Learning from data in the presence of outliers is a fundamental problem in statistics. Until recently, no computationally efficient algorithms were known to compute the mean of a high dimensional distribution under natural assumptions in the presence of even a small fraction of outliers. In this paper, we consider robust statistics in the presence of overwhelming outliers where the majority of the dataset is introduced adversarially. With only an $α< 1/2$ fraction of "inliers" (clean data) the mean of a distribution is unidentifiable. However, in their influential work, [CSV17] introduces a polynomial time algorithm recovering the mean of distributions with bounded covariance by outputting a succinct list of $O(1/α)$ candidate solutions, one of which is guaranteed to be close to the true distributional mean; a direct analog of 'List Decoding' in the theory of error correcting codes. In this work, we develop an algorithm for list decodable mean estimation in the same setting achieving up to constants the information theoretically optimal recovery, optimal sample complexity, and in nearly linear time up to polylogarithmic factors in dimension. Our conceptual innovation is to design a descent style algorithm on a nonconvex landscape, iteratively removing minima to generate a succinct list of solutions. Our runtime bottleneck is a saddle-point optimization for which we design custom primal dual solvers for generalized packing and covering SDP's under Ky-Fan norms, which may be of independent interest.
In the list-decodable learning setup, an overwhelming majority (say a $1-β$-fraction) of the input data consists of outliers and the goal of an algorithm is to output a small list $\mathcal{L}$ of hypotheses such that one of them agrees with inliers. We develop a framework for list-decodable learning via the Sum-of-Squares SDP hierarchy and demonstrate it on two basic statistical estimation problems {\it Linear regression:} Suppose we are given labelled examples $\{(X_i,y_i)\}_{i \in [N]}$ containing a subset $S$ of $βN$ {\it inliers} $\{X_i \}_{i \in S}$ that are drawn i.i.d. from standard Ga…
Learning from data in the presence of outliers is a fundamental problem in statistics. In this work, we study robust statistics in the presence of overwhelming outliers for the fundamental problem of subspace recovery. Given a dataset where an $α$ fraction (less than half) of the data is distributed uniformly in an unknown $k$ dimensional subspace in $d$ dimensions, and with no additional assumptions on the remaining data, the goal is to recover a succinct list of $O(\frac{1}α)$ subspaces one of which is nontrivially correlated with the planted subspace. We provide the first polynomial time al…
University of Texas at Austin（得克萨斯大学奥斯汀分校）；Princeton University（普林斯顿大学）；Institute for Advanced Study（高等研究院）
We give the first polynomial-time algorithm for robust regression in the list-decodable setting where an adversary can corrupt a greater than $1/2$ fraction of examples. For any $α< 1$, our algorithm takes as input a sample $\{(x_i,y_i)\}_{i \leq n}$ of $n$ linear equations where $αn$ of the equations satisfy $y_i = \langle x_i,\ell^*\rangle +ζ$ for some small noise $ζ$ and $(1-α)n$ of the equations are {\em arbitrarily} chosen. It outputs a list $L$ of size $O(1/α)$ - a fixed constant - that contains an $\ell$ that is close to $\ell^*$. Our algorithm succeeds whenever the inliers are chosen f…
University of Wisconsin, Madison（威斯康星大学麦迪逊分校）；University of California, San Diego（加利福尼亚大学圣迭戈分校）；Microsoft Research（微软研究院）；Stanford University（斯坦福大学）
Traditionally, robust statistics has focused on designing estimators tolerant to a minority of contaminated data. Robust list-decodable learning focuses on the more challenging regime where only a minority $\frac 1 k$ fraction of the dataset is drawn from the distribution of interest, and no assumptions are made on the remaining data. We study the fundamental task of list-decodable mean estimation in high dimensions. Our main result is a new list-decodable mean estimation algorithm for bounded covariance distributions with optimal sample complexity and error rate, running in nearly-PCA time. A…
