Ilias Diakonikolas

Ilias Diakonikolas, Daniel M. Kane, Sushrut Karmalkar et al. · cmu

We study the problem of high-dimensional sparse mean estimation in the presence of an $ε$-fraction of adversarial outliers. Prior work obtained sample and computationally efficient algorithms for this task for identity-covariance subgaussian distributions. In this work, we develop the first efficient algorithms for robust sparse mean estimation without a priori knowledge of the covariance. For distributions on $\mathbb R^d$ with "certifiably bounded" $t$-th moments and sufficiently light tails, our algorithm achieves error of $O(ε^{1-1/t})$ with sample complexity $m = (k\log(d))^{O(t)}/ε^{2-2/t}$. For the special case of the Gaussian distribution, our algorithm achieves near-optimal error of $\tilde O(ε)$ with sample complexity $m = O(k^4 \mathrm{polylog}(d))/ε^2$. Our algorithms follow the Sum-of-Squares based, proofs to algorithms approach. We complement our upper bounds with Statistical Query and low-degree polynomial testing lower bounds, providing evidence that the sample-time-error tradeoffs achieved by our algorithms are qualitatively the best possible.

Ilias Diakonikolas, Daniel M. Kane, Ankit Pensia et al. · cmu

We study high-dimensional robust statistics tasks in the streaming model. A recent line of work obtained computationally efficient algorithms for a range of high-dimensional robust estimation tasks. Unfortunately, all previous algorithms require storing the entire dataset, incurring memory at least quadratic in the dimension. In this work, we develop the first efficient streaming algorithms for high-dimensional robust statistics with near-optimal memory requirements (up to logarithmic factors). Our main result is for the task of high-dimensional robust mean estimation in (a strengthening of) Huber's contamination model. We give an efficient single-pass streaming algorithm for this task with near-optimal error guarantees and space complexity nearly-linear in the dimension. As a corollary, we obtain streaming algorithms with near-optimal space complexity for several more complex tasks, including robust covariance estimation, robust regression, and more generally robust stochastic optimization.

Ilias Diakonikolas, Daniel M. Kane, Jasper C. H. Lee et al. · cmu

We study the fundamental task of outlier-robust mean estimation for heavy-tailed distributions in the presence of sparsity. Specifically, given a small number of corrupted samples from a high-dimensional heavy-tailed distribution whose mean $μ$ is guaranteed to be sparse, the goal is to efficiently compute a hypothesis that accurately approximates $μ$ with high probability. Prior work had obtained efficient algorithms for robust sparse mean estimation of light-tailed distributions. In this work, we give the first sample-efficient and polynomial-time robust sparse mean estimator for heavy-tailed distributions under mild moment assumptions. Our algorithm achieves the optimal asymptotic error using a number of samples scaling logarithmically with the ambient dimension. Importantly, the sample complexity of our method is optimal as a function of the failure probability $τ$, having an additive $\log(1/τ)$ dependence. Our algorithm leverages the stability-based approach from the algorithmic robust statistics literature, with crucial (and necessary) adaptations required in our setting. Our analysis may be of independent interest, involving the delicate design of a (non-spectral) decomposition for positive semi-definite matrices satisfying certain sparsity properties.

Ilias Diakonikolas, Daniel M. Kane, Sushrut Karmalkar et al. · cmu

We study the problem of list-decodable sparse mean estimation. Specifically, for a parameter $α\in (0, 1/2)$, we are given $m$ points in $\mathbb{R}^n$, $\lfloor αm \rfloor$ of which are i.i.d. samples from a distribution $D$ with unknown $k$-sparse mean $μ$. No assumptions are made on the remaining points, which form the majority of the dataset. The goal is to return a small list of candidates containing a vector $\widehat μ$ such that $\| \widehat μ- μ\|_2$ is small. Prior work had studied the problem of list-decodable mean estimation in the dense setting. In this work, we develop a novel, conceptually simpler technique for list-decodable mean estimation. As the main application of our approach, we provide the first sample and computationally efficient algorithm for list-decodable sparse mean estimation. In particular, for distributions with "certifiably bounded" $t$-th moments in $k$-sparse directions and sufficiently light tails, our algorithm achieves error of $(1/α)^{O(1/t)}$ with sample complexity $m = (k\log(n))^{O(t)}/α$ and running time $\mathrm{poly}(mn^t)$. For the special case of Gaussian inliers, our algorithm achieves the optimal error guarantee of $Θ(\sqrt{\log(1/α)})$ with quasi-polynomial sample and computational complexity. We complement our upper bounds with nearly-matching statistical query and low-degree polynomial testing lower bounds.

Ilias Diakonikolas, Daniel M. Kane, Ankit Pensia · cmu

We study the following fundamental hypothesis testing problem, which we term Gaussian mean testing. Given i.i.d. samples from a distribution $p$ on $\mathbb{R}^d$, the task is to distinguish, with high probability, between the following cases: (i) $p$ is the standard Gaussian distribution, $\mathcal{N}(0,I_d)$, and (ii) $p$ is a Gaussian $\mathcal{N}(μ,Σ)$ for some unknown covariance $Σ$ and mean $μ\in \mathbb{R}^d$ satisfying $\|μ\|_2 \geq ε$. Recent work gave an algorithm for this testing problem with the optimal sample complexity of $Θ(\sqrt{d}/ε^2)$. Both the previous algorithm and its analysis are quite complicated. Here we give an extremely simple algorithm for Gaussian mean testing with a one-page analysis. Our algorithm is sample optimal and runs in sample linear time.

