. We propose and analyze a distribution learning algorithm for variable memory length Markov processes. These processes can be described by a subclass of probabilistic finite automata which we name Probabilistic Suffix Automata (PSA). Though hardness results are known for learning distributions generated by general probabilistic automata, we prove that the algorithm we present can efficiently learn distributions generated by PSAs. In particular, we show that for any target PSA, the KL-divergence between the distribution generated by the target and the distribution generated by the hypothesis the learning algorithm outputs, can be made small with high confidence in polynomial time and sample complexity. The learning algorithm is motivated by applications in human-machine interaction. Here we present two applications of the algorithm. In the first one we apply the algorithm in order to construct a model of the English language, and use this model to correct corrupted text. In the second ...

. We study online learning algorithms that predict by combining the predictions of several subordinate prediction algorithms, sometimes called "experts." These simple algorithms belong to the multiplicative weights family of algorithms. The performance of these algorithms degrades only logarithmically with the number of experts, making them particularly useful in applications where the number of experts is very large. However, in applications such as text categorization, it is often natural for some of the experts to abstain from making predictions on some of the instances. We show how to transform algorithms that assume that all experts are always awake to algorithms that do not require this assumption. We also show how to derive corresponding loss bounds. Our method is very general, and can be applied to a large family of online learning algorithms. We also give applications to various prediction models including decision graphs and "switching" experts. 1 Introduction We study onlin...

We study estimation in the class of stationary variable length Markov chains (VLMC) on a finite space. The processes in this class are still Markovian of higher order, but with memory of variable length yielding a much bigger and structurally richer class of models than ordinary higher order Markov chains. From a more algorithmic view, the VLMC model class has attracted interest in information theory and machine learning but statistical properties have not been explored very much. Provided that good estimation is available, an additional structural richness of the model class enhances predictive power by finding a better trade-off between model bias and variance and allows better structural description which can be of specific interest. The latter is exemplified with some DNA data. A version of the tree-structured context algorithm, proposed by Rissanen (1983) in an information theoretical set-up, is shown to have new good asymptotic properties for estimation in the class of VLMC's, even when the underlying model increases in dimensionality: consistent estimation of minimal state spaces and mixing properties of fitted models are given. We also propose a new bootstrap scheme based on fitted VLMC's. We show its validity for quite general stationary categorical time series and for a broad range of statistical procedures. AMS 1991 subject classifications. Primary 62M05; secondary 60J10, 62G09, 62M10, 94A15 Key words and phrases. Bootstrap, categorical time series, central limit theorem, context algorithm, data compression, finite-memory sources, FSMX model, Kullback-Leibler distance, model selection, tree model. Short title: Variable Length Markov Chain 1 Research supported in part by the Swiss National Science Foundation. Part of the work has been done while visiting th...

We describe a sequential universal data compression procedure for binary tree sources that performs the "double mixture". Using a context tree, this method weights in an efficient recursive way the coding distributions corresponding to all bounded memory tree sources, and achieves a desirable coding distribution for tree sources with an unknown model and unknown parameters. Computational and storage complexity of the proposed procedure are both linear in the source sequence length. We derive a natural upper bound on the cumulative redundancy of our method for individual sequences. The three terms in this bound can be identified as coding, parameter and model redundancy. The bound holds for all source sequence lengths, not only for asymptotically large lengths. The analysis that leads to this bound is based on standard techniques and turns out to be extremely simple. Our upper bound on the redundancy shows that the proposed context tree weighting procedure is optimal in the sense that i...

We consider adaptive sequential prediction of arbitrary binary sequences when the performance is evaluated using a general loss function. The goal is to predict on each individual sequence nearly as well as the best prediction strategy in a given comparison class of (possibly adaptive) prediction strategies, called experts. By using a general loss function, we generalize previous work on universal prediction, forecasting, and data compression. However, here we restrict ourselves to the case when the comparison class is finite. For a given sequence, we define the regret as the total loss on the entire sequence suffered by the adaptive sequential predictor, minus the total loss suffered by the predictor in the comparison class that performs best on that particular sequence. We show that for a large class of loss functions, the minimax regret is either \Theta(log N) or \Omega\Gamma p ` log N ), depending on the loss function, where N is the number of predictors in the comparison class a...

