23 research outputs found

    Learning circuits with few negations

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    Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in any circuit computing them, a complexity measure that interpolates between monotone functions and the class of all functions. We study this generalization of monotonicity from the vantage point of learning theory, giving near-matching upper and lower bounds on the uniform-distribution learnability of circuits in terms of the number of negations they contain. Our upper bounds are based on a new structural characterization of negation-limited circuits that extends a classical result of A. A. Markov. Our lower bounds, which employ Fourier-analytic tools from hardness amplification, give new results even for circuits with no negations (i.e. monotone functions)

    Learning Circuits with few Negations

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    Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in any circuit computing them, a complexity measure that interpolates between monotone functions and the class of all functions. We study this generalization of monotonicity from the vantage point of learning theory, establishing nearly matching upper and lower bounds on the uniform-distribution learnability of circuits in terms of the number of negations they contain. Our upper bounds are based on a new structural characterization of negation-limited circuits that extends a classical result of A.A. Markov. Our lower bounds, which employ Fourier-analytic tools from hardness amplification, give new results even for circuits with no negations (i.e. monotone functions)

    Monotone Boolean Functions with s Zeros Farthest from Threshold Functions

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    Let TtT_t denote the tt-threshold function on the nn-cube: Tt(x)=1T_t(x) = 1 if ∣{i:xi=1}∣≥t|\{i : x_i=1\}| \geq t, and 00 otherwise. Define the distance between Boolean functions gg and hh, d(g,h)d(g,h), to be the number of points on which gg and hh disagree. We consider the following extremal problem: Over a monotone Boolean function gg on the nn-cube with ss zeros, what is the maximum of d(g,Tt)d(g,T_t)? We show that the following monotone function psp_s maximizes the distance: For x∈{0,1}nx \in \{0,1\}^n, ps(x)=0p_s(x)=0 if and only if N(x)<sN(x) < s, where N(x)N(x) is the integer whose nn-bit binary representation is xx. Our result generalizes the previous work for the case t=⌈n/2⌉t=\lceil n/2 \rceil and s=2n−1s=2^{n-1} by Blum, Burch, and Langford [BBL98-FOCS98], who considered the problem to analyze the behavior of a learning algorithm for monotone Boolean functions, and the previous work for the same tt and ss by Amano and Maruoka [AM02-ALT02]

    A Complete Characterization of Statistical Query Learning with Applications to Evolvability

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    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

    Approximate resilience, monotonicity, and the complexity of agnostic learning

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    A function ff is dd-resilient if all its Fourier coefficients of degree at most dd are zero, i.e., ff is uncorrelated with all low-degree parities. We study the notion of approximate\mathit{approximate} resilience\mathit{resilience} of Boolean functions, where we say that ff is α\alpha-approximately dd-resilient if ff is α\alpha-close to a [−1,1][-1,1]-valued dd-resilient function in ℓ1\ell_1 distance. We show that approximate resilience essentially characterizes the complexity of agnostic learning of a concept class CC over the uniform distribution. Roughly speaking, if all functions in a class CC are far from being dd-resilient then CC can be learned agnostically in time nO(d)n^{O(d)} and conversely, if CC contains a function close to being dd-resilient then agnostic learning of CC in the statistical query (SQ) framework of Kearns has complexity of at least nΩ(d)n^{\Omega(d)}. This characterization is based on the duality between ℓ1\ell_1 approximation by degree-dd polynomials and approximate dd-resilience that we establish. In particular, it implies that ℓ1\ell_1 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 α\alpha-approximately Ω~(αn)\widetilde{\Omega}(\alpha\sqrt{n})-resilient monotone functions for all α>0\alpha>0. Prior to our work, it was conceivable even that every monotone function is Ω(1)\Omega(1)-far from any 11-resilient function. Furthermore, we construct simple, explicit monotone functions based on Tribes{\sf Tribes} and CycleRun{\sf CycleRun} 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

    Convex Influences

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    On the nonlinearity of monotone Boolean functions

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    We first prove the truthfulness of a conjecture on the nonlinearity of monotone Boolean functions in even dimension, proposed in the recent paper ``Cryptographic properties of monotone Boolean functions , by D. Joyner, P. Stanica, D. Tang and the author, to appear in the Journal of Mathematical Cryptology. We prove then an upper bound on such nonlinearity, which is asymptotically much stronger than the conjectured upper bound and than the upper bound proved for odd dimension in this same paper. This bound shows a deep weakness of monotone Boolean functions; they are too closely approximated by affine functions for being usable as nonlinear components in cryptographic applications. We deduce a necessary criterion to be satisfied by a Boolean (resp. vectorial) function for being nonlinear

    KKL, Kruskal-Katona, and Monotone Nets

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