# Computably Continuous Reinforcement-Learning Objectives are PAC-learnable

Cambridge Yang,1 Michael Littman, 2 Michael Carbin 1

###### Abstract

In reinforcement learning, the classic objectives of maximizing discounted and
finite-horizon cumulative rewards are PAC-learnable: There are algorithms that
learn a near-optimal policy with high probability using a finite amount of
samples and computation. In recent years, researchers have introduced
objectives and corresponding reinforcement-learning algorithms beyond the
classic cumulative rewards, such as objectives specified as linear temporal
logic formulas. However, questions about the PAC-learnability of these new
objectives have remained open.

This work demonstrates the PAC-learnability of general reinforcement-learning
objectives through sufficient conditions for PAC-learnability in two analysis
settings. In particular, for the analysis that considers only sample
complexity, we prove that if an objective given as an oracle is uniformly
continuous, then it is PAC-learnable. Further, for the analysis that considers
computational complexity, we prove that if an objective is computable, then it
is PAC-learnable. In other words, if a procedure computes successive
approximations of the objective’s value, then the objective is PAC-learnable.

We give three applications of our condition on objectives from the literature
with previously unknown PAC-learnability and prove that these objectives are
PAC-learnable. Overall, our result helps verify existing objectives’ PAC-
learnability. Also, as some studied objectives that are not uniformly
continuous have been shown to be not PAC-learnable, our results could guide
the design of new PAC-learnable objectives.

##  1 Introduction

In reinforcement learning, we situate an agent in an environment with unknown
dynamics. The agent acts in the environment by executing its current policy.
Executing a policy in an environment induces an infinite-length path of states
and actions. We specify an objective, a function that maps each possible
infinite-length path to a real number—a score—for that path. Moreover, we
request the agent learn a good policy that nearly maximizes the expected score
over the distribution of paths induced by the environment and the policy.

#### PAC-learnability of Objectives

The classic reinforcement-learning objectives include infinite-horizon
discounted cumulative rewards and finite-horizon cumulative rewards. These
objectives are well-studied and have a desirable property: There are
reinforcement-learning algorithms that learn a near-optimal policy with high
probability with number of samples depending only on the parameters known by
the algorithm. We call these algorithms probably approximately correct (PAC),
and these objectives PAC-learnable under reinforcement learning.

PAC-learnability is essential. Specifically, we aim to specify an objective
and let the agent learn a good policy on its own. Thus, we necessarily need
some form of assurance of how close to optimal the learned policy is. If an
objective is not PAC-learnable, then the hope of ensuring learning a near-
optimal policy is lost, and the objective is effectively intractable to learn
under reinforcement learning.

#### General Objectives

In recent years, researchers have introduced various objectives beyond the two
classic rewards objectives (Henriques et al. 2012; Sadigh et al. 2014; Littman
et al. 2017; Hasanbeig et al. 2019; Hahn et al. 2019; Camacho et al. 2019;
Giacomo et al. 2019; Jothimurugan, Alur, and Bastani 2019; Bozkurt et al.
2020; Ronca and De Giacomo 2021). For example:

  * •

Camacho et al. (2019) introduced the reward machine objective. A reward
machine augments classic rewards with an automaton that makes the rewards
history dependent.

  * •

Bozkurt et al. (2020) introduced an objective based on limit deterministic
Buchi Automaton (LDBA).111We summarize works along the same line in
LABEL:sec:summary-rl-automaton-objectives. The objective features history-
dependent discount factors, history-dependent rewards, and an augmented action
space.

Researchers introduced reinforcement-learning algorithms for these objectives
and showed that they empirically learn well-behaving policies with finitely
many samples.

#### PAC-learnability of General Objectives

Despite the advances on empirical algorithms for these objectives, not all
objectives are PAC-learnable: Recent work (Yang, Littman, and Carbin 2022)
proved that infinite-horizon linear temporal logic objectives, a class of
general objectives, are not PAC-learnable. Therefore, to the end of having
assurance on the outcomes of learning, we desire to understand the PAC-
learnability of general objectives.

Some previous work (Ashok, Křetínský, and Weininger 2019; Henriques et al.
2012; Fu and Topcu 2014; Ronca and De Giacomo 2021) address the PAC-
learnability of particular objectives. However, these analyses give
reinforcement-learning algorithms for particular objectives and do not
generalize to other objectives. Previous work (Alur et al. 2021) gave a
framework of reductions between objectives whose flavor of generality is most
similar to our work; however, they did not give a condition for when an
objective is PAC-learnable. To our knowledge, the PAC-learnability of the
objectives in Sadigh et al. (2014); Littman et al. (2017); Hasanbeig et al.
(2019); Hahn et al. (2019); Camacho et al. (2019); Jothimurugan, Alur, and
Bastani (2019); Bozkurt et al. (2020) are not known.

Relevant to model-based reinforcement learning, Bazille et al. (2020) showed
that it is impossible to learn the transitions of a Markov chain such that the
learned and true models agree on all first-order behaviors. However, this
result does not apply to the general reinforcement-learning setting.

We thus raise a research question: When is a reinforcement-learning objective
PAC-learnable?

#### Our Approach

We address the question by giving sufficient conditions for PAC-learnability.
Specifically, we analyze PAC-learnability in both the information-theoretic
setting, that only considers sample complexity, and the computation-theoretic
setting, that also considers computability. We prove that, in the information-
theoretic setting (resp. computation-theoretic setting), an objective is PAC-
learnable if it is uniformly continuous (resp. computable). These conditions
simplify proving objectives’ PAC-learnability. In particular, our conditions
avoid constraints on environments, policies, or reinforcement-learning
algorithms but only require reasoning about the objective itself.

We give example applications of our conditions to three objectives in the
literature whose PAC-learnability was previously unknown and prove that they
are PAC-learnable.