Ilias Diakonikolas, Daniel M. Kane, Lisheng Ren

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 (LWE) problem. Prior hardness results are either qualitatively suboptimal or apply to restricted families of algorithms. Our techniques extend to yield near-optimal lower bounds for related problems, including ReLU regression.

Ilias Diakonikolas, Daniel M. Kane, Pasin Manurangsi et al.

We study the complexity of PAC learning halfspaces in the presence of Massart noise. In this problem, we are given i.i.d. labeled examples $(\mathbf{x}, y) \in \mathbb{R}^N \times \{ \pm 1\}$, where the distribution of $\mathbf{x}$ is arbitrary and the label $y$ is a Massart corruption of $f(\mathbf{x})$, for an unknown halfspace $f: \mathbb{R}^N \to \{ \pm 1\}$, with flipping probability $η(\mathbf{x}) \leq η< 1/2$. The goal of the learner is to compute a hypothesis with small 0-1 error. Our main result is the first computational hardness result for this learning problem. Specifically, assuming the (widely believed) subexponential-time hardness of the Learning with Errors (LWE) problem, we show that no polynomial-time Massart halfspace learner can achieve error better than $Ω(η)$, even if the optimal 0-1 error is small, namely $\mathrm{OPT} = 2^{-\log^{c} (N)}$ for any universal constant $c \in (0, 1)$. Prior work had provided qualitatively similar evidence of hardness in the Statistical Query model. Our computational hardness result essentially resolves the polynomial PAC learnability of Massart halfspaces, by showing that known efficient learning algorithms for the problem are nearly best possible.

Ilias Diakonikolas, Vasilis Kontonis, Christos Tzamos et al.

We study the fundamental problem of learning a single neuron, i.e., a function of the form $\mathbf{x}\mapstoσ(\mathbf{w}\cdot\mathbf{x})$ for monotone activations $σ:\mathbb{R}\mapsto\mathbb{R}$, with respect to the $L_2^2$-loss in the presence of adversarial label noise. Specifically, we are given labeled examples from a distribution $D$ on $(\mathbf{x}, y)\in\mathbb{R}^d \times \mathbb{R}$ such that there exists $\mathbf{w}^\ast\in\mathbb{R}^d$ achieving $F(\mathbf{w}^\ast)=ε$, where $F(\mathbf{w})=\mathbf{E}_{(\mathbf{x},y)\sim D}[(σ(\mathbf{w}\cdot \mathbf{x})-y)^2]$. The goal of the learner is to output a hypothesis vector $\mathbf{w}$ such that $F(\mathbb{w})=C\, ε$ with high probability, where $C>1$ is a universal constant. As our main contribution, we give efficient constant-factor approximate learners for a broad class of distributions (including log-concave distributions) and activation functions. Concretely, for the class of isotropic log-concave distributions, we obtain the following important corollaries: For the logistic activation, we obtain the first polynomial-time constant factor approximation (even under the Gaussian distribution). Our algorithm has sample complexity $\widetilde{O}(d/ε)$, which is tight within polylogarithmic factors. For the ReLU activation, we give an efficient algorithm with sample complexity $\tilde{O}(d\, \polylog(1/ε))$. Prior to our work, the best known constant-factor approximate learner had sample complexity $\tildeΩ(d/ε)$. In both of these settings, our algorithms are simple, performing gradient-descent on the (regularized) $L_2^2$-loss. The correctness of our algorithms relies on novel structural results that we establish, showing that (essentially all) stationary points of the underlying non-convex loss are approximately optimal.

Ilias Diakonikolas, Daniel M. Kane, Vasilis Kontonis et al.

We give the first polynomial-time algorithm for the testable learning of halfspaces in the presence of adversarial label noise under the Gaussian distribution. In the recently introduced testable learning model, one is required to produce 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 tester-learner runs in time $\poly(d/\eps)$ and outputs a halfspace with misclassification error $O(\opt)+\eps$, where $\opt$ is the 0-1 error of the best fitting halfspace. At a technical level, our algorithm employs an iterative soft localization technique enhanced with appropriate testers to ensure that the data distribution is sufficiently similar to a Gaussian.

Ilias Diakonikolas, Christos Tzamos, Daniel M. Kane

The Forster transform is a method of regularizing a dataset by placing it in {\em radial isotropic position} while maintaining some of its essential properties. Forster transforms have played a key role in a diverse range of settings spanning computer science and functional analysis. Prior work had given {\em weakly} polynomial time algorithms for computing Forster transforms, when they exist. Our main result is the first {\em strongly polynomial time} algorithm to compute an approximate Forster transform of a given dataset or certify that no such transformation exists. By leveraging our strongly polynomial Forster algorithm, we obtain the first strongly polynomial time algorithm for {\em distribution-free} PAC learning of halfspaces. This learning result is surprising because {\em proper} PAC learning of halfspaces is {\em equivalent} to linear programming. Our learning approach extends to give a strongly polynomial halfspace learner in the presence of random classification noise and, more generally, Massart noise.

Puqian Wang, Nikos Zarifis, Ilias Diakonikolas et al.

We study the problem of learning a single neuron with respect to the $L_2^2$-loss in the presence of adversarial label noise. We give an efficient algorithm that, for a broad family of activations including ReLUs, approximates the optimal $L_2^2$-error within a constant factor. Our algorithm applies under much milder distributional assumptions compared to prior work. The key ingredient enabling our results is a novel connection to local error bounds from optimization theory.

Ilias Diakonikolas, Daniel M. Kane, Lisheng Ren et al.

