Motivated by real-world scenarios where malicious entities tamper with existing networks, we define a model where an adversary seeks to hide a set of corrupted vertices inside a graph $G^*$. To this end, the adversary can add edges between the corrupted vertices, as well as edges between the corrupted vertices and $G^*$, and its power is then measured by the size of the neighborhood of the corrupted vertices in $G^*$. Our goal is to design an active learning algorithm that efficiently finds the subset of corrupted vertices using a small number of label queries. We devise an efficient algorithm that approximately recovers the corrupted vertices with a query complexity that depends polynomially on both the power of the adversary and the vertex expansion of $G^*$, a fundamental measure of graph connectivity. At the heart of this result is a polynomial-time algorithm, obtained by carefully adapting sum-of-squares algorithms for approximating minimum expansion, that finds a set with small vertex expansion subject to cardinality constraints. To the best of our knowledge, this is the first time that the vertex expansion is shown to play a key role in determining the query complexity of active learning algorithms robust to structural adversarial attacks.

We revisit the task of computing the span of the top \(r\) singular vectors \(u_1, \ldots, u_r\) of a matrix under differential privacy. We show that a simple and efficient algorithm—based on singular value decomposition and standard perturbation mechanisms—returns a private rank-\(r\) approximation whose error depends only on the rank-\(r\) coherence of \(u_1, \ldots, u_r\) and the spectral gap ór \(\sigma_r - \sigma_{r+1}\). This resolves a question posed by Hardt and Roth [HR13]. Our estimator outperforms the state of the art—significantly so in some regimes. In particular, we show that in the dense setting, it achieves the same guarantees for single-spike PCA in the Wishart model as those attained by optimal non-private algorithms, whereas prior private algorithms failed to do so.

In a linear combinatorial optimization problem, we are given a family F ⊆ 2U of feasible subsets of a ground set U of n elements, and our goal is to find S* = argminS ∈ F ⟨ w,1S ⟩. Traditionally, we are either given the weight vector up-front, or else we are given a value oracle which allows us to evaluate w(S) := ⟨ w, 1S ⟩ for any S ∈ F. We consider the weaker and more robust comparison oracle, which for any two feasible sets S, T ∈ F, reveals only if w(S) is less than/equal to/greater than w(T). We ask: When can we find the optimal feasible set S* = argminS ∈ F w(S) using a small number of comparison queries? If so, when can we do this efficiently?

We study a planted clique model introduced by Feige [18] where a complete graph of size c • n is planted uniformly at random in an arbitrary n-vertex graph.We give a simple deterministic algorithm that, in almost linear time, recovers a clique of size (c/3) O(1/c) • n as long as the original graph has maximum degree at most (1 -p)n for some fixed p > 0. The proof hinges on showing that the degrees of the final graph are correlated with the planted clique, in a way similar to (but more intricate than) the classical G(n, 1 /2) + K √ n planted clique model.Our algorithm suggests a separation from the worst-case model, where, assuming the Unique Games Conjecture, no polynomial algorithm can find cliques of size Ω(n) for every fixed c > 0, even if the input graph has maximum degree (1 -p)n.Our techniques extend beyond the planted clique model.For example, when the planted graph is a balanced biclique, we recover a balanced biclique of size larger than the best guarantees known for the worst case. 2012

Whether or not the Sparsest Cut problem admits an efficient $O(1)$-approximation algorithm is a fundamental algorithmic question with connections to geometry and the Unique Games Conjecture. Revisiting spectral algorithms for Sparsest Cut, we present a novel, simple algorithm that combines eigenspace enumeration with a new algorithm for the Cut Improvement problem. The runtime of our algorithm is parametrized by a quantity that we call the solution dimension $\text{SD}_\varepsilon(G)$: the smallest $k$ such that the subspace spanned by the first $k$ Laplacian eigenvectors contains all but $\varepsilon$ fraction of a sparsest cut. Our algorithm matches the guarantees of prior methods based on the threshold-rank paradigm, while also extending beyond them. To illustrate this, we study its performance on low degree Cayley graphs over Abelian groups -- canonical examples of graphs with poor expansion properties. We prove that low degree Abelian Cayley graphs have small solution dimension, yielding an algorithm that computes a $(1+\varepsilon)$-approximation to the uniform Sparsest Cut of a degree-$d$ Cayley graph over an Abelian group of size $n$ in time $n^{O(1)}\cdot\exp(d/\varepsilon)^{O(d)}$. Along the way to bounding the solution dimension of Abelian Cayley graphs, we analyze their sparse cuts and spectra, proving that the collection of $O(1)$-approximate sparsest cuts has an $\varepsilon$-net of size $\exp(d/\varepsilon)^{O(d)}$ and that the multiplicity of $λ_2$ is bounded by $2^{O(d)}$. The latter bound is tight and improves on a previous bound of $2^{O(d^2)}$ by Lee and Makarychev.