#### Contributions

We make the following contributions about reinforcement-learning objectives:

  * •

In the information-theoretic setting, we prove that a uniformly continuous
objective is PAC-learnable.

  * •

In the computation-theoretic setting, we prove that a computable objective is
PAC-learnable.

  * •

We apply our theorem to three objectives (Camacho et al. 2019; Bozkurt et al.
2020; Littman et al. 2017) from the literature whose PAC-learnability was
previously unknown and show that they are PAC-learnable.

Our result makes checking the PAC-learnability of existing objectives easier.
It also potentially guides the design of new PAC-learnable objectives.

##  2 Reinforcement Learning and Objectives

This section reviews reinforcement learning and defines general objectives and
their learnability.

###  2.1 Markov Processes

A Markov decision process (MDP) is a tuple $\mathcal{M}=(S,A,P,s_{0})$, where
$S$ and $A$ are finite sets of states and actions, $P\colon(S\times
A)\to\Delta\left(S\right)$ is a transition probability function that maps a
current state and an action to a distribution over next states, and $s_{0}\in
S$ is an initial state. The MDP is sometimes referred to as the _environment
MDP_ to distinguish it from any specific objective.

A policy for an MDP is a function $\pi\colon(\left(S\times A\right)^{*}\times
S)\to\Delta\left(A\right)$ mapping a history of states and actions to a
distribution over actions. 222 $X^{*}$ denotes all finite-length sequences of
the elements of $X$. $X^{\upomega}$ denotes all infinite-length sequences of
the elements of $X$. An MDP and a policy induce a discrete-time Markov chain
(DTMC). A DTMC is a tuple $\mathcal{D}=(S,P,s_{0})$, where $S$ is the set of
states, $P\colon S\to\Delta\left(S\right)$ is a transition-probability
function mapping states to distributions over next states, and $s_{0}\in S$ is
an initial state. The DTMC induces a probability space over the infinite-
length sequences $w\in S^{\upomega}$.00footnotemark: 0

###  2.2 Objective

#### Environment-specific Objective.

An environment-specific objective for an MDP $(S,A,P,s_{0})$ is a measurable
function $\kappa\colon\left(S\times A\right)^{\upomega}\to\mathbb{R}$. We say
such an objective is environment-specific since it is associated with MDPs
with a fixed set of states and actions.

The value of an environment-specific objective for an MDP $\mathcal{M}$ and a
policy $\pi$ is the expectation of the objective under the probability space
of the DTMC $\mathcal{D}$ induced by $\mathcal{M}$ and $\pi$:
$V^{\pi}_{\mathcal{M},\kappa}=\mathsf{E}_{#1}\left[\kappa\left(w\right)\right].$
We consider only bounded objectives to ensure that the expectation exists and
is finite. The optimal value is the supremum of the values achievable by all
policies: $V^{*}_{\mathcal{M},\kappa}=\sup_{\pi}V^{\pi}_{\mathcal{M},\kappa}$.
A policy $\pi$ is $\epsilon$-optimal if its value is $\epsilon$-close to the
optimal value: $V^{\pi}_{\mathcal{M},\kappa}\geq
V^{*}_{\mathcal{M},\kappa}-\epsilon$.

#### Environment-generic Objective.

An objective defined above is environment-specific because it is associated
with a fixed set of states and actions. However, we would also like to talk
about objectives in a form decoupled from any MDP. For example, the discounted
classical cumulative rewards objective is not bound to any particular reward
function. Further, the objective of “reaching the goal state” in a grid world
environment is not bound to the size of the grid or the allowed actions. Such
decoupling is desirable as it allows one to specify objectives independent of
environments.

To that end, we define environment-generic objectives. The idea of such
objectives is that a labeling function interfaces between the environment and
the environment-generic objective. The definition decouples an environment-
generic objective from environments by requiring different labeling functions
for different environments.

Formally, an environment-generic objective is a measurable function:
$\xi\colon F^{\upomega}\to\mathbb{R}$, where $F$ is a set called features. A
labeling function maps the MDP’s (current) states and actions to the features:
$\mathcal{L}\colon(S\times A)\to F$. Composing $\xi$ and the element-wise
application of $\mathcal{L}$ induces an environment-specific objective. For
example, the discounted cumulative rewards objective
$\xi\colon\mathbb{Q}^{\upomega}\to\mathbb{R}$ is
$\xi(w)=\sum_{i=0}^{\infty}\gamma^{i}\cdot w[i]$. For each MDP, the labeling
function is a classical reward function $\mathcal{L}\colon(S\times
A)\to\mathbb{Q}$. 333For simplicity of analysis, we will let objective
specifications use rationals instead of reals so that they admit a finite
representation. Nonetheless, our analyses also generalize to objective
specifications that contain reals.

The value of $\xi$ for an MDP $\mathcal{M}$, a policy $\pi$, and a labeling
function $\mathcal{L}$ is the value of the environment-specific objective
induced by $\xi$, $\mathcal{M}$, $\pi$, and $\mathcal{L}$.

###  2.3 Learning Model

A reinforcement learning agent has access to a sampler of the MDP’s
transitions but does know the underlying probability values. The agent learns
in two phases: sampling and learning. In the sampling phase, the agent starts
from the initial state and follows a sequence of decision steps to collect
sampled environment transitions. At every step, the agent may  (1) act from
the current state to sample the next state or  (2) reset to the initial state.
In the learning phase, it learns a policy from the collected sampled
transitions.