We study the problem of PAC learning a single neuron in the presence of Massart noise. Specifically, for a known activation function $f: \mathbb{R} \to \mathbb{R}$, the learner is given access to labeled examples $(\mathbf{x}, y) \in \mathbb{R}^d \times \mathbb{R}$, where the marginal distribution of $\mathbf{x}$ is arbitrary and the corresponding label $y$ is a Massart corruption of $f(\langle \mathbf{w}, \mathbf{x} \rangle)$. The goal of the learner is to output a hypothesis $h: \mathbb{R}^d \to \mathbb{R}$ with small squared loss. For a range of activation functions, including ReLUs, we establish super-polynomial Statistical Query (SQ) lower bounds for this learning problem. In more detail, we prove that no efficient SQ algorithm can approximate the optimal error within any constant factor. Our main technical contribution is a novel SQ-hard construction for learning $\{ \pm 1\}$-weight Massart halfspaces on the Boolean hypercube that is interesting on its own right.

Ilias Diakonikolas, Daniel M. Kane

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 $(dk/ε)^{O(k)}$, where $ε>0$ is the target accuracy. Prior work had given an algorithm for this problem with complexity $(dk/ε)^{h(k)}$, 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.

Ilias Diakonikolas, Jelena Diakonikolas, Daniel M. Kane et al.

We study the problem of PAC learning $γ$-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 $\widetildeΘ(1/(γ^2 ε))$. We start by giving a simple efficient algorithm with sample complexity $\widetilde{O}(1/(γ^2 ε^2))$. Our main result is a lower bound for Statistical Query (SQ) algorithms and low-degree polynomial tests suggesting that the quadratic dependence on $1/ε$ in the sample complexity is inherent for computationally efficient algorithms. Specifically, our results imply a lower bound of $\widetildeΩ(1/(γ^{1/2} ε^2))$ on the sample complexity of any efficient SQ learner or low-degree test.

Daniel M. Kane, Ilias Diakonikolas

We prove that for $c>0$ a sufficiently small universal constant that a random set of $c d^2/\log^4(d)$ independent Gaussian random points in $\mathbb{R}^d$ lie on a common ellipsoid with high probability. This nearly establishes a conjecture of~\cite{SaundersonCPW12}, within logarithmic factors. The latter conjecture has attracted significant attention over the past decade, due to its connections to machine learning and sum-of-squares lower bounds for certain statistical problems.

Ilias Diakonikolas, Daniel M. Kane, Sihan Liu

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 $\|\mathbf p-\mathbf q\|_{A_k} > ε$, 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 $O\left((k^{6/7}/ \mathrm{poly}_d(ε)) \log^d(k)\right)$. On the lower bound side, we establish a qualitatively matching sample complexity lower bound of $Ω(k^{6/7}/\mathrm{poly}(ε))$, even for $d=2$. These sample complexity bounds are surprising because the sample complexity of the problem in the univariate setting is $Θ(k^{4/5}/\mathrm{poly}(ε))$. 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.

Alvan Arulandu, Ilias Diakonikolas, Daniel Kane et al.

We study the complexity of two closely related learning problems, one quantum and one classical. In the quantum setting, we consider agnostic tomography for the natural class of product mixed states. Given $N$ copies of an $n$-qubit state $ρ$, the goal is to output a nearly optimal product mixed state approximation in trace distance. While recent work has focused on pure-state ansatz (e.g., product or stabilizer states), no polynomial-time guarantees were previously known for mixed-state ansatz. In the classical setting, we study robust learning of binary product distributions: given samples from an unknown distribution on ${0,1}^n$, the goal is to output a nearly optimal product approximation. Our main contributions are as follows. (1) We give a semi-agnostic tomography algorithm for product mixed states with polynomial sample and computational complexity achieving error $O(\mathrm{opt}\log(1/\mathrm{opt}))$, where $\mathrm{opt}$ is the trace distance to the best product approximation. This is the first efficient algorithm with any nontrivial agnostic guarantee for mixed-state ansatz, using only single-qubit, single-copy measurements. We also prove a Quantum Statistical Query lower bound showing near-optimality, and an unconditional lower bound demonstrating that adaptivity is necessary under single-qubit measurements. (2) We give a semi-agnostic algorithm for robustly learning binary product distributions with matching guarantees and establish a Statistical Query lower bound, essentially resolving the efficient robust learnability of this class and improving on prior work since Diakonikolas et al. (2016).

Daniel M. Kane, Ilias Diakonikolas, Hanshen Xiao et al.

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 $μ_t$ 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 $ε$-fraction of them may instead behave maliciously. The algorithm wishes to compute a good approximation $μ$ to the true mean $μ^\ast := \mathbf{E}[X]$. 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 $(ε,δ)$-stability, we give an efficient online algorithm that outputs estimates $μ_t$, $t \in [T],$ such that with high probability it holds that $\|μ-μ^\ast\|_2 = O(δ\log(T))$, where $μ= (μ_t)_{t \in [T]}$. We note that this error bound is nearly competitive with the best offline algorithms, which would achieve $\ell_2$-error of $O(δ)$. 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.

Ilias Diakonikolas, Daniel M. Kane, Yuxin Sun

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 $n^{\mathrm{poly}(1/Δ) \log(r)}$, where $Δ$ 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.

Ilias Diakonikolas, Vasilis Kontonis, Christos Tzamos et al.

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 $Ω(d \log n)$, even when the points are drawn uniformly from the unit sphere and the learner only needs to predict the labels for $1\%$ of them.

Ilias Diakonikolas, Jelena Diakonikolas, Daniel M. Kane et al.

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 $\widetildeΘ(d/ε)$, where $d$ is the dimension and $ε$ is the excess error. Our positive result is a computationally efficient learning algorithm with sample complexity $\tilde{O}(d/ε+ d/(\max\{p, ε\})^2)$, 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 $Ω(d^{1/2}/(\max\{p, ε\})^2)$. Our lower bound suggests that this quadratic dependence on $1/ε$ is inherent for efficient algorithms.

