arXiv subjects

Aditya Mahajan

Publications and source records attributed to Aditya Mahajan.

At least 37 records · Page 2Linked to original sources

arXiv2022

Robustness and sample complexity of model-based MARL for general-sum Markov games

Multi-agent reinforcement learning (MARL) is often modeled using the framework of Markov games (also called stochastic games or dynamic games). Most of the existing literature on MARL concentrates on zero-sum Markov games but is not applicable to general-sum Markov games. It is known that the best-response dynamics in general-sum Markov games are not a contraction. Therefore, different equilibria in general-sum Markov games can have different values. Moreover, the Q-function is not sufficient to completely characterize the equilibrium. Given these challenges, model based learning is an attractive approach for MARL in general-sum Markov games. In this paper, we investigate the fundamental question of \emph{sample complexity} for model-based MARL algorithms in general-sum Markov games. We show two results. We first use Hoeffding inequality based bounds to show that $\tilde{\mathcal{O}}( (1-γ)^{-4} α^{-2})$ samples per state-action pair are sufficient to obtain a $α$-approximate Markov perfect equilibrium with high probability, where $γ$ is the discount factor, and the $\tilde{\mathcal{O}}(\cdot)$ notation hides logarithmic terms. We then use Bernstein inequality based bounds to show that $\tilde{\mathcal{O}}( (1-γ)^{-1} α^{-2} )$ samples are sufficient. To obtain these results, we study the robustness of Markov perfect equilibrium to model approximations. We show that the Markov perfect equilibrium of an approximate (or perturbed) game is always an approximate Markov perfect equilibrium of the original game and provide explicit bounds on the approximation error. We illustrate the results via a numerical example.

cs.GT↗

arXiv2022

On learning history based policies for controlling Markov decision processes

Reinforcementlearning(RL)folkloresuggeststhathistory-basedfunctionapproximationmethods,suchas recurrent neural nets or history-based state abstraction, perform better than their memory-less counterparts, due to the fact that function approximation in Markov decision processes (MDP) can be viewed as inducing a Partially observable MDP. However, there has been little formal analysis of such history-based algorithms, as most existing frameworks focus exclusively on memory-less features. In this paper, we introduce a theoretical framework for studying the behaviour of RL algorithms that learn to control an MDP using history-based feature abstraction mappings. Furthermore, we use this framework to design a practical RL algorithm and we numerically evaluate its effectiveness on a set of continuous control tasks.

cs.LG↗

arXiv2022

A relaxed technical assumption for posterior sampling-based reinforcement learning for control of unknown linear systems

We revisit the Thompson sampling algorithm to control an unknown linear quadratic (LQ) system recently proposed by Ouyang et al (arXiv:1709.04047). The regret bound of the algorithm was derived under a technical assumption on the induced norm of the closed loop system. In this technical note, we show that by making a minor modification in the algorithm (in particular, ensuring that an episode does not end too soon), this technical assumption on the induced norm can be replaced by a milder assumption in terms of the spectral radius of the closed loop system. The modified algorithm has the same Bayesian regret of $\tilde{\mathcal{O}}(\sqrt{T})$, where $T$ is the time-horizon and the $\tilde{\mathcal{O}}(\cdot)$ notation hides logarithmic terms in~$T$.

eess.SY↗

arXiv2022

Decentralized linear quadratic systems with major and minor agents and non-Gaussian noise

A decentralized linear quadratic system with a major agent and a collection of minor agents is considered. The major agent affects the minor agents, but not vice versa. The state of the major agent is observed by all agents. In addition, the minor agents have a noisy observation of their local state. The noise processes is \emph{not} assumed to be Gaussian. The structures of the optimal strategy and the best linear strategy are characterized. It is shown that major agent's optimal control action is a linear function of the major agent's MMSE (minimum mean squared error) estimate of the system state while the minor agent's optimal control action is a linear function of the major agent's MMSE estimate of the system state and a "correction term" which depends on the difference of the minor agent's MMSE estimate of its local state and the major agent's MMSE estimate of the minor agent's local state. Since the noise is non-Gaussian, the minor agent's MMSE estimate is a non-linear function of its observation. It is shown that replacing the minor agent's MMSE estimate by its LLMS (linear least mean square) estimate gives the best linear control strategy. The results are proved using a direct method based on conditional independence, common-information-based splitting of state and control actions, and simplifying the per-step cost based on conditional independence, orthogonality principle, and completion of squares.

eess.SY↗

arXiv2021

Scalable Operator Allocation for Multi-Robot Assistance: A Restless Bandit Approach

