Platooning has been exploited as a method for vehicles to minimize energy consumption. In this article, we present a constraint-driven optimal control framework that yields emergent platooning behavior for connected and automated vehicles operating in an open transportation system. Our approach combines recent insights in constraint-driven optimal control with the physical aerodynamic interactions between vehicles in a highway setting. The result is a set of equations that describes when platooning is an appropriate strategy, as well as a descriptive optimal control law that yields emergent platooning behavior. Finally, we demonstrate these properties in simulation.
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Physical systems are usually modeled by differential equations, but solving these differential equations analytically is often intractable. Instead, the differential equations can be solved numerically by discretization in a finite computational domain. The discretized equation is reduced to a large linear system, whose solution is typically found using an iterative solver. We start with an initial guess, x_0, and iterate the algorithm to obtain a sequence of solution vectors, x_m. The iterative algorithm is said to converge to solution $x$ if and only if x_m converges to $x$. Accuracy of the numerical solutions is important, especially in the design of safety critical systems such as airplanes, cars, or nuclear power plants. It is therefore important to formally guarantee that the iterative solvers converge to the "true" solution of the original differential equation. In this paper, we first formalize the necessary and sufficient conditions for iterative convergence in the Coq proof assistant. We then extend this result to two classical iterative methods: Gauss-Seidel iteration and Jacobi iteration. We formalize conditions for the convergence of the Gauss--Seidel classical iterative method, based on positive definiteness of the iterative matrix. We then formally state conditions for convergence of Jacobi iteration and instantiate it with an example to demonstrate convergence of iterative solutions to the direct solution of the linear system. We leverage recent developments of the Coq linear algebra and mathcomp library for our formalization.
Developing reinforcement learning algorithms that satisfy safety constraints is becoming increasingly important in real-world applications. In multi-agent reinforcement learning (MARL) settings, policy optimisation with safety awareness is particularly challenging because each individual agent has to not only meet its own safety constraints, but also consider those of others so that their joint behaviour can be guaranteed safe. Despite its importance, the problem of safe multi-agent learning has not been rigorously studied; very few solutions have been proposed, nor a sharable testing environment or benchmarks. To fill these gaps, in this work, we formulate the safe MARL problem as a constrained Markov game and solve it with policy optimisation methods. Our solutions -- Multi-Agent Constrained Policy Optimisation (MACPO) and MAPPO-Lagrangian -- leverage the theories from both constrained policy optimisation and multi-agent trust region learning. Crucially, our methods enjoy theoretical guarantees of both monotonic improvement in reward and satisfaction of safety constraints at every iteration. To examine the effectiveness of our methods, we develop the benchmark suite of Safe Multi-Agent MuJoCo that involves a variety of MARL baselines. Experimental results justify that MACPO/MAPPO-Lagrangian can consistently satisfy safety constraints, meanwhile achieving comparable performance to strong baselines.
Modern networks exhibit a high degree of variability in link rates. Cellular network bandwidth inherently varies with receiver motion and orientation, while class-based packet scheduling in datacenter and service provider networks induces high variability in available capacity for network tenants. Recent work has proposed numerous congestion control protocols to cope with this variability, offering different tradeoffs between link utilization and queuing delay. In this paper, we develop a formal model of congestion control over time-varying links, and we use this model to derive a bound on the performance of any congestion control protocol running over a time-varying link with a given distribution of rate variation. Using the insights from this analysis, we derive an optimal control law that offers a smooth tradeoff between link utilization and queuing delay. We compare the performance of this control law to several existing control algorithms on cellular link traces to show that there is significant room for optimization.
We examine global non-asymptotic convergence properties of policy gradient methods for multi-agent reinforcement learning (RL) problems in Markov potential games (MPG). To learn a Nash equilibrium of an MPG in which the size of state space and/or the number of players can be very large, we propose new independent policy gradient algorithms that are run by all players in tandem. When there is no uncertainty in the gradient evaluation, we show that our algorithm finds an $\epsilon$-Nash equilibrium with $O(1/\epsilon^2)$ iteration complexity which does not explicitly depend on the state space size. When the exact gradient is not available, we establish $O(1/\epsilon^5)$ sample complexity bound in a potentially infinitely large state space for a sample-based algorithm that utilizes function approximation. Moreover, we identify a class of independent policy gradient algorithms that enjoys convergence for both zero-sum Markov games and Markov cooperative games with the players that are oblivious to the types of games being played. Finally, we provide computational experiments to corroborate the merits and the effectiveness of our theoretical developments.