Formally, a reinforcement-learning algorithm is a tuple
$(\mathcal{A}^{S},\mathcal{A}^{L})$, where $\mathcal{A}^{S}$ is a sampling
algorithm that drives how the environment is sampled, and $\mathcal{A}^{L}$ is
a learning algorithm that learns a policy from the samples obtained by the
sampling algorithm. Let $A_{\texttt{reset}}=A\cup\\{\texttt{reset}\\}$ be the
set of actions with an additional reset operation, the sampling algorithm
$\mathcal{A}^{S}\colon\left(\left(S\times A_{\texttt{reset}}\right)^{*}\times
S\right)\to A_{\texttt{reset}}$ maps from sampled transitions to the next
operation. The learning algorithm $\mathcal{A}^{L}\colon\left(\left(S\times
A_{\texttt{reset}}\right)^{*}\times S\right)\to\left(\left(\left(S\times
A\right)^{*}\times S\right)\to\Delta(A)\right)$ maps from the sampled
transitions to the learned policy.

###  2.4 Learnability of Objectives

A good reinforcement-learning algorithm should learn the optimal policy that
maximizes the given objective. In particular, we let the algorithm seek a
near-optimal policy with high probability.

######  Definition 1 (PAC Algorithm for Environment-specific Objective).

Given an objective $\kappa$, a reinforcement-learning algorithm
$(\mathcal{A}^{\text{S}},\mathcal{A}^{\text{L}})$ is $\kappa$-PAC (probably
approximately correct for objective $\kappa$) in an environment MDP
$\mathcal{M}$ with $N$ samples if, with the sequence of transitions $T$ of
length $N$ sampled using the sampling algorithm $\mathcal{A}^{\text{S}}$, the
learning algorithm $\mathcal{A}^{\text{L}}$ outputs an $\epsilon$-optimal
policy with probability at least $1-\delta$ for any given $\epsilon>0$ and
$0<\delta<1$. That is:

$\mathsf{P}_{#1}\left(V^{\mathcal{A}^{\text{L}}\left(T\right)}_{\mathcal{M},\kappa}\geq
V^{*}_{\mathcal{M},\kappa}-\epsilon\right)\geq 1-\delta.$

We use
$\mathsf{T}{\sim}\left\langle\mathcal{M},\mathcal{A}^{\text{S}}\right\rangle_{N}$to
denote that the probability space is over the set of length-$N$ transition
sequences sampled from the environment $\mathcal{M}$ using the sampling
algorithm $\mathcal{A}^{\text{S}}$. We will simply write $\mathsf{P}_{#1}(.)$
when it is clear from the context.

We will consider two settings: the information-theoretic setting that
considers only sample complexity and the computation-theoretic setting that
considers computability.

######  Definition 2 (PAC-learnable Environment-specific Objective).

In the information-theoretic setting (resp. computation-theoretic setting), an
environment-specific objective $\kappa$ is $\kappa$-PAC-learnable if there
exists a function
$C\colon(\mathbb{R}\times\mathbb{R}\times\mathbb{N}\times\mathbb{N})\to\mathbb{N}$
such that, for all consistent environment MDPs for $\kappa$ (i.e., the domain
of $\kappa$ uses the same set of states and actions as the MDP), there exists
a $\kappa$-PAC reinforcement-learning algorithm with less than
$C(\frac{1}{\epsilon},\frac{1}{\delta},|S|,|A|)$ samples (resp. computation
steps).

Our definition focuses on the core tractability issue. Failure to respect our
definition implies that PAC-learning is not achievable with finitely many
samples (in the information-theoretic setting) or not computable (in the
computation-theoretic setting). To that end, we have set the parameters of $C$
to be the only quantities available to an algorithm under the standard
assumptions of reinforcement learning. Specifically, since the transition
dynamics are unknown, they are not parameters of $C$. Moreover, while some
variants of PAC-learnability require $C$ to be a polynomial to capture the
notion of learning efficiency, we have dropped this requirement to focus on
the core tractability issue.

We also define the PAC-learnability of environment-generic objectives, for
both information- and computation-theoretic settings:

######  Definition 3 (PAC-learnable Environment-generic Objective).

An environment-generic objective $\xi$ is $\xi$-PAC-learnable if for all
labeling functions $\mathcal{L}$, the objective $\kappa$ induced by $\xi$ and
$\mathcal{L}$ is $\kappa$-PAC-learnable.

Note that in the information-theoretic setting, we assume that the objectives
$\kappa$ and $\xi$ are given as oracles: they take infinite-length inputs and
return infinite-precision output with no computation overhead.

###  2.5 Established PAC-Learnable Objectives

The standard discounted cumulative rewards objective
$\sum_{i=0}^{\infty}\gamma^{i}w[i]$ and the finite-horizon cumulative rewards
objective $\sum_{i=0}^{H}w[i]$ are known to be PAC-learnable. The folklore
intuition is that these objectives “effectively terminate” in an expected
finite-length horizon, and rewards farther out of the horizon diminish
quickly. This paper formalizes this intuition by connecting it to the standard
definition of the objective function’s uniform continuity and computability.
Later in Section 4, we will prove that uniformly continuous and computable
objectives are PAC-learnable.

##  3 Example: The Reward Machine Objective

This section gives an example general objective: the simple reward machine
objective (Camacho et al. 2019). We will later use this objective as one of
the examples to apply our core theorem and prove its PAC-learnability.

#### The Simple Reward Machine Specification

Simple reward machines generalize from classic Markovian rewards to non-
Markovian rewards. In particular, a simple reward machine is a kind of
deterministic finite automaton. Each automaton transition has a reward value
and a tuple of truth values for a set of propositions about the environment’s
state. The simple reward machine starts from an initial state. As the agent
steps through the environment, a labeling function classifies the
environment’s current state to a tuple of truth values of a set of
propositions. The simple reward machine then transits to the next states
according to the tuple. During each transition, the agent collects a scalar
reward along the transition of the simple reward machine. The overall
objective is to maximize the $\gamma$-discounted sum of collected rewards. The
formal definition of a simple reward machine given by Camacho et al. (2019)
is:

######  Definition 4 (Simple Reward Machine).

Given a finite set $\Pi$ called the propositions, a simple reward machine over
$\Pi$ is a tuple $(U,\delta_{u},\delta_{r},u_{0},\gamma)$, where $U$ is a
finite set of states, $\delta_{u}\colon(U\times 2^{\Pi})\to U$ is a
deterministic state transition function, $\delta_{r}\colon(U\times
U)\to\mathbb{Q}$ is a deterministic reward function, $u_{0}$ is an initial
state, and $\gamma\in\mathbb{Q}$ is a discount factor.