In this paper, we consider the problem of allocating human operators in a system with multiple semi-autonomous robots. Each robot is required to perform an independent sequence of tasks, subjected to a chance of failing and getting stuck in a fault state at every task. If and when required, a human operator can assist or teleoperate a robot. Conventional MDP techniques used to solve such problems face scalability issues due to exponential growth of state and action spaces with the number of robots and operators. In this paper we derive conditions under which the operator allocation problem is indexable, enabling the use of the Whittle index heuristic. The conditions can be easily checked to verify indexability, and we show that they hold for a wide range of problems of interest. Our key insight is to leverage the structure of the value function of individual robots, resulting in conditions that can be verified separately for each state of each robot. We apply these conditions to two types of transitions commonly seen in remote robot supervision systems. Through numerical simulations, we demonstrate the efficacy of Whittle index policy as a near-optimal and scalable approach that outperforms existing scalable methods.

cs.RO↗

arXiv2021

Approximate information state for approximate planning and reinforcement learning in partially observed systems

We propose a theoretical framework for approximate planning and learning in partially observed systems. Our framework is based on the fundamental notion of information state. We provide two equivalent definitions of information state -- i) a function of history which is sufficient to compute the expected reward and predict its next value; ii) equivalently, a function of the history which can be recursively updated and is sufficient to compute the expected reward and predict the next observation. An information state always leads to a dynamic programming decomposition. Our key result is to show that if a function of the history (called approximate information state (AIS)) approximately satisfies the properties of the information state, then there is a corresponding approximate dynamic program. We show that the policy computed using this is approximately optimal with bounded loss of optimality. We show that several approximations in state, observation and action spaces in literature can be viewed as instances of AIS. In some of these cases, we obtain tighter bounds. A salient feature of AIS is that it can be learnt from data. We present AIS based multi-time scale policy gradient algorithms. and detailed numerical experiments with low, moderate and high dimensional environments.

cs.LG↗

arXiv2021

Conditions for indexability of restless bandits and an O(K^3) algorithm to compute Whittle index

Restless bandits are a class of sequential resource allocation problems concerned with allocating one or more resources among several alternative processes where the evolution of the process depends on the resource allocated to them. Such models capture the fundamental trade-offs between exploration and exploitation. In 1988, Whittle developed an index heuristic for restless bandit problems which has emerged as a popular solution approach due to its simplicity and strong empirical performance. The Whittle index heuristic is applicable if the model satisfies a technical condition known as indexability. In this paper, we present two general sufficient conditions for indexability and identify simpler to verify refinements of these conditions. We then revisit a previously proposed algorithm called adaptive greedy algorithm which is known to compute the Whittle index for a subclass of restless bandits. We show that a generalization of the adaptive greedy algorithm computes the Whittle index for all indexable restless bandits. We present an efficient implementation of this algorithm which can compute the Whittle index of a restless bandit with $K$ states in O(K^3) computations. Finally, we present a detailed numerical study which affirms the strong performance of the Whittle index heuristic.

eess.SY↗

arXiv2021

Scalable regret for learning to control network-coupled subsystems with unknown dynamics

We consider the problem of controlling an unknown linear quadratic Gaussian (LQG) system consisting of multiple subsystems connected over a network. Our goal is to minimize and quantify the regret (i.e. loss in performance) of our strategy with respect to an oracle who knows the system model. Viewing the interconnected subsystems globally and directly using existing LQG learning algorithms for the global system results in a regret that increases super-linearly with the number of subsystems. Instead, we propose a new Thompson sampling based learning algorithm which exploits the structure of the underlying network. We show that the expected regret of the proposed algorithm is bounded by $\tilde{\mathcal{O}} \big( n \sqrt{T} \big)$ where $n$ is the number of subsystems, $T$ is the time horizon and the $\tilde{\mathcal{O}}(\cdot)$ notation hides logarithmic terms in $n$ and $T$. Thus, the regret scales linearly with the number of subsystems. We present numerical experiments to illustrate the salient features of the proposed algorithm.

eess.SY↗

arXiv2021

Structure-aware reinforcement learning for node-overload protection in mobile edge computing

