We present Neural Stochastic Contraction Metrics (NSCM), a new design framework for provably-stable robust control and estimation for a class of stochastic nonlinear systems. It uses a spectrally-normalized deep neural network to construct a contraction metric, sampled via simplified convex optimization in the stochastic setting. Spectral normalization constrains the state-derivatives of the metric to be Lipschitz continuous, thereby ensuring exponential boundedness of the mean squared distance of system trajectories under stochastic disturbances. The NSCM framework allows autonomous agents to approximate optimal stable control and estimation policies in real-time, and outperforms existing nonlinear control and estimation techniques including the state-dependent Riccati equation, iterative LQR, EKF, and the deterministic neural contraction metric, as illustrated in simulation results.
This paper presents a new deep learning-based framework for robust nonlinear estimation and control using the concept of a Neural Contraction Metric (NCM). The NCM uses a deep long short-term memory recurrent neural network for a global approximation of an optimal contraction metric, the existence of which is a necessary and sufficient condition for exponential stability of nonlinear systems. The optimality stems from the fact that the contraction metrics sampled offline are the solutions of a convex optimization problem to minimize an upper bound of the steady-state Euclidean distance between…
Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA, USA（加州理工学院工程与应用科学系）；Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA（加州理工学院喷气推进实验室）；Nonlinear Systems Laboratory, Massachusetts Institute of Technology, Cambridge MA, USA（麻省理工学院非线性系统实验室）
Graduate Aerospace Laboratories, Caltech（加州理工学院研究生航空航天实验室）；Nonlinear Systems Laboratory, MIT（麻省理工学院非线性系统实验室）；Reliable Autonomous System Laboratory, MIT（麻省理工学院可靠自主系统实验室）
This paper presents a theoretical overview of a Neural Contraction Metric (NCM): a neural network model of an optimal contraction metric and corresponding differential Lyapunov function, the existence of which is a necessary and sufficient condition for incremental exponential stability of non-autonomous nonlinear system trajectories. Its innovation lies in providing formal robustness guarantees for learning-based control frameworks, utilizing contraction theory as an analytical tool to study the nonlinear stability of learned systems via convex optimization. In particular, we rigorously show …
Despite its experimental success, Model-based Reinforcement Learning still lacks a complete theoretical understanding. To this end, we analyze the error in the cumulative reward using a contraction approach. We consider both stochastic and deterministic state transitions for continuous (non-discrete) state and action spaces. This approach doesn't require strong assumptions and can recover the typical quadratic error to the horizon. We prove that branched rollouts can reduce this error and are essential for deterministic transitions to have a Bellman contraction. Our analysis of policy mismatch…
Harvard University（哈佛大学）；Universidad Nacional Autónoma de México（墨西哥国立自治大学）
We investigate the framework of privacy amplification by iteration, recently proposed by Feldman et al., from an information-theoretic lens. We demonstrate that differential privacy guarantees of iterative mappings can be determined by a direct application of contraction coefficients derived from strong data processing inequalities for $f$-divergences. In particular, by generalizing the Dobrushin's contraction coefficient for total variation distance to an $f$-divergence known as $E_γ$-divergence, we derive tighter bounds on the differential privacy parameters of the projected noisy stochastic…
Department of Brain and Cognitive Sciences, MIT（麻省理工学院脑与认知科学系）；Department of Aerospace and Mechanical Engineering, University of Notre Dame（圣母大学航空航天与机械工程系）；Department of Mechanical Engineering, MIT（麻省理工学院机械工程系）；Google AI（谷歌人工智能）
We prove that Riemannian contraction in a supervised learning setting implies generalization. Specifically, we show that if an optimizer is contracting in some Riemannian metric with rate $λ> 0$, it is uniformly algorithmically stable with rate $\mathcal{O}(1/λn)$, where $n$ is the number of labelled examples in the training set. The results hold for stochastic and deterministic optimization, in both continuous and discrete-time, for convex and non-convex loss surfaces. The associated generalization bounds reduce to well-known results in the particular case of gradient descent over convex or s…
Harvard University（哈佛大学）；Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas (IIMAS), Universidad Nacional Autónoma de México（墨西哥国立自治大学应用数学与系统研究所（IIMAS））
We investigate the contraction coefficients derived from strong data processing inequalities for the $E_γ$-divergence. By generalizing the celebrated Dobrushin's coefficient from total variation distance to $E_γ$-divergence, we derive a closed-form expression for the contraction of $E_γ$-divergence. This result has fundamental consequences in two privacy settings. First, it implies that local differential privacy can be equivalently expressed in terms of the contraction of $E_γ$-divergence. This equivalent formula can be used to precisely quantify the impact of local privacy in (Bayesian and m…
Finite-Time Bounds for Two-Time-Scale Stochastic Approximation with Arbitrary Norm Contractions and Markovian Noise
Department of Electrical Engineering, Stanford University（斯坦福大学电气工程系）；Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院工业与系统工程系）
Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院（KAIST））
Diffusion models have recently attained significant interest within the community owing to their strong performance as generative models. Furthermore, its application to inverse problems have demonstrated state-of-the-art performance. Unfortunately, diffusion models have a critical downside - they are inherently slow to sample from, needing few thousand steps of iteration to generate images from pure Gaussian noise. In this work, we show that starting from Gaussian noise is unnecessary. Instead, starting from a single forward diffusion with better initialization significantly reduces the numbe…
Preventing Model Collapse via Contraction-Conditioned Neural Filters
School of Mathematical Sciences, Tonji University, Shanghai, 200092, P.\,R. China；School of Mathematical Sciences, Nankai University, Tianjin, China；Independent Researcher；Longhua Clinical College, Shanghai University of Traditional Chinese Medicine, Shanghai, China；Dundee International Institute, Central South University, Changsha, China；School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China；School of Life Sciences；Biotechnology, Shanghai Jiao Tong University, Shanghai, China
