Consider a discrete-time Markov chain evolving on a general state-space . The goal in optimal stopping time problems is to minimize over all stopping times , the associated expected cost:
where denotes the per-stage cost, the terminal cost, and is the discount factor. Examples of such a problem arise mostly in financial applications such as derivatives analysis (see Section V), timing of a purchase or sale of an asset, and more generally in sequential analysis problems.
In this work, the optimal decision rule is approximated using reinforcement learning techniques. We propose and analyze an optimal variance algorithm to approximate the value function associated with the optimal stopping rule.
I-a Problem Setup
It is assumed that is compact, and we let denote the associated Borel -algebra. The time-homogeneous Markov chain
is defined on a probability space (), and its distribution is determined by an initial distribution and a transition kernel , defined for each and by
It is assumed that is uniformly ergodic, so in particular it has a unique invariant probability measure denoted .
Denote by the filtration associated with . The Markov property asserts that for bounded measurable functions ,
In this paper a stopping time
is a random variable taking on values in the non-negative integers, with the defining propertyfor each . A stationary policy is a measurable function that defines a stopping time as follows:
The (optimal) value function is defined as the infimum of (1) over all stopping times: for any ,
The associated Q-function is defined by . The optimal stopping problem is a special cast of discounted-cost optimal control . It follows that solves the following Bellman equation: for each ,
Moreover, the optimal stopping rule is defined by the stationary policy,
An optimal stopping time is , using the general definition (2).
The Bellman equation (4) can be expressed as the functional fixed point equation:
where denotes the dynamic programming operator: for any function , and ,
Analysis is framed in the usual Hilbert space of real-valued measurable functions on with inner product:
where the expectation in (7) is with respect to the steady state distribution . It is assumed throughout that the cost functions and are in .
The objective is to approximate using a parameterized family of functions , where
denotes the parameter vector. We restrict to linear parameterization throughout, so that:
where with , , , denotes the basis functions. For any parameter vector , we denote the Bellman error
It is assumed that the basis functions are linearly independent: The covariance matrix is full rank, where
In a finite state-space setting, it is possible to construct a consistent algorithm that computes the Q-function exactly . The Q-learning algorithm of Watkins [18, 19] can be used in this case (see  for a discussion).
In a function approximation setting, we need to relax the goal of solving (4). As in previous works [17, 6, 21], the goal in this paper is to obtain the solution to a Galerkin relaxation of (4): Find such that,
where is adapted to , and the expectation is in steady state. For a given constant , the Zap- algorithm that is proposed in this work intends to solve (11) with
In much of this paper we fix for simplicity. In this special case, we have: , for each , and the goal is to find such that, for each ,
I-C Literature Survey
Obtaining an approximate solution to the original problem (4) using a modified objective (12) was first considered in . The authors propose an extension of the TD() algorithm of , and prove convergence under general conditions. Though it is not obvious at first sight, the algorithm in  is more closely connected to Watkins’ Q-learning algorithm [18, 19], than the TD() algorithm. This is specifically due to a minimum term that appears in (12) (see definition of in (6)), similar to what appears in Q-learning. This is important to note, because Q-learning algorithms are not known to converge under function approximation settings, and this is due to the fact that the dynamic programming operator may not be a contraction in general . The operator defined in (6) is quite special in this sense: it can be shown that it is a contraction with respect to the -norm defined in (8) :
Since , many other algorithms have been proposed to improve the convergence rate. In  the authors propose a matrix gain variant of the algorithm presented in , improving the rate of convergence in numerical studies . In , the authors take a least squares approach to solve the problem, and propose the least squares Q-learning algorithm, that has close resemblance to the least squares policy evaluation algorithm (LSPE () of ). The authors recognize the high computational complexity of the algorithm, and propose variants in . In prior works  and , though a function-approximation setting is considered, the state-space is assumed finite.
More recently, in [8, 7], the authors propose the Zap Q-learning algorithm to solve for a solution to a fixed point equation similar to (but more general than) (4). The proof of convergence is provided only for the tabular case (wherein the ’s span all possible functions), and when the state-action space is finite.