Mobile Edge Computing (MEC) refers to the concept of placing computational capability and applications at the edge of the network, providing benefits such as reduced latency in handling client requests, reduced network congestion, and improved performance of applications. The performance and reliability of MEC are degraded significantly when one or several edge servers in the cluster are overloaded. Especially when a server crashes due to the overload, it causes service failures in MEC. In this work, an adaptive admission control policy to prevent edge node from getting overloaded is presented. This approach is based on a recently-proposed low complexity RL (Reinforcement Learning) algorithm called SALMUT (Structure-Aware Learning for Multiple Thresholds), which exploits the structure of the optimal admission control policy in multi-class queues for an average-cost setting. We extend the framework to work for node overload-protection problem in a discounted-cost setting. The proposed solution is validated using several scenarios mimicking real-world deployments in two different settings - computer simulations and a docker testbed. Our empirical evaluations show that the total discounted cost incurred by SALMUT is similar to state-of-the-art deep RL algorithms such as PPO (Proximal Policy Optimization) and A2C (Advantage Actor Critic) but requires an order of magnitude less time to train, outputs easily interpretable policy, and can be deployed in an online manner.

cs.NI↗

arXiv2021

Multi-agent estimation and filtering for minimizing team mean-squared error

Motivated by estimation problems arising in autonomous vehicles and decentralized control of unmanned aerial vehicles, we consider multi-agent estimation and filtering problems in which multiple agents generate state estimates based on decentralized information and the objective is to minimize a coupled mean-squared error which we call \emph{team mean-square error}. We call the resulting estimates as minimum team mean-squared error (MTMSE) estimates. We show that MTMSE estimates are different from minimum mean-squared error (MMSE) estimates. We derive closed-form expressions for MTMSE estimates, which are linear function of the observations where the corresponding gain depends on the weight matrix that couples the estimation error. We then consider a filtering problem where a linear stochastic process is monitored by multiple agents which can share their observations (with delay) over a communication graph. We derive expressions to recursively compute the MTMSE estimates. To illustrate the effectiveness of the proposed scheme we consider an example of estimating the distances between vehicles in a platoon and show that MTMSE estimates significantly outperform MMSE estimates and consensus Kalman filtering estimates.

eess.SY↗

arXiv2021

Optimal control of network-coupled subsystems: Spectral decomposition and low-dimensional solutions

In this paper, we investigate optimal control of network-coupled subsystems where the dynamics and the cost couplings depend on an underlying undirected weighted graph. The graph coupling matrix in the dynamics may be the adjacency matrix, the Laplacian matrix, or any other symmetric matrix corresponding to the underlying graph. The cost couplings can be any polynomial function of the underlying coupling matrix. We use the spectral decomposition of the graph coupling matrix to decompose the overall system into (L+1) systems with decoupled dynamics and cost, where L is the rank of the coupling matrix. Furthermore, the optimal control input at each subsystem can be computed by solving (Ldist + 1) decoupled Riccati equations where Ldist (Ldist \leq L) is the number of distinct non-zero eigenvalues of the coupling matrix. A salient feature of the result is that the solution complexity depends on the number of distinct eigenvalues of the coupling matrix rather than the size of the network. Therefore, the proposed solution framework provides a scalable method for synthesizing and implementing optimal control laws for large-scale network-coupled subsystems.

eess.SY↗

arXiv2020

Team Optimal Control of Coupled Major-Minor Subsystems with Mean-Field Sharing

In this paper, we investigate team optimal control of coupled major-minor subsystems with mean-field sharing. In such a model, there is one major subsystem that directly influences the dynamics of $n$ homogeneous minor subsystems; however, the minor subsystems influence the dynamics of the major subsystem and each other only through their mean behaviour (indirectly). In this model, the major and the minor subsystems are arbitrarily coupled in the cost and have mean-field sharing information structure. We propose a two-step approach. In the first step, we describe a mean-field sharing model with multiple types of minor subsystems as a generalization of the mean-field sharing model of Arabneydi and Mahajan, CDC 2014. In the second step, we use the results obtained in the first step to construct a dynamic programming decomposition to identify optimal strategies for the major and the minor subsystems. We present an example with numerical results to illustrate the approach.

math.OC↗

arXiv2020

Team Optimal Control of Coupled Subsystems with Mean-Field Sharing

We investigate team optimal control of stochastic subsystems that are weakly coupled in dynamics (through the mean-field of the system) and are arbitrary coupled in the cost. The controller of each subsystem observes its local state and the mean-field of the state of all subsystems. The system has a non-classical information structure. Exploiting the symmetry of the problem, we identify an information state and use that to obtain a dynamic programming decomposition. This dynamic program determines a globally optimal strategy for all controllers. Our solution approach works for arbitrary number of controllers and generalizes to the setup when the mean-field is observed with noise. The size of the information state is time-invariant; thus, the results generalize to the infinite-horizon control setups as well. In addition, when the mean-field is observed without noise, the size of the corresponding information state increases polynomially (rather than exponentially) with the number of controllers which allows us to solve problems with moderate number of controllers. We illustrate our approach by an example motivated by smart grids that consists of $100$ coupled subsystems.