In this paper, we study the parameterized complexity of local search, whose goal is to find a good nearby solution from the given current solution. Formally, given an optimization problem where the goal is to find the largest feasible subset S of a universe U, the new input consists of a current solution P (not necessarily feasible) as well as an ordinary input for the problem. Given the existence of a feasible solution S^*, the goal is to find a feasible solution as good as S^* in parameterized time f(k)⋅n^O(1), where k denotes the distance |PΔ S^*|. This model generalizes numerous classical parameterized optimization problems whose parameter k is the minimum number of elements removed from U to make it feasible, which corresponds to the case P = U. We apply this model to widely studied Constraint Satisfaction Problems (CSPs), where U is the set of constraints, and a subset U' of constraints is feasible if there is an assignment to the variables satisfying all constraints in U'. We give a complete characterization of the parameterized complexity of all boolean-alphabet symmetric CSPs, where the predicate’s acceptance depends on the number of true literals.

We revisit the objective perturbations framework for differential privacy where noise is added to the input $A\in \mathcal{S}$ and the result is then projected back to the space of admissible datasets $\mathcal{S}$. Through this framework, we first design novel efficient algorithms to privately release pair-wise cosine similarities. Second, we derive a novel algorithm to compute $k$-way marginal queries over $n$ features. Prior work could achieve comparable guarantees only for $k$ even. Furthermore, we extend our results to $t$-sparse datasets, where our efficient algorithms yields novel, stronger guarantees whenever $t\le n^{5/6}/\log n.$ Finally, we provide a theoretical perspective on why fast input perturbation algorithms works well in practice. The key technical ingredients behind our results are tight sum-of-squares certificates upper bounding the Gaussian complexity of sets of solutions.

Graph clustering is a central topic in unsupervised learning with a multitude of practical applications. In recent years, multi-view graph clustering has gained a lot of attention for its applicability to real-world instances where one often has access to multiple data sources. In this paper we formalize a new family of models, called multi-view stochastic block models that capture this setting. For this model, we first study efficient algorithms that naively work on the union of multiple graphs. Then, we introduce a new efficient algorithm that provably outperforms previous approaches by analyzing the structure of each graph separately. Finally, we complement our results with an information-theoretic lower bound studying the limits of what can be done in this model.

We consider the semi-random graph model of [Makarychev, Makarychev and Vijayaraghavan, STOC'12], where, given a random bipartite graph with $\alpha$ edges and an unknown bipartition $(A, B)$ of the vertex set, an adversary can add arbitrary edges inside each community and remove arbitrary edges from the cut $(A, B)$ (i.e. all adversarial changes are *monotone* with respect to the bipartition). For this model, a polynomial time algorithm [MMV'12] is known to approximate the Balanced Cut problem up to value $O(\alpha)$ as long as the cut $(A, B)$ has size $\Omega(\alpha)$. However, it consists of slow subroutines requiring optimal solutions for logarithmically many semidefinite programs. We study the fine-grained complexity of the problem and present the first near-linear time algorithm that achieves similar performances to that of [MMV'12]. Our algorithm runs in time $O(|V(G)|^{1+o(1)} + |E(G)|^{1+o(1)})$ and finds a balanced cut of value $O(\alpha).$ Our approach appears easily extendible to related problem, such as Sparsest Cut, and also yields an near-linear time $O(1)$-approximation to Dagupta's objective function for hierarchical clustering [Dasgupta, STOC'16] for the semi-random hierarchical stochastic block model inputs of [Cohen-Addad, Kanade, Mallmann-Trenn, Mathieu, JACM'19].

