This paper introduces a principled approach for the design of a scalable general reinforcement learning agent. This approach is based on a direct approximation of AIXI, a Bayesian optimality notion for general reinforcement learning agents. Previously, it has been unclear whether the theory of AIXI could motivate the design of practical algorithms. We answer this hitherto open question in the affirmative, by providing the first computationally feasible approximation to the AIXI agent. To develop our approximation, we introduce a Monte Carlo Tree Search algorithm along with an agent-specific extension of the Context Tree Weighting algorithm. Empirically, we present a set of encouraging results on a number of stochastic, unknown, and partially observable domains.
We propose a practical integration of logical state abstraction with AIXI, a Bayesian optimality notion for reinforcement learning agents, to significantly expand the model class that AIXI agents can be approximated over to complex history-dependent and structured environments. The state representation and reasoning framework is based on higher-order logic, which can be used to define and enumerate complex features on non-Markovian and structured environments. We address the problem of selecting the right subset of features to form state abstractions by adapting the $Φ$-MDP optimisation criter…
University of New South Wales（新南威尔士大学）；National ICT Australia（澳大利亚国家信息与通信技术研究院）；The Australian National University（澳大利亚国立大学）；Massachusetts Institute of Technology（麻省理工学院）
This paper introduces a principled approach for the design of a scalable general reinforcement learning agent. Our approach is based on a direct approximation of AIXI, a Bayesian optimality notion for general reinforcement learning agents. Previously, it has been unclear whether the theory of AIXI could motivate the design of practical algorithms. We answer this hitherto open question in the affirmative, by providing the first computationally feasible approximation to the AIXI agent. To develop our approximation, we introduce a new Monte-Carlo Tree Search algorithm along with an agent-specific…
Sequential decision theory formally solves the problem of rational agents in uncertain worlds if the true environmental prior probability distribution is known. Solomonoff's theory of universal induction formally solves the problem of sequence prediction for unknown prior distribution. We combine both ideas and get a parameter-free theory of universal Artificial Intelligence. We give strong arguments that the resulting AIXI model is the most intelligent unbiased agent possible. We outline how the AIXI model can formally solve a number of problem classes, including sequence prediction, strategi…
Decision theory formally solves the problem of rational agents in uncertain worlds if the true environmental prior probability distribution is known. Solomonoff's theory of universal induction formally solves the problem of sequence prediction for unknown prior distribution. We combine both ideas and get a parameterless theory of universal Artificial Intelligence. We give strong arguments that the resulting AIXI model is the most intelligent unbiased agent possible. We outline for a number of problem classes, including sequence prediction, strategic games, function minimization, reinforcement …
Natural policy gradient methods are popular reinforcement learning methods that improve the stability of policy gradient methods by utilizing second-order approximations to precondition the gradient with the inverse of the Fisher-information matrix. However, to the best of the authors' knowledge, there has not been a study that has investigated the effects of different second-order approximations in a comprehensive and systematic manner. To address this, five different second-order approximations were studied and compared across multiple key metrics including performance, stability, sample eff…
Reinforcement learning (RL) algorithms based on high-dimensional function approximation have achieved tremendous empirical success in large-scale problems with an enormous number of states. However, most analysis of such algorithms gives rise to error bounds that involve either the number of states or the number of features. This paper considers the situation where the function approximation is made either using the kernel method or the two-layer neural network model, in the context of a fitted Q-iteration algorithm with explicit regularization. We establish an $\tilde{O}(H^3|\mathcal {A}|^{\f…
This paper gives specific divergence examples of value-iteration for several major Reinforcement Learning and Adaptive Dynamic Programming algorithms, when using a function approximator for the value function. These divergence examples differ from previous divergence examples in the literature, in that they are applicable for a greedy policy, i.e. in a "value iteration" scenario. Perhaps surprisingly, with a greedy policy, it is also possible to get divergence for the algorithms TD(1) and Sarsa(1). In addition to these divergences, we also achieve divergence for the Adaptive Dynamic Programmin…
ATR Computational Neuroscience Laboratories（ATR计算神经科学实验室）；Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）
In recent years, neural networks have enjoyed a renaissance as function approximators in reinforcement learning. Two decades after Tesauro's TD-Gammon achieved near top-level human performance in backgammon, the deep reinforcement learning algorithm DQN achieved human-level performance in many Atari 2600 games. The purpose of this study is twofold. First, we propose two activation functions for neural network function approximation in reinforcement learning: the sigmoid-weighted linear unit (SiLU) and its derivative function (dSiLU). The activation of the SiLU is computed by the sigmoid functi…
