89,716 research outputs found
Smoothed Analysis in Unsupervised Learning via Decoupling
Smoothed analysis is a powerful paradigm in overcoming worst-case
intractability in unsupervised learning and high-dimensional data analysis.
While polynomial time smoothed analysis guarantees have been obtained for
worst-case intractable problems like tensor decompositions and learning
mixtures of Gaussians, such guarantees have been hard to obtain for several
other important problems in unsupervised learning. A core technical challenge
in analyzing algorithms is obtaining lower bounds on the least singular value
for random matrix ensembles with dependent entries, that are given by
low-degree polynomials of a few base underlying random variables.
In this work, we address this challenge by obtaining high-confidence lower
bounds on the least singular value of new classes of structured random matrix
ensembles of the above kind. We then use these bounds to design algorithms with
polynomial time smoothed analysis guarantees for the following three important
problems in unsupervised learning:
1. Robust subspace recovery, when the fraction of inliers in the
d-dimensional subspace is at least for any constant integer . This contrasts with the known
worst-case intractability when , and the previous smoothed
analysis result which needed (Hardt and Moitra, 2013).
2. Learning overcomplete hidden markov models, where the size of the state
space is any polynomial in the dimension of the observations. This gives the
first polynomial time guarantees for learning overcomplete HMMs in a smoothed
analysis model.
3. Higher order tensor decompositions, where we generalize the so-called
FOOBI algorithm of Cardoso to find order- rank-one tensors in a subspace.
This allows us to obtain polynomially robust decomposition algorithms for
'th order tensors with rank .Comment: 44 page
Smoothed Analysis of Tensor Decompositions
Low rank tensor decompositions are a powerful tool for learning generative
models, and uniqueness results give them a significant advantage over matrix
decomposition methods. However, tensors pose significant algorithmic challenges
and tensors analogs of much of the matrix algebra toolkit are unlikely to exist
because of hardness results. Efficient decomposition in the overcomplete case
(where rank exceeds dimension) is particularly challenging. We introduce a
smoothed analysis model for studying these questions and develop an efficient
algorithm for tensor decomposition in the highly overcomplete case (rank
polynomial in the dimension). In this setting, we show that our algorithm is
robust to inverse polynomial error -- a crucial property for applications in
learning since we are only allowed a polynomial number of samples. While
algorithms are known for exact tensor decomposition in some overcomplete
settings, our main contribution is in analyzing their stability in the
framework of smoothed analysis.
Our main technical contribution is to show that tensor products of perturbed
vectors are linearly independent in a robust sense (i.e. the associated matrix
has singular values that are at least an inverse polynomial). This key result
paves the way for applying tensor methods to learning problems in the smoothed
setting. In particular, we use it to obtain results for learning multi-view
models and mixtures of axis-aligned Gaussians where there are many more
"components" than dimensions. The assumption here is that the model is not
adversarially chosen, formalized by a perturbation of model parameters. We
believe this an appealing way to analyze realistic instances of learning
problems, since this framework allows us to overcome many of the usual
limitations of using tensor methods.Comment: 32 pages (including appendix
Learning DNF Expressions from Fourier Spectrum
Since its introduction by Valiant in 1984, PAC learning of DNF expressions
remains one of the central problems in learning theory. We consider this
problem in the setting where the underlying distribution is uniform, or more
generally, a product distribution. Kalai, Samorodnitsky and Teng (2009) showed
that in this setting a DNF expression can be efficiently approximated from its
"heavy" low-degree Fourier coefficients alone. This is in contrast to previous
approaches where boosting was used and thus Fourier coefficients of the target
function modified by various distributions were needed. This property is
crucial for learning of DNF expressions over smoothed product distributions, a
learning model introduced by Kalai et al. (2009) and inspired by the seminal
smoothed analysis model of Spielman and Teng (2001).
