97 research outputs found
Distribution-Independent Evolvability of Linear Threshold Functions
Valiant's (2007) model of evolvability models the evolutionary process of
acquiring useful functionality as a restricted form of learning from random
examples. Linear threshold functions and their various subclasses, such as
conjunctions and decision lists, play a fundamental role in learning theory and
hence their evolvability has been the primary focus of research on Valiant's
framework (2007). One of the main open problems regarding the model is whether
conjunctions are evolvable distribution-independently (Feldman and Valiant,
2008). We show that the answer is negative. Our proof is based on a new
combinatorial parameter of a concept class that lower-bounds the complexity of
learning from correlations.
We contrast the lower bound with a proof that linear threshold functions
having a non-negligible margin on the data points are evolvable
distribution-independently via a simple mutation algorithm. Our algorithm
relies on a non-linear loss function being used to select the hypotheses
instead of 0-1 loss in Valiant's (2007) original definition. The proof of
evolvability requires that the loss function satisfies several mild conditions
that are, for example, satisfied by the quadratic loss function studied in
several other works (Michael, 2007; Feldman, 2009; Valiant, 2010). An important
property of our evolution algorithm is monotonicity, that is the algorithm
guarantees evolvability without any decreases in performance. Previously,
monotone evolvability was only shown for conjunctions with quadratic loss
(Feldman, 2009) or when the distribution on the domain is severely restricted
(Michael, 2007; Feldman, 2009; Kanade et al., 2010
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
A Complete Characterization of Statistical Query Learning with Applications to Evolvability
Statistical query (SQ) learning model of Kearns (1993) is a natural
restriction of the PAC learning model in which a learning algorithm is allowed
to obtain estimates of statistical properties of the examples but cannot see
the examples themselves. We describe a new and simple characterization of the
query complexity of learning in the SQ learning model. Unlike the previously
known bounds on SQ learning our characterization preserves the accuracy and the
efficiency of learning. The preservation of accuracy implies that that our
characterization gives the first characterization of SQ learning in the
agnostic learning framework. The preservation of efficiency is achieved using a
new boosting technique and allows us to derive a new approach to the design of
evolutionary algorithms in Valiant's (2006) model of evolvability. We use this
approach to demonstrate the existence of a large class of monotone evolutionary
learning algorithms based on square loss performance estimation. These results
differ significantly from the few known evolutionary algorithms and give
evidence that evolvability in Valiant's model is a more versatile phenomenon
than there had been previous reason to suspect.Comment: Simplified Lemma 3.8 and it's application
Sample Complexity Bounds on Differentially Private Learning via Communication Complexity
In this work we analyze the sample complexity of classification by
differentially private algorithms. Differential privacy is a strong and
well-studied notion of privacy introduced by Dwork et al. (2006) that ensures
that the output of an algorithm leaks little information about the data point
provided by any of the participating individuals. Sample complexity of private
PAC and agnostic learning was studied in a number of prior works starting with
(Kasiviswanathan et al., 2008) but a number of basic questions still remain
open, most notably whether learning with privacy requires more samples than
learning without privacy.
We show that the sample complexity of learning with (pure) differential
privacy can be arbitrarily higher than the sample complexity of learning
without the privacy constraint or the sample complexity of learning with
approximate differential privacy. Our second contribution and the main tool is
an equivalence between the sample complexity of (pure) differentially private
learning of a concept class (or ) and the randomized one-way
communication complexity of the evaluation problem for concepts from . Using
this equivalence we prove the following bounds:
1. , where is the Littlestone's (1987)
dimension characterizing the number of mistakes in the online-mistake-bound
learning model. Known bounds on then imply that can be much
higher than the VC-dimension of .
2. For any , there exists a class such that but .
3. For any , there exists a class such that the sample complexity of
(pure) -differentially private PAC learning is but
the sample complexity of the relaxed -differentially private
PAC learning is . This resolves an open problem of
Beimel et al. (2013b).Comment: Extended abstract appears in Conference on Learning Theory (COLT)
201
Optimal Bounds on Approximation of Submodular and XOS Functions by Juntas
We investigate the approximability of several classes of real-valued
functions by functions of a small number of variables ({\em juntas}). Our main
results are tight bounds on the number of variables required to approximate a
function within -error over
the uniform distribution: 1. If is submodular, then it is -close
to a function of variables.
This is an exponential improvement over previously known results. We note that
variables are necessary even for linear
functions. 2. If is fractionally subadditive (XOS) it is -close
to a function of variables. This result holds for all
functions with low total -influence and is a real-valued analogue of
Friedgut's theorem for boolean functions. We show that
variables are necessary even for XOS functions.
As applications of these results, we provide learning algorithms over the
uniform distribution. For XOS functions, we give a PAC learning algorithm that
runs in time . For submodular functions we give
an algorithm in the more demanding PMAC learning model (Balcan and Harvey,
2011) which requires a multiplicative factor approximation with
probability at least over the target distribution. Our uniform
distribution algorithm runs in time .
This is the first algorithm in the PMAC model that over the uniform
distribution can achieve a constant approximation factor arbitrarily close to 1
for all submodular functions. As follows from the lower bounds in (Feldman et
al., 2013) both of these algorithms are close to optimal. We also give
applications for proper learning, testing and agnostic learning with value
queries of these classes.Comment: Extended abstract appears in proceedings of FOCS 201
Agnostic Learning of Disjunctions on Symmetric Distributions
We consider the problem of approximating and learning disjunctions (or
equivalently, conjunctions) on symmetric distributions over .
Symmetric distributions are distributions whose PDF is invariant under any
permutation of the variables. We give a simple proof that for every symmetric
distribution , there exists a set of
functions , such that for every disjunction , there is function
, expressible as a linear combination of functions in , such
that -approximates in distance on or
. This directly
gives an agnostic learning algorithm for disjunctions on symmetric
distributions that runs in time . The best known
previous bound is and follows from approximation of the
more general class of halfspaces (Wimmer, 2010). We also show that there exists
a symmetric distribution , such that the minimum degree of a
polynomial that -approximates the disjunction of all variables is
distance on is . Therefore the
learning result above cannot be achieved via -regression with a
polynomial basis used in most other agnostic learning algorithms.
Our technique also gives a simple proof that for any product distribution
and every disjunction , there exists a polynomial of
degree such that -approximates in
distance on . This was first proved by Blais et al.
(2008) via a more involved argument
Learning Coverage Functions and Private Release of Marginals
We study the problem of approximating and learning coverage functions. A
function is a coverage function, if
there exists a universe with non-negative weights for each
and subsets of such that . Alternatively, coverage functions can be described
as non-negative linear combinations of monotone disjunctions. They are a
natural subclass of submodular functions and arise in a number of applications.
We give an algorithm that for any , given random and uniform
examples of an unknown coverage function , finds a function that
approximates within factor on all but -fraction of the
points in time . This is the first fully-polynomial
algorithm for learning an interesting class of functions in the demanding PMAC
model of Balcan and Harvey (2011). Our algorithms are based on several new
structural properties of coverage functions. Using the results in (Feldman and
Kothari, 2014), we also show that coverage functions are learnable agnostically
with excess -error over all product and symmetric
distributions in time . In contrast, we show that,
without assumptions on the distribution, learning coverage functions is at
least as hard as learning polynomial-size disjoint DNF formulas, a class of
functions for which the best known algorithm runs in time
(Klivans and Servedio, 2004).
As an application of our learning results, we give simple
differentially-private algorithms for releasing monotone conjunction counting
queries with low average error. In particular, for any , we obtain
private release of -way marginals with average error in time
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