Learning intersections of halfspaces is a central problem in Computational Learning Theory. Even for just two halfspaces, it remains a major open question whether learning is possible in polynomial time with respect to the margin $\gamma$ of the data points and their dimensionality $d$. The best-known algorithms run in quasi-polynomial time $d^{O( \log{1/\gamma} )}$, and it has been shown that this complexity is unavoidable for any algorithm relying solely on correlational statistical queries (CSQ). In this work, we introduce a novel algorithm that provably circumvents the CSQ hardness barrier. Our approach applies to a broad class of distributions satisfying a natural, previously studied, factorizability assumption. Factorizable distributions lie between the distribution-specific and distribution-free settings, and significantly extend previously known tractable cases. For these distributions, we show that CSQ-based methods still require quasipolynomial time even for weak learning. Our main result is a learning algorithm for intersections of two margin halfspaces under factorizable distributions that achieves $\text{poly}(d,1/\gamma)$ time by leveraging more general statistical queries (SQ). As a corollary, we establish a strong separation between CSQ and SQ for this fundamental PAC learning problem. Our main result is grounded in a rigorous analysis utilizing a novel duality framework that characterizes the moment tensor structure induced by the marginal distributions. Building on these structural insights, our learning algorithm combines a refined variant of Jennrich’s Algorithm with PCA over random projections of the moment tensor, along with a gradient-descent-based non-convex optimization framework.

We study the task of Multiclass Linear Classification (MLC) in the distribution-free PAC model with Random Classification Noise (RCN). Specifically, the learner is given a set of labeled examples $(x, y)$, where $x$ is drawn from an unknown distribution on $R^d$ and the labels are generated by a multiclass linear classifier corrupted with RCN. That is, the label $y$ is flipped from $i$ to $j$ with probability $H_{ij}$ according to a known noise matrix $H$ with non-negative separation $\sigma: = \min_{i \neq j} H_{ii}-H_{ij}$. The goal is to compute a hypothesis with small 0-1 error. For the special case of two labels, prior work has given polynomial-time algorithms achieving the optimal error. Surprisingly, little is known about the complexity of this task even for three labels. As our main contribution, we show that the complexity of MLC with RCN becomes drastically different in the presence of three or more labels. Specifically, we prove super-polynomial Statistical Query (SQ) lower bounds for this problem. In more detail, even for three labels and constant separation, we give a super-polynomial lower bound on the complexity of any SQ algorithm achieving optimal error. For a larger number of labels and smaller separation, we show a super-polynomial SQ lower bound even for the weaker goal of achieving any constant factor approximation to the optimal loss or even beating the trivial hypothesis.

We study the task of agnostically learning general (as opposed to homogeneous) ReLUs under the Gaussian distribution with respect to the squared loss. In the passive learning setting, recent work gave a computationally efficient algorithm that uses $poly(d,1/\epsilon)$ labeled examples and outputs a hypothesis with error $O(opt)+\epsilon$, where $opt$ is the squared loss of the best fit ReLU. Here we focus on the interactive setting, where the learner has some form of query access to the labels of unlabeled examples. Our main result is the first computationally efficient learner that uses $d polylog(1/\epsilon)+\tilde{O}(\min\{1/p, 1/\epsilon\})$ black-box label queries, where $p$ is the bias of the target function, and achieves error $O(opt)+\epsilon$. We complement our algorithmic result by showing that its query complexity bound is qualitatively near-optimal, even ignoring computational constraints. Finally, we establish that query access is essentially necessary to improve on the label complexity of passive learning. Specifically, for pool-based active learning, any active learner requires $\tilde{\Omega}(d/\epsilon)$ labels, unless it draws a super-polynomial number of unlabeled examples.

