Testing Closeness of Multivariate Distributions via Ramsey Theory

We investigate the statistical task of closeness (or equivalence) testing for multidimensional distributions. Specifically, given sample access to two unknown distributions $\mathbf p, \mathbf q$ on $\mathbb R^d$, we want to distinguish between the case that $\mathbf p=\mathbf q$ versus $\|\mathbf p-\mathbf q\|_{A_k} > \epsilon$, where $\|\mathbf p-\mathbf q\|_{A_k}$ denotes the generalized ${A}_k$ distance between $\mathbf p$ and $\mathbf q$ -- measuring the maximum discrepancy between the distributions over any collection of $k$ disjoint, axis-aligned rectangles. Our main result is the first closeness tester for this problem with {\em sub-learning} sample complexity in any fixed dimension and a nearly-matching sample complexity lower bound. In more detail, we provide a computationally efficient closeness tester with sample complexity $O\left((k^{6/7}/ \mathrm{poly}_d(\epsilon)) \log^d(k)\right)$. On the lower bound side, we establish a qualitatively matching sample complexity lower bound of $\Omega(k^{6/7}/\mathrm{poly}(\epsilon))$, even for $d=2$. These sample complexity bounds are surprising because the sample complexity of the problem in the univariate setting is $\Theta(k^{4/5}/\mathrm{poly}(\epsilon))$. This has the interesting consequence that the jump from one to two dimensions leads to a substantial increase in sample complexity, while increases beyond that do not. As a corollary of our general $A_k$ tester, we obtain $d_{\mathrm TV}$-closeness testers for pairs of $k$-histograms on $\mathbb R^d$ over a common unknown partition, and pairs of uniform distributions supported on the union of $k$ unknown disjoint axis-aligned rectangles. Both our algorithm and our lower bound make essential use of tools from Ramsey theory.

Online Robust Mean Estimation

We study the problem of high-dimensional robust mean estimation in an online setting. Specifically, we consider a scenario where $n$ sensors are measuring some common, ongoing phenomenon. At each time step $t=1,2,\ldots,T$, the $i^{th}$ sensor reports its readings $x^{(i)}_t$ for that time step. The algorithm must then commit to its estimate $\mu_t$ for the true mean value of the process at time $t$. We assume that most of the sensors observe independent samples from some common distribution $X$, but an $\epsilon$-fraction of them may instead behave maliciously. The algorithm wishes to compute a good approximation $\mu$ to the true mean $\mu^\ast := \mathbf{E}[X]$. We note that if the algorithm is allowed to wait until time $T$ to report its estimate, this reduces to the well-studied problem of robust mean estimation. However, the requirement that our algorithm produces partial estimates as the data is coming in substantially complicates the situation. We prove two main results about online robust mean estimation in this model. First, if the uncorrupted samples satisfy the standard condition of $(\epsilon,\delta)$-stability, we give an efficient online algorithm that outputs estimates $\mu_t$, $t \in [T],$ such that with high probability it holds that $\|\mu-\mu^\ast\|_2 = O(\delta \log(T))$, where $\mu = (\mu_t)_{t \in [T]}$. We note that this error bound is nearly competitive with the best offline algorithms, which would achieve $\ell_2$-error of $O(\delta)$. Our second main result shows that with additional assumptions on the input (most notably that $X$ is a product distribution) there are inefficient algorithms whose error does not depend on $T$ at all.

SQ Lower Bounds for Learning Mixtures of Linear Classifiers

We study the problem of learning mixtures of linear classifiers under Gaussian covariates. Given sample access to a mixture of $r$ distributions on $\mathbb{R}^n$ of the form $(\mathbf{x},y_{\ell})$, $\ell\in [r]$, where $\mathbf{x}\sim\mathcal{N}(0,\mathbf{I}_n)$ and $y_\ell=\mathrm{sign}(\langle\mathbf{v}_\ell,\mathbf{x}\rangle)$ for an unknown unit vector $\mathbf{v}_\ell$, the goal is to learn the underlying distribution in total variation distance. Our main result is a Statistical Query (SQ) lower bound suggesting that known algorithms for this problem are essentially best possible, even for the special case of uniform mixtures. In particular, we show that the complexity of any SQ algorithm for the problem is $n^{\mathrm{poly}(1/\Delta) \log(r)}$, where $\Delta$ is a lower bound on the pairwise $\ell_2$-separation between the $\mathbf{v}_\ell$'s. The key technical ingredient underlying our result is a new construction of spherical designs that may be of independent interest.