Clément L. Canonne, Ilias Diakonikolas, Daniel M. Kane et al.

We investigate the problem of testing whether a discrete probability distribution over an ordered domain is a histogram on a specified number of bins. One of the most common tools for the succinct approximation of data, $k$-histograms over $[n]$, are probability distributions that are piecewise constant over a set of $k$ intervals. The histogram testing problem is the following: Given samples from an unknown distribution $\mathbf{p}$ on $[n]$, we want to distinguish between the cases that $\mathbf{p}$ is a $k$-histogram versus $\varepsilon$-far from any $k$-histogram, in total variation distance. Our main result is a sample near-optimal and computationally efficient algorithm for this testing problem, and a nearly-matching (within logarithmic factors) sample complexity lower bound. Specifically, we show that the histogram testing problem has sample complexity $\widetilde Θ(\sqrt{nk} / \varepsilon + k / \varepsilon^2 + \sqrt{n} / \varepsilon^2)$.

Ilias Diakonikolas, Daniel M. Kane, Yuxin Sun

We establish optimal Statistical Query (SQ) lower bounds for robustly learning certain families of discrete high-dimensional distributions. In particular, we show that no efficient SQ algorithm with access to an $ε$-corrupted binary product distribution can learn its mean within $\ell_2$-error $o(ε\sqrt{\log(1/ε)})$. Similarly, we show that no efficient SQ algorithm with access to an $ε$-corrupted ferromagnetic high-temperature Ising model can learn the model to total variation distance $o(ε\log(1/ε))$. Our SQ lower bounds match the error guarantees of known algorithms for these problems, providing evidence that current upper bounds for these tasks are best possible. At the technical level, we develop a generic SQ lower bound for discrete high-dimensional distributions starting from low dimensional moment matching constructions that we believe will find other applications. Additionally, we introduce new ideas to analyze these moment-matching constructions for discrete univariate distributions.

Ilias Diakonikolas, Sushrut Karmalkar, Jongho Park et al.

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} $y = g(w^* \cdot x) + ξ+ ε$, where $ξ$ is the oblivious noise drawn independently of $x$ \new{and satisfies} $\Pr[ξ= 0] \geq o(1)$, and $ε\sim \mathcal N(0, σ^2)$. 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 $ξ+ ε= 0$ 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.

Ilias Diakonikolas, Daniel M. Kane, Thanasis Pittas et al.

We study the complexity of learning mixtures of separated Gaussians with common unknown bounded covariance matrix. Specifically, we focus on learning Gaussian mixture models (GMMs) on $\mathbb{R}^d$ of the form $P= \sum_{i=1}^k w_i \mathcal{N}(\boldsymbol μ_i,\mathbf Σ_i)$, where $\mathbf Σ_i = \mathbf Σ\preceq \mathbf I$ and $\min_{i \neq j} \| \boldsymbol μ_i - \boldsymbol μ_j\|_2 \geq k^ε$ for some $ε>0$. Known learning algorithms for this family of GMMs have complexity $(dk)^{O(1/ε)}$. In this work, we prove that any Statistical Query (SQ) algorithm for this problem requires complexity at least $d^{Ω(1/ε)}$. In the special case where the separation is on the order of $k^{1/2}$, we additionally obtain fine-grained SQ lower bounds with the correct exponent. Our SQ lower bounds imply similar lower bounds for low-degree polynomial tests. Conceptually, our results provide evidence that known algorithms for this problem are nearly best possible.

Ilias Diakonikolas, Mingchen Ma, Lisheng Ren et al.

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 $γ$ of the data points and their dimensionality $d$. The best-known algorithms run in quasi-polynomial time $d^{O(\log(1/γ))}$, 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 distribution-specific and distribution-free settings, and significantly extend previously known tractable cases. Under these distributions, we show that CSQ-based methods still require quasipolynomial time even for weakly learning, whereas our algorithm achieves $poly(d,1/γ)$ time by leveraging more general statistical queries (SQ), establishing a strong separation between CSQ and SQ for this simple realizable PAC learning problem. Our 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, we propose new, efficient learning algorithms. These algorithms combine 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.

Xuefeng Du, Zhen Fang, Ilias Diakonikolas et al.

Using unlabeled data to regularize the machine learning models has demonstrated promise for improving safety and reliability in detecting out-of-distribution (OOD) data. Harnessing the power of unlabeled in-the-wild data is non-trivial due to the heterogeneity of both in-distribution (ID) and OOD data. This lack of a clean set of OOD samples poses significant challenges in learning an optimal OOD classifier. Currently, there is a lack of research on formally understanding how unlabeled data helps OOD detection. This paper bridges the gap by introducing a new learning framework SAL (Separate And Learn) that offers both strong theoretical guarantees and empirical effectiveness. The framework separates candidate outliers from the unlabeled data and then trains an OOD classifier using the candidate outliers and the labeled ID data. Theoretically, we provide rigorous error bounds from the lens of separability and learnability, formally justifying the two components in our algorithm. Our theory shows that SAL can separate the candidate outliers with small error rates, which leads to a generalization guarantee for the learned OOD classifier. Empirically, SAL achieves state-of-the-art performance on common benchmarks, reinforcing our theoretical insights. Code is publicly available at https://github.com/deeplearning-wisc/sal.

Ilias Diakonikolas, Daniel M. Kane, Sihan Liu et al.

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.