We give the first polynomial time algorithm for \emph{list-decodable covariance estimation}. For any $α> 0$, our algorithm takes input a sample $Y \subseteq \mathbb{R}^d$ of size $n\geq d^{\mathsf{poly}(1/α)}$ obtained by adversarially corrupting an $(1-α)n$ points in an i.i.d. sample $X$ of size $n$ from the Gaussian distribution with unknown mean $μ_*$ and covariance $Σ_*$. In $n^{\mathsf{poly}(1/α)}$ time, it outputs a constant-size list of $k = k(α)= (1/α)^{\mathsf{poly}(1/α)}$ candidate parameters that, with high probability, contains a $(\hatμ,\hatΣ)$ such that the total variation distan…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of California, San Diego（加州大学圣地亚哥分校）；Web 3 Foundation（Web3基金会）
We study the problem of list-decodable linear regression, where an adversary can corrupt a majority of the examples. Specifically, we are given a set $T$ of labeled examples $(x, y) \in \mathbb{R}^d \times \mathbb{R}$ and a parameter $0< α<1/2$ such that an $α$-fraction of the points in $T$ are i.i.d. samples from a linear regression model with Gaussian covariates, and the remaining $(1-α)$-fraction of the points are drawn from an arbitrary noise distribution. The goal is to output a small list of hypothesis vectors such that at least one of them is close to the target regression vector. Our m…
University of Wisconsin, Madison（威斯康星大学麦迪逊分校）；University of California, San Diego（加利福尼亚大学圣迭戈分校）
We study the problem of {\em list-decodable mean estimation} for bounded covariance distributions. Specifically, we are given a set $T$ of points in $\mathbb{R}^d$ with the promise that an unknown $α$-fraction of points in $T$, where $0< α< 1/2$, are drawn from an unknown mean and bounded covariance distribution $D$, and no assumptions are made on the remaining points. The goal is to output a small list of hypothesis vectors such that at least one of them is close to the mean of $D$. We give the first practically viable estimator for this problem. In more detail, our algorithm is sample and co…
University of Wisconsin, Madison（威斯康星大学麦迪逊分校）；University of California, San Diego（加利福尼亚大学圣迭戈分校）；Microsoft Research（微软研究院）；Stanford University（斯坦福大学）
We study the problem of list-decodable mean estimation, where an adversary can corrupt a majority of the dataset. Specifically, we are given a set $T$ of $n$ points in $\mathbb{R}^d$ and a parameter $0< α<\frac 1 2$ such that an $α$-fraction of the points in $T$ are i.i.d. samples from a well-behaved distribution $\mathcal{D}$ and the remaining $(1-α)$-fraction are arbitrary. The goal is to output a small list of vectors, at least one of which is close to the mean of $\mathcal{D}$. We develop new algorithms for list-decodable mean estimation, achieving nearly-optimal statistical guarantees, wi…
University of Southern California（南加州大学）；University of California, San Diego（加利福尼亚大学圣迭戈分校）
We study the problem of list-decodable Gaussian mean estimation and the related problem of learning mixtures of separated spherical Gaussians. We develop a set of techniques that yield new efficient algorithms with significantly improved guarantees for these problems. {\bf List-Decodable Mean Estimation.} Fix any $d \in \mathbb{Z}_+$ and $0< α<1/2$. We design an algorithm with runtime $O (\mathrm{poly}(n/α)^{d})$ that outputs a list of $O(1/α)$ many candidate vectors such that with high probability one of the candidates is within $\ell_2$-distance $O(α^{-1/(2d)})$ from the true mean. The only …
In list-decodable subspace recovery, the input is a collection of $n$ points $αn$ (for some $α\ll 1/2$) of which are drawn i.i.d. from a distribution $\mathcal{D}$ with a isotropic rank $r$ covariance $Π_*$ (the \emph{inliers}) and the rest are arbitrary, potential adversarial outliers. The goal is to recover a $O(1/α)$ size list of candidate covariances that contains a $\hatΠ$ close to $Π_*$. Two recent independent works (Raghavendra-Yau, Bakshi-Kothari 2020) gave the first efficient algorithm for this problem. These results, however, obtain an error that grows with the dimension (linearly in…
