We present a unified framework for learning continuous control policies using backpropagation. It supports stochastic control by treating stochasticity in the Bellman equation as a deterministic function of exogenous noise. The product is a spectrum of general policy gradient algorithms that range from model-free methods with value functions to model-based methods without value functions. We use learned models but only require observations from the environment in- stead of observations from model-predicted trajectories, minimizing the impact of compounded model errors. We apply these algorithms first to a toy stochastic control problem and then to several physics-based control problems in simulation. One of these variants, SVG(1), shows the effectiveness of learning models, value functions, and policies simultaneously in continuous domains.
Writing the noise variables as exogenous inputs to the system to allow direct differentiation with respect to the system state (equation ) is a known device in control theory @cite_12 @cite_7 where the model is given analytically. The idea of using a model to optimize a parametric policy around real trajectories is presented heuristically in @cite_29 and @cite_17 for deterministic policies and models. Also in the limit of deterministic policies and models, the recursions we have derived in Algorithm reduce to those of @cite_14 . Werbos defines an actor-critic algorithm called Heuristic Dynamic Programming that uses a deterministic model to roll-forward one step to produce a state prediction that is evaluated by a value function @cite_1 . have used Gaussian process models to compute policy gradients that are sensitive to model-uncertainty @cite_24 , and have optimized impressive policies with the aid of a non-parametric trajectory optimizer and locally-linear models @cite_20 . Our work in contrast has focused on using global, neural network models conjoined to value function approximators.
@cite_14 @cite_7 @cite_29 @cite_1 @cite_24 @cite_20 @cite_12 @cite_17
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We present an algorithm that probabilistically covers a bounded region of the state space of a nonlinear system with a sparse tree of feedback stabilized trajectories leading to a goal state. The generated tree serves as a lookup table control policy to get any reachable initial condition within that region to the goal. The approach combines motion planning with reasoning about the set of states around a trajectory for which the feedback policy of the trajectory is able to stabilize the system. The key idea is to use a random sample from the bounded region for both motion planning and approximation of the stabilizable sets by falsification; this keeps the number of samples and simulations needed to generate covering policies reasonably low. We simulate the nonlinear system to falsify the stabilizable sets, which allows enforcing input and state constraints. Compared to the algebraic verification using sums of squares optimization in our previous work, the simulation-based approximation of the stabilizable set is less exact, but considerably easier to implement and can be applied to a broader range of nonlinear systems. We show simulation results obtained with model systems and study the performance and robustness of the generated policies. This paper develops the application of the alternating direction method of multipliers (ADMM) to optimize a dynamic objective function in a decentralized multi-agent system. At each time slot, agents in the network observe local functions and cooperate to track the optimal time-varying argument of the sum objective. This cooperation is based on maintaining local primal variables that estimate the value of the optimal argument and auxiliary dual variables that encourage proximity with neighboring estimates. Primal and dual variables are updated by an ADMM iteration that can be implemented in a distributed manner whereby local updates require access to local variables and the most recent primal variables from adjacent agents. For objective functions that are strongly convex and have Lipschitz continuous gradients, the distances between the primal and dual iterates to their corresponding time-varying optimal values are shown to converge to a steady state gap. This gap is explicitly characterized in terms of the condition number of the objective function, the condition number of the network that is defined as the ratio between the largest and smallest nonzero Laplacian eigenvalues, and a bound on the drifts of the optimal primal variables and the optimal gradients. Numerical experiments corroborate theoretical findings and show that the results also hold for non-differentiable and non-strongly convex primal objectives. I. Introduction.- 1.1. General Remarks.- 1.2. The Robbins-Monro Process.- 1.3. A "Continuous" Process Version of Section 2.- 1.4. Regulation of a Dynamical System a simple example.- 1.5. Function Minimization: The Kiefer-Wolfowitz Procedure.- 1.6. Constrained Problems.- 1.7. An Economics Example.- II. Convergence w.p.1 for Unconstrained Systems.- 2.1. Preliminaries and Motivation.- 2.2. The Robbins-Monro and Kiefer-Wolfowitz Algorithms: Conditions and Discussion.