Optimal Reinforcement Learning for Gaussian Systems

The exploration-exploitation trade-off is among the central challenges of reinforcement learning. The optimal Bayesian solution is intractable in general. This paper studies to what extent analytic statements about optimal learning are possible if all beliefs are Gaussian processes. A first order approximation of learning of both loss and dynamics, for nonlinear, time-varying systems in continuous time and space, subject to a relatively weak restriction on the dynamics, is described by an infinite-dimensional partial differential equation. An approximate finite-dimensional projection gives an impression for how this result may be helpful.Comment: final pre-conference version of this NIPS 2011 paper. Once again, please note some nontrivial changes to exposition and interpretation of the results, in particular in Equation (9) and Eqs. 11-14. The algorithm and results have remained the same, but their theoretical interpretation has change

Data-Efficient Reinforcement Learning with Probabilistic Model Predictive Control

Trial-and-error based reinforcement learning (RL) has seen rapid advancements in recent times, especially with the advent of deep neural networks. However, the majority of autonomous RL algorithms require a large number of interactions with the environment. A large number of interactions may be impractical in many real-world applications, such as robotics, and many practical systems have to obey limitations in the form of state space or control constraints. To reduce the number of system interactions while simultaneously handling constraints, we propose a model-based RL framework based on probabilistic Model Predictive Control (MPC). In particular, we propose to learn a probabilistic transition model using Gaussian Processes (GPs) to incorporate model uncertainty into long-term predictions, thereby, reducing the impact of model errors. We then use MPC to find a control sequence that minimises the expected long-term cost. We provide theoretical guarantees for first-order optimality in the GP-based transition models with deterministic approximate inference for long-term planning. We demonstrate that our approach does not only achieve state-of-the-art data efficiency, but also is a principled way for RL in constrained environments.Comment: Accepted at AISTATS 2018

Bayesian Learning-Based Adaptive Control for Safety Critical Systems

Deep learning has enjoyed much recent success, and applying state-of-the-art model learning methods to controls is an exciting prospect. However, there is a strong reluctance to use these methods on safety-critical systems, which have constraints on safety, stability, and real-time performance. We propose a framework which satisfies these constraints while allowing the use of deep neural networks for learning model uncertainties. Central to our method is the use of Bayesian model learning, which provides an avenue for maintaining appropriate degrees of caution in the face of the unknown. In the proposed approach, we develop an adaptive control framework leveraging the theory of stochastic CLFs (Control Lyapunov Functions) and stochastic CBFs (Control Barrier Functions) along with tractable Bayesian model learning via Gaussian Processes or Bayesian neural networks. Under reasonable assumptions, we guarantee stability and safety while adapting to unknown dynamics with probability 1. We demonstrate this architecture for high-speed terrestrial mobility targeting potential applications in safety-critical high-speed Mars rover missions.Comment: Corrected an error in section II, where previously the problem was introduced in a non-stochastic setting and wrongly assumed the solution to an ODE with Gaussian distributed parametric uncertainty was equivalent to an SDE with a learned diffusion term. See Lew, T et al. "On the Problem of Reformulating Systems with Uncertain Dynamics as a Stochastic Differential Equation

Gaussian Max-Value Entropy Search for Multi-Agent Bayesian Optimization

We study the multi-agent Bayesian optimization (BO) problem, where multiple agents maximize a black-box function via iterative queries. We focus on Entropy Search (ES), a sample-efficient BO algorithm that selects queries to maximize the mutual information about the maximum of the black-box function. One of the main challenges of ES is that calculating the mutual information requires computationally-costly approximation techniques. For multi-agent BO problems, the computational cost of ES is exponential in the number of agents. To address this challenge, we propose the Gaussian Max-value Entropy Search, a multi-agent BO algorithm with favorable sample and computational efficiency. The key to our idea is to use a normal distribution to approximate the function maximum and calculate its mutual information accordingly. The resulting approximation allows queries to be cast as the solution of a closed-form optimization problem which, in turn, can be solved via a modified gradient ascent algorithm and scaled to a large number of agents. We demonstrate the effectiveness of Gaussian max-value Entropy Search through numerical experiments on standard test functions and real-robot experiments on the source-seeking problem. Results show that the proposed algorithm outperforms the multi-agent BO baselines in the numerical experiments and can stably seek the source with a limited number of noisy observations on real robots.Comment: 10 pages, 9 figure

Probabilistic models for data efficient reinforcement learning

Trial-and-error based reinforcement learning (RL) has seen rapid advancements in recent times, especially with the advent of deep neural networks. However, the standard deep learning methods often overlook the progress made in control theory by treating systems as black-box. We propose a model-based RL framework based on probabilistic Model Predictive Control (MPC). In particular, we propose to learn a probabilistic transition model using Gaussian Processes (GPs) to incorporate model uncertainty into long-term predictions, thereby, reducing the impact of model errors. We provide theoretical guarantees for first-order optimality in the GP-based transition models with deterministic approximate inference for long-term planning. We demonstrate that our approach not only achieves the state-of-the-art data efficiency, but also is a principled way for RL in constrained environments. When the true state of the dynamical system cannot be fully observed the standard model based methods cannot be directly applied. For these systems an additional step of state estimation is needed. We propose distributed message passing for state estimation in non-linear dynamical systems. In particular, we propose to use expectation propagation (EP) to iteratively refine the state estimate, i.e., the Gaussian posterior distribution on the latent state. We show two things: (a) Classical Rauch-Tung-Striebel (RTS) smoothers, such as the extended Kalman smoother (EKS) or the unscented Kalman smoother (UKS), are special cases of our message passing scheme; (b) running the message passing scheme more than once can lead to significant improvements over the classical RTS smoothers. We show the explicit connection between message passing with EP and well-known RTS smoothers and provide a practical implementation of the suggested algorithm. Furthermore, we address convergence issues of EP by generalising this framework to damped updates and the consideration of general -divergences. Probabilistic models can also be used to generate synthetic data. In model based RL we use ’synthetic’ data as a proxy to real environments and in order to achieve high data efficiency. The ability to generate high-fidelity synthetic data is crucial when available (real) data is limited as in RL or where privacy and data protection standards allow only for limited use of the given data, e.g., in medical and financial data-sets. Current state-of-the-art methods for synthetic data generation are based on generative models, such as Generative Adversarial Networks (GANs). Even though GANs have achieved remarkable results in synthetic data generation, they are often challenging to interpret. Furthermore, GAN-based methods can suffer when used with mixed real and categorical variables. Moreover, the loss function (discriminator loss) design itself is problem specific, i.e., the generative model may not be useful for tasks it was not explicitly trained for. In this paper, we propose to use a probabilistic model as a synthetic data generator. Learning the probabilistic model for the data is equivalent to estimating the density of the data. Based on the copula theory, we divide the density estimation task into two parts, i.e., estimating univariate marginals and estimating the multivariate copula density over the univariate marginals. We use normalising flows to learn both the copula density and univariate marginals. We benchmark our method on both simulated and real data-sets in terms of density estimation as well as the ability to generate high-fidelity synthetic data.Open Acces