7,303 research outputs found
Approximate resilience, monotonicity, and the complexity of agnostic learning
A function is -resilient if all its Fourier coefficients of degree at
most are zero, i.e., is uncorrelated with all low-degree parities. We
study the notion of of Boolean
functions, where we say that is -approximately -resilient if
is -close to a -valued -resilient function in
distance. We show that approximate resilience essentially characterizes the
complexity of agnostic learning of a concept class over the uniform
distribution. Roughly speaking, if all functions in a class are far from
being -resilient then can be learned agnostically in time and
conversely, if contains a function close to being -resilient then
agnostic learning of in the statistical query (SQ) framework of Kearns has
complexity of at least . This characterization is based on the
duality between approximation by degree- polynomials and
approximate -resilience that we establish. In particular, it implies that
approximation by low-degree polynomials, known to be sufficient for
agnostic learning over product distributions, is in fact necessary.
Focusing on monotone Boolean functions, we exhibit the existence of
near-optimal -approximately
-resilient monotone functions for all
. Prior to our work, it was conceivable even that every monotone
function is -far from any -resilient function. Furthermore, we
construct simple, explicit monotone functions based on and that are close to highly resilient functions. Our constructions are
based on a fairly general resilience analysis and amplification. These
structural results, together with the characterization, imply nearly optimal
lower bounds for agnostic learning of monotone juntas
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
Privately Releasing Conjunctions and the Statistical Query Barrier
Suppose we would like to know all answers to a set of statistical queries C
on a data set up to small error, but we can only access the data itself using
statistical queries. A trivial solution is to exhaustively ask all queries in
C. Can we do any better?
+ We show that the number of statistical queries necessary and sufficient for
this task is---up to polynomial factors---equal to the agnostic learning
complexity of C in Kearns' statistical query (SQ) model. This gives a complete
answer to the question when running time is not a concern.
+ We then show that the problem can be solved efficiently (allowing arbitrary
error on a small fraction of queries) whenever the answers to C can be
described by a submodular function. This includes many natural concept classes,
such as graph cuts and Boolean disjunctions and conjunctions.
While interesting from a learning theoretic point of view, our main
applications are in privacy-preserving data analysis:
Here, our second result leads to the first algorithm that efficiently
releases differentially private answers to of all Boolean conjunctions with 1%
average error. This presents significant progress on a key open problem in
privacy-preserving data analysis.
Our first result on the other hand gives unconditional lower bounds on any
differentially private algorithm that admits a (potentially
non-privacy-preserving) implementation using only statistical queries. Not only
our algorithms, but also most known private algorithms can be implemented using
only statistical queries, and hence are constrained by these lower bounds. Our
result therefore isolates the complexity of agnostic learning in the SQ-model
as a new barrier in the design of differentially private algorithms
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
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Local search: A guide for the information retrieval practitioner
There are a number of combinatorial optimisation problems in information retrieval in which the use of local search methods are worthwhile. The purpose of this paper is to show how local search can be used to solve some well known tasks in information retrieval (IR), how previous research in the field is piecemeal, bereft of a structure and methodologically flawed, and to suggest more rigorous ways of applying local search methods to solve IR problems. We provide a query based taxonomy for analysing the use of local search in IR tasks and an overview of issues such as fitness functions, statistical significance and test collections when conducting experiments on combinatorial optimisation problems. The paper gives a guide on the pitfalls and problems for IR practitioners who wish to use local search to solve their research issues, and gives practical advice on the use of such methods. The query based taxonomy is a novel structure which can be used by the IR practitioner in order to examine the use of local search in IR
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