Reliable Learning of Halfspaces under Gaussian Marginals

We study the problem of PAC learning halfspaces in the reliable agnostic model of Kalai et al. (2012). The reliable PAC model captures learning scenarios where one type of error is costlier than the others. Our main positive result is a new algorithm for reliable learning of Gaussian halfspaces on $\mathbb{R}^d$ with sample and computational complexity $$d^{O(\log (\min\{1/\alpha, 1/\epsilon\}))}\min (2^{\log(1/\epsilon)^{O(\log (1/\alpha))}},2^{\mathrm{poly}(1/\epsilon)})\;,$$ where $\epsilon$ is the excess error and $\alpha$ is the bias of the optimal halfspace. We complement our upper bound with a Statistical Query lower bound suggesting that the $d^{\Omega(\log (1/\alpha))}$ dependence is best possible. Conceptually, our results imply a strong computational separation between reliable agnostic learning and standard agnostic learning of halfspaces in the Gaussian setting.

Sample and Computationally Efficient Robust Learning of Gaussian Single-Index Models

A single-index model (SIM) is a function of the form $\sigma(\mathbf{w}^{\ast} \cdot \mathbf{x})$, where $\sigma: \mathbb{R} \to \mathbb{R}$ is a known link function and $\mathbf{w}^{\ast}$ is a hidden unit vector. We study the task of learning SIMs in the agnostic (a.k.a. adversarial label noise) model with respect to the $L^2_2$-loss under the Gaussian distribution. Our main result is a sample and computationally efficient agnostic proper learner that attains $L^2_2$-error of $O(\mathrm{OPT})+\epsilon$, where $\mathrm{OPT}$ is the optimal loss. The sample complexity of our algorithm is $\tilde{O}(d^{\lceil k^{\ast}/2\rceil}+d/\epsilon)$, where $k^{\ast}$ is the information-exponent of $\sigma$ corresponding to the degree of its first non-zero Hermite coefficient. This sample bound nearly matches known CSQ lower bounds, even in the realizable setting. Prior algorithmic work in this setting had focused on learning in the realizable case or in the presence of semi-random noise. Prior computationally efficient robust learners required significantly stronger assumptions on the link function.

Efficient Testable Learning of General Halfspaces with Adversarial Label Noise

We study the task of testable learning of general -- not necessarily homogeneous -- halfspaces with adversarial label noise with respect to the Gaussian distribution. In the testable learning framework, the goal is to develop 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 main result is the first polynomial time tester-learner for general halfspaces that achieves dimension-independent misclassification error. At the heart of our approach is a new methodology to reduce testable learning of general halfspaces to testable learning of nearly homogeneous halfspaces that may be of broader interest.

Online Learning of Halfspaces with Massart Noise

We study the task of online learning in the presence of Massart noise. Instead of assuming that the online adversary chooses an arbitrary sequence of labels, we assume that the context $\mathbf{x}$ is selected adversarially but the label $y$ presented to the learner disagrees with the ground-truth label of $\mathbf{x}$ with unknown probability at most $\eta$. We study the fundamental class of $\gamma$-margin linear classifiers and present a computationally efficient algorithm that achieves mistake bound $\eta T + o(T)$. Our mistake bound is qualitatively tight for efficient algorithms: it is known that even in the offline setting achieving classification error better than $\eta$ requires super-polynomial time in the SQ model. We extend our online learning model to a $k$-arm contextual bandit setting where the rewards -- instead of satisfying commonly used realizability assumptions -- are consistent (in expectation) with some linear ranking function with weight vector $\mathbf{w}^\ast$. Given a list of contexts $\mathbf{x}_1,\ldots \mathbf{x}_k$, if $\mathbf{w}^*\cdot \mathbf{x}_i > \mathbf{w}^* \cdot \mathbf{x}_j$, the expected reward of action $i$ must be larger than that of $j$ by at least $\Delta$. We use our Massart online learner to design an efficient bandit algorithm that obtains expected reward at least $(1-1/k)~ \Delta T - o(T)$ bigger than choosing a random action at every round.

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. Our main algorithmic result is a polynomial-time PAC learning algorithm for this concept class in the strong contamination model under the Gaussian distribution with error guarantee $O_{d, c}(\text{opt}^{1-c})$, for any desired constant $c>0$, where $\text{opt}$ is the fraction of corruptions. In the strong contamination model, an omniscient adversary can arbitrarily corrupt an $\text{opt}$-fraction of the data points and their labels. This model generalizes the malicious noise model and the adversarial label noise model. Prior to our work, known polynomial-time algorithms in this corruption model (or even in the weaker adversarial label noise model) achieved error $\tilde{O}_d(\text{opt}^{1/(d+1)})$, which deteriorates significantly as a function of the degree $d$. Our algorithm employs an iterative approach inspired by localization techniques previously used in the context of learning linear threshold functions. Specifically, we use a robust perceptron algorithm to compute a good partial classifier and then iterate on the unclassified points. In order to achieve this, we need to take a set defined by a number of polynomial inequalities and partition it into several well-behaved subsets. To this end, we develop new polynomial decomposition techniques that may be of independent interest.