Ilias Diakonikolas, Jerry Li, Anastasia Voloshinov

We study the fundamental problem of fixed design {\em multidimensional segmented regression}: Given noisy samples from a function $f$, promised to be piecewise linear on an unknown set of $k$ rectangles, we want to recover $f$ up to a desired accuracy in mean-squared error. We provide the first sample and computationally efficient algorithm for this problem in any fixed dimension. Our algorithm relies on a simple iterative merging approach, which is novel in the multidimensional setting. Our experimental evaluation on both synthetic and real datasets shows that our algorithm is competitive and in some cases outperforms state-of-the-art heuristics. Code of our implementation is available at \url{https://github.com/avoloshinov/multidimensional-segmented-regression}.

Ilias Diakonikolas, Daniel Kane, Lisheng Ren et al.

We study the complexity of Non-Gaussian Component Analysis (NGCA) in the Statistical Query (SQ) model. Prior work developed a general methodology to prove SQ lower bounds for this task that have been applicable to a wide range of contexts. In particular, it was known that for any univariate distribution $A$ satisfying certain conditions, distinguishing between a standard multivariate Gaussian and a distribution that behaves like $A$ in a random hidden direction and like a standard Gaussian in the orthogonal complement, is SQ-hard. The required conditions were that (1) $A$ matches many low-order moments with the standard univariate Gaussian, and (2) the chi-squared norm of $A$ with respect to the standard Gaussian is finite. While the moment-matching condition is necessary for hardness, the chi-squared condition was only required for technical reasons. In this work, we establish that the latter condition is indeed not necessary. In particular, we prove near-optimal SQ lower bounds for NGCA under the moment-matching condition only. Our result naturally generalizes to the setting of a hidden subspace. Leveraging our general SQ lower bound, we obtain near-optimal SQ lower bounds for a range of concrete estimation tasks where existing techniques provide sub-optimal or even vacuous guarantees.

Nikos Zarifis, Puqian Wang, Ilias Diakonikolas et al.

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.

Ilias Diakonikolas, Samuel B. Hopkins, Ankit Pensia et al.

We prove that there is a universal constant $C>0$ so that for every $d \in \mathbb N$, every centered subgaussian distribution $\mathcal D$ on $\mathbb R^d$, and every even $p \in \mathbb N$, the $d$-variate polynomial $(Cp)^{p/2} \cdot \|v\|_{2}^p - \mathbb E_{X \sim \mathcal D} \langle v,X\rangle^p$ is a sum of square polynomials. This establishes that every subgaussian distribution is \emph{SoS-certifiably subgaussian} -- a condition that yields efficient learning algorithms for a wide variety of high-dimensional statistical tasks. As a direct corollary, we obtain computationally efficient algorithms with near-optimal guarantees for the following tasks, when given samples from an arbitrary subgaussian distribution: robust mean estimation, list-decodable mean estimation, clustering mean-separated mixture models, robust covariance-aware mean estimation, robust covariance estimation, and robust linear regression. Our proof makes essential use of Talagrand's generic chaining/majorizing measures theorem.

Ilias Diakonikolas, Daniel M. Kane, Vasilis Kontonis et al.

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 $\mathrm{opt}+ε$, 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/ε)} \; \mathrm{poly}(d) $. In contrast, algorithms that rely only on random examples inherently require $d^{\mathrm{poly}(1/ε)}$ 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.

Ilias Diakonikolas, Daniel M. Kane

We study the task of learning latent-variable models. A common algorithmic technique for this task is the method of moments. Unfortunately, moment-based approaches are hampered by the fact that the moment tensors of super-constant degree cannot even be written down in polynomial time. Motivated by such learning applications, we develop a general efficient algorithm for {\em implicit moment tensor computation}. Our framework generalizes the work of~\cite{LL21-opt} which developed an efficient algorithm for the specific moment tensors that arise in clustering mixtures of spherical Gaussians. By leveraging our implicit moment estimation algorithm, we obtain the first $\mathrm{poly}(d, k)$-time learning algorithms for the following models. * {\bf Mixtures of Linear Regressions} We give a $\mathrm{poly}(d, k, 1/ε)$-time algorithm for this task, where $ε$ is the desired error. * {\bf Mixtures of Spherical Gaussians} For density estimation, we give a $\mathrm{poly}(d, k, 1/ε)$-time learning algorithm, where $ε$ is the desired total variation error, under the condition that the means lie in a ball of radius $O(\sqrt{\log k})$. For parameter estimation, we give a $\mathrm{poly}(d, k, 1/ε)$-time algorithm under the {\em optimal} mean separation of $Ω(\log^{1/2}(k/ε))$. * {\bf Positive Linear Combinations of Non-Linear Activations} We give a general algorithm for this task with complexity $\mathrm{poly}(d, k) g(ε)$, where $ε$ is the desired error and the function $g$ depends on the Hermite concentration of the target class of functions. Specifically, for positive linear combinations of ReLU activations, our algorithm has complexity $\mathrm{poly}(d, k) 2^{\mathrm{poly}(1/ε)}$.

Ilias Diakonikolas, Daniel M. Kane, Ankit Pensia et al. · cmu

We study the fundamental problems of Gaussian mean estimation and linear regression with Gaussian covariates in the presence of Huber contamination. Our main contribution is the design of the first sample near-optimal and almost linear-time algorithms with optimal error guarantees for both of these problems. Specifically, for Gaussian robust mean estimation on $\mathbb{R}^d$ with contamination parameter $ε\in (0, ε_0)$ for a small absolute constant $ε_0$, we give an algorithm with sample complexity $n = \tilde{O}(d/ε^2)$ and almost linear runtime that approximates the target mean within $\ell_2$-error $O(ε)$. This improves on prior work that achieved this error guarantee with polynomially suboptimal sample and time complexity. For robust linear regression, we give the first algorithm with sample complexity $n = \tilde{O}(d/ε^2)$ and almost linear runtime that approximates the target regressor within $\ell_2$-error $O(ε)$. This is the first polynomial sample and time algorithm achieving the optimal error guarantee, answering an open question in the literature. At the technical level, we develop a methodology that yields almost-linear time algorithms for multi-directional filtering that may be of broader interest.

