Over the past decade, the low-degree heuristic has been used to estimate the algorithmic thresholds for a wide range of average-case planted vs null distinguishing problems. Such results rely on the hypothesis that if the low-degree moments of the planted and null distributions are sufficiently close, then no efficient (noise-tolerant) algorithm should be able to distinguish between them. This hypothesis is appealing due to the simplicity of calculating the low-degree likelihood ratio (LDLR), a quantity that measures the similarity between low-degree moments. However, despite sustained interest in the area, it remains unclear whether low-degree indistinguishability actually rules out any interesting class of algorithms.

We study the problem of robustly estimating the edge density of Erdos Renyi random graphs $\mathbb{G}(n, d^\circ/n)$ when an adversary can arbitrarily add or remove edges incident to an $\eta$-fraction of the nodes. We develop the first polynomial-time algorithm for this problem that estimates $d^\circ$ up to an additive error $O\left({[\sqrt{\log(n) / n} + \eta\sqrt{\log(1/\eta)} ] \cdot \sqrt{d^\circ} + \eta \log(1/\eta)}\right)$. Our error guarantee matches information-theoretic lower bounds up to factors of $\log(1/\eta)$. Moreover, our estimator works for all $d^\circ \geq \Omega(1)$ and achieves optimal breakdown point $\eta = 1/2$. Previous algorithms [Acharya et al 2022, Chen et al 2024], including inefficient ones, incur significantly suboptimal errors. Furthermore, even admitting suboptimal error guarantees, only inefficient algorithms achieve optimal breakdown point. Our algorithm is based on the sum-of-squares (SoS) hierarchy. A key ingredient is to construct constant-degree SoS certificates for concentration of the number of edges incident to small sets in $\mathbb{G}(n, d^\circ/n)$. Crucially, we show that these certificates also exist in the sparse regime, when $d^\circ = o(\log n)$, a regime in which the performance of previous algorithms was significantly suboptimal.

STOC ’25, Prague, Czechia

We study computational-statistical gaps for improper learning in sparse linear regression. More specifically, given $n$ samples from a $k$-sparse linear model in dimension $d$, we ask what is the minimum sample complexity to efficiently (in time polynomial in $d$, $k$, and $n$) find a potentially dense estimate for the regression vector that achieves non-trivial prediction error on the $n$ samples. Information-theoretically this can be achieved using $\Theta(k \log (d/k))$ samples. Yet, despite its prominence in the literature, there is no polynomial-time algorithm known to achieve the same guarantees using less than $\Theta(d)$ samples without additional restrictions on the model. Similarly, existing hardness results are either restricted to the proper setting, in which the estimate must be sparse as well, or only apply to specific algorithms. We give evidence that efficient algorithms for this task require at least (roughly) $\Omega(k^2)$ samples. In particular, we show that an improper learning algorithm for sparse linear regression can be used to solve sparse PCA problems (with a negative spike) in their Wishart form, in regimes in which efficient algorithms are widely believed to require at least $\Omega(k^2)$ samples. We complement our reduction with low-degree and statistical query lower bounds for the sparse PCA problems from which we reduce. Our hardness results apply to the (correlated) random design setting in which the covariates are drawn i.i.d. from a mean-zero Gaussian distribution with unknown covariance.

We show strong (and surprisingly simple) lower bounds for weakly learning intersections of halfspaces in the improper setting. Strikingly little is known about this problem. For instance, it is not even known if there is a polynomial-time algorithm for learning the intersection of only two halfspaces. On the other hand, lower bounds based on well-established assumptions (such as approximating worst-case lattice problems or variants of Feige’s 3SAT hypothesis) are only known (or are implied by existing results) for the intersection of super-logarithmically many halfspaces (KS06, KS09, DS16). With intersections of fewer halfspaces being only ruled out under less standard assumptions (DV21) (such as the existence of local pseudo-random generators with large stretch). We significantly narrow this gap by showing that even learning $\omega(\log \log N)$ halfspaces in dimension $N$ takes super-polynomial time under standard assumptions on worst-case lattice problems (namely that SVP and SIVP are hard to approximate within polynomial factors). Further, we give unconditional hardness results in the statistical query framework. Specifically, we show that for any $k$ (even constant), learning $k$ halfspaces in dimension $N$ requires accuracy $N^{-\Omega(k)}$, or exponentially many queries – in particular ruling out SQ algorithms with polynomial accuracy for $\omega(1)$ halfspaces. To the best of our knowledge this is the first unconditional hardness result for learning a super-constant number of halfspaces. Our lower bounds are obtained in a unified way via a novel connection we make between intersections of halfspaces and the so-called parallel pancakes distribution (DKS17, PLBR19, BRST21) that has been at the heart of many lower bound constructions in (robust) high-dimensional statistics in the past few years.

