Probabilistic latent-variable models are a powerful tool for modelling structured data. However, traditional expectation-maximization methods of learning such models are both computationally expensive and prone to local-minima. In contrast to these traditional methods, recently developed learning algorithms based upon the method of moments are both computationally efficient and provide strong statistical guarantees. In this work we provide a unified presentation and empirical comparison of three general moment-based methods in the context of modelling stochastic languages. By rephrasing these methods upon a common theoretical ground, introducing novel theoretical results where necessary, we provide a clear comparison, making explicit the statistical assumptions upon which each method relies. With this theoretical grounding, we then provide an in-depth empirical analysis of the methods on both real and synthetic data with the goal of elucidating performance trends and highlighting important implementation details.

/pdf/methods-of-moments-for-learning-stochastic-languages-unified-2hiq6hv56c.pdf

Methods of Moments for Learning Stochastic Languages: Unified Presentation and Empirical Comparison

Considering either two independent i.i.d. samples, or two independent samples generated from a heteroscedastic regression model, or two independent Poisson processes, we address the question of testing equality of their respective distributions. We first propose single testing procedures based on a general symmetric kernel. The corresponding critical values are chosen from a wild or permutation bootstrap approach, and the obtained tests are exactly (and not just asymptotically) of level . We then introduce an aggregation method, which enables to overcome the difficulty of choosing a kernel and/or the parameters of the kernel. We derive non-asymptotic properties for the aggregated tests, proving that they may be optimal in a classical statistical sense.

Kernels based tests with non-asymptotic bootstrap approaches for two-sample problem

Continuous-time Markov chains (CTMCs) have been widely used to determine system performance and dependability characteristics. Their analysis most often concerns the computation of steady-state and transient-state probabilities. This paper introduces a branching temporal logic for expressing real-time probabilistic properties on CTMCs and presents approximate model checking algorithms for this logic. The logic, an extension of the continuous stochastic logic CSL of Aziz et al. (1995, 2000), contains a time-bounded until operator to express probabilistic timing properties over paths as well as an operator to express steady-state probabilities. We show that the model checking problem for this logic reduces to a system of linear equations (for unbounded until and the steady-state operator) and a Volterra integral equation system (for time-bounded until). We then show that the problem of model-checking time-bounded until properties can be reduced to the problem of computing transient state probabilities for CTMCs. This allows the verification of probabilistic timing properties by efficient techniques for transient analysis for CTMCs such as uniformization. Finally, we show that a variant of lumping equivalence (bisimulation), a well-known notion for aggregating CTMCs, preserves the validity of all formulas in the logic.

Model-Checking Algorithms for Continuous-Time Markov Chains

Models are used in both Software Engineering (SE) and Artificial Intelligence (AI). SE models may specify the architecture at different levels of abstraction and for addressing different concerns at various stages of the software development life-cycle, from early conceptualization and design, to verification, implementation, testing and evolution. However, AI models may provide smart capabilities, such as prediction and decision-making support. For instance, in Machine Learning (ML), which is currently the most popular sub-discipline of AI, mathematical models may learn useful patterns in the observed data and can become capable of making predictions. The goal of this work is to create synergy by bringing models in the said communities together and proposing a holistic approach to model-driven software development for intelligent systems that require ML. We illustrate how software models can become capable of creating and dealing with ML models in a seamless manner. The main focus is on the domain of the Internet of Things (IoT), where both ML and model-driven SE play a key role. In the context of the need to take a Cyber-Physical System-of-Systems perspective of the targeted architecture, an integrated design environment for both SE and ML sub-systems would best support the optimization and overall efficiency of the implementation of the resulting system. In particular, we implement the proposed approach, called ML-Quadrat, based on ThingML, and validate it using a case study from the IoT domain, as well as through an empirical user evaluation. It transpires that the proposed approach is not only feasible, but may also contribute to the performance leap of software development for smart Cyber-Physical Systems (CPS) which are connected to the IoT, as well as an enhanced user experience of the practitioners who use the proposed modeling solution.

A Model-Driven Approach to Machine Learning and Software Modeling for the IoT.

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.

/pdf/learning-probability-distributions-generated-by-finite-state-4bgxjjrcd8.pdf

Learning probability distributions generated by finite-state machines

The present thesis addresses several machine learning problems on generative and predictive models on sequential data. All the models considered have in common that they can be de ned in terms of nite-state machines. On one line of work we study algorithms for learning the probabilistic analog of Deterministic Finite Automata (DFA). This provides a fairly expressive generative model for sequences with very interesting algorithmic properties. State-merging algorithms for learning these models can be interpreted as a divisive clustering scheme where the "dependency graph" between clusters is not necessarily a tree. We characterize these algorithms in terms of statistical queries and a use this characterization for proving a lower bound with an explicit dependency on the distinguishability of the target machine. In a more realistic setting, we give an adaptive state-merging algorithm satisfying the stringent algorithmic constraints of the data streams computing paradigm. Our algorithms come with strict PAC learning guarantees. At the heart of state-merging algorithms lies a statistical test for distribution similarity. In the streaming version this is replaced with a bootstrap-based test which yields faster convergence in many situations. We also studied a wider class of models for which the state-merging paradigm also yield PAC learning algorithms. Applications of this method are given to continuous-time Markovian models and stochastic transducers on pairs of aligned sequences. The main tools used for obtaining these results include a variety of concentration inequalities and sketching algorithms. In another line of work we contribute to the rapidly growing body of spectral learning algorithms. The main virtues of this type of algorithms include the possibility of proving nite-sample error bounds in the realizable case and enormous savings on computing time over iterative methods like Expectation-Maximization. In this thesis we give the rst application of this method for learning conditional distributions over pairs of aligned sequences de ned by probabilistic nite-state transducers. We also prove that the method can learn the whole class of probabilistic automata, thus extending the class of models previously known to be learnable with this approach. In the last two chapters we present works combining spectral learning with methods from convex optimization and matrix completion. Respectively, these yield an alternative interpretation of spectral learning and an extension to cases with missing data. In the latter case we used a novel joint stability analysis of matrix completion and spectral learning to prove the rst generalization bound for this type of algorithms that holds in the non-realizable case. Work in this area has been motivated by connections between spectral learning, classic automata theory, and statistical learning; tools from these three areas have been used.

/pdf/learning-finite-state-machines-statistical-and-algorithmic-40npykhwvo.pdf

Learning finite-state machines: statistical and algorithmic aspects

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Learning finite-state machines: statistical and algorithmic aspects