113 research outputs found
Learning circuits with few negations
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)
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
Learning Circuits with few Negations
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)
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
Stabbing Planes
We introduce and develop a new semi-algebraic proof system, called Stabbing Planes that is in the style of DPLL-based modern SAT solvers. As with DPLL, there is only one rule: the current polytope can be subdivided by branching on an inequality and its "integer negation." That is, we can (nondeterministically choose) a hyperplane a x >= b with integer coefficients, which partitions the polytope into three pieces: the points in the polytope satisfying a x >= b, the points satisfying a x <= b-1, and the middle slab b-1 < a x < b. Since the middle slab contains no integer points it can be safely discarded, and the algorithm proceeds recursively on the other two branches. Each path terminates when the current polytope is empty, which is polynomial-time checkable. Among our results, we show somewhat surprisingly that Stabbing Planes can efficiently simulate Cutting Planes, and moreover, is strictly stronger than Cutting Planes under a reasonable conjecture. We prove linear lower bounds on the rank of Stabbing Planes refutations, by adapting
a lifting argument in communication complexity
Weighted Polynomial Approximations: Limits for Learning and Pseudorandomness
Polynomial approximations to boolean functions have led to many positive
results in computer science. In particular, polynomial approximations to the
sign function underly algorithms for agnostically learning halfspaces, as well
as pseudorandom generators for halfspaces. In this work, we investigate the
limits of these techniques by proving inapproximability results for the sign
function.
Firstly, the polynomial regression algorithm of Kalai et al. (SIAM J. Comput.
2008) shows that halfspaces can be learned with respect to log-concave
distributions on in the challenging agnostic learning model. The
power of this algorithm relies on the fact that under log-concave
distributions, halfspaces can be approximated arbitrarily well by low-degree
polynomials. We ask whether this technique can be extended beyond log-concave
distributions, and establish a negative result. We show that polynomials of any
degree cannot approximate the sign function to within arbitrarily low error for
a large class of non-log-concave distributions on the real line, including
those with densities proportional to .
Secondly, we investigate the derandomization of Chernoff-type concentration
inequalities. Chernoff-type tail bounds on sums of independent random variables
have pervasive applications in theoretical computer science. Schmidt et al.
(SIAM J. Discrete Math. 1995) showed that these inequalities can be established
for sums of random variables with only -wise independence,
for a tail probability of . We show that their results are tight up to
constant factors.
These results rely on techniques from weighted approximation theory, which
studies how well functions on the real line can be approximated by polynomials
under various distributions. We believe that these techniques will have further
applications in other areas of computer science.Comment: 22 page
Recommended from our members
Unconditional Lower Bounds in Complexity Theory
This work investigates the hardness of solving natural computational problems according to different complexity measures. Our results and techniques span several areas in theoretical computer science and discrete mathematics. They have in common the following aspects: (i) the results are unconditional, i.e., they rely on no unproven hardness assumption from complexity theory; (ii) the corresponding lower bounds are essentially optimal. Among our contributions, we highlight the following results.
Constraint Satisfaction Problems and Monotone Complexity. We introduce a natural formulation of the satisfiability problem as a monotone function, and prove a near-optimal 2^{Ω (n/log n)} lower bound on the size of monotone formulas solving k-SAT on n-variable instances (for a large enough k ∈ ℕ). More generally, we investigate constraint satisfaction problems according to the geometry of their constraints, i.e., as a function of the hypergraph describing which variables appear in each constraint. Our results show in a certain technical sense that the monotone circuit depth complexity of the satisfiability problem is polynomially related to the tree-width of the corresponding graphs.
Interactive Protocols and Communication Complexity. We investigate interactive compression protocols, a hybrid model between computational complexity and communication complexity. We prove that the communication complexity of the Majority function on n-bit inputs with respect to Boolean circuits of size s and depth d extended with modulo p gates is precisely n/log^{ϴ(d)} s, where p is a fixed prime number, and d ∈ ℕ. Further, we establish a strong round-separation theorem for bounded-depth circuits, showing that (r+1)-round protocols can be substantially more efficient than r-round protocols, for every r ∈ ℕ.