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012301

Figure 1: Left: simple reward machine. Right: environment.

Figure 1 shows an example simple reward machine and an accompanying grid
environment. The states of the simple reward machine are
$\\{u_{1},u_{2},u_{3}\\}$. The labeling function for this particular
environment maps grid locations $(1,0)$ and $(2,0)$ to “fire” ( ) and $(3,0)$
to “goal” ( ). Each transition of the simple reward machine is labeled by a
tuple of truth values of the two propositions (“fire” and “goal”). For
example, $u_{1}$ transits to $u_{2}$ and produces a reward of $1$ if the
environment’s current state is labeled as “goal” but not “fire”.

#### The Simple Reward Machine Objective

Formally, a simple reward machine $R$ specifies an environment-generic
objective $\llbracket
R\rrbracket\colon\left(2^{\Pi}\right)^{\upomega}\to\mathbb{R}$ given by:

$\llbracket
R\rrbracket(w)=\sum_{k=1}^{\infty}\gamma^{k}\delta_{r}(u_{k},u_{k+1}),\quad\forall
k\geq 0.\,u_{k+1}=\delta_{u}(u_{k},w[k]).$

The set of features $F$ corresponding to this environment-generic objective is
the possible truth values of the propositions $\Pi$, that is $F=2^{\Pi}$. The
labeling function classifies each environment’s current state and action to
these features.

##  4 Condition for PAC-Learnability

This section presents our main result: sufficient conditions for an
objective’s learnability. The first two subsections analyze learnability in
the information-theoretic setting. Specifically, we show that an objective
given as an oracle is PAC-learnable if it is uniformly continuous. The next
two subsections analyze learnability in the computation-theoretic setting.
Specifically, using a standard result from computational analysis, we show
that a computable objective is PAC-learnable. LABEL:sec:proof-of-unnecessity
complements our result by showing that our conditions are sufficient but not
necessary.

###  4.1 Uniform Continuity

We first recall the following standard definition of a uniformly continuous
function.

######  Definition 5 (Uniformly Continuous Function).

A function $f\colon X\to Y$ with metric spaces $(X,d_{X})$ and $(Y,d_{Y})$ is
uniformly continuous if, for any $\epsilon>0$, there exists $\delta>0$ so that
$f$ maps $\delta$-close elements in the domain to $\epsilon$-close elements in
the image 444Note that textbook definitions commonly use ${<}$ instead of
${\leq}$: our definition is equivalent. We use $\leq$ to match with the
comparison operators in the PAC definitions. :

$\begin{split}\forall\epsilon>0.\exists\delta>0.\forall x_{1}\in X.\forall
x_{2}\in X:\\\ d_{X}(x_{1},x_{2})\leq\delta\Rightarrow
d_{Y}(f(x_{1}),f(x_{2}))\leq\epsilon.\end{split}$

To specialize the above definition to an objective, we next note the metric
space of the domain of an objective. An objective’s domain is the set of
infinite-length sequences $X^{\upomega}$, where $X=(S\times A)$ for an
environment-specific objective and $X=F$ for an environment-generic objective.
The domain forms a metric space by the standard distance function
$d_{X^{\omega}}(w_{1},w_{2})=2^{-L_{\text{prefix}}(w_{1},w_{2})}$, where
$L_{\text{prefix}}(w_{1},w_{2})$ is the length of the longest common prefix of
$w_{1}$ and $w_{2}$ (Manna and Pnueli 1987). We are now ready to specialize
the definition of uniform continuity to objectives.

######  Definition 6 (Uniformly Continuous Objective).

An objective (environment-specific or environment-generic) $f\colon
X^{\upomega}\to\mathbb{R}$ is uniformly continuous if, for any
$\epsilon\,{>}\,0$, there exists a finite horizon $H$ so that the objective
maps all infinite-length sequences sharing the same prefix of length $H$ to
$\epsilon$-close values:

$\begin{split}\forall\epsilon>0.\exists H\in\mathbb{N}.\forall w\in
X^{\omega}.\forall w^{\prime}\in X^{\omega}:\\\
L_{\text{prefix}}(w,w^{\prime})\geq
H\Rightarrow|f(w)-f(w^{\prime})|\leq\epsilon.\end{split}$

Note that since the domain of an objective is compact, the Heine–Cantor
theorem guarantees that a continuous objective is also uniformly continuous.
This paper only uses uniform continuity since it is more relevant to our
theorem and proof. Nonetheless, theorems presented in the following section
also hold for continuous objectives.

###  4.2 Continuity Implies PAC-learnability

#### Environment-specific Objectives.

We give a sufficient condition for a learnable environment-specific objective:

######  Theorem 1.

An environment-specific objective $\kappa$ is $\kappa$-PAC-learnable in the
information-theoretic setting if it is uniformly continuous.

We will prove the theorem by constructing a $\kappa$-PAC reinforcement-
learning algorithm for any uniformly continuous $\kappa$. To that end, we
reduce $\kappa$ to a finite-horizon cumulative rewards problem; we then prove
the theorem by invoking an existing PAC reinforcement-learning algorithm for
finite-horizon cumulative rewards problems.

###### Proof of Theorem 1.

For any $\epsilon^{\prime}>0$, since $\kappa$ is uniformly continuous, there
exists a bound $H$ such that infinite-length sequences sharing a length-$H$
prefix are all mapped to $\epsilon^{\prime}$-close values.

