arXiv subjects

Evangelos A. Theodorou

Publications and source records attributed to Evangelos A. Theodorou.

At least 91 records · Page 5Linked to original sources

arXiv2021

Dynamic Game Theoretic Neural Optimizer

The connection between training deep neural networks (DNNs) and optimal control theory (OCT) has attracted considerable attention as a principled tool of algorithmic design. Despite few attempts being made, they have been limited to architectures where the layer propagation resembles a Markovian dynamical system. This casts doubts on their flexibility to modern networks that heavily rely on non-Markovian dependencies between layers (e.g. skip connections in residual networks). In this work, we propose a novel dynamic game perspective by viewing each layer as a player in a dynamic game characterized by the DNN itself. Through this lens, different classes of optimizers can be seen as matching different types of Nash equilibria, depending on the implicit information structure of each (p)layer. The resulting method, called Dynamic Game Theoretic Neural Optimizer (DGNOpt), not only generalizes OCT-inspired optimizers to richer network class; it also motivates a new training principle by solving a multi-player cooperative game. DGNOpt shows convergence improvements over existing methods on image classification datasets with residual and inception networks. Our work marries strengths from both OCT and game theory, paving ways to new algorithmic opportunities from robust optimal control and bandit-based optimization.

cs.LG↗

arXiv2021

Leveraging Stochasticity for Open Loop and Model Predictive Control of Complex Fluid Systems

Stochastic Spatio-Temporal processes are prevalent across domains ranging from modeling of plasma to the turbulence in fluids to the wave function of quantum systems. This letter studies a measure-theoretic description of such systems by describing them as evolutionary processes on Hilbert spaces, and in doing so, derives a framework for spatio-temporal manipulation from fundamental thermodynamic principles. This approach yields a variational optimization framework for controlling stochastic fields. The resulting scheme is applicable to a wide class of spatio-temporal processes and can be used for optimizing parameterized control policies. Our simulated experiments explore the application of two forms of this approach on four stochastic spatio-temporal processes, with results that suggest new perspectives and directions for studying stochastic control problems for spatio-temporal systems.

math.OC↗

arXiv2021

Large-Scale Multi-Agent Deep FBSDEs

In this paper we present a scalable deep learning framework for finding Markovian Nash Equilibria in multi-agent stochastic games using fictitious play. The motivation is inspired by theoretical analysis of Forward Backward Stochastic Differential Equations (FBSDE) and their implementation in a deep learning setting, which is the source of our algorithm's sample efficiency improvement. By taking advantage of the permutation-invariant property of agents in symmetric games, the scalability and performance is further enhanced significantly. We showcase superior performance of our framework over the state-of-the-art deep fictitious play algorithm on an inter-bank lending/borrowing problem in terms of multiple metrics. More importantly, our approach scales up to 3000 agents in simulation, a scale which, to the best of our knowledge, represents a new state-of-the-art. We also demonstrate the applicability of our framework in robotics on a belief space autonomous racing problem.

cs.AI↗

arXiv2021

DDPNOpt: Differential Dynamic Programming Neural Optimizer

Interpretation of Deep Neural Networks (DNNs) training as an optimal control problem with nonlinear dynamical systems has received considerable attention recently, yet the algorithmic development remains relatively limited. In this work, we make an attempt along this line by reformulating the training procedure from the trajectory optimization perspective. We first show that most widely-used algorithms for training DNNs can be linked to the Differential Dynamic Programming (DDP), a celebrated second-order method rooted in the Approximate Dynamic Programming. In this vein, we propose a new class of optimizer, DDP Neural Optimizer (DDPNOpt), for training feedforward and convolution networks. DDPNOpt features layer-wise feedback policies which improve convergence and reduce sensitivity to hyper-parameter over existing methods. It outperforms other optimal-control inspired training methods in both convergence and complexity, and is competitive against state-of-the-art first and second order methods. We also observe DDPNOpt has surprising benefit in preventing gradient vanishing. Our work opens up new avenues for principled algorithmic design built upon the optimal control theory.