Ilias Diakonikolas, Sushrut Karmalkar, Shuo Pang et al.

Non-Gaussian Component Analysis (NGCA) is the statistical task of finding a non-Gaussian direction in a high-dimensional dataset. Specifically, given i.i.d.\ samples from a distribution $P^A_{v}$ on $\mathbb{R}^n$ that behaves like a known distribution $A$ in a hidden direction $v$ and like a standard Gaussian in the orthogonal complement, the goal is to approximate the hidden direction. The standard formulation posits that the first $k-1$ moments of $A$ match those of the standard Gaussian and the $k$-th moment differs. Under mild assumptions, this problem has sample complexity $O(n)$. On the other hand, all known efficient algorithms require $Ω(n^{k/2})$ samples. Prior work developed sharp Statistical Query and low-degree testing lower bounds suggesting an information-computation tradeoff for this problem. Here we study the complexity of NGCA in the Sum-of-Squares (SoS) framework. Our main contribution is the first super-constant degree SoS lower bound for NGCA. Specifically, we show that if the non-Gaussian distribution $A$ matches the first $(k-1)$ moments of $\mathcal{N}(0, 1)$ and satisfies other mild conditions, then with fewer than $n^{(1 - \varepsilon)k/2}$ many samples from the normal distribution, with high probability, degree $(\log n)^{{1\over 2}-o_n(1)}$ SoS fails to refute the existence of such a direction $v$. Our result significantly strengthens prior work by establishing a super-polynomial information-computation tradeoff against a broader family of algorithms. As corollaries, we obtain SoS lower bounds for several problems in robust statistics and the learning of mixture models. Our SoS lower bound proof introduces a novel technique, that we believe may be of broader interest, and a number of refinements over existing methods.

Ilias Diakonikolas, Nikos Zarifis

We study the problem of PAC learning $γ$-margin halfspaces in the presence of Massart noise. Without computational considerations, the sample complexity of this learning problem is known to be $\widetildeΘ(1/(γ^2 ε))$. Prior computationally efficient algorithms for the problem incur sample complexity $\tilde{O}(1/(γ^4 ε^3))$ and achieve 0-1 error of $η+ε$, where $η<1/2$ is the upper bound on the noise rate. Recent work gave evidence of an information-computation tradeoff, suggesting that a quadratic dependence on $1/ε$ is required for computationally efficient algorithms. Our main result is a computationally efficient learner with sample complexity $\widetildeΘ(1/(γ^2 ε^2))$, nearly matching this lower bound. In addition, our algorithm is simple and practical, relying on online SGD on a carefully selected sequence of convex losses.

Ilias Diakonikolas, Giannis Iakovidis, Daniel M. Kane et al.

We study the task of learning Multi-Index Models (MIMs) with label noise under the Gaussian distribution. A $K$-MIM is any function $f$ that only depends on a $K$-dimensional subspace. We focus on well-behaved MIMs with finite ranges that satisfy certain regularity properties. Our main contribution is a general robust learner that is qualitatively optimal in the Statistical Query (SQ) model. Our algorithm iteratively constructs better approximations to the defining subspace by computing low-degree moments conditional on the projection to the subspace computed thus far, and adding directions with relatively large empirical moments. This procedure efficiently finds a subspace $V$ so that $f(\mathbf{x})$ is close to a function of the projection of $\mathbf{x}$ onto $V$. Conversely, for functions for which these conditional moments do not help, we prove an SQ lower bound suggesting that no efficient learner exists. As applications, we provide faster robust learners for the following concept classes: * {\bf Multiclass Linear Classifiers} We give a constant-factor approximate agnostic learner with sample complexity $N = O(d) 2^{\mathrm{poly}(K/ε)}$ and computational complexity $\mathrm{poly}(N ,d)$. This is the first constant-factor agnostic learner for this class whose complexity is a fixed-degree polynomial in $d$. * {\bf Intersections of Halfspaces} We give an approximate agnostic learner for this class achieving 0-1 error $K \tilde{O}(\mathrm{OPT}) + ε$ with sample complexity $N=O(d^2) 2^{\mathrm{poly}(K/ε)}$ and computational complexity $\mathrm{poly}(N ,d)$. This is the first agnostic learner for this class with near-linear error dependence and complexity a fixed-degree polynomial in $d$. Furthermore, we show that in the presence of random classification noise, the complexity of our algorithm scales polynomially with $1/ε$.

Ilias Diakonikolas, Samuel B. Hopkins, Ankit Pensia et al.

We study $\textit{sparse singular value certificates}$ for random rectangular matrices. If $M$ is an $n \times d$ matrix with independent Gaussian entries, we give a new family of polynomial-time algorithms which can certify upper bounds on the maximum of $\|M u\|$, where $u$ is a unit vector with at most $ηn$ nonzero entries for a given $η\in (0,1)$. This basic algorithmic primitive lies at the heart of a wide range of problems across algorithmic statistics and theoretical computer science. Our algorithms certify a bound which is asymptotically smaller than the naive one, given by the maximum singular value of $M$, for nearly the widest-possible range of $n,d,$ and $η$. Efficiently certifying such a bound for a range of $n,d$ and $η$ which is larger by any polynomial factor than what is achieved by our algorithm would violate lower bounds in the SQ and low-degree polynomials models. Our certification algorithm makes essential use of the Sum-of-Squares hierarchy. To prove the correctness of our algorithm, we develop a new combinatorial connection between the graph matrix approach to analyze random matrices with dependent entries, and the Efron-Stein decomposition of functions of independent random variables. As applications of our certification algorithm, we obtain new efficient algorithms for a wide range of well-studied algorithmic tasks. In algorithmic robust statistics, we obtain new algorithms for robust mean and covariance estimation with tradeoffs between breakdown point and sample complexity, which are nearly matched by SQ and low-degree polynomial lower bounds (that we establish). We also obtain new polynomial-time guarantees for certification of $\ell_1/\ell_2$ distortion of random subspaces of $\mathbb{R}^n$ (also with nearly matching lower bounds), sparse principal component analysis, and certification of the $2\rightarrow p$ norm of a random matrix.

