140 research outputs found
Robust Estimators in High Dimensions without the Computational Intractability
We study high-dimensional distribution learning in an agnostic setting where an adversary is allowed to arbitrarily corrupt an epsilon fraction of the samples. Such questions have a rich history spanning statistics, machine learning and theoretical computer science. Even in the most basic settings, the only known approaches are either computationally inefficient or lose dimension dependent factors in their error guarantees. This raises the following question: Is high-dimensional agnostic distribution learning even possible, algorithmically? In this work, we obtain the first computationally efficient algorithms for agnostically learning several fundamental classes of high-dimensional distributions: (1) a single Gaussian, (2) a product distribution on the hypercube, (3) mixtures of two product distributions (under a natural balancedness condition), and (4) mixtures of k Gaussians with identical spherical covariances. All our algorithms achieve error that is independent of the dimension, and in many cases depends nearly-linearly on the fraction of adversarially corrupted samples. Moreover, we develop a general recipe for detecting and correcting corruptions in high-dimensions, that may be applicable to many other problems.United States. Office of Naval Research (Grant N00014-12-1-0999)National Science Foundation (U.S.) (CAREER Award CCF-1453261)National Science Foundation (U.S.) (CAREER Award CCF-0953960)Google (Firm) (Faculty Research Award)National Science Foundation (U.S.). Graduate Research Fellowship ProgramNEC Corporatio
Resilient Distributed Optimization Algorithms for Resource Allocation
Distributed algorithms provide flexibility over centralized algorithms for
resource allocation problems, e.g., cyber-physical systems. However, the
distributed nature of these algorithms often makes the systems susceptible to
man-in-the-middle attacks, especially when messages are transmitted between
price-taking agents and a central coordinator. We propose a resilient strategy
for distributed algorithms under the framework of primal-dual distributed
optimization. We formulate a robust optimization model that accounts for
Byzantine attacks on the communication channels between agents and coordinator.
We propose a resilient primal-dual algorithm using state-of-the-art robust
statistics methods. The proposed algorithm is shown to converge to a
neighborhood of the robust optimization model, where the neighborhood's radius
is proportional to the fraction of attacked channels.Comment: 15 pages, 1 figure, accepted to CDC 201
Spectral Signatures in Backdoor Attacks
A recent line of work has uncovered a new form of data poisoning: so-called
\emph{backdoor} attacks. These attacks are particularly dangerous because they
do not affect a network's behavior on typical, benign data. Rather, the network
only deviates from its expected output when triggered by a perturbation planted
by an adversary.
In this paper, we identify a new property of all known backdoor attacks,
which we call \emph{spectral signatures}. This property allows us to utilize
tools from robust statistics to thwart the attacks. We demonstrate the efficacy
of these signatures in detecting and removing poisoned examples on real image
sets and state of the art neural network architectures. We believe that
understanding spectral signatures is a crucial first step towards designing ML
systems secure against such backdoor attacksComment: 16 pages, accepted to NIPS 201
Sample-Efficient Learning of Mixtures
We consider PAC learning of probability distributions (a.k.a. density
estimation), where we are given an i.i.d. sample generated from an unknown
target distribution, and want to output a distribution that is close to the
target in total variation distance. Let be an arbitrary class of
probability distributions, and let denote the class of
-mixtures of elements of . Assuming the existence of a method
for learning with sample complexity ,
we provide a method for learning with sample complexity
. Our mixture
learning algorithm has the property that, if the -learner is
proper/agnostic, then the -learner would be proper/agnostic as
well.
This general result enables us to improve the best known sample complexity
upper bounds for a variety of important mixture classes. First, we show that
the class of mixtures of axis-aligned Gaussians in is
PAC-learnable in the agnostic setting with
samples, which is tight in and up to logarithmic factors. Second, we
show that the class of mixtures of Gaussians in is
PAC-learnable in the agnostic setting with sample complexity
, which improves the previous known
bounds of and
in its dependence on and . Finally,
we show that the class of mixtures of log-concave distributions over
is PAC-learnable using
samples.Comment: A bug from the previous version, which appeared in AAAI 2018
proceedings, is fixed. 18 page
All-In-One Robust Estimator of the Gaussian Mean
The goal of this paper is to show that a single robust estimator of the mean
of a multivariate Gaussian distribution can enjoy five desirable properties.
First, it is computationally tractable in the sense that it can be computed in
a time which is at most polynomial in dimension, sample size and the logarithm
of the inverse of the contamination rate. Second, it is equivariant by
translations, uniform scaling and orthogonal transformations. Third, it has a
high breakdown point equal to , and a nearly-minimax-rate-breakdown point
approximately equal to . Fourth, it is minimax rate optimal, up to a
logarithmic factor, when data consists of independent observations corrupted by
adversarially chosen outliers. Fifth, it is asymptotically efficient when the
rate of contamination tends to zero. The estimator is obtained by an iterative
reweighting approach. Each sample point is assigned a weight that is
iteratively updated by solving a convex optimization problem. We also establish
a dimension-free non-asymptotic risk bound for the expected error of the
proposed estimator. It is the first result of this kind in the literature and
involves only the effective rank of the covariance matrix. Finally, we show
that the obtained results can be extended to sub-Gaussian distributions, as
well as to the cases of unknown rate of contamination or unknown covariance
matrix.Comment: 41 pages, 5 figures; added sub-Gaussian case with unknown Sigma or
ep
Robust polynomial regression up to the information theoretic limit
We consider the problem of robust polynomial regression, where one receives
samples that are usually within of a polynomial , but have a chance of being arbitrary adversarial outliers.
Previously, it was known how to efficiently estimate only when . We give an algorithm that works for the entire feasible
range of , while simultaneously improving other parameters of the
problem. We complement our algorithm, which gives a factor 2 approximation,
with impossibility results that show, for example, that a approximation
is impossible even with infinitely many samples.Comment: 19 Pages. To appear in FOCS 201
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