Distribution-Independent Regression for Generalized Linear Models with Oblivious Corruptions

We demonstrate the first algorithms for the problem of regression for generalized linear models (GLMs) in the presence of additive oblivious noise. We assume we have sample access to examples $(x, y)$ where $y$ is a noisy measurement of $g(w^* \cdot x)$. In particular, \new{the noisy labels are of the form} $y = g(w^* \cdot x) + \xi + \epsilon$, where $\xi$ is the oblivious noise drawn independently of $x$ \new{and satisfies} $\Pr[\xi = 0] \geq o(1)$, and $\epsilon \sim \mathcal N(0, \sigma^2)$. Our goal is to accurately recover a \new{parameter vector $w$ such that the} function $g(w \cdot x)$ \new{has} arbitrarily small error when compared to the true values $g(w^* \cdot x)$, rather than the noisy measurements $y$. We present an algorithm that tackles \new{this} problem in its most general distribution-independent setting, where the solution may not \new{even} be identifiable. \new{Our} algorithm returns \new{an accurate estimate of} the solution if it is identifiable, and otherwise returns a small list of candidates, one of which is close to the true solution. Furthermore, we \new{provide} a necessary and sufficient condition for identifiability, which holds in broad settings. \new{Specifically,} the problem is identifiable when the quantile at which $\xi + \epsilon = 0$ is known, or when the family of hypotheses does not contain candidates that are nearly equal to a translated $g(w^* \cdot x) + A$ for some real number $A$, while also having large error when compared to $g(w^* \cdot x)$. This is the first \new{algorithmic} result for GLM regression \new{with oblivious noise} which can handle more than half the samples being arbitrarily corrupted. Prior work focused largely on the setting of linear regression, and gave algorithms under restrictive assumptions.

GLM Regression with Oblivious Corruptions

We demonstrate the first algorithms for the problem of regression for generalized linear models (GLMs) in the presence of additive oblivious noise. We assume we have sample access to examples $(x, y)$ where $y$ is a noisy measurement of $g(w^* \cdot x)$. In particular, \new{the noisy labels are of the form} $y = g(w^* \cdot x) + \xi + \epsilon$, where $\xi$ is the oblivious noise drawn independently of $x$ \new{and satisfies} $\Pr[\xi = 0] \geq o(1)$, and $\epsilon \sim \mathcal N(0, \sigma^2)$. Our goal is to accurately recover a \new{parameter vector $w$ such that the} function $g(w \cdot x)$ \new{has} arbitrarily small error when compared to the true values $g(w^* \cdot x)$, rather than the noisy measurements $y$. We present an algorithm that tackles \new{this} problem in its most general distribution-independent setting, where the solution may not \new{even} be identifiable. \new{Our} algorithm returns \new{an accurate estimate of} the solution if it is identifiable, and otherwise returns a small list of candidates, one of which is close to the true solution. Furthermore, we \new{provide} a necessary and sufficient condition for identifiability, which holds in broad settings. \new{Specifically,} the problem is identifiable when the quantile at which $\xi + \epsilon = 0$ is known, or when the family of hypotheses does not contain candidates that are nearly equal to a translated $g(w^* \cdot x) + A$ for some real number $A$, while also having large error when compared to $g(w^* \cdot x)$. This is the first \new{algorithmic} result for GLM regression \new{with oblivious noise} which can handle more than half the samples being arbitrarily corrupted. Prior work focused largely on the setting of linear regression, and gave algorithms under restrictive assumptions.

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.

Efficiently Learning One-Hidden-Layer ReLU Networks via Schur Polynomials

We study the problem of PAC learning a linear combination of $k$ ReLU activations under the standard Gaussian distribution on $\mathbb{R}^d$ with respect to the square loss. Our main result is an efficient algorithm for this learning task with sample and computational complexity $(dk/\epsilon)^{O(k)}$, where $\epsilon>0$ is the target accuracy. Prior work had given an algorithm for this problem with complexity $(dk/\epsilon)^{h(k)}$, where the function $h(k)$ scales super-polynomially in $k$. Interestingly, the complexity of our algorithm is near-optimal within the class of Correlational Statistical Query algorithms. At a high-level, our algorithm uses tensor decomposition to identify a subspace such that all the $O(k)$-order moments are small in the orthogonal directions. Its analysis makes essential use of the theory of Schur polynomials to show that the higher-moment error tensors are small given that the lower-order ones are.

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.

Information-Computation Tradeoffs for Learning Margin Halfspaces with Random Classification Noise

We study the problem of PAC learning $\gamma$-margin halfspaces with Random Classification Noise. We establish an information-computation tradeoff suggesting an inherent gap between the sample complexity of the problem and the sample complexity of computationally efficient algorithms. Concretely, the sample complexity of the problem is $\widetilde{\Theta}(1/(\gamma^2 \epsilon))$. We start by giving a simple efficient algorithm with sample complexity $\widetilde{O}(1/(\gamma^2 \epsilon^2))$. Our main result is a lower bound for Statistical Query (SQ) algorithms and low-degree polynomial tests suggesting that the quadratic dependence on $1/\epsilon$ in the sample complexity is inherent for computationally efficient algorithms. Specifically, our results imply a lower bound of $\widetilde{\Omega}(1/(\gamma^{1/2} \epsilon^2))$ on the sample complexity of any efficient SQ learner or low-degree test.