The remainder of the paper is organized as follows: Section II contains the approximation architecture, and introduces the Zap Q-learning algorithm. The assumptions and main results are contained in Section III. Section IV provides a high-level proof of the results, numerical results are collected together in Section V, and conclusions in Section VI. Full proofs are available in an extended version of this paper, available on arXiv .
Ii Q-learning for Optimal Stopping
The following notation will be used throughout. Define for each , the corresponding policy ,
For any function with domain , two operators are defined as the simple products,
Observe that .
We then denote a matrix, two -dimensional vectors as follows:
The objective (12) can be expressed:
Ii-B Zap Q-Learning
It is useful to first look at a more general class of “matrix gain Q-learning” algorithms. Given a matrix gain sequence with each invertible, and a scalar step-size sequence , the corresponding matrix gain Q-learning algorithm for optimal stopping is given by the following recursion:
with denoting the “temporal difference” sequence:
The algorithm proposed in  is (19), with (the identity matrix). This is similar to the TD() algorithm [16, 13].
The fixed point Kalman filter
fixed point Kalman filteralgorithm of  can also be written as a special case of (19): Each is an estimate of the matrix defined in (10), which can be obtained recursively:
In the Zap Q() algorithm, the matrix gain sequence is designed so that the asymptotic covariance of the resulting algorithm is minimized (see Section III for details). Similar to the Zap-Q algorithm of , it uses matrix gain (the projected pseudo-inverse of ); an estimate of , with defined in (15).
The term inside the expectation in (15), following the substitution , is denoted
Using (21), the matrix is recursively estimated using Monte Carlo in the Zap Q() algorithm. For simplicity we give details only for :
The gain sequences and in the above algorithm are chosen as: for some ,
For each , consider the following terms:
The vector is analogous to in (18), and (25b) recalls the Bellman equation (4). Prop. II.1 follows from these definitions. It shows that is the “projection” of the cost function , similar to how is related to through (16).
For each , we have:
where the expectation is in steady state. In particular,
Iii Assumptions and Main Results
Preliminary results are summarized here that will be useful in establishing the main results. We start with the contraction property of the dynamic programming operator defined in (6). This is a result directly obtained from  (Lemma on p. ).
The dynamic programming operator defined in (6) satisfies:
Furthermore, is the unique fixed point of in .
Recall that is defined in (9). Similar to the operator , for each we define operators and that operate on functions as follows:
The following Lemma is a slight extension of Lemma III.1.
For each , the operator satisfies:
Based on the definition (28), we have:
where the first inequality follows from the fact that (with the induced operator norm in ). The last inequality is true because:
The next result is a direct consequence of Lemma III.2.
The mapping is Lipschitz: For some , and each ,
The following assumptions are made throughout the paper:
Assumption A1: is a uniformly ergodic Markov chain on the compact state space . Its unique invariant probability measure is denoted (see  for definitions).
Assumption A2: There exists a unique solution to the objective (12).
Assumption A3: The conditional distribution of given has a density, . This density is also assumed to have uniformly bounded likelihood ratio with respect to the Gaussian density . Consequently, .
It is assumed moreover that the function is in the span of .
Assumption A4: The parameter sequence is bounded a.s..
Assumption (A3) consists of technical conditions in the proof. The density assumption is imposed to ensure that the conditional expectation given of functions such as are smooth as a function of .
Iii-C Main Result
The main result of this paper establishes convergence of iterates obtained using Algorithm 1:
Suppose that Assumptions A1-A4 hold. Then,
The parameter sequence obtained using the Zap-Q(0) algorithm converges to a.s., where satisfies (12).
An ODE approximation holds for the sequences by continuous time functions satisfying
The term ODE approximation is standard in the SA literature: For , let denote the solution to:
for some . We say that the ODE approximation:
holds for the sequence if the following is true for any :
where denotes the continuous time process constructed from the sequence , whose meaning will be made precise in Sec IV-B. The optimality of the algorithm in terms of the asymptotic variance is discussed next.
