Zhenjie Ren

Bio: Zhenjie Ren is an academic researcher from École Polytechnique. The author has contributed to research in topics: Stochastic control & Viscosity solution. The author has an hindex of 9, co-authored 34 publications receiving 287 citations. Previous affiliations of Zhenjie Ren include Paris Dauphine University & CEREMADE.

Mean-Field Langevin Dynamics and Energy Landscape of Neural Networks

Abstract: Our work is motivated by a desire to study the theoretical underpinning for the convergence of stochastic gradient type algorithms widely used for non-convex learning tasks such as training of neural networks. The key insight, already observed in the works of Mei, Montanari and Nguyen (2018), Chizat and Bach (2018) as well as Rotskoff and Vanden-Eijnden (2018), is that a certain class of the finite-dimensional non-convex problems becomes convex when lifted to infinite-dimensional space of measures. We leverage this observation and show that the corresponding energy functional defined on the space of probability measures has a unique minimiser which can be characterised by a first-order condition using the notion of linear functional derivative. Next, we study the corresponding gradient flow structure in 2-Wasserstein metric, which we call Mean-Field Langevin Dynamics (MFLD), and show that the flow of marginal laws induced by the gradient flow converges to a stationary distribution, which is exactly the minimiser of the energy functional. We observe that this convergence is exponential under conditions that are satisfied for highly regularised learning tasks. Our proof of convergence to stationary probability measure is novel and it relies on a generalisation of LaSalle's invariance principle combined with HWI inequality. Importantly, we assume neither that interaction potential of MFLD is of convolution type nor that it has any particular symmetric structure. Furthermore, we allow for the general convex objective function, unlike, most papers in the literature that focus on quadratic loss. Finally, we show that the error between finite-dimensional optimisation problem and its infinite-dimensional limit is of order one over the number of parameters.

Comparison of Viscosity Solutions of Fully Nonlinear Degenerate Parabolic Path-Dependent PDEs

Abstract: We prove a comparison result for viscosity solutions of (possibly degenerate) parabolic fully nonlinear path-dependent PDEs. In contrast with the previous result in Ekren, Touzi, and Zhang [Ann. Probab., 44 (2016), pp. 2507--2553], our conditions are easier to check and allow for the degenerate case, thus including first order path-dependent PDEs. Our argument follows the regularization method as introduced by Jensen, Lions, and Souganidis [Proc. Amer. Math. Soc., 102, (1988)] in the corresponding finite-dimensional PDE setting. The present argument significantly simplifies the comparison proof of Ekren, Touzi, and Zhang but requires an $\mathbb{L}^p$-type of continuity (with respect to the path) for the viscosity semisolutions and for the nonlinearity defining the equation.

An overview of Viscosity Solutions of Path-Dependent PDEs

Abstract: This paper provides an overview of the recently developed notion of viscosity solutions of path-dependent partial differential equations. We start by a quick review of the Crandall-Ishii notion of viscosity solutions, so as to motivate the relevance of our definition in the path-dependent case. We focus on the wellposedness theory of such equations. In particular, we provide a simple presentation of the current existence and uniqueness arguments in the semilinear case. We also review the stability property of this notion of solutions, including the adaptation of the Barles-Souganidis monotonic scheme approximation method. Our results rely crucially on the theory of optimal stopping under nonlinear expectation. In the dominated case, we provide a self-contained presentation of all required results. The fully nonlinear case is more involved and is addressed in [12].

Comparison of Viscosity Solutions of Semi-linear Path-Dependent PDEs

Abstract: This paper provides a probabilistic proof of the comparison result for viscosity solutions of path-dependent semilinear PDEs. We consider the notion of viscosity solutions introduced in \cite{EKTZ} which considers as test functions all those smooth processes which are tangent in mean. When restricted to the Markovian case, this definition induces a larger set of test functions, and reduces to the notion of stochastic viscosity solutions analyzed in \cite{BayraktarSirbu1,BayraktarSirbu2}. Our main result takes advantage of this enlargement of the test functions, and provides an easier proof of comparison. This is most remarkable in the context of the linear path-dependent heat equation. As a key ingredient for our methodology, we introduce a notion of punctual differentiation, similar to the corresponding concept in the standard viscosity solutions \cite{CaffarelliCabre}, and we prove that semimartingales are almost everywhere punctually differentiable. This smoothness result can be viewed as the counterpart of the Aleksandroff smoothness result for convex functions. A similar comparison result was established earlier in \cite{EKTZ}. The result of this paper is more general and, more importantly, the arguments that we develop do not rely on any representation of the solution.

Viscosity solutions of fully nonlinear elliptic path dependent partial differential equations.

Abstract: This paper extends the recent work on path-dependent PDEs to elliptic equations with Dirichlet boundary conditions. We propose a notion of viscosity solution in the same spirit as [Ann. Probab. 44 (2016) 1212–1253, Part 1; Ekren, Touzi and Zhang (2016), Part 2], relying on the theory of optimal stopping under nonlinear expectation. We prove a comparison result implying the uniqueness of viscosity solution, and the existence follows from a Perron-type construction using path-frozen PDEs. We also provide an application to a time homogeneous stochastic control problem motivated by an application in finance.

Convergence of probability measures

Abstract: The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

The Stochastic Process π

Abstract: The principal aim of this chapter is to familiarize the reader with the fact that the conditional distribution of the signal can be viewed as a stochastic process with values in the space of probability measures. While it is true that this chapter sets the scene for the subsequent chapters, it can be skipped by those readers whose interests are biased towards the applied aspects of the subject. The gist of the chapter can be summarized by the following.