Authors:
(1) Jongmin Lee, Department of Mathematical Science, Seoul National University;
(2) Ernest K. Ryu, Department of Mathematical Science, Seoul National University and Interdisciplinary Program in Artificial Intelligence, Seoul National University. Abstract and 1 Introduction 1.1 Notations and preliminaries 1.2 Prior works 2 Anchored Value Iteration 2.1 Accelerated rate for Bellman consistency operator 2.2 Accelerated rate for Bellman optimality opera 3 Convergence when y=1 4 Complexity lower bound 5 Approximate Anchored Value Iteration 6 Gauss–Seidel Anchored Value Iteration 7 Conclusion, Acknowledgments and Disclosure of Funding and References A Preliminaries B Omitted proofs in Section 2 C Omitted proofs in Section 3 D Omitted proofs in Section 4 E Omitted proofs in Section 5 F Omitted proofs in Section 6 G Broader Impacts H Limitations 1.2 Prior works Value Iteration. Value iteration (VI) was first introduced in the DP literature [8] for finding optimal value function, and its variant approximate VI [11, 19, 30, 32, 56, 81, 90] considers approximate evaluations of the Bellman optimality operator. In RL, VI and approximate VI have served as the basis of RL algorithms such as fitted value iteration [29, 36, 50, 52, 57, 87] and temporal difference learning [41, 54, 80, 89, 94]. There is a line of research that emulates VI by learning a model of the MDP dynamics [62, 83, 85] and applying a modified Bellman operator [7, 33]. Asynchronous VI, another variation of VI updating the coordinate of value function in asynchronous manner, has also been studied in both RL and DP literature [9, 11, 88, 100]. Acceleration. Since Nesterov’s seminal work [61], there has been a large body of research on acceleration in convex minimization. Gradient descent [15] can be accelerated to efficiently reduce function value and squared gradient magnitude for smooth convex minimization problems [21, 44– 46, 60, 61, 102] and smooth strongly convex minimization problems [59, 64, 73, 86, 91]. Motivated by Nesterov acceleration, inertial fixed-point iterations [22, 42, 51, 70, 75] have also been suggested to accelerate fixed-point iterations. Anderson acceleration [2], another acceleration scheme for fixedpoint iterations, has recently been studied with interest [6, 74, 93, 101]. In DP and RL, prioritized sweeping [55] is a well-known method that changes the order of updates to accelerate convergence, and several variants [3, 18, 53, 68, 95] have been proposed. Speedy Q-learning [4] modifies the update rule of Q-learning and uses aggressive learning rates for acceleration. Recently, there has been a line of research that applies acceleration techniques of other areas to VI: [27, 28, 34, 67, 76, 79] uses Anderson acceleration of fixed-point iterations, [1, 12, 37, 38, 92] uses Nesterov acceleration of convex optimization, and [31] uses ideas inspired by PID controllers in control theory. Among those works, [1, 37, 38] applied Nesterov acceleration to obtain theoretically accelerated convergence rates, but those analyses require certain reversibility conditions or restrictions on eigenvalues of the transition probability induced by the policy. The anchor acceleration, a new acceleration mechanism distinct from Nesterov’s, lately gained attention in convex optimization and fixed-point theory. The anchoring mechanism, which retracts iterates towards the initial point, has been used to accelerate algorithms for minimax optimization and fixed-point problems [20, 43, 47, 65, 71, 78, 98, 99], and we focus on it in this paper. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Jongmin Lee, Department of Mathematical Science, Seoul National University; (2) Ernest K. Ryu, Department of Mathematical Science, Seoul National University and Interdisciplinary Program in Artificial Intelligence, Seoul National University. Authors: Authors: (1) Jongmin Lee, Department of Mathematical Science, Seoul National University; (2) Ernest K. Ryu, Department of Mathematical Science, Seoul National University and Interdisciplinary Program in Artificial Intelligence, Seoul National University. Abstract and 1 Introduction Abstract and 1 Introduction 1.1 Notations and preliminaries 1.1 Notations and preliminaries 1.2 Prior works 1.2 Prior works 2 Anchored Value Iteration 2 Anchored Value Iteration 2.1 Accelerated rate for Bellman consistency operator 2.1 Accelerated rate for Bellman consistency operator 2.2 Accelerated rate for Bellman optimality opera 2.2 Accelerated rate for Bellman optimality opera 3 Convergence when y=1 3 Convergence when y=1 4 Complexity lower bound 4 Complexity lower bound 5 Approximate Anchored Value Iteration 5 Approximate Anchored Value Iteration 6 Gauss–Seidel Anchored Value Iteration 6 Gauss–Seidel Anchored Value Iteration 7 Conclusion, Acknowledgments and Disclosure of Funding and References 7 Conclusion, Acknowledgments and Disclosure of Funding and References A Preliminaries A Preliminaries B Omitted proofs in Section 2 B Omitted proofs in Section 2 C Omitted proofs in Section 3 C Omitted proofs in Section 3 D Omitted proofs in Section 4 D Omitted proofs in Section 4 E Omitted proofs in Section 5 E Omitted proofs in Section 5 F Omitted proofs in Section 6 F Omitted proofs in Section 6 G Broader Impacts G Broader Impacts H Limitations H Limitations 1.2 Prior works Value Iteration. Value iteration (VI) was first introduced in the DP literature [8] for finding optimal value function, and its variant approximate VI [11, 19, 30, 32, 56, 81, 90] considers approximate evaluations of the Bellman optimality operator. In RL, VI and approximate VI have served as the basis of RL algorithms such as fitted value iteration [29, 36, 50, 52, 57, 87] and temporal difference learning [41, 54, 80, 89, 94]. There is a line of research that emulates VI by learning a model of the MDP dynamics [62, 83, 85] and applying a modified Bellman operator [7, 33]. Asynchronous VI, another variation of VI updating the coordinate of value function in asynchronous manner, has also been studied in both RL and DP literature [9, 11, 88, 100]. Value Iteration. Acceleration. Since Nesterov’s seminal work [61], there has been a large body of research on acceleration in convex minimization. Gradient descent [15] can be accelerated to efficiently reduce function value and squared gradient magnitude for smooth convex minimization problems [21, 44– 46, 60, 61, 102] and smooth strongly convex minimization problems [59, 64, 73, 86, 91]. Motivated by Nesterov acceleration, inertial fixed-point iterations [22, 42, 51, 70, 75] have also been suggested to accelerate fixed-point iterations. Anderson acceleration [2], another acceleration scheme for fixedpoint iterations, has recently been studied with interest [6, 74, 93, 101]. Acceleration. In DP and RL, prioritized sweeping [55] is a well-known method that changes the order of updates to accelerate convergence, and several variants [3, 18, 53, 68, 95] have been proposed. Speedy Q-learning [4] modifies the update rule of Q-learning and uses aggressive learning rates for acceleration. Recently, there has been a line of research that applies acceleration techniques of other areas to VI: [27, 28, 34, 67, 76, 79] uses Anderson acceleration of fixed-point iterations, [1, 12, 37, 38, 92] uses Nesterov acceleration of convex optimization, and [31] uses ideas inspired by PID controllers in control theory. Among those works, [1, 37, 38] applied Nesterov acceleration to obtain theoretically accelerated convergence rates, but those analyses require certain reversibility conditions or restrictions on eigenvalues of the transition probability induced by the policy. The anchor acceleration, a new acceleration mechanism distinct from Nesterov’s, lately gained attention in convex optimization and fixed-point theory. The anchoring mechanism, which retracts iterates towards the initial point, has been used to accelerate algorithms for minimax optimization and fixed-point problems [20, 43, 47, 65, 71, 78, 98, 99], and we focus on it in this paper. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

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