math.OC↗

arXiv2020

Reinforcement Learning in Decentralized Stochastic Control Systems with Partial History Sharing

In this paper, we are interested in systems with multiple agents that wish to collaborate in order to accomplish a common task while a) agents have different information (decentralized information) and b) agents do not know the model of the system completely i.e., they may know the model partially or may not know it at all. The agents must learn the optimal strategies by interacting with their environment i.e., by decentralized Reinforcement Learning (RL). The presence of multiple agents with different information makes decentralized reinforcement learning conceptually more difficult than centralized reinforcement learning. In this paper, we develop a decentralized reinforcement learning algorithm that learns $ε$-team-optimal solution for partial history sharing information structure, which encompasses a large class of decentralized control systems including delayed sharing, control sharing, mean field sharing, etc. Our approach consists of two main steps. In the first step, we convert the decentralized control system to an equivalent centralized POMDP (Partially Observable Markov Decision Process) using an existing approach called common information approach. However, the resultant POMDP requires the complete knowledge of system model. To circumvent this requirement, in the second step, we introduce a new concept called "Incrementally Expanding Representation" using which we construct a finite-state RL algorithm whose approximation error converges to zero exponentially fast. We illustrate the proposed approach and verify it numerically by obtaining a decentralized Q-learning algorithm for two-user Multi Access Broadcast Channel (MABC) which is a benchmark example for decentralized control systems.

math.OC↗

arXiv2020

Team-Optimal Solution of Finite Number of Mean-Field Coupled LQG Subsystems

A decentralized control system with linear dynamics, quadratic cost, and Gaussian disturbances is considered. The system consists of a finite number of subsystems whose dynamics and per-step cost function are coupled through their mean-field (empirical average). The system has mean-field sharing information structure, i.e., each controller observes the state of its local subsystem (either perfectly or with noise) and the mean-field. It is shown that the optimal control law is unique, linear, and identical across all subsystems. Moreover, the optimal gains are computed by solving two decoupled Riccati equations in the full observation model and by solving an additional filter Riccati equation in the noisy observation model. These Riccati equations do not depend on the number of subsystems. It is also shown that the optimal decentralized performance is the same as the optimal centralized performance. An example, motivated by smart grids, is presented to illustrate the result.

math.OC↗

arXiv2020

Thompson sampling for linear quadratic mean-field teams

We consider optimal control of an unknown multi-agent linear quadratic (LQ) system where the dynamics and the cost are coupled across the agents through the mean-field (i.e., empirical mean) of the states and controls. Directly using single-agent LQ learning algorithms in such models results in regret which increases polynomially with the number of agents. We propose a new Thompson sampling based learning algorithm which exploits the structure of the system model and show that the expected Bayesian regret of our proposed algorithm for a system with agents of $|M|$ different types at time horizon $T$ is $\tilde{\mathcal{O}} \big( |M|^{1.5} \sqrt{T} \big)$ irrespective of the total number of agents, where the $\tilde{\mathcal{O}}$ notation hides logarithmic factors in $T$. We present detailed numerical experiments to illustrate the salient features of the proposed algorithm.

eess.SY↗

arXiv2020

Counterexamples on the monotonicity of delay optimal strategies for energy harvesting transmitters

We consider cross-layer design of delay optimal transmission strategies for energy harvesting transmitters where the data and energy arrival processes are stochastic. Using Markov decision theory, we show that the value function is weakly increasing in the queue state and weakly decreasing in the battery state. It is natural to expect that the delay optimal policy should be weakly increasing in the queue and battery states. We show via counterexamples that this is not the case. In fact, we show that for some sample scenarios the delay optimal policy may perform 5-13% better than the best monotone policy.

cs.IT↗

arXiv2019

Optimal local and remote controllers with unreliable uplink channels: An elementary proof

Recently, a model of a decentralized control system with local and remote controllers connected over unreliable channels was presented in [1]. The model has a non-classical information structure that is not partially nested. Nonetheless, it is shown in [1] that the optimal control strategies are linear functions of the state estimate (which is a non-linear function of the observations). Their proof is based on a fairly sophisticated dynamic programming argument. In this note, we present an alternative and elementary proof of the result which uses common information based conditional independence and completion of squares.

eess.SY↗