Learning DNFs under product distributions via {\mu}-biased quantum Fourier sampling

We show that DNF formulae can be quantum PAC-learned in polynomial time under product distributions using a quantum example oracle. The best classical algorithm (without access to membership queries) runs in superpolynomial time. Our result extends the work by Bshouty and Jackson (1998) that proved that DNF formulae are efficiently learnable under the uniform distribution using a quantum example oracle. Our proof is based on a new quantum algorithm that efficiently samples the coefficients of a {\mu}-biased Fourier transform.Comment: 17 pages; v3 based on journal version; minor corrections and clarification

Pac-Learning Recursive Logic Programs: Efficient Algorithms

We present algorithms that learn certain classes of function-free recursive logic programs in polynomial time from equivalence queries. In particular, we show that a single k-ary recursive constant-depth determinate clause is learnable. Two-clause programs consisting of one learnable recursive clause and one constant-depth determinate non-recursive clause are also learnable, if an additional ``basecase'' oracle is assumed. These results immediately imply the pac-learnability of these classes. Although these classes of learnable recursive programs are very constrained, it is shown in a companion paper that they are maximally general, in that generalizing either class in any natural way leads to a computationally difficult learning problem. Thus, taken together with its companion paper, this paper establishes a boundary of efficient learnability for recursive logic programs.Comment: See http://www.jair.org/ for any accompanying file

Complexity of Equivalence and Learning for Multiplicity Tree Automata

We consider the complexity of equivalence and learning for multiplicity tree automata, i.e., weighted tree automata over a field. We first show that the equivalence problem is logspace equivalent to polynomial identity testing, the complexity of which is a longstanding open problem. Secondly, we derive lower bounds on the number of queries needed to learn multiplicity tree automata in Angluin's exact learning model, over both arbitrary and fixed fields. Habrard and Oncina (2006) give an exact learning algorithm for multiplicity tree automata, in which the number of queries is proportional to the size of the target automaton and the size of a largest counterexample, represented as a tree, that is returned by the Teacher. However, the smallest tree-counterexample may be exponential in the size of the target automaton. Thus the above algorithm does not run in time polynomial in the size of the target automaton, and has query complexity exponential in the lower bound. Assuming a Teacher that returns minimal DAG representations of counterexamples, we give a new exact learning algorithm whose query complexity is quadratic in the target automaton size, almost matching the lower bound, and improving the best previously-known algorithm by an exponential factor

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

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

Learning probability distributions generated by finite-state machines

We review methods for inference of probability distributions generated by probabilistic automata and related models for sequence generation. We focus on methods that can be proved to learn in the inference in the limit and PAC formal models. The methods we review are state merging and state splitting methods for probabilistic deterministic automata and the recently developed spectral method for nondeterministic probabilistic automata. In both cases, we derive them from a high-level algorithm described in terms of the Hankel matrix of the distribution to be learned, given as an oracle, and then describe how to adapt that algorithm to account for the error introduced by a finite sample.Peer ReviewedPostprint (author's final draft

The Consistency dimension and distribution-dependent learning from queries

We prove a new combinatorial characterization of polynomial learnability from equivalence queries, and state some of its consequences relating the learnability of a class with the learnability via equivalence and membership queries of its subclasses obtained by restricting the instance space. Then we propose and study two models of query learning in which there is a probability distribution on the instance space, both as an application of the tools developed from the combinatorial characterization and as models of independent interest.Postprint (published version