Iii-D Asymptotic Variance
The asymptotic covariance of any algorithm is defined to be the following limit:
Consider the matrix gain Q-learning algorithm (19), and suppose the matrix sequence is constant: . Also, suppose that all eigenvalues of satisfy . Following standard analysis (see Section 2.2 of  and references therein), it can be shown that, under general assumptions, the asymptotic covariance of the algorithm (19) can be obtained as a solution to the Lyapunov equation:
where is the “noise covariance matrix”, that is defined as follows.
A “noise sequence” is defined as
and defined in (16). For the matrix gain algorithm with , the algorithm would be deterministic in the ideal case .
The noise covariance matrix is defined as the limit
in which , and the expectation is in steady state.
Optimality of the asymptotic covariance
The asymptotic covariance can be obtained as a solution to (33) only when all eigenvalues satisfy . If there exists at least one eigenvalue such that , then, under general conditions, it can be shown that the asymptotic covariance is not finite. This implies that the rate of convergence of is slower than .
It is possible to optimize the covariance over all matrix gains using (33). Specifically, it can be shown that letting will result in the minimum asymptotic covariance , where
The Zap Q algorithm is specifically designed to achieve the optimal asymptotic covariance. A full proof of optimality will require extra effort. Thm. III.5 tells us that we have the required convergence for this algorithm. Provided we can obtain additional tightness bounds for the scaled error
, we obtain a functional Central Limit Theorem with optimal covariance. Minor additional bounds ensure convergence of (32) to the optimal covariance .
The next section is dedicated to the proof of Thm. III.5.
Iv Proof of Theorem iii.5
Iv-a Overview of the Proof
Unlike martingale difference assumptions in standard stochastic approximation , the noise in our algorithm is Markovian. The first part of this section establishes that our noise sequence satisfies the so called ODE friendly property , such that their asymptotic effect over the parameter update is zero. This enables the argument that the gain matrices are close to their equilibrium over the fast time scale defined by . We then go back to the slow time scale defined by , and obtain the ODE approximations for and the expected projected cost .
Iv-B ODE Analysis
The remainder of this section is devoted to the proof of the ODE approximation (30). The construction of an approximating ODE involves first defining a continuous time process . Denote
and define at these time points, with the definition extended to
via linear interpolation.
Along with the piecewise linear continuous-time process , denote by the piecewise linear continuous-time process defined similarly, with , . Furthermore, for each , denote
The following definition is needed in Lemma IV.1:
ODE-friendly sequence: A vector-valued sequence of random variables will be called ODE-friendly if it admits the decomposition,
is a martingale-difference sequence satisfying a.s. for all
is a bounded sequence
The final sequence is bounded and satisfies:
Lemma IV.1 establishes that the error sequences that appear in the updates for and are ODE friendly.
The argument proceeds by decomposing noise sequences into tractable terms, each of which is shown to be ODE-friendly by standard arguments based on solutions to Poisson’s equation . We only present the treatment for here through Lemma IV.2 to IV.5 since same technique can be applied to and .
Denote for ,
For noise sequence in (41b), we have
Suppose that is a bounded function on with zero mean. Then the sequence is ODE-friendly, with equal to zero, and , with the solution to Poisson’s equation with forcing function .
The sequence is then ODE-friendly since it is a bounded over state space with zero mean. Before we show the same for sequences , we need two preliminary results: the Lipschitz continuity of , and that this continuity is preserved in associated Poisson equation solutions.
There is a deterministic constant such that, with probability one,
The second result is similar to Lemma IV.2. Both are consequences of the fact that the fundamental kernel defined in Section 20.1 of  is a bounded linear operator on when the Markov chain is uniformly ergodic.
There is a constant such that the following holds: For any family of zero mean functions satisfying for some constants , ,
for all , then there are solutions to corresponding Poisson’s equation, , with zero mean, and satisfying
Now we are ready to claim:
The sequences are ODE friendly.
The following Lemma shows that the matrix gain , recursively obtained by (22), approximates the mean , with .
Suppose the sequence is ODE-friendly. Then,
Consequently, only finitely often, and
With the definition of ODE approximation below (31), we have:
The ODE approximation for holds: with probability one, asymptotically tracks the ODE:
For a fixed but arbitrary time horizon , we define two families of uniformly bounded and uniformly Lipschitz continuous functions: and . Sub-sequential limits of and are denoted and respectively.
