强化学习入门经典：Reinforcement Learning_An Introduction

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《强化学习：一个介绍》(Reinforcement Learning: An Introduction) 是一本由 Richard S. Sutton 和 Andrew G. Barto 所著的英文专业书籍，该书属于第二版，但仍在修订中。截止日期为2017年11月5日。本书的主要内容涵盖了强化学习的基本概念、实例分析以及其核心要素。在第一章的"Introduction"中，作者首先定义了强化学习（Reinforcement Learning），这是一种人工智能的学习方式，其中智能体通过与环境的交互来学习最有效的行为策略，以最大化长期累积奖励。这种学习方法强调实践中的反馈，不同于传统的监督学习，它不需要预先标记的数据。书中列举了丰富的例子，旨在帮助读者理解强化学习的实际应用，比如游戏策略（如棋类游戏）、机器人控制、自然语言处理和经济决策等领域。通过这些实例，读者可以直观地看到强化学习如何解决实际问题。接着，作者详细介绍了强化学习的几个关键元素，包括状态空间（State Space）、动作空间（Action Space）、状态转移函数（State Transition Function）、奖励函数（Reward Function）以及价值函数（Value Functions）。这些都是设计和实现强化学习算法的基础。在"Limitations and Scope"部分，作者讨论了强化学习的局限性，包括可能的训练时间较长、对环境的依赖性、以及在某些情况下难以估计最优策略等问题。同时，也明确了本书的范围，即专注于基于模型的强化学习方法，而不是无模型或深度学习领域的最新进展。最后，"An Extended Example"可能是本书中深入探讨的一个案例研究，可能会展示强化学习在某个复杂环境中的完整应用过程，以便读者更好地理解理论在实际场景中的操作和优化。该书作为一本权威的教材，对强化学习的研究者和从业者来说具有很高的参考价值。由于还在修订阶段，作者鼓励读者发现并报告任何错误，以确保最终版本的准确性。此外，他们还欢迎读者分享可能被遗漏的重要文献，以扩展和深化对这一领域的理解。《强化学习：一个介绍》提供了全面而深入的强化学习理论框架，适合于那些希望了解和应用强化学习技术的专业人员。随着科技的发展，该书将持续更新和完善，以适应人工智能领域的快速发展。

xvi Summary of Notation

|S| number of elements in set S

t discrete time step

T, T (t) ﬁnal time step of an episode, or of the episode including time step t

action at time t

state at time t, typically due, stochastically, to S

t−1

and A

t−1

reward at time t, typically due, stochastically, to S

t−1

and A

t−1

π policy, decision-making rule

π(s) action taken in state s under deterministic policy π

π(a|s) probability of taking action a in state s under stochastic policy π

π(a|s, θ) probability of taking action a in state s given parameter θ

return (cumulative discounted reward) following time t (Section 3.3)

t:h

ﬂat return (uncorrected, undiscounted) from t + 1 to h (Section 5.8)

λs

λ-return, corrected by estimated state values (Section 12.1)

λa

λ-return, corrected by estimated action values (Section 12.1)

λs

t:h

truncated, corrected λ-return, with state values (Section 12.3)

λa

t:h

truncated, corrected λ-return, with action values (Section 12.3)

p(s

, r|s, a) probability of transition to state s

with reward r, from state s and action a

p(s

|s, a) probability of transition to state s

, from state s taking action a

r(s, a, s

) expected immediate reward on transition from s to s

under action a

(s) value of state s under policy π (expected return)

∗

(s) value of state s under the optimal policy

(s, a) value of taking action a in state s under policy π

∗

(s, a) value of taking action a in state s under the optimal policy

V, V

array estimates of state-value function v

or v

∗

Q, Q

array estimates of action-value function q

or q

∗

temporal-diﬀerence error at t (a random variable) (Section 6.1)

w, w

d-vector of weights underlying an approximate value function

, w

t,i

ith component of learnable weight vector

d dimensionality—the number of components of w

alternate dimensionality—the number of components of θ

m number of 1s in a sparse binary feature vector

ˆv(s,w) approximate value of state s given weight vector w

(s) alternate notation for ˆv(s,w)

ˆq(s, a, w) approximate value of state–action pair s, a given weight vector w

x(s) vector of features visible when in state s

x(s, a) vector of features visible when in state s taking action a

(s), x

(s, a) ith component of vector x(s) or x(s, a)

shorthand for x(S

) or x(S

, A

)

x inner product of vectors, w

; e.g., ˆv(s,w)

= w

x(s)

µ(s) onpolicy distribution over states (Section 9.2)

µ |S|-vector of the µ(s)

kxk

µ-weighted norm of any vector x(s), i.e.,

µ(s)x(s)

(Section 11.4)

v, v

secondary d-vector of weights, used to learn w (Chapter 11)

d-vector of eligibility traces at time t (Chapter 12)

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Chapter 1

Introduction

The idea that we learn by interacting with our environment is probably the ﬁrst to occur to us when

we think about the nature of learning. When an infant plays, waves its arms, or looks about, it has no

explicit teacher, but it does have a direct sensorimotor connection to its environment. Exercising this

connection produces a wealth of information about cause and eﬀect, about the consequences of actions,

and about what to do in order to achieve goals. Throughout our lives, such interactions are undoubtedly

a major source of knowledge about our environment and ourselves. Whether we are learning to drive a

car or to hold a conversation, we are acutely aware of how our environment responds to what we do, and

we seek to inﬂuence what happens through our behavior. Learning from interaction is a foundational

idea underlying nearly all theories of learning and intelligence.