We study the problem of estimating the means of well-separated mixtures when an adversary may add arbitrary outliers. While strong guarantees are available when the outlier fraction is significantly smaller than the minimum mixing weight, much less is known when outliers may crowd out low-weight clusters – a setting we refer to as list-decodable mixture learning (LD-ML). In this case, adversarial outliers can simulate additional spurious mixture components. Hence, if all means of the mixture must be recovered up to a small error in the output list, the list size needs to be larger than the number of (true) components. We propose an algorithm that obtains order-optimal error guarantees for each mixture mean with a minimal list-size overhead, significantly improving upon list-decodable mean estimation, the only existing method that is applicable for LD-ML. Although improvements are observed even when the mixture is non-separated, our algorithm achieves particularly strong guarantees when the mixture is separated: it can leverage the mixture structure to partially cluster the samples before carefully iterating a base learner for list-decodable mean estimation at different scales.

Rubinfeld \& Vasilyan recently introduced the framework of *testable learning* as an extension of the classical agnostic model. It relaxes distributional assumptions which are difficult to verify by conditions that can be checked efficiently by a *tester*. The tester has to accept whenever the data truly satisfies the original assumptions, and the learner has to succeed whenever the tester accepts. We focus on the setting where the tester has to accept standard Gaussian data. There, it is known that basic concept classes such as halfspaces can be learned testably with the same time complexity as in the (distribution-specific) agnostic model. In this work, we ask whether there is a price to pay for testably learning more complex concept classes. In particular, we consider polynomial threshold functions (PTFs), which naturally generalize halfspaces. We show that PTFs of arbitrary constant degree can be testably learned up to excess error $\varepsilon > 0$ in time $n^{\mathrm{poly}(1/\varepsilon)}$. This qualitatively matches the best known guarantees in the agnostic model. Our results build on a connection between testable learning and *fooling*. In particular, we show that distributions that approximately match at least $\mathrm{poly}(1/\varepsilon)$ moments of the standard Gaussian fool constant-degree PTFs (up to error $\varepsilon$). As a secondary result, we prove that a direct approach to show testable learning (without fooling), which was successfully used for halfspaces, cannot work for PTFs.

We show hardness of improperly learning halfspaces in the agnostic model, both in the distribution-independent as well as the distribution-specific setting, based on the assumption that worst-case lattice problems, e.g., approximating shortest vectors within polynomial factors, are hard.In particular, we show that under this assumption there is no efficient algorithm that outputs any binary hypothesis, not necessarily a halfspace, achieving misclassfication error better than $\frac 1 2 - \gamma$ even if the optimal misclassification error is as small is as small as $\delta$.Here, $\gamma$ can be smaller than the inverse of any polynomial in the dimension and $\delta$ as small as $\exp(-\Omega(\log^{1-c}(d)))$, where $0 < c < 1$ is an arbitrary constant and $d$ is the dimension.For the distribution-specific setting, we show that if the marginal distribution is standard Gaussian, for any $\beta > 0$ learning halfspaces up to error $\OPT_\LTF + \varepsilon$ takes time at least $d^{\tilde{\Omega}(1/\varepsilon^{2-\beta})}$ under the same hardness assumptions.Similarly, we show that learning degree-$\ell$ polynomial threshold functions up to error $\OPT_{\PTF_\ell} + \e$ takes time at least $d^{\tilde{\Omega}(\ell^{2-\beta}/\varepsilon^{4-2\beta})}$.$\OPT_\LTF$ and $\OPT_{\PTF_\ell}$ denote the best error achievable by any halfspace or polynomial threshold function, respectively.Our lower bounds qualitively match algorithmic guarantees and (nearly) recover known lower bounds based on non-worst-case assumptions.Previously, such hardness results [D16, DKPZ21] were based on average-case complexity assumptions, specifically, variants of Feige’s random 3SAT hypothesis, or restricted to the statistical query model.Our work gives the first hardness results basing these fundamental learning problems on well-understood worst-case complexity assumption.It is inspired by a sequence of recent works showing hardness of learning well-separated Gaussian mixtures based on worst-case lattice problems.

