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，我一直致力于概率系统的研究，最初的目的是对这种系统进行正式的验证。 在过去的15年里，我越来越多地与机器学习领域的同事接触，并越来越多地与他们合作研究理论课题。 我的研究计划集中在：（i）基于度量的工具，用于推理强化学习算法，（ii）新的逻辑结构，使推理更加模块化，（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进展，这些进展与我的建议的主题有关。 这些是：（1）定量方程逻辑的发展，它允许人们将联合收割机代数和度量结合起来，并为以规范方式出现的Wasserstein度量等概念提供了新的见解，（2）扩散过程等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）利用概率分布和耦合参数之间的度量来推理随机逼近算法的收敛性;（5）发展了加权自动机的近似最小化概念。 我已经开始在上述所有领域开展工作。 在（i）中，我们已经得到了有希望的结果，表明各种不同的收敛参数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度量作为我们定义的某个方程理论的“自由代数”出现。 这使它的普遍属性，可能会成为有用的计算it. Under主题（二），我们已经开发了一种新的类型的语义随机的布尔值集的基础上，微积分。 要将这一点与实际使用的语言联系起来，还有许多工作要做。 我们还开发了石型对偶马尔可夫过程的模态逻辑推理马尔可夫过程的完整性定理。 课题（iv）对我来说是一个新的尝试，我们已经做的工作为简化复杂的自动机提供了一些强有力的新工具。 我们希望将这些想法应用于自动机学习。 在传统的自动机学习中，人们准确地学习正确的确定性自动机。 我们希望近似地学习一个概率自动机。 度量和互模拟的思想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结合使用将特别有趣;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.