We introduce a new approach to learning (or approximating) a polynomial
threshold functions which is based on creating a function with range [-1,1]
that approximately agrees with the unknown function on low-degree Fourier
coefficients. We then describe conditions under which this is sufficient for
learning polynomial threshold functions. Our approach yields a new, simple
algorithm for approximating any polynomial-size DNF expression from its "heavy"
low-degree Fourier coefficients alone. Our algorithm greatly simplifies the
proof of learnability of DNF expressions over smoothed product distributions.
We also describe an application of our algorithm to learning monotone DNF
expressions over product distributions. Building on the work of Servedio
(2001), we give an algorithm that runs in time \poly((s \cdot
\log{(s/\eps)})^{\log{(s/\eps)}}, n), where is the size of the target DNF
expression and \eps is the accuracy. This improves on \poly((s \cdot
\log{(ns/\eps)})^{\log{(s/\eps)} \cdot \log{(1/\eps)}}, n) bound of Servedio
(2001).Comment: Appears in Conference on Learning Theory (COLT) 201
Learning Mixtures of Gaussians in High Dimensions
Efficiently learning mixture of Gaussians is a fundamental problem in
statistics and learning theory. Given samples coming from a random one out of k
Gaussian distributions in Rn, the learning problem asks to estimate the means
and the covariance matrices of these Gaussians. This learning problem arises in
many areas ranging from the natural sciences to the social sciences, and has
also found many machine learning applications. Unfortunately, learning mixture
of Gaussians is an information theoretically hard problem: in order to learn
the parameters up to a reasonable accuracy, the number of samples required is
exponential in the number of Gaussian components in the worst case. In this
work, we show that provided we are in high enough dimensions, the class of
Gaussian mixtures is learnable in its most general form under a smoothed
analysis framework, where the parameters are randomly perturbed from an
adversarial starting point. In particular, given samples from a mixture of
Gaussians with randomly perturbed parameters, when n > {\Omega}(k^2), we give
an algorithm that learns the parameters with polynomial running time and using
polynomial number of samples. The central algorithmic ideas consist of new ways
to decompose the moment tensor of the Gaussian mixture by exploiting its
structural properties. The symmetries of this tensor are derived from the
combinatorial structure of higher order moments of Gaussian distributions
(sometimes referred to as Isserlis' theorem or Wick's theorem). We also develop
new tools for bounding smallest singular values of structured random matrices,
which could be useful in other smoothed analysis settings
Polynomial-time Tensor Decompositions with Sum-of-Squares
We give new algorithms based on the sum-of-squares method for tensor
decomposition. Our results improve the best known running times from
quasi-polynomial to polynomial for several problems, including decomposing
random overcomplete 3-tensors and learning overcomplete dictionaries with
constant relative sparsity. We also give the first robust analysis for
decomposing overcomplete 4-tensors in the smoothed analysis model. A key
ingredient of our analysis is to establish small spectral gaps in moment
matrices derived from solutions to sum-of-squares relaxations. To enable this
analysis we augment sum-of-squares relaxations with spectral analogs of maximum
entropy constraints.Comment: to appear in FOCS 201
Efficiently Learning Neural Networks: What Assumptions May Suffice?
Understanding when neural networks can be learned efficiently is a
fundamental question in learning theory. Existing hardness results suggest that
assumptions on both the input distribution and the network's weights are
necessary for obtaining efficient algorithms. Moreover, it was previously shown
that depth- networks can be efficiently learned under the assumptions that
the input distribution is Gaussian, and the weight matrix is non-degenerate. In
this work, we study whether such assumptions may suffice for learning deeper
networks and prove negative results. We show that learning depth- ReLU
networks under the Gaussian input distribution is hard even in the
smoothed-analysis framework, where a random noise is added to the network's
parameters. It implies that learning depth- ReLU networks under the Gaussian
distribution is hard even if the weight matrices are non-degenerate. Moreover,
we consider depth- networks, and show hardness of learning in the
smoothed-analysis framework, where both the network parameters and the input
distribution are smoothed. Our hardness results are under a well-studied
assumption on the existence of local pseudorandom generators.Comment: arXiv admin note: text overlap with arXiv:2101.0830
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