We consider an active learning setting where a learner is presented with a pool $S$ of $n$ unlabeled examples belonging to a domain $\mathcal X$ and asks queries to find the underlying labeling that agrees with a target concept $h^\ast \in \mathcal H$. In contrast to traditional active learning that queries a single example for its label, we study more general \emph{region queries} that allow the learner to pick a subset of the domain $T \subset \mathcal X$ and a target label $y$ and ask a labeler whether $h^\ast(x) = y $ for every example in the set $T \cap S$. Such more powerful queries allow us to bypass the limitations of traditional active learning and use significantly fewer rounds of interactions to learn but can potentially lead to a significantly more complex query language. Our main contribution is quantifying the trade-off between the number of queries and the complexity of the query language used by the learner. We measure the complexity of the region queries via the VC dimension of the family of regions. We show that given any hypothesis class $\H$ with VC dimension $d$, one can design a region query family $Q$ with VC dimension $6d$ such that for every set of $n$ examples $S \subset \X$ and every $h^* \in \H$, a learner can submit $O(d\log n)$ queries from $Q$ to a labeler and perfectly label $S$. We show a matching lower bound by designing a hypothesis class $\H$ with VC dimension $d$ and a dataset $S \subset \X$ of size $n$ such that any learning algorithm using any query class with VC dimension $(d-2)/3$ must make $\poly(n)$ queries to label $S$ perfectly. Finally, we focus on well-studied hypothesis classes including unions of intervals, high-dimensional boxes, and $d$-dimensional halfspaces, and obtain stronger results. In particular, we design learning algorithms that (i) are computationally efficient and (ii) work even when the queries are not answered based on the learner’s pool of examples $S$ but on some unknown superset $L$ of $S$.

Co-training is a classical semi-supervised learning method which only requires a small number of labeled examples for learning, under reasonable assumptions. Despite extensive literature on the topic, very few hypothesis classes are known to be provably efficiently learnable via co-training, even under very strong distributional assumptions. In this work, we study the co-training problem in the stream-based active learning model. We show that a range of natural concept classes are efficiently learnable via co-training, in terms of both label efficiency and computational efficiency. We provide an efficient reduction of co-training under the standard assumption of weak dependence, in the stream-based active model, to online classification. As a corollary, we obtain efficient co-training algorithms with error independent label complexity for every concept class class efficiently learnable in the mistake bound online model. Our framework also gives co-training algorithms with label complexity $\tilde{O}(d\log (1/\epsilon))$ for any concept class with VC dimension $d$, though in general this reduction is not computationally efficient. Finally, using additional ideas from online learning, we design the first efficient co-training algorithms with label complexity $\tilde{O}(d^2\log (1/\epsilon))$ for several concept classes, including unions of intervals and homogeneous halfspaces.

We study the problem of learning general (i.e., not necessarily homogeneous) halfspaces under the Gaussian distribution on $\mathbb{R}^d$ in the presence of some form of query access. In the classical pool-based active learning model, where the algorithm is allowed to make adaptive label queries to previously sampled points, we establish a strong information-theoretic lower bound ruling out non-trivial improvements over the passive setting. Specifically, we show that any active learner requires label complexity of $\tilde{\Omega}(d/(\log(m)\epsilon))$, where $m$ is the number of unlabeled examples. Specifically, to beat the passive label complexity of $\tilde{O}(d/\epsilon)$, an active learner requires a pool of $2^{\mathrm{poly}(d)}$ unlabeled samples. On the positive side, we show that this lower bound can be circumvented with membership query access, even in the agnostic model. Specifically, we give a computationally efficient learner with query complexity of $\tilde{O}(\min(1/p, 1/\epsilon) + d\mathrm{polylog}(1/\epsilon))$ achieving error guarantee of $O(\mathrm{opt}+\epsilon)$. Here $p \in [0, 1/2]$ is the bias and $\mathrm{opt}$ is the 0-1 loss of the optimal halfspace. As a corollary, we obtain a strong separation between the active and membership query models. Taken together, our results characterize the complexity of learning general halfspaces under Gaussian marginals in these models.