For concreteness, let us pick any $\dot{s}\,{\in}\,S$ and any
$\dot{a}\,{\in}\,A$. For each length-$H$ sequence $u\,{\in}\,(S\times A)^{H}$,
we pick the representative infinite-length sequence
$[u;(\dot{s},\dot{a})^{\upomega}]$ that starts with the prefix $u$ and ends in
an infinite repetition of $(\dot{s},\dot{a})$. Using these representatives, we
construct a finite-horizon rewards objective
$\tilde{\kappa}_{\epsilon^{\prime}}$ of horizon $H$. The construction assigns
each infinite-length sequence with the value of the original $\kappa$ at the
chosen representative. That is, let $w[{:}H]$ denote the length-$H$ prefix of
$w$, we define $\tilde{\kappa}_{\epsilon^{\prime}}$ as:

$\tilde{\kappa}_{\epsilon^{\prime}}(w)\triangleq\kappa([w[{:}H];(\dot{s},\dot{a})^{\upomega}]),\,\,\forall
w\in(S\times A)^{\upomega}.$

By construction, $\tilde{\kappa}_{\epsilon^{\prime}}$ is
$\epsilon^{\prime}$-close to $\kappa$, meaning that for any infinite-length
input, their evaluations differ by at most $\epsilon^{\prime}$:

$|\tilde{\kappa}_{\epsilon^{\prime}}(w)-\kappa(w)|\leq\epsilon^{\prime},\,\,\forall
w\in(S\times A)^{\upomega}.$

Thus, an $\epsilon^{\prime}$-optimal policy for
$\tilde{\kappa}_{\epsilon^{\prime}}$ is $2\epsilon^{\prime}$-optimal for
$\kappa$.

We then reduce the approximated objective $\tilde{\kappa}_{\epsilon}$, which
assigns a history-dependent reward at the horizon $H$, into a finite-horizon
cumulative rewards objective, which assigns a history-independent reward at
each step. To that end, we lift the state space to
$U\,{=}\,\bigcup_{t=1}^{H}(S\times A)^{t}$. Each state $u_{t}\,{=}\,(S\times
A)^{t}$at step $t$ in the lifted state space is the length-$t$ history of
states and actions encountered in the environment. For any state before step
$H$, we assign a reward of zero. For any state $u_{H}\,{=}\,(S\times A)^{H}$
at step $H$, we assign a reward of
$\tilde{\kappa}_{\epsilon}([u_{H};(\dot{s},\dot{a})^{\upomega}])$. The lifted
state space and the history-independent reward function above form the desired
finite-horizon cumulative rewards problem.

Dann et al. (2019) introduced ORLC, a PAC reinforcement-learning algorithm for
finite-horizon cumulative rewards problems.555ORLC provides an individual
policy certificates (IPOC) guarantee. Dann et al. showed that IPOC implies our
PAC definition, which they called “supervised-learning style PAC”. Applying
ORLC to the above finite-horizon cumulative rewards problem produces an
$2\epsilon^{\prime}$-optimal policy for $\kappa$. Finally, for any $\epsilon$,
choosing $\epsilon^{\prime}\;{=}\;\frac{\epsilon}{2}$ gives a $\kappa$-PAC
reinforcement-learning algorithm for $\kappa$. ∎

#### Environment-generic Objectives.

Theorem 1 states a sufficient condition for when an environment-specific
objective is PAC-learnable. The following corollary generalizes the condition
to environment-generic objectives.

######  Corollary 2.

An environment-generic objective $\xi$ is $\xi$-PAC-learnable in the
information-theoretic setting if $\xi$ is uniformly continuous.

To the end of proving Corollary 2, we first observe the following lemma, which
we prove in LABEL:sec:label_preserves_uniform_continuity_proof.

######  Lemma 3.

If an environment-generic objective is uniformly continuous, then, for all
labeling functions, the induced environment-specific objective is also
uniformly continuous.

With Lemma 3, Corollary 2 is straightforward. Since each induced environment-
specific objective $\kappa$ is uniformly continuous, each $\kappa$ is
$\kappa$-PAC-learnable by Theorem 1. Thus, the objective $\xi$ is $\xi$-PAC-
learnable by definition.

###  4.3 Computability

We now define the computability of an objective $f\colon
X^{\upomega}\to\mathbb{R}$. The standard definition of computability of such
functions depends on Type-2 Turing machines (Weihrauch 2000, Chapter 2,
Definition 2) and a representation of the reals by an infinite sequence of
rational approximations, called the Cauchy-representation (Weihrauch 2000,
Chapter 3, Definition 3). Informally, a Type-2 Turing machine is a Turing
machine with an infinite-length input tape and a one-way infinite-length
output tape. The machine reads the input tape and computes forever writing to
the output tape.

######  Definition 7 (Computable Objective).

An objective $f$ is computable if a Type-2 Turing machine reads $w$ from the
input tape and writes to the output tape a fast-converging Cauchy sequence
$[q_{0},q_{1},\dots]\in\mathbb{Q}^{\upomega}$ of rational approximations to
$f(w)$, that is: $\forall n\in\mathbb{N}$, $|f(w)-q_{n}|\leq 2^{-n}$.

When proving computability, this definition is tedious to work with since it
requires implementing the function on a Turing machine. Instead, we will use
pseudocode to formulate an algorithm that takes in an infinite-stream input
$w$ and a natural number $n$ and outputs the $n$-th rational approximation
$q_{n}$. Repeatedly invoking the algorithm by enumerating $n$ produces the
Cauchy sequence of rational approximations.