In this book we explore a computational approach to learning from interaction. Rather than directly

theorizing about how people or animals learn, we explore idealized learning situations and evaluate the

eﬀectiveness of various learning methods. That is, we adopt the perspective of an artiﬁcial intelligence

researcher or engineer. We explore designs for machines that are eﬀective in solving learning problems of

scientiﬁc or economic interest, evaluating the designs through mathematical analysis or computational

experiments. The approach we explore, called reinforcement learning, is much more focused on goal-

directed learning from interaction than are other approaches to machine learning.

1.1 Reinforcement Learning

Reinforcement learning is learning what to do—how to map situations to actions—so as to maximize

a numerical reward signal. The learner is not told which actions to take, but instead must discover

which actions yield the most reward by trying them. In the most interesting and challenging cases,

actions may aﬀect not only the immediate reward but also the next situation and, through that, all

subsequent rewards. These two characteristics—trial-and-error search and delayed reward—are the two

most important distinguishing features of reinforcement learning.

Reinforcement learning, like many topics whose names end with “ing,” such as machine learning

and mountaineering, is simultaneously a problem, a class of solution methods that work well on the

problem, and the ﬁeld that studies this problems and its solution methods. It is convenient to use a

single name for all three things, but at the same time essential to keep the three conceptually separate.

In particular, the distinction between problems and solution methods is very important in reinforcement

learning; failing to make this distinction is the source of a many confusions.

We formalize the problem of reinforcement learning using ideas from dynamical systems theory,

speciﬁcally, as the optimal control of incompletely-known Markov decision processes. The details of this

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2 CHAPTER 1. INTRODUCTION

formalization must wait until Chapter 3, but the basic idea is simply to capture the most important

aspects of the real problem facing a learning agent interacting over time with its environment to achieve

a goal. A learning agent must be able to sense the state of its environment to some extent and must be

able to take actions that aﬀect the state. The agent also must have a goal or goals relating to the state of

the environment. Markov decision processes are intended to include just these three aspects—sensation,

action, and goal—in their simplest possible forms without trivializing any of them. Any method that

is well suited to solving such problems we consider to be a reinforcement learning method.

Reinforcement learning is diﬀerent from supervised learning, the kind of learning studied in most

current research in the ﬁeld of machine learning. Supervised learning is learning from a training set

of labeled examples provided by a knowledgable external supervisor. Each example is a description of

a situation together with a speciﬁcation—the label—of the correct action the system should take to

that situation, which is often to identify a category to which the situation belongs. The object of this

kind of learning is for the system to extrapolate, or generalize, its responses so that it acts correctly

in situations not present in the training set. This is an important kind of learning, but alone it is

not adequate for learning from interaction. In interactive problems it is often impractical to obtain

examples of desired behavior that are both correct and representative of all the situations in which the

agent has to act. In uncharted territory—where one would expect learning to be most beneﬁcial—an

agent must be able to learn from its own experience.

Reinforcement learning is also diﬀerent from what machine learning researchers call unsupervised

learning, which is typically about ﬁnding structure hidden in collections of unlabeled data. The terms

supervised learning and unsupervised learning would seem to exhaustively classify machine learning

paradigms, but they do not. Although one might be tempted to think of reinforcement learning as a

kind of unsupervised learning because it does not rely on examples of correct behavior, reinforcement

learning is trying to maximize a reward signal instead of trying to ﬁnd hidden structure. Uncovering

structure in an agent’s experience can certainly be useful in reinforcement learning, but by itself does

not address the reinforcement learning problem of maximizing a reward signal. We therefore consider

reinforcement learning to be a third machine learning paradigm, alongside supervised learning and

unsupervised learning and perhaps other paradigms as well.

One of the challenges that arise in reinforcement learning, and not in other kinds of learning, is the

trade-oﬀ between exploration and exploitation. To obtain a lot of reward, a reinforcement learning

agent must prefer actions that it has tried in the past and found to be eﬀective in producing reward.

But to discover such actions, it has to try actions that it has not selected before. The agent has to

exploit what it has already experienced in order to obtain reward, but it also has to explore in order to

make better action selections in the future. The dilemma is that neither exploration nor exploitation

can be pursued exclusively without failing at the task. The agent must try a variety of actions and

progressively favor those that appear to be best. On a stochastic task, each action must be tried many

times to gain a reliable estimate of its expected reward. The exploration–exploitation dilemma has been

intensively studied by mathematicians for many decades, yet remains unresolved. For now, we simply

note that the entire issue of balancing exploration and exploitation does not even arise in supervised

and unsupervised learning, at least in their purest forms.

Another key feature of reinforcement learning is that it explicitly considers the whole problem of a

goal-directed agent interacting with an uncertain environment. This is in contrast to many approaches

that consider subproblems without addressing how they might ﬁt into a larger picture. For example, we

have mentioned that much of machine learning research is concerned with supervised learning without

explicitly specifying how such an ability would ﬁnally be useful. Other researchers have developed

theories of planning with general goals, but without considering planning’s role in real-time decision

making, or the question of where the predictive models necessary for planning would come from. Al-

though these approaches have yielded many useful results, their focus on isolated subproblems is a

signiﬁcant limitation.

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