Negations in Computational Learning Theory. We study the learnability of circuits containing a given number of negation gates, a measure that interpolates between monotone functions, and the class of all functions. Let C^t_n be the class of Boolean functions on n input variables that can be computed by Boolean circuits with at most t negations. We prove that any algorithm that learns every f ∈ C^t_n with membership queries according to the uniform distribution to accuracy ε has query complexity 2^{Ω (2^t sqrt(n)/ε)} (for a large range of these parameters). Moreover, we give an algorithm that learns C^t_n from random examples only, and with a running time that essentially matches this information-theoretic lower bound.
Negations in Theory of Cryptography. We investigate the power of negation gates in cryptography and related areas, and prove that many basic cryptographic primitives require essentially the maximum number of negations among all Boolean functions. In other words, cryptography is highly non-monotone. Our results rely on a variety of techniques, and give near-optimal lower bounds for pseudorandom functions, error-correcting codes, hardcore predicates, randomness extractors, and small-bias generators.
Algorithms versus Circuit Lower Bounds. We strengthen a few connections between algorithms and circuit lower bounds. We show that the design of faster algorithms in some widely investigated learning models would imply new unconditional lower bounds in complexity theory. In addition, we prove that the existence of non-trivial satisfiability algorithms for certain classes of Boolean circuits of depth d+2 leads to lower bounds for the corresponding class of circuits of depth d. These results show that either there are no faster algorithms for some computational tasks, or certain circuit lower bounds hold
Lower Bounds for Secret-Sharing Schemes for k-Hypergraphs
A secret-sharing scheme enables a dealer, holding a secret string, to distribute shares to parties such that only pre-defined authorized subsets of parties can reconstruct the secret. The collection of authorized sets is called an access structure. There is a huge gap between the best known upper bounds on the share size of a secret-sharing scheme realizing an arbitrary access structure and the best known lower bounds on the size of these shares. For an arbitrary -party access structure, the best known upper bound on the share size is . On the other hand, the best known lower bound on the total share size is much smaller, i.e., [Csirmaz, \emph{Studia Sci. Math. Hungar.}]. This lower bound was proved more than 25 years ago and no major progress has been made since.
In this paper, we study secret-sharing schemes for -hypergraphs, i.e., for access structures where all minimal authorized sets are of size exactly (however, unauthorized sets can be larger). We consider the case where is small, i.e., constant or at most . The trivial upper bound for these access structures is and this can be slightly improved. If there were efficient secret-sharing schemes for such -hypergraphs (e.g., -hypergraphs or -hypergraphs), then we would be able to construct secret-sharing schemes for arbitrary access structures that are better than the best known schemes. Thus, understanding the share size required for -hypergraphs is important. Prior to our work, the best known lower bound for these access structures was , which holds already for graphs (i.e., -hypergraphs). We improve this lower bound, proving a lower bound of on the total share size for some explicit -hypergraphs, where . For example, for -hypergraphs we prove a lower bound of . For -hypergraphs, we prove a lower bound of , i.e., we show that the lower bound of Csirmaz holds already when all minimal authorized sets are of size . Our proof is simple and shows that the lower bound of Csirmaz holds for a simple variant of the access structure considered by Csirmaz. Using our results, we prove a near quadratic separation between the required share size for realizing an explicit access structure and the monotone circuit size describing the access structure,i.e., the share size in and the monotone circuit size is (where the circuit has depth )
Theoretical and Practical Approaches for Hardness Amplification of PUFs
The era of PUFs has been characterized by the efforts put into research and the development of PUFs that are robust against attacks, in particular, machine learning (ML) attacks. In the lack of systematic and provable methods for this purpose, we have witnessed the ever-continuing competition between PUF designers/ manufacturers, cryptanalysts, and of course, adversaries that maliciously break the security of PUFs. This is despite a series of acknowledged principles developed in cryptography and complexity theory, under the umbrella term ``hardness amplification. The goal of studies on the hardness amplification is to build a strongly secure construction out of considerably weaker primitives. This paper aims at narrowing the gap between these studies and hardware security, specifically for applications in the domain of PUFs. To this end, we first review an example of practical efforts made to construct more secure PUFs, namely the concept of rolling PUFs. Based on what can be learned from this and central insights provided by the ML and complexity theory, we propose a new PUF-based scheme built around the idea of using a new function, namely, the Tribes function, which combines the outputs of a set of PUFs to generate the final response. Our theoretical findings are discussed in an exhaustive manner and supported by the results of experiments, conducted extensively on real-world PUFs
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
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