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

Categorical invariance and structural complexity in human concept learning

An alternative account of human concept learning based on an invariance measure of the categorical\ud stimulus is proposed. The categorical invariance model (CIM) characterizes the degree of structural\ud complexity of a Boolean category as a function of its inherent degree of invariance and its cardinality or\ud size. To do this we introduce a mathematical framework based on the notion of a Boolean differential\ud operator on Boolean categories that generates the degrees of invariance (i.e., logical manifold) of the\ud category in respect to its dimensions. Using this framework, we propose that the structural complexity\ud of a Boolean category is indirectly proportional to its degree of categorical invariance and directly\ud proportional to its cardinality or size. Consequently, complexity and invariance notions are formally\ud unified to account for concept learning difficulty. Beyond developing the above unifying mathematical\ud framework, the CIM is significant in that: (1) it precisely predicts the key learning difficulty ordering of\ud the SHJ [Shepard, R. N., Hovland, C. L.,&Jenkins, H. M. (1961). Learning and memorization of classifications.\ud Psychological Monographs: General and Applied, 75(13), 1-42] Boolean category types consisting of three\ud binary dimensions and four positive examples; (2) it is, in general, a good quantitative predictor of the\ud degree of learning difficulty of a large class of categories (in particular, the 41 category types studied\ud by Feldman [Feldman, J. (2000). Minimization of Boolean complexity in human concept learning. Nature,\ud 407, 630-633]); (3) it is, in general, a good quantitative predictor of parity effects for this large class of\ud categories; (4) it does all of the above without free parameters; and (5) it is cognitively plausible (e.g.,\ud cognitively tractable)

Improve SAT-solving with Machine Learning

In this project, we aimed to improve the runtime of Minisat, a Conflict-Driven Clause Learning (CDCL) solver that solves the Propositional Boolean Satisfiability (SAT) problem. We first used a logistic regression model to predict the satisfiability of propositional boolean formulae after fixing the values of a certain fraction of the variables in each formula. We then applied the logistic model and added a preprocessing period to Minisat to determine the preferable initial value (either true or false) of each boolean variable using a Monte-Carlo approach. Concretely, for each Monte-Carlo trial, we fixed the values of a certain ratio of randomly selected variables, and calculated the confidence that the resulting sub-formula is satisfiable with our logistic regression model. The initial value of each variable was set based on the mean confidence scores of the trials that started from the literals of that variable. We were particularly interested in setting the initial values of the backbone variables correctly, which are variables that have the same value in all solutions of a SAT formula. Our Monte-Carlo method was able to set 78% of the backbones correctly. Excluding the preprocessing time, compared with the default setting of Minisat, the runtime of Minisat for satisfiable formulae decreased by 23%. However, our method did not outperform vanilla Minisat in runtime, as the decrease in the conflicts was outweighed by the long runtime of the preprocessing period.Comment: 2 pages, SIGCSE SRC 201

Learning DNF Expressions from Fourier Spectrum

Since its introduction by Valiant in 1984, PAC learning of DNF expressions remains one of the central problems in learning theory. We consider this problem in the setting where the underlying distribution is uniform, or more generally, a product distribution. Kalai, Samorodnitsky and Teng (2009) showed that in this setting a DNF expression can be efficiently approximated from its "heavy" low-degree Fourier coefficients alone. This is in contrast to previous approaches where boosting was used and thus Fourier coefficients of the target function modified by various distributions were needed. This property is crucial for learning of DNF expressions over smoothed product distributions, a learning model introduced by Kalai et al. (2009) and inspired by the seminal smoothed analysis model of Spielman and Teng (2001). We introduce a new approach to learning (or approximating) a polynomial threshold functions which is based on creating a function with range [-1,1] that approximately agrees with the unknown function on low-degree Fourier coefficients. We then describe conditions under which this is sufficient for learning polynomial threshold functions. Our approach yields a new, simple algorithm for approximating any polynomial-size DNF expression from its "heavy" low-degree Fourier coefficients alone. Our algorithm greatly simplifies the proof of learnability of DNF expressions over smoothed product distributions. We also describe an application of our algorithm to learning monotone DNF expressions over product distributions. Building on the work of Servedio (2001), we give an algorithm that runs in time \poly((s \cdot \log{(s/\eps)})^{\log{(s/\eps)}}, n), where $s$ is the size of the target DNF expression and \eps is the accuracy. This improves on \poly((s \cdot \log{(ns/\eps)})^{\log{(s/\eps)} \cdot \log{(1/\eps)}}, n) bound of Servedio (2001).Comment: Appears in Conference on Learning Theory (COLT) 201

Towards a Law of Invariance in Human Concept Learning

Invariance principles underlie many key theories in modern science. They provide the explanatory and predictive framework necessary for the rigorous study of natural phenomena ranging from the structure of crystals, to magnetism, to relativistic mechanics. Vigo (2008, 2009)introduced a new general notion and principle of invariance from which two parameter-free (ratio and exponential) models were derived to account for human conceptual behavior. Here we introduce a new parameterized \ud exponential “law” based on the same invariance principle. The law accurately predicts the subjective degree of difficulty that humans experience when learning different types of concepts. In addition, it precisely fits the data from a large-scale experiment which examined a total of 84 category structures across 10 category families (R-Squared =.97, p < .0001; r= .98, p < .0001). Moreover, it overcomes seven key challenges that had, hitherto, been grave obstacles for theories of concept learning

Formal Modeling of Connectionism using Concurrency Theory, an Approach Based on Automata and Model Checking

This paper illustrates a framework for applying formal methods techniques, which are symbolic in nature, to specifying and verifying neural networks, which are sub-symbolic in nature. The paper describes a communicating automata [Bowman &amp; Gomez, 2006] model of neural networks. We also implement the model using timed automata [Alur &amp; Dill, 1994] and then undertake a verification of these models using the model checker Uppaal [Pettersson, 2000] in order to evaluate the performance of learning algorithms. This paper also presents discussion of a number of broad issues concerning cognitive neuroscience and the debate as to whether symbolic processing or connectionism is a suitable representation of cognitive systems. Additionally, the issue of integrating symbolic techniques, such as formal methods, with complex neural networks is discussed. We then argue that symbolic verifications may give theoretically well-founded ways to evaluate and justify neural learning systems in the field of both theoretical research and real world applications

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