We introduce general tools for designing efficient private estimation algorithms, in the high-dimensional settings, whose statistical guarantees almost match those of the best known non-private algorithms. To illustrate our techniques, we consider two problems: recovery of stochastic block models and learning mixtures of spherical Gaussians. For the former, we present the first efficient $(\epsilon, \delta)$-differentially private algorithm for both weak recovery and exact recovery. Previously known algorithms achieving comparable guarantees required quasi-polynomial time. For the latter, we design an $(\epsilon, \delta)$-differentially private algorithm that recovers the centers of the $k$-mixture when the minimum separation is at least $ O(k^{1/t}\sqrt{t})$. For all choices of $t$, this algorithm requires sample complexity $n\geq k^{O(1)}d^{O(t)}$ and time complexity $(nd)^{O(t)}$. Prior work required either an additional additive $\Omega(\sqrt{\log n})$ term in the minimum separation or an explicit upper bound on the Euclidean norm of the centers.

We study the problem of robustly estimating the mean or location parameter without moment assumptions. Known computationally efficient algorithms rely on strong distributional assumptions, such as sub-Gaussianity, or (certifiably) bounded moments. Moreover, the guarantees that they achieve in the heavy-tailed setting are weaker than those for sub-Gaussian distributions with known covariance. In this work, we show that such a tradeoff, between error guarantees and heavy-tails, is not necessary for symmetric distributions. We show that for a large class of symmetric distributions, the same error as in the Gaussian setting can be achieved efficiently. The distributions we study include products of arbitrary symmetric one-dimensional distributions, such as product Cauchy distributions, as well as elliptical distributions, a vast generalization of the Gaussian distribution. For product distributions and elliptical distributions with known scatter (covariance) matrix, we show that given an $\varepsilon$-corrupted sample, we can with probability at least $1-\delta$ estimate its location up to error $O(\varepsilon \sqrt{\log(1/\varepsilon)})$ using $\tfrac{d\log(d) + \log(1/\delta)}{\varepsilon^2 \log(1/\varepsilon)}$ samples. This result matches the best-known guarantees for the Gaussian distribution and known SQ lower bounds (up to the $\log(d)$ factor). For elliptical distributions with unknown scatter (covariance) matrix, we propose a sequence of efficient algorithms that approaches this optimal error. Specifically, for every $k \in \mathbb{N}$, we design an estimator using time and samples $\tilde{O}({d^k})$ achieving error $O(\varepsilon^{1-\frac{1}{2k}})$. This matches the error and running time guarantees when assuming certifiably bounded moments of order up to $k$. For unknown covariance, such error bounds of $o(\sqrt{\varepsilon})$ are not even known for (general) sub-Gaussian distributions. Our algorithms are based on a generalization of the well-known filtering technique [DK22]. More specifically, we show how this machinery can be combined with Huber-loss-based techniques to work with projections of the noise that behave more nicely than the initial noise. Moreover, we show how sum-of-squares proofs can be used to obtain algorithmic guarantees even for distributions without a first moment. We believe that this approach may find other applications in future works.