cs.LG↗

arXiv2021

Approximate Inverse Reinforcement Learning from Vision-based Imitation Learning

In this work, we present a method for obtaining an implicit objective function for vision-based navigation. The proposed methodology relies on Imitation Learning, Model Predictive Control (MPC), and an interpretation technique used in Deep Neural Networks. We use Imitation Learning as a means to do Inverse Reinforcement Learning in order to create an approximate cost function generator for a visual navigation challenge. The resulting cost function, the costmap, is used in conjunction with MPC for real-time control and outperforms other state-of-the-art costmap generators in novel environments. The proposed process allows for simple training and robustness to out-of-sample data. We apply our method to the task of vision-based autonomous driving in multiple real and simulated environments and show its generalizability.

cs.RO↗

arXiv2021

Spatio-Temporal Differential Dynamic Programming for Control of Fields

We consider the optimal control problem of a general nonlinear spatio-temporal system described by Partial Differential Equations (PDEs). Theory and algorithms for control of spatio-temporal systems are of rising interest among the automatic control community and exhibit numerous challenging characteristic from a control standpoint. Recent methods focus on finite-dimensional optimization techniques of a discretized finite dimensional ODE approximation of the infinite dimensional PDE system. In this paper, we derive a differential dynamic programming (DDP) framework for distributed and boundary control of spatio-temporal systems in infinite dimensions that is shown to generalize both the spatio-temporal LQR solution, and modern finite dimensional DDP frameworks. We analyze the convergence behavior and provide a proof of global convergence for the resulting system of continuous-time forward-backward equations. We explore and develop numerical approaches to handle sensitivities that arise during implementation, and apply the resulting STDDP algorithm to a linear and nonlinear spatio-temporal PDE system. Our framework is derived in infinite dimensional Hilbert spaces, and represents a discretization-agnostic framework for control of nonlinear spatio-temporal PDE systems.

math.OC↗

arXiv2021

NOVAS: Non-convex Optimization via Adaptive Stochastic Search for End-to-End Learning and Control

In this work we propose the use of adaptive stochastic search as a building block for general, non-convex optimization operations within deep neural network architectures. Specifically, for an objective function located at some layer in the network and parameterized by some network parameters, we employ adaptive stochastic search to perform optimization over its output. This operation is differentiable and does not obstruct the passing of gradients during backpropagation, thus enabling us to incorporate it as a component in end-to-end learning. We study the proposed optimization module's properties and benchmark it against two existing alternatives on a synthetic energy-based structured prediction task, and further showcase its use in stochastic optimal control applications.

cs.LG↗

arXiv2021

Variational Inference MPC using Tsallis Divergence

In this paper, we provide a generalized framework for Variational Inference-Stochastic Optimal Control by using thenon-extensive Tsallis divergence. By incorporating the deformed exponential function into the optimality likelihood function, a novel Tsallis Variational Inference-Model Predictive Control algorithm is derived, which includes prior works such as Variational Inference-Model Predictive Control, Model Predictive PathIntegral Control, Cross Entropy Method, and Stein VariationalInference Model Predictive Control as special cases. The proposed algorithm allows for effective control of the cost/reward transform and is characterized by superior performance in terms of mean and variance reduction of the associated cost. The aforementioned features are supported by a theoretical and numerical analysis on the level of risk sensitivity of the proposed algorithm as well as simulation experiments on 5 different robotic systems with 3 different policy parameterizations.

cs.LG↗

arXiv2021

Compositionality of Linearly Solvable Optimal Control in Networked Multi-Agent Systems

In this paper, we discuss the methodology of generalizing the optimal control law from learned component tasks to unlearned composite tasks on Multi-Agent Systems (MASs), by using the linearity composition principle of linearly solvable optimal control (LSOC) problems. The proposed approach achieves both the compositionality and optimality of control actions simultaneously within the cooperative MAS framework in both discrete- and continuous-time in a sample-efficient manner, which reduces the burden of re-computation of the optimal control solutions for the new task on the MASs. We investigate the application of the proposed approach on the MAS with coordination between agents. The experiments show feasible results in investigated scenarios, including both discrete and continuous dynamical systems for task generalization without resampling.

