems, such as GapSVP or SIVP, are hard. In particular, we show that under this assumption there is no efficient algorithm that outputs any binary hypothesis, not necessarily a halfspace, achieving misclassfication error better than $\frac 1 2 - γ$ even if the optimal misclassification error is as small is as small as $δ$. Here, $γ$ can be smaller than the inverse of any polynomial in the …
A remarkable recent paper by Rubinfeld and Vasilyan (2022) initiated the study of \emph{testable learning}, where the goal is to replace hard-to-verify distributional assumptions (such as Gaussianity) with efficiently testable ones and to require that the learner succeed whenever the unknown distribution passes the corresponding test. In this model, they gave an efficient algorithm for learning halfspaces under testable assumptions that are provably satisfied by Gaussians. In this paper we give a powerful new approach for developing algorithms for testable learning using tools from moment matc…
Polynomial approximations to boolean functions have led to many positive results in computer science. In particular, polynomial approximations to the sign function underly algorithms for agnostically learning halfspaces, as well as pseudorandom generators for halfspaces. In this work, we investigate the limits of these techniques by proving inapproximability results for the sign function. Firstly, the polynomial regression algorithm of Kalai et al. (SIAM J. Comput. 2008) shows that halfspaces can be learned with respect to log-concave distributions on $\mathbb{R}^n$ in the challenging agnostic…
IBM Research - Almaden（IBM阿尔马登研究中心）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
We consider the problem of approximating and learning disjunctions (or equivalently, conjunctions) on symmetric distributions over $\{0,1\}^n$. Symmetric distributions are distributions whose PDF is invariant under any permutation of the variables. We give a simple proof that for every symmetric distribution $\mathcal{D}$, there exists a set of $n^{O(\log{(1/ε)})}$ functions $\mathcal{S}$, such that for every disjunction $c$, there is function $p$, expressible as a linear combination of functions in $\mathcal{S}$, such that $p$ $ε$-approximates $c$ in $\ell_1$ distance on $\mathcal{D}$ or $\ma…