Google Inc.（谷歌公司）；Microsoft Research, Redmond（微软研究院（雷德蒙德））；University of Illinois at Urbana Champaign（伊利诺伊大学厄巴纳-香槟分校）；Tencent AI Lab, Bellevue（腾讯人工智能实验室（贝尔维尤））
When function approximation is used, solving the Bellman optimality equation with stability guarantees has remained a major open problem in reinforcement learning for decades. The fundamental difficulty is that the Bellman operator may become an expansion in general, resulting in oscillating and even divergent behavior of popular algorithms like Q-learning. In this paper, we revisit the Bellman equation, and reformulate it into a novel primal-dual optimization problem using Nesterov's smoothing technique and the Legendre-Fenchel transformation. We then develop a new algorithm, called Smoothed …
We propose a successive convex approximation based off-policy optimization (SCAOPO) algorithm to solve the general constrained reinforcement learning problem, which is formulated as a constrained Markov decision process (CMDP) in the context of average cost. The SCAOPO is based on solving a sequence of convex objective/feasibility optimization problems obtained by replacing the objective and constraint functions in the original problems with convex surrogate functions. At each iteration, the convex surrogate problem can be efficiently solved by Lagrange dual method even the policy is parameter…
Motivated by the wide adoption of reinforcement learning (RL) in real-world personalized services, where users' sensitive and private information needs to be protected, we study regret minimization in finite-horizon Markov decision processes (MDPs) under the constraints of differential privacy (DP). Compared to existing private RL algorithms that work only on tabular finite-state, finite-actions MDPs, we take the first step towards privacy-preserving learning in MDPs with large state and action spaces. Specifically, we consider MDPs with linear function approximation (in particular linear mixt…
University of Toronto（多伦多大学）；New York University（纽约大学）；Vector Institute（向量研究所）
In this work, we propose to apply trust region optimization to deep reinforcement learning using a recently proposed Kronecker-factored approximation to the curvature. We extend the framework of natural policy gradient and propose to optimize both the actor and the critic using Kronecker-factored approximate curvature (K-FAC) with trust region; hence we call our method Actor Critic using Kronecker-Factored Trust Region (ACKTR). To the best of our knowledge, this is the first scalable trust region natural gradient method for actor-critic methods. It is also a method that learns non-trivial task…
Boston University（波士顿大学）；The University of Sydney（悉尼大学）
We consider the regret minimization problem in reinforcement learning (RL) in the episodic setting. In many real-world RL environments, the state and action spaces are continuous or very large. Existing approaches establish regret guarantees by either a low-dimensional representation of the stochastic transition model or an approximation of the $Q$-functions. However, the understanding of function approximation schemes for state-value functions largely remains missing. In this paper, we propose an online model-based RL algorithm, namely the CME-RL, that learns representations of transition dis…
We study reinforcement learning with linear function approximation where the transition probability and reward functions are linear with respect to a feature mapping $\boldsymbolϕ(s,a)$. Specifically, we consider the episodic inhomogeneous linear Markov Decision Process (MDP), and propose a novel computation-efficient algorithm, LSVI-UCB$^+$, which achieves an $\widetilde{O}(Hd\sqrt{T})$ regret bound where $H$ is the episode length, $d$ is the feature dimension, and $T$ is the number of steps. LSVI-UCB$^+$ builds on weighted ridge regression and upper confidence value iteration with a Bernstei…
Johns Hopkins University（约翰斯·霍普金斯大学）；UC Santa Barbara（加州大学圣巴巴拉分校）；Deakin University（迪肯大学）
Sample-efficient offline reinforcement learning (RL) with linear function approximation has recently been studied extensively. Much of prior work has yielded the minimax-optimal bound of $\tilde{\mathcal{O}}(\frac{1}{\sqrt{K}})$, with $K$ being the number of episodes in the offline data. In this work, we seek to understand instance-dependent bounds for offline RL with function approximation. We present an algorithm called Bootstrapped and Constrained Pessimistic Value Iteration (BCP-VI), which leverages data bootstrapping and constrained optimization on top of pessimism. We show that under a p…
Google Research（谷歌研究院）；Tel Aviv University（特拉维夫大学）；Courant Institute of Mathematical Sciences（柯朗数学科学研究所）；Cornell University（康奈尔大学）
Myopic exploration policies such as epsilon-greedy, softmax, or Gaussian noise fail to explore efficiently in some reinforcement learning tasks and yet, they perform well in many others. In fact, in practice, they are often selected as the top choices, due to their simplicity. But, for what tasks do such policies succeed? Can we give theoretical guarantees for their favorable performance? These crucial questions have been scarcely investigated, despite the prominent practical importance of these policies. This paper presents a theoretical analysis of such policies and provides the first regret…
We study human-in-the-loop reinforcement learning (RL) with trajectory preferences, where instead of receiving a numeric reward at each step, the agent only receives preferences over trajectory pairs from a human overseer. The goal of the agent is to learn the optimal policy which is most preferred by the human overseer. Despite the empirical successes, the theoretical understanding of preference-based RL (PbRL) is only limited to the tabular case. In this paper, we propose the first optimistic model-based algorithm for PbRL with general function approximation, which estimates the model using …