A classic result in computable analysis is that computable functions are
continuous (Weihrauch 2000, Theorem 2.5 and 4.3). Since an objective’s domain
is compact, by the Heine-Cantor theorem, this result also holds for uniform
continuity. Even stronger, the following theorem, modified from Weihrauch
(2000, Theorem 6.4) for our context, guarantees that for a computable
objective, for any rational $\epsilon>0$, we can compute a horizon $H$ that
satisfies the definition of uniform continuity. Define the modulus of
continuity of an objective as a function $m\colon\mathbb{Q}\to\mathbb{N}$ that
satisfies $\forall\epsilon\in\mathbb{Q}$, $\forall w_{1}\in
X^{\upomega},\forall w_{2}\in
X^{\upomega}:L_{\text{prefix}}\left(w_{1},w_{2}\right)\geq
m(\epsilon)\Rightarrow\left|f\left(w_{1}\right)-f\left(w_{2}\right)\right|\leq\epsilon$.
Then, we have:

######  Theorem 4.

A computable objective is uniformly continuous. Further, its modulus of
continuity $m$ is computable.

For completeness, LABEL:sec:computing-modulus-of-continuity gives pseudocode
that computes the modulus of continuity for any computable objective specified
by the interface described above.

###  4.4 Computability Implies PAC-learnability

We now extend our result in Section 4.2 from the information-theoretic to the
computation-theoretic setting.

######  Theorem 5.

An (environment-generic or environment-specific) objective $f$ is $f$-PAC-
learnable in the computation-theoretic setting if $f$ is computable.

###### Proof.

Combining theorems in Section 4.2 and Theorem 4, a computable objective $f$ is
uniformly continuous, therefore $f$-PAC-learnable in the information-theoretic
setting. In the computational-theoretic setting, we need to further construct
a computable reinforcement-learning algorithm. Note that our proof of
Corollary 2 is already constructive of an algorithm. However, we need to:

  * •

compute the bound $H$ from the given $\epsilon^{\prime}$ and

  * •

computably evaluate the approximated objective
$\tilde{\kappa}_{\epsilon^{\prime}}$.

A computable objective resolves both points:

  * •

By Theorem 4, the bound $H$ is computable for any $\epsilon$.

  * •

Evaluating the approximate objective is computable, since the approximated
objective only depends on the length-$H$ prefix of the input.∎

LABEL:sec:concrete-pac-rl-algorithm provides pseudocode for an $f$-PAC
reinforcement-learning for any computable objective $f$.

##  5 Theorem Applications

This section applies the core theorem and corollary to two objectives in the
existing literature and proves each objective’s PAC-learnability. Due to
space, we give the third objective from Littman et al. (2017) and prove its
PAC-learnability in LABEL:sec:gltl.

###  5.1 Reward Machine

#### Proof of PAC-learnability

We prove that the reward-machine objective reviewed in Section 3 is PAC-
learnable.

######  Proposition 6.

The objective $\llbracket R\rrbracket$ of a simple reward machine $R$ is
$\llbracket R\rrbracket$-PAC-learnable.

###### Proof.

By Theorem 5, it is sufficient to show that a simple reward-machine objective
is computable. Consider the pseudocode with Python-like syntax in
LABEL:lst:simple-reward-machine-objective-impl.

Listing 1: Computation of the simple reward objective

⬇

# Given reward machine $(U,\delta_{u},\delta_{r},u_{0},\gamma)$

def SRM(w: $(2^{\Pi})^{\upomega}$, n: $\mathbb{N}$) $\rightarrow$
$\mathbb{Q}$:

u: $U$, value: $\mathbb{Q}$ = $u_{0}$, 0

rmax: $\mathbb{Q}$ = max(abs($\delta_{r}(\text{u1},\text{u2})$) for u1, u2 in
$U^{2}$))

H: $\mathbb{N}$ = (log2floor(1 - $\gamma$) -n - log2ceil(rmax))) \

/ log2ceil($\gamma$)

for k in range(H):

u’ = $\delta_{u}$(u, w[k])

value += $\gamma$**k * $\delta_{r}$(u, u’)

u = u’

return value’

LABEL:lst:simple-reward-machine-objective-impl defines an algorithm for
computing the simple reward-machine objective. It first initializes the state
variable u to the initial state $u_{0}$. It then computes a horizon
$H=\left(\lfloor\log_{2}(1-\gamma)\rfloor-n-\lceil\log_{2}r_{\max}\rceil\right)/\lceil\log_{2}\gamma\rceil$,
where $r_{\max}=\max\left(\left|\delta_{r}(\cdot)\right|\right)$ is the
maximum magnitude of all possible rewards. It iterates through the first $H$
indices of the input and transits the reward machine’s state according to the
transitions $\delta_{u}$. For each input $w$ and $n$, the algorithm
accumulates the discounted cumulative rewards truncated to the first
$H$-terms: $\sum_{k=0}^{H-1}\gamma^{k}\delta_{r}(u_{k},u_{k+1})$.

By definition of a computable objective, we need to show that the returned
values for all $n$ form a fast-converging Cauchy sequence: $\forall
n\in\mathbb{N},|\texttt{SRM}(w,n)-\llbracket R\rrbracket(w)|\leq 2^{-n}$. To
see this, let $\Delta\triangleq\left|\texttt{SRM}(w,n)-\llbracket
R\rrbracket\left(w\right)\right|=\left|\sum_{k=H}^{\infty}\gamma^{k}\delta_{r}(u_{k},u_{k+1})\right|$.
Then, we have $\Delta\leq\nicefrac{{r_{\max}\cdot\gamma^{H}}}{{1-\gamma}}$ by
upper bounding the rewards by $r_{\max}$, then simplifying the infinite sum
into a closed form. By plugging in the value of $H$ and simplifying the
inequality, we have $\Delta\leq 2^{-n}$. Thus, the objective is computable and
$\llbracket R\rrbracket$-PAC-learnable.∎

###  5.2 LTL-in-the-limit Objectives

Linear temporal logic (LTL) objectives are measurable Boolean objectives that
live in the first two-and-half levels of the Borel hierarchy (Manna and Pnueli
1987). Various works (Hasanbeig et al. 2019; Hahn et al. 2019; Bozkurt et al.
2020) considered LTL objectives for reinforcement learning and empirical
algorithms for learning. A common pattern of these algorithms is that they
convert a given LTL formula to an intermediate specification that takes in
additional hyper-parameters. They show that in an unreachable limit of these
hyper-parameters, the optimal policy for this intermediate specification
becomes the optimal policy for the given LTL formula. We call such
intermediate specifications LTL-in-the-limit specifications. Due to space, we
will focus on Bozkurt et al. (2020) and give the objective specified by their
LTL-in-the-limit-specification. We will show that this objective is PAC-
learnable. The same process, namely writing down the LTL-in-the-limit
specification and then proving that the specified objective is PAC-learnable,
also applies to the approaches in Sadigh et al. (2014); Hasanbeig et al.
(2019); Hahn et al. (2019).