We recast the ODE limit of the projected cost as follows:
For any sub-sequential limits ,
They satisfy .
For a.e. ,
Proof of Thm. iii.5
V Numerical Results
In this section we illustrate the performance of the Zap Q-learning algorithm in comparison with existing techniques, on a finance problem that has been studied in prior work [6, 17]. We observe that the Zap algorithm performs very well, despite the fact that some of the technical assumptions made in Section III do not hold.
V-a Finance model
The following finance example is used in [6, 17] to evaluate the performance of their algorithms for optimal stopping. The reader is referred to these references for complete details of the problem set-up.
The Markovian state process considered is the vector of ratios: , , in which is a geometric Brownian motion (derived from an exogenous price-process). This uncontrolled Markov chain is positive Harris recurrent on the state space , so is not compact. The Markov chain is uniformly ergodic.
The “time to exercise” is modeled as a stopping time . The associated expected reward is defined as , with and fixed. The objective of finding a policy that maximizes the expected reward is modeled as an optimal stopping time problem.
The value function is defined to be the infimum (3), with and (the objective in Section I is to minimize the expected cost, while here, the objective is to maximize the expected reward). The associated Q-function is defined using (4), and the associated optimal policy using (5):
When the Q-function is linearly approximated using (9), for a fixed parameter vector , the associated value function can be expressed:
Given a parameter estimate and the initial state , the corresponding average reward was estimated using Monte-Carlo in the numerical experiments that follow.
V-B Approximation & Algorithms
Along with Zap Q-learning algorithm we also implement the fixed point Kalman filter algorithm of  to estimate . This algorithm is given by the update equations (19) and (20). The computational as well as storage complexities of the least squares Q-learning algorithm (and its variants)  are too high for implementation.
V-C Implementation Details
The experimental setting of [6, 17] is used to define the set of basis functions and other parameters. We choose the dimension of the parameter vector , with the basis functions defined in . The objective here is to compare the performances of the fixed point Kalman filter algorithm with the Zap-Q learning algorithm in terms of the resulting average reward (45).
Recall that the step-size for the Zap Q-learning algorithm is given in (24). We set for all implementations of the Zap algorithm, but similar to what is done in , we experiment with different choices for . Specifically, in addition to , we let:
with and experiment with and . In addition, we also implement Zap with . Based on the discussion in Section III-D, we expect this choice of step-size sequences to result in infinite asymptotic variance.
In the implementation of the fixed point Kalman filter algorithm, as suggested by the authors, we choose step-size for the matrix gain update rule in (20), and step-size of the form (47) for the parameter update in (19). Specifically, we let , and and .
The number of iterations for each of the algorithm is fixed to be .
V-D Experimental Results
The average reward histogram was obtained by the following steps: We simulate parallel simulations of each of the algorithms to obtain as many estimates of . Each of these estimates defines a policy defined in (46). We then estimate the corresponding average reward defined in (45), with .
Along with the average discounted rewards, we also plot the histograms to visualize the asymptotic variance (32), for each of the algorithms. The theoretical values of the covariance matrices and were estimated through the following steps: The matrices and (the limit of the matrix gain used in ) were estimated via Monte-Carlo. Estimation of requires an estimate of ; this was taken to be obtained using the Zap-Q algorithm with and . This estimate of was also used to estimate the covariance matrix defined in (36) using the batch means method. The matrices and were then obtained using (33) and (37), respectively.
It was observed that the eigenvalues of the matrix have a wide spread: The condition-number is of the order . Despite a badly conditioned matrix gain, it is observed in Fig. 1, that the average rewards of the Zap-Q algorithms are better than its competitors. It is also observed that the algorithm is robust to the choice of step-sizes. In Fig. 2 we observe that the asymptotic behavior of the algorithms are a close match to the theoretical prediction. Specifically, large variance of Zap-Q with step-size confirms that the asymptotic variance is very large (ideally, infinity), if the eigenvalues of the matrix .
In this paper, we extend the theory for the Zap Q-learning algorithm to a linear function approximation setting, with application to optimal stopping. We prove