arXiv subjects

Tommaso d'Orsi

Publications and source records attributed to Tommaso d'Orsi.

23 records · Page 2Linked to original sources

arXiv2021

Robust recovery for stochastic block models

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 presence 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.

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arXiv2021

Consistent Estimation for PCA and Sparse Regression with Oblivious Outliers

We develop machinery to design efficiently computable and 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 $α$ 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 $n\gtrsim (k\log d)/α^2$ and optimal error rate $O(\sqrt{(k\log d)/(n\cdot α^2)})$ 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 $α$ 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. [dNS21]. 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 [dNS21] in a novel way to analyze the loss function. Our machinery appears to be easily applicable to a wide range of estimation problems.

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arXiv2021

The Complexity of Sparse Tensor PCA

We study the problem of sparse tensor principal component analysis: given a tensor $\pmb Y = \pmb W + λx^{\otimes p}$ 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 $λ\geq \tilde{\mathcal{O}} (\sqrt{t} \cdot (k/t)^{p/2})$ 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 $λ\geq \tilde{\mathcal{O}}(k \cdot r)$ while our algorithms require $λ\geq \tilde{\mathcal{O}}(k)$. 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 $p$.

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arXiv2021

Consistent regression when oblivious outliers overwhelm

We consider a robust linear regression model $y=Xβ^* + η$, where an adversary oblivious to the design $X\in \mathbb{R}^{n\times d}$ may choose $η$ to corrupt all but an $α$ fraction of the observations $y$ in an arbitrary way. Prior to our work, even for Gaussian $X$, no estimator for $β^*$ was known to be consistent in this model except for quadratic sample size $n \gtrsim (d/α)^2$ or for logarithmic inlier fraction $α\ge 1/\log n$. 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 $n= ω(d/α^2)$ and achieves an error rate of $O(d/α^2n)^{1/2}$. 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 optimal guarantees in nearly-linear time, and that can exploit sparsity of $β^*$. The model studied here also captures heavy-tailed noise distributions that may not even have a first moment.

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arXiv2020

Sparse PCA: Algorithms, Adversarial Perturbations and Certificates

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 \emph{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 \emph{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.

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