####  Bozkurt et al.’s LTL-in-the-limit Specification

Given an LTL formula, Bozkurt et al. first convert the formula into a limit
deterministic Buchi Automaton (LDBA) by a standard conversion algorithm
(Sickert et al. 2016) with two additional discount factor parameters. An LDBA
is a non-deterministic finite automaton. It is bipartite by two sets of
states, those in an initial component and those in an accepting component.
Transitions in the automaton can only go from the initial component to the
accepting component, but not the reverse. An LDBA is “deterministic in the
limit”: it only has non-deterministic $\epsilon$-transitions in the initial
component, but it is deterministic in the accepting component. The formal
definition of LDBA is:

######  Definition 8 (LDBA).

For an LTL formula over propositions $\Pi$, an LDBA converted from the formula
is a tuple $(U,\mathcal{E},\delta_{u},u_{0},B)$, where $U$ is a finite set of
states, $\delta_{u}\colon\left(U\times(2^{\Pi}\cup\\{\epsilon\\})\right)\to
2^{U}$ is a non-deterministic transition function, $u_{0}$ is an initial
state, and $B\subseteq U$ is a set of accepting states. Additionally, $U$ has
a bi-partition of an initial component with states $U_{I}$ and an accepting
component with states $U_{B}$. An LDBA satisfies the conditions:  (1)
$\delta_{u}(u,\epsilon)=\emptyset$for all $u\in U_{B}$,  (2)
$\delta_{u}(u,2^{\Pi})\subseteq U_{B}$for all $u\in U_{B}$, and  (3)
$B\subseteq U_{B}$.

The agent and environment models are similar to a simple reward machine: At
each step, the agent chooses an environment’s action and steps in the
environment. A labeling function classifies the current state of the
environment to a tuple of truth values of the set of propositions $\Pi$.

At each step, an LDBA takes either a non-deterministic $\epsilon$-transitions
(if such transition is available) or the transition along the tuple of the
truth values of the propositions. Each time the LDBA enters an accepting
state, the agent receives a reward of $1-\gamma_{1}$, and discounts all future
rewards by $\gamma_{1}$. Each time the LDBA enters a non-accepting state, the
agent receives no reward and discounts all future rewards by $\gamma_{2}$. An
oracle controls the $\epsilon$-transitions. In words, the objective is to
maximize the (state-dependent) discounted cumulative rewards, assuming the
oracle always makes the optimal choice that helps to maximize the cumulative
rewards.

####  Bozkurt et al.’s LTL-in-the-limit Objective

Bozkurt et al.’s LTL-in-the-limit specification is a tuple
$(L,\gamma_{1},\gamma_{2})$: the LDBA $L$ and the two hyper-parameters
$\gamma_{1},\gamma_{2}\in\mathbb{Q}$. It specifies an environment-generic
objective
$\llbracket(L,\gamma_{1},\gamma_{2})\rrbracket\colon(2^{\Pi})^{\upomega}\to\mathbb{R}$.
Let $\mathcal{E}^{+}=\mathcal{E}\cup\\{\bot\\}$, where $\mathcal{E}$ is the
set of $\epsilon$-transitions and $\bot$ is a non-$\epsilon$-transition (i.e.,
following a transition with a tuple classified by the labeling function), the
objective is:

$\begin{split}&\llbracket(L,\gamma_{1},\gamma_{2})\rrbracket(w)=\max_{w_{\mathcal{E}}\in\mathcal{E}^{\upomega}}g(w_{\mathcal{E}},w)\quad\text{where
}\\\
&g(w_{\mathcal{E}},w)=\sum_{i=1}^{\infty}R(u_{i})\prod_{j=1}^{i-1}\Gamma(u_{j}),\\\
&R(u)=(1-\gamma_{1})\mathbb{1}\\{{u\in B}\\},\\\
&\Gamma(u)=\gamma_{1}\mathbb{1}\\{{u\in B}\\}+\gamma_{2}\mathbb{1}\\{{u\not\in
B}\\},\\\ &\forall k\geq 0\colon u_{k+1}=\delta_{u}(u_{k},w^{+}_{k}),\\\
&\quad t_{k}=\sum_{i=1}^{k}\mathbb{1}\\{{w_{\mathcal{E}}[i]=\bot\text{ or
}(u_{k},w_{\mathcal{E}}[i])\not\in\delta_{u}}\\},\\\ &\quad
w^{+}_{k}=\begin{cases}w[t_{k}]&\text{if }w_{\mathcal{E}}[k]=\bot\text{ or
}(u_{k},w_{\mathcal{E}}[k])\not\in\delta_{u}\\\
w_{\mathcal{E}}[k]&\text{otherwise}\end{cases}\end{split}.$ (1)

Here, $t_{k}$ is the step count of the environment when the LDBA takes its
$k$-th step. Note that $t_{k}\leq k$, since the environment does not step when
LDBA takes an $\epsilon$-transition. The value $w[t_{k}]$ is the tuple of
truth values of the input infinite-length sequence $w$ at $t_{k}$. We define
$w^{+}_{k}$ as the transition label taken by the LDBA at the $k$-th step: It
is either  (1) a tuple of truth values $w[k]$, if $w_{\mathcal{E}}[k]$ is a
non-$\epsilon$-transition or an $\epsilon$-transition that is not available
from the current LDBA state $u_{k}$, or  (2) the $\epsilon$-transition
$w_{\mathcal{E}}[k]$.  By its definition, $w^{+}_{k}$ is always a valid
transition of the LDBA, and it always leads to a deterministic next state.
Therefore, we write $u_{k+1}=\delta_{u}(u_{k},w^{+}_{k})$ to denote that the
LDBA state $u_{k+1}$ follows this deterministic transition to the next state.