In recent years, there has been a surge of interest in the use of machine-learned predictions to bypass worst-case lower bounds for classical problems in combinatorial optimization. So far, the focus has mostly been on online algorithms, where information-theoretic barriers are overcome using predictions about the unknown future. In this paper, we consider the complementary question of using learned information to overcome computational barriers in the form of approximation hardness of polynomial-time algorithms for NP-hard (offline) problems. We show that noisy predictions about the optimal solution can be used to break classical hardness results for maximization problems such as the max-cut problem and more generally, maximization versions of constraint satisfaction problems (CSPs).

We develop the first pure node-differentially-private algorithms for learning stochastic block models and for graphon estimation with polynomial running time for any constant number of blocks. The statistical utility guarantees match those of the previous best information-theoretic (exponential-time) node-private mechanisms for these problems. The algorithm is based on an exponential mech- anism for a score function defined in terms of a sum-of-squares relaxation whose level depends on the number of blocks. The key ingredients of our results are (1) a characterization of the distance between the block graphons in terms of a quadratic optimization over the polytope of doubly stochastic matrices, (2) a general sum-of-squares convergence result for polynomial op- timization over arbitrary polytopes, and (3) a general approach to perform Lipschitz extensions of score functions as part of the sum-of-squares algorithmic paradigm.

We define a novel notion of "non-backtracking" matrix associated to any symmetric matrix, and we prove a "Ihara-Bass" type formula for it. We use this theory to prove new results on polynomial-time strong refutations of random constraint satisfaction problems with k variables per constraints (k-CSPs). For a random k-CSP instance constructed out of a constraint that is satisfied by a p fraction of assignments, if the instance contains n variables and n^{k/2} / ε² constraints, we can efficiently compute a certificate that the optimum satisfies at most a p+O_k(ε) fraction of constraints. Previously, this was known for even k, but for odd k one needed n^{k/2} (log n)^{O(1)} / ε² random constraints to achieve the same conclusion. Although the improvement is only polylogarithmic, it overcomes a significant barrier to these types of results. Strong refutation results based on current approaches construct a certificate that a certain matrix associated to the k-CSP instance is quasirandom. Such certificate can come from a Feige-Ofek type argument, from an application of Grothendieck’s inequality, or from a spectral bound obtained with a trace argument. The first two approaches require a union bound that cannot work when the number of constraints is o(n^⌈k/2⌉) and the third one cannot work when the number of constraints is o(n^{k/2} √{log n}). We further apply our techniques to obtain a new PTAS finding assignments for k-CSP instances with n^{k/2} / ε² constraints in the semi-random settings where the constraints are random, but the sign patterns are adversarial.

We study robust community detection in the context of node-corrupted stochastic block model, where an adversary can arbitrarily modify all the edges incident to a fraction of the n vertices. We present the first polynomial-time algorithm that achieves weak recovery at the Kesten-Stigum threshold even in the presence of a small constant fraction of corrupted nodes. Prior to this work, even state-of-the-art robust algorithms were known to break under such node corruption adversaries, when close to the Kesten-Stigum threshold.We further extend our techniques to the $Z_2$ synchronization problem, where our algorithm reaches the optimal recovery threshold in the presence of similar strong adversarial perturbations.The key ingredient of our algorithm is a novel identifiability proof that leverages the push-out effect of the Grothendieck norm of principal submatrices.