The paper outlines a framework for autonomous control of a CRM (customer relationship management) system. First, it explores how a modified version of the widely accepted Recency-Frequency-Monetary Value system of metrics can be used to define the state space of clients or donors. Second, it describes a procedure to determine the optimal direct marketing action in discrete and continuous action space for the given individual, based on his position in the state space. The procedure involves the use of model-free Q-learning to train a deep neural network that relates a client's position in the s…
UC Berkeley（加州大学伯克利分校）；University of Washington（华盛顿大学）；Princeton University（普林斯顿大学）
Value function approximation is important in modern reinforcement learning (RL) problems especially when the state space is (infinitely) large. Despite the importance and wide applicability of value function approximation, its theoretical understanding is still not as sophisticated as its empirical success, especially in the context of general function approximation. In this paper, we propose a provably efficient RL algorithm (both computationally and statistically) with general value function approximations. We show that if the value functions can be approximated by a function class that sati…
Technion（以色列理工学院）；Indian Institute of Science Bangalore（印度科学学院班加罗尔分院）；IISC Bangalore（印度科学学院班加罗尔分院）
In this paper we obtain several informative error bounds on function approximation for the policy evaluation algorithm proposed by Basu et al. when the aim is to find the risk-sensitive cost represented using exponential utility. The main idea is to use classical Bapat's inequality and to use Perron-Frobenius eigenvectors (exists if we assume irreducible Markov chain) to get the new bounds. The novelty of our approach is that we use the irreduciblity of Markov chain to get the new bounds whereas the earlier work by Basu et al. used spectral variation bound which is true for any matrix. We also…
École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
This paper provides a theoretical study of deep neural function approximation in reinforcement learning (RL) with the $ε$-greedy exploration under the online setting. This problem setting is motivated by the successful deep Q-networks (DQN) framework that falls in this regime. In this work, we provide an initial attempt on theoretical understanding deep RL from the perspective of function class and neural networks architectures (e.g., width and depth) beyond the ``linear'' regime. To be specific, we focus on the value based algorithm with the $ε$-greedy exploration via deep (and two-layer) neu…
We show how an action-dependent baseline can be used by the policy gradient theorem using function approximation, originally presented with action-independent baselines by (Sutton et al. 2000).
In this paper, we propose a generic framework for devising an adaptive approximation scheme for value function approximation in reinforcement learning, which introduces multiscale approximation. The two basic ingredients are multiresolution analysis as well as tree approximation. Starting from simple refinable functions, multiresolution analysis enables us to construct a wavelet system from which the basis functions are selected adaptively, resulting in a tree structure. Furthermore, we present the convergence rate of our multiscale approximation which does not depend on the regularity of basi…
Reinforcement learning is commonly used with function approximation. However, very few positive results are known about the convergence of function approximation based RL control algorithms. In this paper we show that TD(0) and Sarsa(0) with linear function approximation is convergent for a simple class of problems, where the system is linear and the costs are quadratic (the LQ control problem). Furthermore, we show that for systems with Gaussian noise and non-completely observable states (the LQG problem), the mentioned RL algorithms are still convergent, if they are combined with Kalman filt…
This works handles the inverse reinforcement learning problem in high-dimensional state spaces, which relies on an efficient solution of model-based high-dimensional reinforcement learning problems. To solve the computationally expensive reinforcement learning problems, we propose a function approximation method to ensure that the Bellman Optimality Equation always holds, and then estimate a function based on the observed human actions for inverse reinforcement learning problems. The time complexity of the proposed method is linearly proportional to the cardinality of the action set, thus it c…
University of Florida（佛罗里达大学）；Carnegie Mellon University（卡内基梅隆大学）；Institute for Advanced Studies（高等研究院）；Microsoft Research（微软研究院）
We design a new provably efficient algorithm for episodic reinforcement learning with generalized linear function approximation. We analyze the algorithm under a new expressivity assumption that we call "optimistic closure," which is strictly weaker than assumptions from prior analyses for the linear setting. With optimistic closure, we prove that our algorithm enjoys a regret bound of $\tilde{O}(\sqrt{d^3 T})$ where $d$ is the dimensionality of the state-action features and $T$ is the number of episodes. This is the first statistically and computationally efficient algorithm for reinforcement…
Using a martingale concentration inequality, concentration bounds `from time $n_0$ on' are derived for stochastic approximation algorithms with contractive maps and both martingale difference and Markov noises. These are applied to reinforcement learning algorithms, in particular to asynchronous Q-learning and TD(0).
