Probabilistic reasoning and machine learning

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Machine learning has had many spectacular successes recently which have sparked interest in understanding the reasons for, and the limitations of, these achievements.  I have worked on probabilistic systems for the past 25 years originally with a view to working on formal verification of such systems.  In the last 15 years I have been more and more in contact with machine learning colleagues with whom I have increasingly collaborated on research on theoretical topics.  My research proposal focuses on: (i) metric-based tools for reasoning about reinforcement learning algorithms, (ii) new logical structures that will make reasoning more modular, (iii) theoretical results about quantitative logics and (iv) automata learning.  There have been a number of developments in my research in the last five years that are relevant to the subject of my proposal.  These are: (1) the development of quantitative equational logic which allows one to combine algebras and metrics and which gives new insights into concepts like the Wasserstein metric which emerge in a canonical way, (2) the development of bisimulation concepts for continuous-time systems like diffusion processes, (3) semantics for higher-order probabilistic programming languages which are playing an important role in Bayesian inference, (4) the use of metrics between probability distributions and coupling arguments to reason about convergence of stochastic approximation algorithms, and (5) the the development of a notion of approximate minimization of weighted automata.  I have begun work on all the areas mentioned above.  In (i) we have obtained promising results showing that a variety of different convergence arguments are amenable to our technique and we are working to extend it to new examples.  In (iii) we have shown that the Wasserstein metric emerges as the "free algebra" of a certain equational theory that we have defined.  This gives it universal properties that may turn out to be useful in computing it.  Under topic (ii) We have developed a new type of semantics for a stochastic lambda-calculus based on Boolean-valued sets.  Much remains to be done to link this to languages used in practice.  We have also developed Stone-type dualities for Markov processes which give completeness theorems for modal logics for reasoning about Markov processes.  Topic (iv) is a new venture for me.  The work we have already done gives some powerful new tools to simplify complicated automata.  We are hoping to apply such ideas to automata learning.  In traditional automata learning one learns exactly the right deterministic automaton.  We are hoping to approximately learn a probabilistic automaton.  Ideas from metrics and bisimulation will certainly be useful here since our metrics measure behavioural similarity of automata.  It will be particularly interesting to use this in conjunction with the extraction of automata from recurrent neural nets; a topic which is gaining currency.