eess.SY↗

arXiv2021

Cooperative Path Integral Control for Stochastic Multi-Agent Systems

A distributed stochastic optimal control solution is presented for cooperative multi-agent systems. The network of agents is partitioned into multiple factorial subsystems, each of which consists of a central agent and neighboring agents. Local control actions that rely only on agents' local observations are designed to optimize the joint cost functions of subsystems. When solving for the local control actions, the joint optimality equation for each subsystem is cast as a linear partial differential equation and solved using the Feynman-Kac formula. The solution and the optimal control action are then formulated as path integrals and approximated by a Monte-Carlo method. Numerical verification is provided through a simulation example consisting of a team of cooperative UAVs.

eess.SY↗

arXiv2021

Learning Deep Stochastic Optimal Control Policies using Forward-Backward SDEs

In this paper we propose a new methodology for decision-making under uncertainty using recent advancements in the areas of nonlinear stochastic optimal control theory, applied mathematics, and machine learning. Grounded on the fundamental relation between certain nonlinear partial differential equations and forward-backward stochastic differential equations, we develop a control framework that is scalable and applicable to general classes of stochastic systems and decision-making problem formulations in robotics and autonomy. The proposed deep neural network architectures for stochastic control consist of recurrent and fully connected layers. The performance and scalability of the aforementioned algorithm are investigated in three non-linear systems in simulation with and without control constraints. We conclude with a discussion on future directions and their implications to robotics.

cs.RO↗

arXiv2021

Distributed Algorithms for Linearly-Solvable Optimal Control in Networked Multi-Agent Systems

Distributed algorithms for both discrete-time and continuous-time linearly solvable optimal control (LSOC) problems of networked multi-agent systems (MASs) are investigated in this paper. A distributed framework is proposed to partition the optimal control problem of a networked MAS into several local optimal control problems in factorial subsystems, such that each (central) agent behaves optimally to minimize the joint cost function of a subsystem that comprises a central agent and its neighboring agents, and the local control actions (policies) only rely on the knowledge of local observations. Under this framework, we not only preserve the correlations between neighboring agents, but moderate the communication and computational complexities by decentralizing the sampling and computational processes over the network. For discrete-time systems modeled by Markov decision processes, the joint Bellman equation of each subsystem is transformed into a system of linear equations and solved using parallel programming. For continuous-time systems modeled by Itô diffusion processes, the joint optimality equation of each subsystem is converted into a linear partial differential equation, whose solution is approximated by a path integral formulation and a sample-efficient relative entropy policy search algorithm, respectively. The learned control policies are generalized to solve the unlearned tasks by resorting to the compositionality principle, and illustrative examples of cooperative UAV teams are provided to verify the effectiveness and advantages of these algorithms.

cs.LG↗

arXiv2021

Stochastic Spatio-Temporal Optimization for Control and Co-Design of Systems in Robotics and Applied Physics

Correlated with the trend of increasing degrees of freedom in robotic systems is a similar trend of rising interest in Spatio-Temporal systems described by Partial Differential Equations (PDEs) among the robotics and control communities. These systems often exhibit dramatic under-actuation, high dimensionality, bifurcations, and multimodal instabilities. Their control represents many of the current-day challenges facing the robotics and automation communities. Not only are these systems challenging to control, but the design of their actuation is an NP-hard problem on its own. Recent methods either discretize the space before optimization, or apply tools from linear systems theory under restrictive linearity assumptions in order to arrive at a control solution. This manuscript provides a novel sampling-based stochastic optimization framework based entirely in Hilbert spaces suitable for the general class of \textit{semi-linear} SPDEs which describes many systems in robotics and applied physics. This framework is utilized for simultaneous policy optimization and actuator co-design optimization. The resulting algorithm is based on variational optimization, and performs joint episodic optimization of the feedback control law and the actuation design over episodes. We study first and second order systems, and in doing so, extend several results to the case of second order SPDEs. Finally, we demonstrate the efficacy of the proposed approach with several simulated experiments on a variety of SPDEs in robotics and applied physics including an infinite degree-of-freedom soft robotic manipulator.

cs.RO↗

arXiv2021

Robust Model Predictive Path Integral Control: Analysis and Performance Guarantees

