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 isallowed to make adaptive label queries to previously sampled points, we establish a strong information-theoretic lower bound ruling out non-trivialimprovements over the passive setting. Specifically, we show thatany 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.

It is widely believed that complex machine learning models generally encode features through linear representations, but these features exist in superposition, making them challenging to recover. We study the following fundamental setting for learning features in superposition from black-box query access: we are given query access to a function \[ f(x)=\sum_{i=1}^n a_i\,σ_i(v_i^\top x), \] where each unit vector $v_i$ encodes a feature direction and $σ_i:\mathbb{R} \rightarrow \mathbb{R}$ is an arbitrary response function and our goal is to recover the $v_i$ and the function $f$. In learning-theoretic terms, superposition refers to the overcomplete regime, when the number of features is larger than the underlying dimension (i.e. $n > d$), which has proven especially challenging for typical algorithmic approaches. Our main result is an efficient query algorithm that, from noisy oracle access to $f$, identifies all feature directions whose responses are non-degenerate and reconstructs the function $f$. Crucially, our algorithm works in a significantly more general setting than all related prior results -- we allow for essentially arbitrary superpositions, only requiring that $v_i, v_j$ are not nearly identical for $i \neq j$, and general response functions $σ_i$. At a high level, our algorithm introduces an approach for searching in Fourier space by iteratively refining the search space to locate the hidden directions $v_i$.

We initiate an investigation of learning tasks in a setting where the learner is given access to two competing provers, only one of which is honest. Specifically, we consider the power of such learners in assessing purported properties of opaque models. Following prior work that considers the power of competing provers in different settings, we call this setting refereed learning. After formulating a general definition of refereed learning tasks, we show refereed learning protocols that obtain a level of accuracy that far exceeds what is obtainable at comparable cost without provers, or even with a single prover. We concentrate on the task of choosing the better one out of two black-box models, with respect to some ground truth. While we consider a range of parameters, perhaps our most notable result is in the high-precision range: For all $\varepsilon>0$ and ambient dimension $d$, our learner makes only one query to the ground truth function, communicates only $(1+\frac{1}{\varepsilon^2})\cdot\text{poly}(d)$ bits with the provers, and outputs a model whose loss is within a multiplicative factor of $(1+\varepsilon)$ of the best model's loss. Obtaining comparable loss with a single prover would require the learner to access the ground truth at almost all of the points in the domain. To obtain this bound, we develop a technique that allows the learner to sample, using the provers, from a distribution that is not efficiently samplable to begin with. We find this technique to be of independent interest. We also present lower bounds that demonstrate the optimality of our protocols in a number of respects, including prover complexity, number of samples, and need for query access.

In the classic point location problem, one is given an arbitrary dataset $X \subset \mathbb{R}^d$ of $n$ points with query access to an unknown halfspace $f : \mathbb{R}^d \to \{0,1\}$, and the goal is to learn the label of every point in $X$. This problem is extremely well-studied and a nearly-optimal $\widetilde{O}(d \log n)$ query algorithm is known due to Hopkins-Kane-Lovett-Mahajan (FOCS 2020). However, their algorithm is granted the power to query arbitrary points outside of $X$ (point synthesis), and in fact without this power there is an $Ω(n)$ query lower bound due to Dasgupta (NeurIPS 2004). In this work our goal is to design efficient algorithms for learning halfspaces without point synthesis. To circumvent the $Ω(n)$ lower bound, we consider learning halfspaces whose normal vectors come from a set of size $D$, and show tight bounds of $Θ(D + \log n)$. As a corollary, we obtain an optimal $O(d + \log n)$ query deterministic learner for axis-aligned halfspaces, closing a previous gap of $O(d \log n)$ vs. $Ω(d + \log n)$. In fact, our algorithm solves the more general problem of learning a Boolean function $f$ over $n$ elements which is monotone under at least one of $D$ provided orderings. Our technical insight is to exploit the structure in these orderings to perform a binary search in parallel rather than considering each ordering sequentially, and we believe our approach may be of broader interest. Furthermore, we use our exact learning algorithm to obtain nearly optimal algorithms for PAC-learning. We show that $O(\min(D + \log(1/\varepsilon), 1/\varepsilon) \cdot \log D)$ queries suffice to learn $f$ within error $\varepsilon$, even in a setting when $f$ can be adversarially corrupted on a $c\varepsilon$-fraction of points, for a sufficiently small constant $c$. This bound is optimal up to a $\log D$ factor, including in the realizable setting.

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 isallowed to make adaptive label queries to previously sampled points, we establish a strong information-theoretic lower bound ruling out non-trivialimprovements over the passive setting. Specifically, we show thatany 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. 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.

Counterfactual explanations (CFEs) are sets of actions that an agent with a negative classification could take to achieve a (desired) positive classification, for consequential decisions such as loan applications, hiring, admissions, etc. In this work, we consider settings where optimal CFEs correspond to solutions of weighted set cover problems. In particular, there is a collection of actions that agents can perform that each have their own cost and each provide the agent with different sets of capabilities. The agent wants to perform the cheapest subset of actions that together provide all the needed capabilities to achieve a positive classification. Since this is an NP-hard optimization problem, we are interested in the question: can we, from training data (instances of agents and their optimal CFEs) learn a CFE generator that will quickly provide optimal sets of actions for new agents? In this work, we provide a deep-network learning procedure that we show experimentally is able to achieve strong performance at this task. We consider several problem formulations, including formulations in which the underlying "capabilities" and effects of actions are not explicitly provided, and so there is an informational challenge in addition to the computational challenge. Our problem can also be viewed as one of learning an optimal policy in a family of large but deterministic Markov Decision Processes (MDPs).

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.

We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang's breakthrough quantum-inspired algorithm for recommendation systems [STOC'19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gily\'en, Su, Low, and Wiebe [STOC'19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffice to generalize all recent results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: $\ell^2$-norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive.

Learning and equilibrium computation in games are fundamental problems across computer science and economics, with applications ranging from politics to machine learning. Much of the work in this area revolves around a simple algorithm termed \emph{randomized weighted majority} (RWM), also known as "Hedge" or "Multiplicative Weights Update," which is well known to achieve statistically optimal rates in adversarial settings (Littlestone and Warmuth '94, Freund and Schapire '99). Unfortunately, RWM comes with an inherent computational barrier: it requires maintaining and sampling from a distribution over all possible actions. In typical settings of interest the action space is exponentially large, seemingly rendering RWM useless in practice. In this work, we refute this notion for a broad variety of \emph{structured} games, showing it is possible to efficiently (approximately) sample the action space in RWM in \emph{polylogarithmic} time. This gives the first efficient no-regret algorithms for problems such as the \emph{(discrete) Colonel Blotto game}, \emph{matroid congestion}, \emph{matroid security}, and basic \emph{dueling games}. As an immediate corollary, we give a polylogarithmic time meta-algorithm to compute approximate Nash Equilibria for these games that is exponentially faster than prior methods in several important settings. Further, our algorithm is the first to efficiently compute equilibria for more involved variants of these games with general sums, more than two players, and, for Colonel Blotto, multiple resource types.