The \emph{turnpike property} in contemporary macroeconomics asserts that if an economic planner seeks to move an economy from one level of capital to another, then the most efficient path, as long as the planner has enough time, is to rapidly move stock to a level close to the optimal stationary or constant path, then allow for capital to develop along that path until the desired term is nearly reached, at which point the stock ought to be moved to the final target. Motivated in part by its nature as a resource allocation strategy, over the past decade, the turnpike property has also been shown to hold for several classes of partial differential equations arising in mechanics. When formalized mathematically, the turnpike theory corroborates the insights from economics: for an optimal control problem set in a finite-time horizon, optimal controls and corresponding states, are close (often exponentially), during most of the time, except near the initial and final time, to the optimal control and corresponding state for the associated stationary optimal control problem. In particular, the former are mostly constant over time. This fact provides a rigorous meaning to the asymptotic simplification that some optimal control problems appear to enjoy over long time intervals, allowing the consideration of the corresponding stationary problem for computing and applications. We review a slice of the theory developed over the past decade --the controllability of the underlying system is an important ingredient, and can even be used to devise simple turnpike-like strategies which are nearly optimal--, and present several novel applications, including, among many others, the characterization of Hamilton-Jacobi-Bellman asymptotics, and stability estimates in deep learning via residual neural networks.
Active inference is a unifying theory for perception and action resting upon the idea that the brain maintains an internal model of the world by minimizing free energy. From a behavioral perspective, active inference agents can be seen as self-evidencing beings that act to fulfill their optimistic predictions, namely preferred outcomes or goals. In contrast, reinforcement learning requires human-designed rewards to accomplish any desired outcome. Although active inference could provide a more natural self-supervised objective for control, its applicability has been limited because of the shortcomings in scaling the approach to complex environments. In this work, we propose a contrastive objective for active inference that strongly reduces the computational burden in learning the agent's generative model and planning future actions. Our method performs notably better than likelihood-based active inference in image-based tasks, while also being computationally cheaper and easier to train. We compare to reinforcement learning agents that have access to human-designed reward functions, showing that our approach closely matches their performance. Finally, we also show that contrastive methods perform significantly better in the case of distractors in the environment and that our method is able to generalize goals to variations in the background.
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 trajectory optimization algorithm rooted in the Approximate Dynamic Programming. In this vein, we propose a new variant of DDP that can accept batch optimization for training feedforward networks, while integrating naturally with the recent progress in curvature approximation. The resulting algorithm features layer-wise feedback policies which improve convergence rate and reduce sensitivity to hyper-parameter over existing methods. We show that the algorithm is competitive against state-ofthe-art first and second order methods. Our work opens up new avenues for principled algorithmic design built upon the optimal control theory.
To make deliberate progress towards more intelligent and more human-like artificial systems, we need to be following an appropriate feedback signal: we need to be able to define and evaluate intelligence in a way that enables comparisons between two systems, as well as comparisons with humans. Over the past hundred years, there has been an abundance of attempts to define and measure intelligence, across both the fields of psychology and AI. We summarize and critically assess these definitions and evaluation approaches, while making apparent the two historical conceptions of intelligence that have implicitly guided them. We note that in practice, the contemporary AI community still gravitates towards benchmarking intelligence by comparing the skill exhibited by AIs and humans at specific tasks such as board games and video games. We argue that solely measuring skill at any given task falls short of measuring intelligence, because skill is heavily modulated by prior knowledge and experience: unlimited priors or unlimited training data allow experimenters to "buy" arbitrary levels of skills for a system, in a way that masks the system's own generalization power. We then articulate a new formal definition of intelligence based on Algorithmic Information Theory, describing intelligence as skill-acquisition efficiency and highlighting the concepts of scope, generalization difficulty, priors, and experience. Using this definition, we propose a set of guidelines for what a general AI benchmark should look like. Finally, we present a benchmark closely following these guidelines, the Abstraction and Reasoning Corpus (ARC), built upon an explicit set of priors designed to be as close as possible to innate human priors. We argue that ARC can be used to measure a human-like form of general fluid intelligence and that it enables fair general intelligence comparisons between AI systems and humans.
Reinforcement learning (RL) algorithms have been around for decades and been employed to solve various sequential decision-making problems. These algorithms however have faced great challenges when dealing with high-dimensional environments. The recent development of deep learning has enabled RL methods to drive optimal policies for sophisticated and capable agents, which can perform efficiently in these challenging environments. This paper addresses an important aspect of deep RL related to situations that demand multiple agents to communicate and cooperate to solve complex tasks. A survey of different approaches to problems related to multi-agent deep RL (MADRL) is presented, including non-stationarity, partial observability, continuous state and action spaces, multi-agent training schemes, multi-agent transfer learning. The merits and demerits of the reviewed methods will be analyzed and discussed, with their corresponding applications explored. It is envisaged that this review provides insights about various MADRL methods and can lead to future development of more robust and highly useful multi-agent learning methods for solving real-world problems.
In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.
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