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 consider estimation models of the form Y = X* + N, where X* is some m-dimensional structured signal we wish to recover, and N is symmetrically distributed noise that may be unbounded in all but a small α fraction of the entries. This setting captures problems such as (sparse) linear regression, (sparse) principal component analysis (PCA), and tensor PCA, even in the presence of oblivious outliers and heavy-tailed noise. We introduce a family of algorithms that under mild assumptions recover the signal X* in all estimation problems for which there exists a sum-of-squares algorithm that succeeds in recovering the signal X* when the noise N is Gaussian. This essentially shows that it is enough to design a sum-of-squares algorithm for an estimation problem with Gaussian additive noise in order to get the algorithm that works with the symmetric noise model. Our framework extends far beyond previous results on symmetric noise models and is even robust to an ε-fraction of adversarial perturbations. As concrete examples, we investigate two problems for which no efficient algorithms were known to work for heavy-tailed noise: tensor PCA and sparse PCA. For the former, our algorithm recovers the principal component in polynomial time when the signal-to-noise ratio is at least Õ(np/4/ α), that matches (up to logarithmic factors) current best known algorithmic guarantees for Gaussian noise. For the latter, our algorithm runs in quasipolynomial time and matches the state-of-the-art guarantees for quasipolynomial time algorithms in the case of Gaussian noise. Using a reduction from the planted clique problem, we provide evidence that the quasipolynomial time is likely to be necessary for sparse PCA with symmetric noise. In our proofs we use bounds on the covering numbers of sets of pseudo-expectations, which we obtain by certifying in sum-of-squares upper bounds on the Gaussian complexities of sets of solutions. This approach for bounding the covering numbers of sets of pseudo-expectations may be interesting in its own right and may find other application in future works. * This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 815464).

We consider the robust linear regression model $\bm{y} = X\beta^* + \bm{\eta}$, where an adversary oblivious to the design $X \in \R^{n \times d}$ may choose $\bm{\eta}$ to corrupt all but a (possibly vanishing) fraction of the observations $\bm{y}$ in an arbitrary way. Recent work \cite{d2021consistent, d2021consistentICML} has introduced efficient algorithms for consistent recovery of the parameter vector. These algorithms crucially rely on the design matrix being well-spread (a matrix is well-spread if its column span is far from any sparse vector). In this paper, we show that there exists a family of design matrices lacking well-spreadness such that consistent recovery of the parameter vector in the above robust linear regression model is information-theoretically impossible. We further investigate the average-case time complexity of certifying well-spreadness of random matrices. We show that it is possible to efficiently certify whether a given $n$-by-$d$ Gaussian matrix is well-spread if the number of observations is quadratic in the ambient dimension. We complement this result by showing rigorous evidence —in the form of a lower bound against low-degree polynomials— of the computational hardness of this same certification problem when the number of observations is $o(d^2)$.

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 develop an efficient algorithm for weak recovery in a robust version of the stochastic block model. The algorithm matches the statistical guarantees of the best known algorithms for the vanilla version of the stochastic block model. In this sense, our results show that there is no price of robustness in the stochastic block model. Our work is heavily inspired by recent work of Banks, Mohanty, and Raghavendra (SODA 2021) that provided an efficient algorithm for the corresponding distinguishing problem. Our algorithm and its analysis significantly depart from previous ones for robust recovery. A key challenge is the peculiar optimization landscape underlying our algorithm: The planted partition may be far from optimal in the sense that completely unrelated solutions could achieve the same objective value. This phenomenon is related to the push-out effect at the BBP phase transition for PCA. To the best of our knowledge, our algorithm is the first to achieve robust recovery in the presense of such a push-out effect in a non-asymptotic setting. Our algorithm is an instantiation of a framework based on convex optimization (related to but distinct from sum-of-squares), which may be useful for other robust matrix estimation problems. A by-product of our analysis is a general technique that boosts the probability of success (over the randomness of the input) of an arbitrary robust weak-recovery algorithm from constant (or slowly vanishing) probability to exponentially high probability.