We study pool-based active learning, where a learner has a large pool $S$ of unlabeled examples and can adaptively ask a labeler questions to learn these labels. The goal of the learner is to output a labeling for $S$ that can compete with the best hypothesis from a given hypothesis class $\mathcal{H}$. We focus on halfspace learning, one of the most important problems in active learning. It is well known that in the standard active learning model, learning the labels of an arbitrary pool of examples labeled by some halfspace up to error $\epsilon$ requires at least $\Omega(1/\epsilon)$ queries. To overcome this difficulty, previous work designs simple but powerful query languages to achieve $O(\log(1/\epsilon))$ query complexity, but only focuses on the realizable setting where data are perfectly labeled by some halfspace. However, when labels are noisy, such queries are too fragile and lead to high query complexity even under the simple random classification noise model. In this work, we propose a new query language called threshold statistical queries and study their power for learning under various noise models. Our main algorithmic result is the first query-efficient algorithm for learning halfspaces under the popular Massart noise model. With an arbitrary dataset corrupted with Massart noise at noise rate $\eta$, our algorithm uses only $\mathrm{polylog(1/\epsilon)}$ threshold statistical queries and computes an $(\eta + \epsilon)$-accurate labeling in polynomial time. For the harder case of agnostic noise, we show that it is impossible to beat $O(1/\epsilon)$ query complexity even for the much simpler problem of learning singleton functions (and thus for learning halfspaces) using a reduction from agnostic distributed learning.

Stochastic optimization is one of the central problems in Machine Learning and Theoretical Computer Science. In the standard model, the algorithm is given a fixed distribution known in advance. In practice though, one may acquire at a cost extra information to make better decisions. In this paper, we study how to buy information for stochastic optimization and formulate this question as an online learning problem. Assuming the learner has an oracle for the original optimization problem, we design a $2$-competitive deterministic algorithm and a $e/(e-1)$-competitive randomized algorithm for buying information. We show that this ratio is tight as the problem is equivalent to a robust generalization of the ski-rental problem, which we call super-martingale stopping. We also consider an adaptive setting where the learner can choose to buy information after taking some actions for the underlying optimization problem. We focus on the classic optimization problem, Min-Sum Set Cover, where the goal is to quickly find an action that covers a given request drawn from a known distribution. We provide an $8$-competitive algorithm running in polynomial time that chooses actions and decides when to buy information about the underlying request.

We study fixed-design online learning where the learner is allowed to choose the order of the datapoints in order to minimize their regret (aka self-directed online learning). We focus on the fundamental task of online linear regression: the learner is given a dataset $X$ with $n$ examples in $d$ dimensions and at step $t$ they select a point $x_t \in X$, predict a value $\widetilde y_t$, and suffer loss $(\widetilde y_t - w^\ast \cdot x_t)^2$. The goal is to design algorithms that order the examples and achieve better regret than random- or worst-order online algorithms. For an arbitrary dataset $X$, we show that, under the Exponential Time Hypothesis, no efficient algorithm can approximate the optimal (best-order) regret within a factor of $d^{1/\poly(\log \log d)}$. We then show that, for structured datasets, we can bypass the above hardness result and achieve nearly optimal regret. When the examples of $X$ are drawn i.i.d.\ from the uniform distribution on the sphere, we present an algorithm based on the greedy heuristic of selecting ``easiest'' examples first that achieves a $\log d$-approximation of the optimal regret.

The seminal paper by Mazumdar and Saha (2017a) introduced an extensive line of work on clustering with noisy queries. Yet, despite significant progress on the problem, the proposed methods depend crucially on knowing the exact probabilities of errors of the underlying fully-random oracle. In this work, we develop robust learning methods that tolerate general semi-random noise obtaining qualitatively the same guarantees as the best possible methods in the fully-random model. More specifically, given a set of n points with an unknown underlying partition, we are allowed to query pairs of points u,v to check if they are in the same cluster, but with probability p, the answer may be adversarially chosen. We show that information theoretically O(nk log n /(1-2p)^2) queries suffice to learn any cluster of sufficiently large size. Our main result is a computationally efficient algorithm that can identify large clusters with O(nk log n/ (1-2p)^2) + poly(log n, k, 1/(1-2p)) queries, matching the guarantees of the best known algorithms in the fully-random model. As a corollary of our approach, we develop the first parameter-free algorithm for the fully-random model, answering an open question in Mazumdar and Saha (2017a).