Columbia University（哥伦比亚大学）；Point Zero One Technology（Point Zero One Technology（Point Zero One科技公司））
With the increasing power of computers and the rapid development of self-learning methodologies such as machine learning and artificial intelligence, the problem of constructing an automatic Financial Trading Systems (FTFs) becomes an increasingly attractive research topic. An intuitive way of developing such a trading algorithm is to use Reinforcement Learning (RL) algorithms, which does not require model-building. In this paper, we dive into the RL algorithms and illustrate the definitions of the reward function, actions and policy functions in details, as well as introducing algorithms that…
This paper introduces a new method for inverse reinforcement learning in large-scale and high-dimensional state spaces. To avoid solving the computationally expensive reinforcement learning problems in reward learning, we propose a function approximation method to ensure that the Bellman Optimality Equation always holds, and then estimate a function to maximize the likelihood of the observed motion. The time complexity of the proposed method is linearly proportional to the cardinality of the action set, thus it can handle large state spaces efficiently. We test the proposed method in a simulat…
In this paper, we provide two new stable online algorithms for the problem of prediction in reinforcement learning, \emph{i.e.}, estimating the value function of a model-free Markov reward process using the linear function approximation architecture and with memory and computation costs scaling quadratically in the size of the feature set. The algorithms employ the multi-timescale stochastic approximation variant of the very popular cross entropy (CE) optimization method which is a model based search method to find the global optimum of a real-valued function. A proof of convergence of the alg…
University of New South Wales（新南威尔士大学）；University of Amsterdam（阿姆斯特丹大学）
We present Nonparametric Approximation of Inter-Trace returns (NAIT), a Reinforcement Learning algorithm for discrete action, pixel-based environments that is both highly sample and computation efficient. NAIT is a lazy-learning approach with an update that is equivalent to episodic Monte-Carlo on episode completion, but that allows the stable incorporation of rewards while an episode is ongoing. We make use of a fixed domain-agnostic representation, simple distance based exploration and a proximity graph-based lookup to facilitate extremely fast execution. We empirically evaluate NAIT on both…
Full-sampling (e.g., Q-learning) and pure-expectation (e.g., Expected Sarsa) algorithms are efficient and frequently used techniques in reinforcement learning. Q$(σ,λ)$ is the first approach unifies them with eligibility trace through the sampling degree $σ$. However, it is limited to the tabular case, for large-scale learning, the Q$(σ,λ)$ is too expensive to require a huge volume of tables to accurately storage value functions. To address above problem, we propose a GQ$(σ,λ)$ that extends tabular Q$(σ,λ)$ with linear function approximation. We prove the convergence of GQ$(σ,λ)$. Empirical re…
Tel Aviv University（特拉维夫大学）；Google Research（谷歌研究院）
We study reinforcement learning with linear function approximation and adversarially changing cost functions, a setup that has mostly been considered under simplifying assumptions such as full information feedback or exploratory conditions.We present a computationally efficient policy optimization algorithm for the challenging general setting of unknown dynamics and bandit feedback, featuring a combination of mirror-descent and least squares policy evaluation in an auxiliary MDP used to compute exploration bonuses.Our algorithm obtains an $\widetilde O(K^{6/7})$ regret bound, improving signifi…
Offline reinforcement learning, which aims at optimizing sequential decision-making strategies with historical data, has been extensively applied in real-life applications. State-Of-The-Art algorithms usually leverage powerful function approximators (e.g. neural networks) to alleviate the sample complexity hurdle for better empirical performances. Despite the successes, a more systematic understanding of the statistical complexity for function approximation remains lacking. Towards bridging the gap, we take a step by considering offline reinforcement learning with differentiable function class…
Motivated by broad applications in reinforcement learning and federated learning, we study local stochastic approximation over a network of agents, where their goal is to find the root of an operator composed of the local operators at the agents. Our focus is to characterize the finite-time performance of this method when the data at each agent are generated from Markov processes, and hence they are dependent. In particular, we provide the convergence rates of local stochastic approximation for both constant and time-varying step sizes. Our results show that these rates are within a logarithmi…
In an ever expanding set of research and application areas, deep neural networks (DNNs) set the bar for algorithm performance. However, depending upon additional constraints such as processing power and execution time limits, or requirements such as verifiable safety guarantees, it may not be feasible to actually use such high-performing DNNs in practice. Many techniques have been developed in recent years to compress or distill complex DNNs into smaller, faster or more understandable models and controllers. This work seeks to identify reduced models that not only preserve a desired performanc…