Ilias Diakonikolas, Daniel M. Kane, Thanasis Pittas et al.

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 $\mathcal{N}(\boldsymbol{ μ}, \mathbf{ I})$ truncated to the set $S$. The goal is to estimate $\boldsymbolμ$ within accuracy $ε>0$ 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 $d^{\mathrm{poly}(1/ε)}$, even when the class $\mathcal{C}$ is simple so that $\mathrm{poly}(d/ε)$ 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.

Ilias Diakonikolas, Giannis Iakovidis, Daniel M. Kane et al.

We study the complexity of learning real-valued Multi-Index Models (MIMs) under the Gaussian distribution. A $K$-MIM is a function $f:\mathbb{R}^d\to \mathbb{R}$ that depends only on the projection of its input onto a $K$-dimensional subspace. We give a general algorithm for PAC learning a broad class of MIMs with respect to the square loss, even in the presence of adversarial label noise. Moreover, we establish a nearly matching Statistical Query (SQ) lower bound, providing evidence that the complexity of our algorithm is qualitatively optimal as a function of the dimension. Specifically, we consider the class of bounded variation MIMs with the property that degree at most $m$ distinguishing moments exist with respect to projections onto any subspace. In the presence of adversarial label noise, the complexity of our learning algorithm is $d^{O(m)}2^{\mathrm{poly}(K/ε)}$. For the realizable and independent noise settings, our algorithm incurs complexity $d^{O(m)}2^{\mathrm{poly}(K)}(1/ε)^{O(K)}$. To complement our upper bound, we show that if for some subspace degree-$m$ distinguishing moments do not exist, then any SQ learner for the corresponding class of MIMs requires complexity $d^{Ω(m)}$. As an application, we give the first efficient learner for the class of positive-homogeneous $L$-Lipschitz $K$-MIMs. The resulting algorithm has complexity $\mathrm{poly}(d) 2^{\mathrm{poly}(KL/ε)}$. This gives a new PAC learning algorithm for Lipschitz homogeneous ReLU networks with complexity independent of the network size, removing the exponential dependence incurred in prior work.

Shuyao Li, Yu Cheng, Ilias Diakonikolas et al.

Finding an approximate second-order stationary point (SOSP) is a well-studied and fundamental problem in stochastic nonconvex optimization with many applications in machine learning. However, this problem is poorly understood in the presence of outliers, limiting the use of existing nonconvex algorithms in adversarial settings. In this paper, we study the problem of finding SOSPs in the strong contamination model, where a constant fraction of datapoints are arbitrarily corrupted. We introduce a general framework for efficiently finding an approximate SOSP with \emph{dimension-independent} accuracy guarantees, using $\widetilde{O}({D^2}/ε)$ samples where $D$ is the ambient dimension and $ε$ is the fraction of corrupted datapoints. As a concrete application of our framework, we apply it to the problem of low rank matrix sensing, developing efficient and provably robust algorithms that can tolerate corruptions in both the sensing matrices and the measurements. In addition, we establish a Statistical Query lower bound providing evidence that the quadratic dependence on $D$ in the sample complexity is necessary for computationally efficient algorithms.

Shuyao Li, Sushrut Karmalkar, Ilias Diakonikolas et al.

We study the problem of learning a single neuron with respect to the $L_2^2$-loss in the presence of adversarial distribution shifts, where the labels can be arbitrary, and the goal is to find a ``best-fit'' function. More precisely, given training samples from a reference distribution $\mathcal{p}_0$, the goal is to approximate the vector $\mathbf{w}^*$ which minimizes the squared loss with respect to the worst-case distribution that is close in $χ^2$-divergence to $\mathcal{p}_{0}$. We design a computationally efficient algorithm that recovers a vector $ \hat{\mathbf{w}}$ satisfying $\mathbb{E}_{\mathcal{p}^*} (σ(\hat{\mathbf{w}} \cdot \mathbf{x}) - y)^2 \leq C \, \mathbb{E}_{\mathcal{p}^*} (σ(\mathbf{w}^* \cdot \mathbf{x}) - y)^2 + ε$, where $C>1$ is a dimension-independent constant and $(\mathbf{w}^*, \mathcal{p}^*)$ is the witness attaining the min-max risk $\min_{\mathbf{w}~:~\|\mathbf{w}\| \leq W} \max_{\mathcal{p}} \mathbb{E}_{(\mathbf{x}, y) \sim \mathcal{p}} (σ(\mathbf{w} \cdot \mathbf{x}) - y)^2 - νχ^2(\mathcal{p}, \mathcal{p}_0)$. Our algorithm follows a primal-dual framework and is designed by directly bounding the risk with respect to the original, nonconvex $L_2^2$ loss. From an optimization standpoint, our work opens new avenues for the design of primal-dual algorithms under structured nonconvexity.

Puqian Wang, Nikos Zarifis, Ilias Diakonikolas et al.