- 2.3. Convergence Proofs for RM and KW-like Procedures.- 2.3.1. A Basic RM-like Procedure.- 2.3.2. One Dimensional RM and Accelerated RM Procedures.- 2.3.3. A Continuous Parameter RM Procedure.- 2.3.4. The Basic Kiefer-Wolfowitz Procedure.- 2.3.5. Random Directions KW Methods.- 2.4. A General Robbins-Monro Process: "Exogenous Noise".- 2.4.1. The Case of Bounded h(*,*).- 2.4.2. Unbounded h(*,*): Exogenous Noise.- 2.5. A General RM Process State Dependent Noise.- 2.5.1. Extensions and Localizations of Theorem 2.5.2.- 2.6. Some Applications.- 2.7. Mensov-Rademacher Estimates.- III. Weak Convergence of Probability Measures.- IV. Weak Convergence for Unconstrained Systems.- 4.1. Conditions and General Discussion.- 4.2. The Robbins-Monro and Kiefer-Wolfowitz Procedures.- 4.2.1. The Basic Robbins-Monro Procedure.- 4.2.2. The One-Dimensional Robbins-Monro Procedure.- 4.2.3. The Kiefer-Wolfowitz Procedure.- 4.2.4. A Case Where the Limit Satisfies a Generalized ODE.- 4.2.5. A Continuous Parameter KW Procedure.- 4.3. A General Robbins-Monro Process: Exogenous Noise.- 4.4. A General RM Process: State Dependent Noise.- 4.5. The Identification Problem.- 4.6. A Counter-Example to Tightness.- 4.7. Boundedness of Xn and Tightness of Xn(*) .- V. Convergence w.p.1 For Constrained Systems.- 5.1. A Penalty-Multiplier Algorithm for Equality Constraints.- 5.1.1. A Basic RM-like Algorithm, Conditions and Discussion.- 5.1.2. The Noise Condition, Discussion and Generalization.- 5.1.3. Boundedness of Xn .- 5.1.4. Proof of the Main Theorem.- 5.1.5. Constrained Function Minimization and Other Extensions.- 5.2. A Lagrangian Method for Inequality Constraints.- 5.2.1. The Algorithm and Conditions.- 5.2.2. The Convergence Theorem 18.- 5.2.3. A Non-Convergent but Useful Algorithm.- 5.2.4. An Application to the Identification Problem.- 5.3. A Projection Algorithm.- 5.4. A Penalty-Multiplier Method for Inequality Constraints.- VI. Weak Convergence: Constrained Systems.- 6.1. A Multiplier Type Algorithm for Equality Constraints.- 6.1.1. Boundedness of Xn .- 6.1.2. The Noise Condition, Discussion.- 6.1.3. The Convergence Theorem.- 6.2. The Lagrangian Method.- 6.3. A Projection Algorithm.- 6.4. A Penalty-Multiplier Algorithm for Inequality Constraints.- VII. Rates of Convergence.- 7.1. The Problem Formulation.- 7.2. Conditions and Discussions.- 7.3. Rates of Convergence for Case 1, the KW Algorithm.- 7.4. Discussion of Rates of Convergence for Two KW Algorithms. Optimizing shared vehicle systems (bike-sharing car-sharing ride-sharing) is more challenging compared to traditional resource allocation settings due to the presence of complex network externalities. In particular, changes in the demand supply at any location (via dynamic pricing, rebalancing of empty vehicles, etc.) affect future supply throughout the system within short timescales. Such externalities are well captured by steady-state Markovian models, which are therefore widely used to analyze and design shared vehicle systems. However, using such models to design pricing control policies is computationally difficult since the resulting optimization problems are high-dimensional and non-convex. To this end, we develop a general approximation framework for designing pricing policies in shared vehicle systems, based on a novel convex relaxation which we term elevated flow relaxation. Our approach provides the first efficient algorithms with rigorous approximation guarantees for a wide range of objective functions (throughput, revenue, welfare). For any shared vehicle system with @math stations and m vehicles, our framework provides a pricing policy with an approximation ratio of 1+(n-1) m. This guarantee is particularly meaningful when m n, the average number of vehicles per station is large, as is often the case in practice. Further, the simplicity of our approach allows us to extend it to more complex settings. Apart from pricing, shared vehicle systems enable other control levers for modulating demand and supply, e.g. rebalancing empty vehicles, redirecting riders to nearby vehicles, etc. Our approach yields efficient algorithms with the same approximation guarantees for all these problems, and in the process, obtains as special cases several existing heuristics and asymptotic guarantees. We also extend our approach to obtain bi-criterion guarantees in multi-objective settings; we illustrate this with the example of Ramsey pricing. From a technical perspective, our work develops a new approach for obtaining control policies with approximation guarantees in steady-state Markovian models. Our approach can be distilled into the following three-step program: (i) construct an upper bound via a relaxation to the original problem that encodes essential conservation laws of the system, (ii) identify a family of control policies inducing known steady-state distributions that achieve the value of the relaxed solution in an appropriate scaling limit (in our case, state-independent policies in the limit m++), and (iii) characterize the performance loss between the finite system (i.e. fixed m) and the scaling limit. This technique may be of independent interest for other settings.