Probabilistic reasoning and machine learning

机器学习最近取得了许多惊人的成功，这引发了人们对理解这些成就的原因和局限性的兴趣。在过去的25年里，我一直在研究概率系统，最初的目的是致力于此类系统的形式验证。在过去的15年里，我与机器学习的同事们接触越来越多，他们越来越多地合作研究理论主题。我的研究建议集中在：(I)用于关于强化学习算法的基于度量的推理工具，(Ii)将使推理更加模块化的新逻辑结构，(Iii)关于定量逻辑和(Iv)自动机学习的理论结果。在过去五年中，我的研究中有一些与我提议的主题相关的发展。这些是：(1)定量方程逻辑的发展，它允许一个人将代数和度量结合起来，并给出了对以规范方式出现的像Wasserstein度量这样的概念的新见解，(2)像扩散过程这样的连续时间系统的互模拟概念的发展，(3)在贝叶斯推理中发挥重要作用的高阶概率编程语言的语义学，(4)使用概率分布和耦合变元之间的度量来推断随机近似算法的收敛，以及(5)加权自动机近似最小化概念的发展。我已经开始了上述所有领域的工作。在(I)中，我们获得了很有希望的结果，表明各种不同的收敛论点都适用于我们的技术，我们正在努力将其推广到新的例子。在(Iii)中，我们证明了Wasserstein度量作为我们定义的某个方程理论的自由代数出现。这给了它普遍的性质，这些性质可能在计算它时被证明是有用的。在主题(Ii)下，我们发展了一个新的例子基于布尔值集合的随机Lambda演算的语义类型。要将它与实际使用的语言联系起来，还有很多工作要做。我们还开发了马尔可夫过程的Stone-type对偶，它给出了关于马尔可夫过程的推理的模式逻辑的完备性定理。对我来说，主题(Iv)是一个新的尝试。我们所做的工作给出了一些强大的新工具来简化复杂的自动机。我们希望将这些想法应用到自动机学习中。在传统的自动机学习中，一个人学习的是正确的确定自动机。我们希望学习一个近似的概率自动机来自度量和互模拟的想法在这里肯定是有用的，因为我们的度量度量自动机的行为相似性。将其与从递归神经网络中提取自动机一起使用将是特别有趣的；这是一个越来越流行的话题。

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.