#### Proof of PAC-learnability

We now prove that the objective specified by an LTL-in-the-limit specification
in Bozkurt et al. (2020) is PAC-learnable. Although this section aims to show
an example, as we mentioned, the proof strategy here also applies to the
approaches in Sadigh et al. (2014); Hasanbeig et al. (2019); Hahn et al.
(2019).

######  Proposition 7.

Bozkurt et al.’s LTL-in-the-limit objective
$\llbracket(L,\gamma_{1},\gamma_{2})\rrbracket$ is
$\llbracket(L,\gamma_{1},\gamma_{2})\rrbracket$-PAC-learnable.

###### Proof.

By Theorem 5, it is sufficient to show that Bozkurt et al.’s objective is
computable. Consider the pseudocode with Python-like syntax in LABEL:lst:ldba-
objective-impl.

Listing 2: Computation of Bozkurt et al.’s objective

⬇

# Given LDBA $(U,\mathcal{E},\delta_{u},u_{0},B)$ and $\gamma_{1},\gamma_{2}$

def bozkurt_objective(w: $(2^{\Pi})^{\upomega}$, n: $\mathbb{N}$)
$\rightarrow$ $\mathbb{Q}$:

gamma_max: $\mathbb{Q}$ = max($\gamma_{1}$, $\gamma_{1}$)

H: $\mathbb{N}$ = (log2floor(1 - gamma_max) - n) \

/ log2ceil(gamma_max)

v: $\mathbb{Q}$ = 0

for w_e in $\mathcal{E}^{\text{H}}$:

v = max(v, bozkurt_helper(H, w_e, w))

return v

def bozkurt_helper(H: $\mathbb{N}$, w_e: $\mathcal{E}^{\text{H}}$, w:
$S^{\omega}$) $\rightarrow$ $\mathbb{Q}$:

v: $\mathbb{Q}$, u: $U$, discount: $\mathbb{Q}$ = 0, $u_{0}$, 1

for k in range(H):

if u in $B$:

reward, gamma = 1, $\gamma_{1}$

else:

reward, gamma = 0, $\gamma_{2}$

v += reward * discount

discount *= gamma

if w_e[k] == $\bot$ or (u, w_e[k]) not in $\delta_{u}$:

w_k_plus = w[k]

else:

w_k_plus = we[k]

u = $\delta_{u}$(u, w_k_plus)

return v

LABEL:lst:ldba-objective-impl gives pseudocode for computing Bozkurt et al.’s
objective. The pseudocode contains two procedures. The procedure
bozkurt_helper computes $g$ but truncates the sum to the first
$H=\left(\lfloor\log_{2}(1-\max(\gamma_{1},\gamma_{2})\rfloor-n\right)/\lceil\log_{2}\max(\gamma_{1},\gamma_{2})\rceil$
terms. The procedure bozkurt_objective then computes the $n$-th rational
approximation of the objective’s value. It invokes the helper function for all
$\hat{w_{\epsilon}}\in\mathcal{E}^{n}$ and calculates the value of
$\max_{\hat{w_{\epsilon}}\in\mathcal{E}^{n}}\texttt{bozkurt\\_helper}(\hat{w_{\epsilon}},w,n)$.

LABEL:sec:bozkurt_objective_pac_proof proves that the return values of
bozkurt_objective for all $n\in\mathbb{N}$ form a fast-converging Cauchy
sequence:

$|\texttt{bozkurt\\_objective}(w,n)-\llbracket(L,\gamma_{1},\gamma_{2})\rrbracket(w)|\leq
2^{-n}.$

Therefore, the objective is computable and consequently
$\llbracket(L,\gamma_{1},\gamma_{2})\rrbracket$-PAC-learnable. ∎

##  6 Conclusion

This work studies the PAC-learnability of general reinforcement-learning
objectives and gives the first sufficient condition of PAC-learnability of an
objective. We use examples to show the applicability of our condition on
various existing objectives whose learnability were previously unknown.

#### Applications to Existing Objectives

Although we only demonstrated three examples, our theorem also applies to
other objectives in the literature. Some examples are  (1) modifications to
the simple reward machine such as the (standard) reward machine (Camacho et
al. 2019) (where rewards depend on not only the reward machine’s state but
also the environment’s state) and the stochastic reward machine (Corazza,
Gavran, and Neider 2022),  (2) other LTL-in-the-limit objectives (Sadigh et
al. 2014; Hahn et al. 2019; Hasanbeig et al. 2019), and  (3) various finite-
horizon objectives (Henriques et al. 2012; Jothimurugan, Alur, and Bastani
2019; Giacomo et al. 2019).

Moreover, we gave an example objective in LABEL:sec:proof-of-unnecessity
showing our condition is sufficient but not necessary. However, to our
knowledge, no previous objective has a similar pattern to our example.
Therefore, we conjecture that our condition applies to most, if not all,
existing PAC-learnable objectives in the literature. Nonetheless, verifying
each objective’s PAC-learnability is out of scope of this work.

#### Guiding The Design of New Objectives

Our main result could also help the design of new objectives. With our
sufficient condition, researchers can create continuous and computable
objectives by design, and our condition will ensure the PAC-learnability of
such objectives.

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