In this paper we propose a novel decision making architecture for Robust Model Predictive Path Integral control (RMPPI) and investigate its performance guarantees and applicability to off-road navigation. Key building blocks of the proposed architecture are an augmented state space representation of the system consisting of nominal and actual dynamics, a placeholder for different types of tracking controllers, a safety logic for nominal state propagation, and an importance sampling scheme that takes into account the capabilities of the underlying tracking control. Using these ingredients, we derive a bound on the free energy growth of the dynamical system which is a function of task constraint satisfaction level, the performance of the underlying tracking controller, and the sampling error of the stochastic optimization used within RMPPI. To validate the bound on free energy growth, we perform experiments in simulation using two types of tracking controllers, namely the iterative Linear Quadratic Gaussian and Contraction-Metric based control. We further demonstrate the applicability of RMPPI in real hardware using the GT AutoRally vehicle. Our experiments demonstrate that RMPPI outperforms MPPI and Tube-MPPI by alleviating issues of the aforementioned model predictive controllers related to either lack of robustness or excessive conservatism. RMPPI provides the best of the two worlds in terms of agility and robustness to disturbances.

eess.SY↗

arXiv2021

Adaptive Risk Sensitive Model Predictive Control with Stochastic Search

We present a general framework for optimizing the Conditional Value-at-Risk for dynamical systems using stochastic search. The framework is capable of handling the uncertainty from the initial condition, stochastic dynamics, and uncertain parameters in the model. The algorithm is compared against a risk-sensitive distributional reinforcement learning framework and demonstrates outperformance on a pendulum and cartpole with stochastic dynamics. We also showcase the applicability of the framework to robotics as an adaptive risk-sensitive controller by optimizing with respect to the fully nonlinear belief provided by a particle filter on a pendulum, cartpole, and quadcopter in simulation.

math.OC↗

arXiv2020

Stabilizing Optimal Density Control of Nonlinear Agents with Multiplicative Noise

Control of continuous time dynamics with multiplicative noise is a classic topic in stochastic optimal control. This work addresses the problem of designing infinite horizon optimal controls with stability guarantees for \textit{a single agent or large population systems} of identical, non-cooperative and non-networked agents, with multi-dimensional and nonlinear stochastic dynamics excited by multiplicative noise. For agent dynamics belonging to the class of reversible diffusion processes, we provide constraints on the state and control cost functions which guarantee stability of the closed-loop system under the action of the individual optimal controls. A condition relating the state-dependent control cost and volatility is introduced to prove the stability of the equilibrium density. This condition is a special case of the constraint required to use the path integral Feynman-Kac formula for computing the control. We investigate the connection between the stabilizing optimal control and the path integral formalism, leading us to a control law formulation expressed exclusively in terms of the desired equilibrium density.

math.OC↗

arXiv2020

Open-loop Deterministic Density Control of Marked Jump Diffusions

The standard practice in modeling dynamics and optimal control of a large population, ensemble, multi-agent system represented by it's continuum density, is to model individual decision making using local feedback information. In comparison to a closed-loop optimal control scheme, an open-loop strategy, in which a centralized controller broadcasts identical control signals to the ensemble of agents, mitigates the computational and infrastructure requirements for such systems. This work considers the open-loop, deterministic and optimal control synthesis for the density control of agents governed by marked jump diffusion stochastic diffusion equations. The density evolves according to a forward-in-time Chapman-Kolmogorov partial integro-differential equation and the necessary optimality conditions are obtained using the infinite dimensional minimum principle (IDMP). We establish the relationship between the IDMP and the dynamic programming principle as well as the IDMP and stochastic dynamic programming for the synthesized controller. Using the linear Feynman-Kac lemma, a sampling-based algorithm to compute the control is presented and demonstrated for agent dynamics with non-affine and nonlinear drift as well as noise terms.

math.OC↗

arXiv2020

Safe Optimal Control Using Stochastic Barrier Functions and Deep Forward-Backward SDEs

This paper introduces a new formulation for stochastic optimal control and stochastic dynamic optimization that ensures safety with respect to state and control constraints. The proposed methodology brings together concepts such as Forward-Backward Stochastic Differential Equations, Stochastic Barrier Functions, Differentiable Convex Optimization and Deep Learning. Using the aforementioned concepts, a Neural Network architecture is designed for safe trajectory optimization in which learning can be performed in an end-to-end fashion. Simulations are performed on three systems to show the efficacy of the proposed methodology.

eess.SY↗