9 research outputs found
Quantum Algorithms for Learning and Testing Juntas
In this article we develop quantum algorithms for learning and testing
juntas, i.e. Boolean functions which depend only on an unknown set of k out of
n input variables. Our aim is to develop efficient algorithms:
- whose sample complexity has no dependence on n, the dimension of the domain
the Boolean functions are defined over;
- with no access to any classical or quantum membership ("black-box")
queries. Instead, our algorithms use only classical examples generated
uniformly at random and fixed quantum superpositions of such classical
examples;
- which require only a few quantum examples but possibly many classical
random examples (which are considered quite "cheap" relative to quantum
examples).
Our quantum algorithms are based on a subroutine FS which enables sampling
according to the Fourier spectrum of f; the FS subroutine was used in earlier
work of Bshouty and Jackson on quantum learning. Our results are as follows:
- We give an algorithm for testing k-juntas to accuracy that uses
quantum examples. This improves on the number of examples used
by the best known classical algorithm.
- We establish the following lower bound: any FS-based k-junta testing
algorithm requires queries.
- We give an algorithm for learning -juntas to accuracy that
uses quantum examples and
random examples. We show that this learning algorithms is close to optimal by
giving a related lower bound.Comment: 15 pages, 1 figure. Uses synttree package. To appear in Quantum
Information Processin
Detecting controlling nodes of boolean regulatory networks
Boolean models of regulatory networks are assumed to be tolerant to perturbations. That qualitatively implies that each function can only depend on a few nodes. Biologically motivated constraints further show that functions found in Boolean regulatory networks belong to certain classes of functions, for example, the unate functions. It turns out that these classes have specific properties in the Fourier domain. That motivates us to study the problem of detecting controlling nodes in classes of Boolean networks using spectral techniques. We consider networks with unbalanced functions and functions of an average sensitivity less than 23k, where k is the number of controlling variables for a function. Further, we consider the class of 1-low networks which include unate networks, linear threshold networks, and networks with nested canalyzing functions. We show that the application of spectral learning algorithms leads to both better time and sample complexity for the detection of controlling nodes compared with algorithms based on exhaustive search. For a particular algorithm, we state analytical upper bounds on the number of samples needed to find the controlling nodes of the Boolean functions. Further, improved algorithms for detecting controlling nodes in large-scale unate networks are given and numerically studied
Multiple Random Oracles Are Better Than One
We study the problem of learning k-juntas given access to examples drawn from a number of different product distributions. Thus we wish to learn a function f: {ā1, 1}n ā {ā1, 1} that depends on k (unknown) coordinates. While the best-known algorithms for the general problem of learning a k-junta require running times of nk poly(n, 2k), we show that, given access to k different product distributions with biases separated by Ī³ \u3e 0, the functions may be learned in time poly(n, 2k, Ī³āk). More generally, given access to t ā¤ k different product distributions, the functions may be learned in time nk/tpoly(n, 2k, Ī³āk). Our techniques involve novel results in Fourier analysis, relating Fourier expansions with respect to different biases, and a generalization of Russo\u27s formula
Nearly Optimal Algorithms for Testing and Learning Quantum Junta Channels
We consider the problems of testing and learning quantum -junta channels,
which are -qubit to -qubit quantum channels acting non-trivially on at
most out of qubits and leaving the rest of qubits unchanged. We show
the following.
1. An -query algorithm to distinguish
whether the given channel is -junta channel or is far from any -junta
channels, and a lower bound on the number of
queries;
2. An -query algorithm to learn a -junta
channel, and a lower bound on the number of queries.
This answers an open problem raised by Chen et al. (2023). In order to settle
these problems, we develop a Fourier analysis framework over the space of
superoperators and prove several fundamental properties, which extends the
Fourier analysis over the space of operators introduced in Montanaro and
Osborne (2010)
A Strong Composition Theorem for Junta Complexity and the Boosting of Property Testers
We prove a strong composition theorem for junta complexity and show how such
theorems can be used to generically boost the performance of property testers.
The -approximate junta complexity of a function is the
smallest integer such that is -close to a function that
depends only on variables. A strong composition theorem states that if
has large -approximate junta complexity, then has even
larger -approximate junta complexity, even for . We develop a fairly complete understanding of this behavior,
proving that the junta complexity of is characterized by that of
along with the multivariate noise sensitivity of . For the important
case of symmetric functions , we relate their multivariate noise sensitivity
to the simpler and well-studied case of univariate noise sensitivity.
We then show how strong composition theorems yield boosting algorithms for
property testers: with a strong composition theorem for any class of functions,
a large-distance tester for that class is immediately upgraded into one for
small distances. Combining our contributions yields a booster for junta
testers, and with it new implications for junta testing. This is the first
boosting-type result in property testing, and we hope that the connection to
composition theorems adds compelling motivation to the study of both topics.Comment: 44 pages, 1 figure, FOCS 202
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Combinatorics, Probability and Computing
One of the exciting phenomena in mathematics in recent years has been the widespread and surprisingly eļ¬ective use of probabilistic methods in diverse areas. The probabilistic point of view has turned out to b
Learning Juntas in the Presence of Noise
We investigate the combination of two major challenges in computational learning: dealing with huge amounts of irrelevant information and learning from noisy data. It is shown that large classes of Boolean concepts that depend only on a small fraction of their variablesāso-called juntasācan be learned efficiently from uniformly distributed examples that are corrupted by random attribute and classification noise. We present solutions to cope with the manifold problems that inhibit a straightforward generalization of the noise-free case. Additionally, we extend our methods to non-uniformly distributed examples and derive new results for monotone juntas and for parity juntas in this setting. It is assumed that the attribute noise is generated by a product distribution. Without any restrictions of the attribute noise distribution, learning in the presence of noise is in general impossible. This follows from our construction of a noise distribution P and a concept class C such that it is impossible to learn C under P-noise