Probabilistic reasoning and machine learning

机器学习最近取得了许多令人瞩目的成功，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正式验证。在过去的15年里，我与机器学习的同事接触越来越多，我与他们在理论主题的研究上进行了越来越多的合作。我的研究建议集中在：(I)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的逻辑结构，(Iii)关于定量逻辑的理论结果和(Iv)自动机学习。在过去的五年里，我的研究取得了一些与我的提案主题相关的进展。它们是：(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；(4)在概率分布和耦合变元之间的度量用于推理随机逼近算法的收敛；以及(5)加权自动机近似最小化的概念的发展。我已经开始了上面提到的所有领域的工作。在(I)中，我们已经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，我们正在努力将其扩展到新的例子。在(Iii)中，我们已经证明了Wasserstein度规作为我们定义的某一方程理论的“自由代数”出现。这赋予了它可能被证明在计算中有用的普遍性质。在主题(Ii)下，我们发展了基于布尔值集的随机Lambda演算的一种新类型的语义。要将这一点与实践中使用的语言联系起来，还有很多工作要做。我们还发展了马尔可夫过程的Stone-型对偶，它给出了关于马尔可夫过程推理的模式逻辑的完备性定理。题目(四)对我来说是一次新的冒险。我们已经做的工作提供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，一个人准确地学习正确的确定自动机。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，因为我们的度量衡量自动机的行为相似性。将它与从递归神经网络中提取自动机结合使用将是特别有趣的；这是一个正在流行的话题。

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.