We consider a robust linear regression model $y=X\beta^* + \eta$, where an adversary oblivious to the design $X\in \mathbb{R}^{n\times d}$ may choose $\eta$ to corrupt all but an $\alpha$ fraction of the observations $y$ in an arbitrary way. Prior to our work, even for Gaussian $X$, no estimator for $\beta^*$ was known to be consistent in this model except for quadratic sample size $n \gtrsim (d/\alpha)^2$ or for logarithmic inlier fraction $\alpha\ge 1/\log n$. We show that consistent estimation is possible with nearly linear sample size and inverse-polynomial inlier fraction. Concretely, we show that the Huber loss estimator is consistent for every sample size $n= \omega(d/\alpha^2)$ and achieves an error rate of $O(d/\alpha^2n)^{1/2}$ (both bounds are optimal up to constant factors). Our results extend to designs far beyond the Gaussian case and only require the column span of $X$ to not contain approximately sparse vectors (similar to the kind of assumption commonly made about the kernel space for compressed sensing). We provide two technically similar proofs. One proof is phrased in terms of strong convexity, extending work of [Tsakonas et al. ’14], and particularly short. The other proof highlights a connection between the Huber loss estimator and high-dimensional median computations. In the special case of Gaussian designs, this connection leads us to a strikingly simple algorithm based on computing coordinate-wise medians that achieves nearly optimal guarantees in linear time, and that can exploit sparsity of $\beta^*$. The model studied here also captures heavy-tailed noise distributions that may not even have a first moment.

We study the problem of sparse tensor principal component analysis: given a tensor $\pmb Y = \pmb W + \lambda x^{\otimes p}$ with $\pmb W \in \otimes^p \mathbb{R}^n$ having i.i.d. Gaussian entries, the goal is to recover the $k$-sparse unit vector $x \in \mathbb{R}^n$. The model captures both sparse PCA (in its Wigner form) and tensor PCA.For the highly sparse regime of $k \leq \sqrt{n}$, we present a family of algorithms that smoothly interpolates between a simple polynomial-time algorithm and the exponential-time exhaustive search algorithm. For any $1 \leq t \leq k$, our algorithms recovers the sparse vector for signal-to-noise ratio $\lambda \geq \tilde{\mathcal{O}} (\sqrt{t} \cdot (k/t)^{p/2})$ in time $\tilde{\mathcal{O}}(n^{p+t})$, capturing the state-of-the-art guarantees for the matrix settings (in both the polynomial-time and sub-exponential time regimes).Our results naturally extend to the case of $r$ distinct $k$-sparse signals with disjoint supports, with guarantees that are independent of the number of spikes. Even in the restricted case of sparse PCA, known algorithms only recover the sparse vectors for $\lambda \geq \tilde{\mathcal{O}}(k \cdot r)$ while our algorithms require $\lambda \geq \tilde{\mathcal{O}}(k)$.Finally, by analyzing the low-degree likelihood ratio, we complement these algorithmic results with rigorous evidence illustrating the trade-offs between signal-to-noise ratio and running time. This lower bound captures the known lower bounds for both sparse PCA and tensor PCA. In this general model, we observe a more intricate three-way trade-off between the number of samples $n$, the sparsity $k$, and the tensor power $p$.

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 study efficient algorithms for Sparse PCA in standard statistical models (spiked covariance in its Wishart form). Our goal is to achieve optimal recovery guarantees while being resilient to small perturbations. Despite a long history of prior works, including explicit studies of perturbation resilience, the best known algorithmic guarantees for Sparse PCA are fragile and break down under small adversarial perturbations. We observe a basic connection between perturbation resilience and certifying algorithms that are based on certificates of upper bounds on sparse eigenvalues of random matrices. In contrast to other techniques, such certifying algorithms, including the brute-force maximum likelihood estimator, are automatically robust against small adversarial perturbation. We use this connection to obtain the first polynomial-time algorithms for this problem that are resilient against additive adversarial perturbations by obtaining new efficient certificates for upper bounds on sparse eigenvalues of random matrices. Our algorithms are based either on basic semidefinite programming or on its low-degree sum-of-squares strengthening depending on the parameter regimes. Their guarantees either match or approach the best known guarantees of fragile algorithms in terms of sparsity of the unknown vector, number of samples and the ambient dimension. To complement our algorithmic results, we prove rigorous lower bounds matching the gap between fragile and robust polynomial-time algorithms in a natural computational model based on low-degree polynomials (closely related to the pseudo-calibration technique for sum-of-squares lower bounds) that is known to capture the best known guarantees for related statistical estimation problems. The combination of these results provides formal evidence of an inherent price to pay to achieve robustness. Beyond these issues of perturbation resilience, our analysis also leads to new algorithms for the fragile setting, whose guarantees improve over best previous results in some parameter regimes (e.g. if the sample size is polynomially smaller than the dimension).