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

PAC Classification based on PAC Estimates of Label Class Distributions

A standard approach in pattern classification is to estimate the distributions of the label classes, and then to apply the Bayes classifier to the estimates of the distributions in order to classify unlabeled examples. As one might expect, the better our estimates of the label class distributions, the better the resulting classifier will be. In this paper we make this observation precise by identifying risk bounds of a classifier in terms of the quality of the estimates of the label class distributions. We show how PAC learnability relates to estimates of the distributions that have a PAC guarantee on their $L_1$ distance from the true distribution, and we bound the increase in negative log likelihood risk in terms of PAC bounds on the KL-divergence. We give an inefficient but general-purpose smoothing method for converting an estimated distribution that is good under the $L_1$ metric into a distribution that is good under the KL-divergence.Comment: 14 page

Bootstrapping and learning PDFA in data streams

Best Student Paper ICGI 2012Markovian models with hidden state are widely-used formalisms for modeling sequential phenomena. Learnability of these models has been well studied when the sample is given in batch mode, and algorithms with PAC-like learning guarantees exist for specic classes of models such as Probabilistic Deterministic Finite Automata (PDFA). Here we focus on PDFA and give an algorithm for infering models in this class under the stringent data stream scenario: unlike existing methods, our algorithm works incrementally and in one pass, uses memory sublinear in the stream length, and processes input items in amortized constant time. We provide rigorous PAC-like bounds for all of the above, as well as an evaluation on synthetic data showing that the algorithm performs well in practice. Our algorithm makes a key usage of several old and new sketching techniques. In particular, we develop a new sketch for implementing bootstrapping in a streaming setting which may be of independent interest. In experiments we have observed that this sketch yields important reductions in the examples required for performing some crucial statistical tests in our algorithm.Peer ReviewedAward-winningPostprint (published version

Pattern classification via unsupervised learners

We consider classification problems in a variant of the Probably Approximately Correct (PAC)-learning framework, in which an unsupervised learner creates a discriminant function over each class and observations are labeled by the learner returning the highest value associated with that observation. Consideration is given to whether this approach gains significant advantage over traditional discriminant techniques. It is shown that PAC-learning distributions over class labels under Ll distance or KL-divergence implies PAC classification in this framework. We give bounds on the regret associated with the resulting classifier, taking into account the possibility of variable misclassification penalties. We demonstrate the advantage of estimating the a posteriori probability distributions over class labels in the setting of Optical Character Recognition. We show that unsupervised learners can be used to learn a class of probabilistic concepts (stochastic rules denoting the probability that an observation has a positive label in a 2-class setting). This demonstrates a situation where unsupervised learners can be used even when it is hard to learn distributions over class labels - in this case the discriminant functions do not estimate the class probability densities. We use a standard state-merging technique to PAC-learn a class of probabilistic automata and show that by learning the distribution over outputs under the weaker L1 distance rather than KL-divergence we are able to learn without knowledge of the expected length of an output. It is also shown that for a restricted class of these automata learning under L1 distance is equivalent to learning under KL-divergence

On the Learnability of Shuffle Ideals

PAC learning of unrestricted regular languages is long known to be a difficult problem. The class of shuffle ideals is a very restricted subclass of regular languages, where the shuffle ideal generated by a string u is the collection of all strings containing u as a subsequence. This fundamental language family is of theoretical interest in its own right and provides the building blocks for other important language families. Despite its apparent simplicity, the class of shuffle ideals appears quite difficult to learn. In particular, just as for unrestricted regular languages, the class is not properly PAC learnable in polynomial time if RP 6= NP, and PAC learning the class improperly in polynomial time would imply polynomial time algorithms for certain fundamental problems in cryptography. In the positive direction, we give an efficient algorithm for properly learning shuffle ideals in the statistical query (and therefore also PAC) model under the uniform distribution.T-Party Projec

Pattern classification via unsupervised learners

We consider classification problems in a variant of the Probably Approximately Correct (PAC)-learning framework, in which an unsupervised learner creates a discriminant function over each class and observations are labeled by the learner returning the highest value associated with that observation. Consideration is given to whether this approach gains significant advantage over traditional discriminant techniques. It is shown that PAC-learning distributions over class labels under Ll distance or KL-divergence implies PAC classification in this framework. We give bounds on the regret associated with the resulting classifier, taking into account the possibility of variable misclassification penalties. We demonstrate the advantage of estimating the a posteriori probability distributions over class labels in the setting of Optical Character Recognition. We show that unsupervised learners can be used to learn a class of probabilistic concepts (stochastic rules denoting the probability that an observation has a positive label in a 2-class setting). This demonstrates a situation where unsupervised learners can be used even when it is hard to learn distributions over class labels - in this case the discriminant functions do not estimate the class probability densities. We use a standard state-merging technique to PAC-learn a class of probabilistic automata and show that by learning the distribution over outputs under the weaker L1 distance rather than KL-divergence we are able to learn without knowledge of the expected length of an output. It is also shown that for a restricted class of these automata learning under L1 distance is equivalent to learning under KL-divergence.EThOS - Electronic Theses Online ServiceEngineering and Physical Sciences Research Council (Great Britain) (EPSRC) (GR/R86188/01)GBUnited Kingdo

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