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. Specifically, for a fixed but unknown truncation set $S \subseteq \mathbb{R}^d$, we are given access to samples from the distribution $\mathcal{N}(\boldsymbol{ \mu}, \mathbf{ I})$ truncated to the set $S$. The goal is to estimate $\boldsymbol\mu$ within accuracy $\epsilon>0$ in $\ell_2$-norm. Our main result is a Statistical Query (SQ) lower bound suggesting a super-polynomial information-computation gap for this task. In more detail, we show that the complexity of any SQ algorithm for this problem is $d^{\mathrm{poly}(1/\epsilon)}$, even when the class $\mathcal{C}$ is simple so that $\mathrm{poly}(d/\epsilon)$ samples information-theoretically suffice. Concretely, our SQ lower bound applies when $\mathcal{C}$ is a union of a bounded number of rectangles whose VC dimension and Gaussian surface are small. As a corollary of our construction, it also follows that the complexity of the previously known algorithm for this task is qualitatively best possible.

Robustly Learning Single-Index Models via Alignment Sharpness

We study the problem of learning Single-Index Models under the $L_2^2$ loss in the agnostic model. We give an efficient learning algorithm, achieving a constant factor approximation to the optimal loss, that succeeds under a range of distributions (including log-concave distributions) and a broad class of monotone and Lipschitz link functions. This is the first efficient constant factor approximate agnostic learner, even for Gaussian data and for any nontrivial class of link functions. Prior work for the case of unknown link function either works in the realizable setting or does not attain constant factor approximation. The main technical ingredient enabling our algorithm and analysis is a novel notion of a local error bound in optimization that we term alignment sharpness and that may be of broader interest.

Agnostically Learning Multi-index Models with Queries

We study the power of query access for the task of agnostic learning under the Gaussian distribution. In the agnostic model, no assumptions are made on the labels and the goal is to compute a hypothesis that is competitive with the {\em best-fit} function in a known class, i.e., it achieves error $\mathrm{opt}+\epsilon$, where $\mathrm{opt}$ is the error of the best function in the class. We focus on a general family of Multi-Index Models (MIMs), which are $d$-variate functions that depend only on few relevant directions, i.e., have the form $g(\mathbf{W} \mathbf{x})$ for an unknown link function $g$ and a $k \times d$ matrix $\mathbf{W}$. Multi-index models cover a wide range of commonly studied function classes, including constant-depth neural networks with ReLU activations, and intersections of halfspaces. Our main result shows that query access gives significant runtime improvements over random examples for agnostically learning MIMs. Under standard regularity assumptions for the link function (namely, bounded variation or surface area), we give an agnostic query learner for MIMs with complexity $O(k)^{\mathrm{poly}(1/\epsilon)} \; \mathrm{poly}(d) $. 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. Our algorithmic result establishes a strong computational separation between the agnostic PAC and the agnostic PAC+Query models under the Gaussian distribution. Prior to our work, no such separation was known -- even for the special case of agnostically learning a single halfspace, for which it was an open problem first posed by Feldman. Our results are enabled by a general dimension-reduction technique that leverages query access to estimate gradients of (a smoothed version of) the underlying label function.

Self-Directed Linear Classification

In online classification, a learner is presented with a sequence of examples and aims to predict their labels in an online fashion so as to minimize the total number of mistakes. In the self-directed variant, the learner knows in advance the pool of examples and can adaptively choose the order in which predictions are made. Here we study the power of choosing the prediction order and establish the first strong separation between worst-order and random-order learning for the fundamental task of linear classification. Prior to our work, such a separation was known only for very restricted concept classes, e.g., one-dimensional thresholds or axis-aligned rectangles. We present two main results. If $X$ is a dataset of $n$ points drawn uniformly at random from the $d$-dimensional unit sphere, we design an efficient self-directed learner that makes $O(d \log \log(n))$ mistakes and classifies the entire dataset. If $X$ is an arbitrary $d$-dimensional dataset of size $n$, we design an efficient self-directed learner that predicts the labels of $99\%$ of the points in $X$ with mistake bound independent of $n$. In contrast, under a worst- or random-ordering, the number of mistakes must be at least $\Omega(d \log n)$, even when the points are drawn uniformly from the unit sphere and the learner only needs to predict the labels for $1\%$ of them.

Near-Optimal Bounds for Learning Gaussian Halfspaces with Random Classification Noise

We study the problem of learning general (i.e., not necessarily homogeneous) halfspaces with Random Classification Noise under the Gaussian distribution. We establish nearly-matching algorithmic and Statistical Query (SQ) lower bound results revealing a surprising information-computation gap for this basic problem. Specifically, the sample complexity of this learning problem is $\widetilde{\Theta}(d/\epsilon)$, where $d$ is the dimension and $\epsilon$ is the excess error. Our positive result is a computationally efficient learning algorithm with sample complexity $\tilde{O}(d/\epsilon + d/(\max\{p, \epsilon\})^2)$, where $p$ quantifies the bias of the target halfspace. On the lower bound side, we show that any efficient SQ algorithm (or low-degree test) for the problem requires sample complexity at least $\Omega(d^{1/2}/(\max\{p, \epsilon\})^2)$. Our lower bound suggests that this quadratic dependence on $1/\epsilon$ is inherent for efficient algorithms.