We give a complete answer to the following basic question: "What is the maximal fraction of deletions or insertions tolerable by $q$-ary list-decodable codes with non-vanishing information rate?" This question has been open even for binary codes, including the restriction to the binary insertion-only setting, where the best-known result was that a $γ\leq 0.707$ fraction of insertions is tolerable by some binary code family. For any desired $ε> 0$, we construct a family of binary codes of positive rate which can be efficiently list-decoded from any combination of $γ$ fraction of insertions and …
Robust Mixture Learning when Outliers Overwhelm Small Groups
ETH Zurich（苏黎世联邦理工学院）；TU Munich（慕尼黑工业大学）；Lucerne School of Computer Science and Information Technology（卢塞恩计算机科学与信息技术学院）；University of Copenhagen（哥本哈根大学）
A hypergraph $\mathcal{H}$ on $n$ vertices and $m$ edges is said to be {\it nearly-intersecting} if every edge of $\mathcal{H}$ intersects all but at most polylogarthmically many (in $m$ and $n$) other edges. Given lists of colors $\mathcal{L}(v)$, for each vertex $v\in V$, $\mathcal{H}$ is said to be $\mathcal{L}$-(list) colorable, if each vertex can be assigned a color from its list such that no edge in $\mathcal{H}$ is monochromatic. We show that list-colorability for any nearly intersecting hypergraph, and lists drawn from a set of constant size, can be checked in quasi-polynomial time in …
We introduce fast-decodable indexing schemes for edit distance which can be used to speed up edit distance computations to near-linear time if one of the strings is indexed by an indexing string $I$. In particular, for every length $n$ and every $\varepsilon >0$, one can in near linear time construct a string $I \in Σ'^n$ with $|Σ'| = O_{\varepsilon}(1)$, such that, indexing any string $S \in Σ^n$, symbol-by-symbol, with $I$ results in a string $S' \in Σ''^n$ where $Σ'' = Σ\times Σ'$ for which edit distance computations are easy, i.e., one can compute a $(1+\varepsilon)$-approximation of the e…
We strengthen the notion of "double samplers", first introduced by Dinur and Kaufman [Proc. 58th FOCS, 2017], which are samplers with additional combinatorial properties, and whose existence we prove using high dimensional expanders. The ABNNR code construction [IEEE Trans. Inform. Theory, 38(2):509--516, 1992] achieves large distance by starting with a base code $C$ with moderate distance, and then amplifying the distance using a sampler. We show that if the sampler is part of a larger double sampler then the construction has an efficient list-decoding algorithm. Our algorithm works even if t…
In the long-studied problem of combinatorial group testing, one is asked to detect a set of $k$ defective items out of a population of size $n$, using $m \ll n$ disjunctive measurements. In the non-adaptive setting, the most widely used combinatorial objects are disjunct and list-disjunct matrices, which define incidence matrices of test schemes. Disjunct matrices allow the identification of the exact set of defectives, whereas list disjunct matrices identify a small superset of the defectives. Apart from the combinatorial guarantees, it is often of key interest to equip measurement designs wi…
In the sparse polynomial multiplication problem, one is asked to multiply two sparse polynomials f and g in time that is proportional to the size of the input plus the size of the output. The polynomials are given via lists of their coefficients F and G, respectively. Cole and Hariharan (STOC 02) have given a nearly optimal algorithm when the coefficients are positive, and Arnold and Roche (ISSAC 15) devised an algorithm running in time proportional to the "structural sparsity" of the product, i.e. the set supp(F)+supp(G). The latter algorithm is particularly efficient when there not "too many…
The degeneracy of an $n$-vertex graph $G$ is the smallest number $d$ such that every subgraph of $G$ contains a vertex of degree at most $d$. We show that there exists a nearly-optimal fixed-parameter tractable algorithm for enumerating all maximal cliques, parametrized by degeneracy. To achieve this result, we modify the classic Bron--Kerbosch algorithm and show that it runs in time $O(dn3^{d/3})$. We also provide matching upper and lower bounds showing that the largest possible number of maximal cliques in an $n$-vertex graph with degeneracy $d$ (when $d$ is a multiple of 3 and $n\ge d+3$) i…