A single-index model (SIM) is a function of the form $σ(\mathbf{w}^{\ast} \cdot \mathbf{x})$, where $σ: \mathbb{R} \to \mathbb{R}$ 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 $O(\mathrm{OPT})+ε$, where $\mathrm{OPT}$ is the optimal loss. The sample complexity of our algorithm is $\tilde{O}(d^{\lceil k^{\ast}/2\rceil}+d/ε)$, where $k^{\ast}$ is the information-exponent of $σ$ 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.

Ilias Diakonikolas, Vasilis Kontonis, Christos Tzamos et al.

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 $η$. We study the fundamental class of $γ$-margin linear classifiers and present a computationally efficient algorithm that achieves mistake bound $ηT + o(T)$. Our mistake bound is qualitatively tight for efficient algorithms: it is known that even in the offline setting achieving classification error better than $η$ 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 $Δ$. We use our Massart online learner to design an efficient bandit algorithm that obtains expected reward at least $(1-1/k)~ ΔT - o(T)$ bigger than choosing a random action at every round.

Ilias Diakonikolas, Giannis Iakovidis, Daniel M. Kane et al.

We study the algorithmic problem of robust mean estimation of an identity covariance Gaussian in the presence of mean-shift contamination. In this contamination model, we are given a set of points in $\mathbb{R}^d$ generated i.i.d. via the following process. For a parameter $α<1/2$, the $i$-th sample $x_i$ is obtained as follows: with probability $1-α$, $x_i$ is drawn from $\mathcal{N}(μ, I)$, where $μ\in \mathbb{R}^d$ is the target mean; and with probability $α$, $x_i$ is drawn from $\mathcal{N}(z_i, I)$, where $z_i$ is unknown and potentially arbitrary. Prior work characterized the information-theoretic limits of this task. Specifically, it was shown that, in contrast to Huber contamination, in the presence of mean-shift contamination consistent estimation is possible. On the other hand, all known robust estimators in the mean-shift model have running times exponential in the dimension. Here we give the first computationally efficient algorithm for high-dimensional robust mean estimation with mean-shift contamination that can tolerate a constant fraction of outliers. In particular, our algorithm has near-optimal sample complexity, runs in sample-polynomial time, and approximates the target mean to any desired accuracy. Conceptually, our result contributes to a growing body of work that studies inference with respect to natural noise models lying in between fully adversarial and random settings.

Ilias Diakonikolas, Mingchen Ma, Lisheng Ren et al.

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 $σ: = \min_{i \neq j} H_{ii}-H_{ij}$. 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.

Nikos Zarifis, Puqian Wang, Ilias Diakonikolas et al.

We study the task of learning Generalized Linear models (GLMs) in the agnostic model under the Gaussian distribution. We give the first polynomial-time algorithm that achieves a constant-factor approximation for \textit{any} monotone Lipschitz activation. Prior constant-factor GLM learners succeed for a substantially smaller class of activations. Our work resolves a well-known open problem, by developing a robust counterpart to the classical GLMtron algorithm (Kakade et al., 2011). Our robust learner applies more generally, encompassing all monotone activations with bounded $(2+ζ)$-moments, for any fixed $ζ>0$ -- a condition that is essentially necessary. To obtain our results, we leverage a novel data augmentation technique with decreasing Gaussian noise injection and prove a number of structural results that may be useful in other settings.

Ilias Diakonikolas, Daniel M. Kane, Sihan Liu et al.

We study the task of high-dimensional entangled mean estimation in the subset-of-signals model. Specifically, given $N$ independent random points $x_1,\ldots,x_N$ in $\mathbb{R}^D$ and a parameter $α\in (0, 1)$ such that each $x_i$ is drawn from a Gaussian with mean $μ$ and unknown covariance, and an unknown $α$-fraction of the points have identity-bounded covariances, the goal is to estimate the common mean $μ$. The one-dimensional version of this task has received significant attention in theoretical computer science and statistics over the past decades. Recent work [LY20; CV24] has given near-optimal upper and lower bounds for the one-dimensional setting. On the other hand, our understanding of even the information-theoretic aspects of the multivariate setting has remained limited. In this work, we design a computationally efficient algorithm achieving an information-theoretically near-optimal error. Specifically, we show that the optimal error (up to polylogarithmic factors) is $f(α,N) + \sqrt{D/(αN)}$, where the term $f(α,N)$ is the error of the one-dimensional problem and the second term is the sub-Gaussian error rate. Our algorithmic approach employs an iterative refinement strategy, whereby we progressively learn more accurate approximations $\hat μ$ to $μ$. This is achieved via a novel rejection sampling procedure that removes points significantly deviating from $\hat μ$, as an attempt to filter out unusually noisy samples. A complication that arises is that rejection sampling introduces bias in the distribution of the remaining points. To address this issue, we perform a careful analysis of the bias, develop an iterative dimension-reduction strategy, and employ a novel subroutine inspired by list-decodable learning that leverages the one-dimensional result.

Ilias Diakonikolas, Daniel M. Kane, Mingchen Ma

We study the problem of learning general (i.e., not necessarily homogeneous) halfspaces under the Gaussian distribution on $R^d$ 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 $\tildeΩ(d/(\log(m)ε))$, where $m$ is the number of unlabeled examples. Specifically, to beat the passive label complexity of $\tilde{O} (d/ε)$, an active learner requires a pool of $2^{poly(d)}$ 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 $\tilde{O}(\min\{1/p, 1/ε\} + d\cdot polylog(1/ε))$ achieving error guarantee of $O(opt)+ε$. Here $p \in [0, 1/2]$ is the bias and $opt$ 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.