The Burrows Wheeler Transform (BWT) is a reversible sequence transformation used in a variety of practical lossless source-coding algorithms. In each, the BWT is followed by a lossless source code that attempts to exploit the natural ordering of the BWT coefficients. BWT-based compression schemes are widely touted as low-complexity algorithms giving lossless coding rates better than those of the Ziv--Lempel codes (commonly known as LZ'77 and LZ'78) and almost as good as those achieved by prediction by partial matching (PPM) algorithms. To date, the coding performance claims have been made primarily on the basis of experimental results. This work gives a theoretical evaluation of BWT-based coding. The main results of this theoretical evaluation include: 1) statistical characterizations of the BWT output on both finite strings and sequences of length , 2) a variety of very simple new techniques for BWT-based lossless source coding, and 3) proofs of the universality and bounds on the rates of convergence of both new and existing BWT-based codes for finite-memory and stationary ergodic sources. The end result is a theoretical justification and validation of the experimentally derived conclusions: BWT-based lossless source codes achieve universal lossless coding performance that converges to the optimal coding performance more quickly than the rate of convergence observed in Ziv--Lempel style codes and, for some BWT-based codes, within a constant factor of the optimal rate of convergence for finite-memory sources.

This paper explains how to significantly improve the compression performance of any PPM variant

A grammar transform is a transformation that converts any data sequence to be compressed into a grammar from which the original data sequence can be fully reconstructed. In a grammar-based code, a data sequence is first converted into a grammar by a grammar transform and then losslessly encoded. In this paper, a greedy grammar transform is first presented; this grammar transform constructs sequentially a sequence of irreducible grammars from which the original data sequence can be recovered incrementally. Based on this grammar transform, three universal lossless data compression algorithms, a sequential algorithm, an improved sequential algorithm, and a hierarchical algorithm, are then developed. These algorithms combine the power of arithmetic coding with that of string matching. It is shown that these algorithms are all universal in the sense that they can achieve asymptotically the entropy rate of any stationary, ergodic source. Moreover, it is proved that their worst case redundancies among all individual sequences of length are upper-bounded by �� � �� � �� � , where is a constant. Simulation results show that the proposed algorithms outperform the Unix Compress and Gzip algorithms, which are based on LZ78 and LZ77, respectively.

The concept of context tree, usually defined for finite memory processes, is extended to arbitrary stationary ergodic processes (with finite alphabet). These context trees are not necessarily complete, and may be of infinite depth. The familiar BIC and MDL principles are shown to provide strongly consistent estimators of the context tree, via optimization of a criterion for hypothetical context trees of finite depth, allowed to grow with the sample size n as o(log n). Algorithms are provided to compute these estimators in O(n) time, and to compute them on-line for all i ≤ n in o(n log n) time.

The ability to predict the future based upon the past in finite-alphabet sequences has many applications, including communications, data security, pattern recognition, and natural language processing. By Shannon's theory and the breakthrough development of arithmetic coding, any sequence, a 1 a 2 \Delta \Delta \Delta a n , can be encoded in a number of bits that is essentially equal to the minimal information-lossless codelength, P i \Gamma log 2 p(a i ja 1 \Delta \Delta \Delta a i\Gamma1 ). The goal of universal on-line modeling, and therefore of universal data compression, is to deduce the model of the input sequence a 1 a 2 \Delta \Delta \Delta a n that can estimate each p(a i ja 1 \Delta \Delta \Delta a i\Gamma1 ) knowing only a 1 a 2 \Delta \Delta \Delta a i\Gamma1 so that the ex...