We develop the first fast spectral algorithm to decompose a random third-order tensor over of rank up to $$O(d^{3/2}/polylog(d))$$. Our algorithm only involves simple linear algebra operations and can recover all components in time $$O(d^{6.05})$$ under the current matrix multiplication time. Prior to this work, comparable guarantees could only be achieved via sum-of-squares [Ma, Shi, Steurer 2016]. In contrast, fast algorithms [Hopkins, Schramm, Shi, Steurer 2016] could only decompose tensors of rank at most $$O(d^{4/3}/polylog(d))$$. Our algorithmic result rests on two key ingredients. A clean lifting of the third-order tensor to a sixth-order tensor, which can be expressed in the language of tensor networks. A careful decomposition of the tensor network into a sequence of rectangular matrix multiplications, which allows us to have a fast implementation of the algorithm.

We give tight statistical query (SQ) lower bounds for learnining halfspaces in the presence of Massart noise. In particular, suppose that all labels are corrupted with probability at most $\eta$. We show that for arbitrary $\eta \in [0,1/2]$ every SQ algorithm achieving misclassification error better than $\eta$ requires queries of superpolynomial accuracy or at least a superpolynomial number of queries. Further, this continues to hold even if the information-theoretically optimal error $\OPT$ is as small as $\exp\Paren{-\log^c(d)}$, where $d$ is the dimension and $0 < c < 1$ is an arbitrary absolute constant, and an overwhelming fraction of examples are noiseless. Our lower bound matches known polynomial time algorithms, which are also implementable in the SQ framework. Previously, such lower bounds only ruled out algorithms achieving error $\OPT + \e$ or error better than $\Omega(\eta)$ or, if $\eta$ is close to $1/2$, error $\eta - o_\eta(1)$, where the term $o_\eta(1)$ is constant in $d$ but going to 0 for $\eta$ approaching $1/2$. As a consequence, we also show that achieving misclassification error better than $1/2$ in the $(A,\alpha)$-Tsybakov model is SQ-hard for $A$ constant and $\alpha$ bounded away from 1.

We develop machinery to design efficiently computable and \emph{consistent} estimators, achieving estimation error approaching zero as the number of observations grows, when facing an oblivious adversary that may corrupt responses in all but an $\alpha$ fraction of the samples.As concrete examples, we investigate two problems: sparse regression and principal component analysis (PCA).For sparse regression, we achieve consistency for optimal sample size $n\gtrsim (k\log d)/\alpha^2$ and optimal error rate $O(\sqrt{(k\log d)/(n\cdot \alpha^2)})$where $n$ is the number of observations, $d$ is the number of dimensions and $k$ is the sparsity of the parameter vector, allowing the fraction of inliers to be inverse-polynomial in the number of samples.Prior to this work, no estimator was known to be consistent when the fraction of inliers $\alpha$ is $o(1/\log \log n)$, even for (non-spherical) Gaussian design matrices.Results holding under weak design assumptions and in the presence of such general noise have only been shown in dense setting (i.e., general linear regression) very recently by d'Orsi et al.~\cite{ICML-linear-regression}.In the context of PCA, we attain optimal error guarantees under broad spikiness assumptions on the parameter matrix (usually used in matrix completion). Previous works could obtain non-trivial guarantees only under the assumptions that the measurement noise corresponding to the inliers is polynomially small in $n$ (e.g., Gaussian with variance $1/n^2$).To devise our estimators, we equip the Huber loss with non-smooth regularizers such as the $\ell_1$ norm or the nuclear norm, and extend d'Orsi et al.'s approach~\cite{ICML-linear-regression} in a novel way to analyze the loss function.Our machinery appears to be easily applicable to a wide range of estimation problems.We complement these algorithmic results with statistical lower bounds showing that the fraction of inliers that our PCA estimator can deal with is optimal up to a constant factor.

We develop a general framework to significantly reduce the degree of sum-of-squares proofs by introducing new variables. To illustrate the power of this framework, we use it to speed up previous algorithms based on sum-of-squares for two important estimation problems, clustering and robust moment estimation. The resulting algorithms offer the same statistical guarantees as the previous best algorithms but have significantly faster running times. Roughly speaking, given a sample of n points in dimension d, our algorithms can exploit order-ℓ moments in time dO(ℓ) · nO(1), whereas a naive implementation requires time (d · n)O(ℓ). Since for the aforementioned applications, the typical sample size is dΘ(ℓ), our framework improves running times from to dO(ℓ).