G RADIENT DESCENT TEMPORAL DIFFERENCE - DIFFERENCE LEARNING
Anonymous authors
Paper under double-blind review
A BSTRACT
Off-policy methods deal with for learning a value function from behavior generated by a different policy.
However, even for simple convex problems such as linear value function approximation, these algorithms are not guaranteed to be stable.
To address this, alternative algorithms that are provably convergent in such cases have been introduced, the most well known being gradient descent temporal difference (GTD) learning.
This algorithm and others like it, however, tend to converge much more slowly than conventional temporal difference learning.
In this paper we propose gradient descent temporal difference-difference (Gradient-DD) learning in order to improve GTD learning by introducing second-order differences in successive parameter updates.
We investigate this algorithm in the framework of linear value function approximation, analytically showing its improvement over GTD learning.
Studying the model empirically on the random walk and Boyanchain prediction tasks, we ﬁnd substantial improvement over GTD learning and, in several cases, better performance even than conventional TD learning.
I NTRODUCTION
Off-policy is the problem of learning a value function from behavior generated by a different policy.
However, because off-policy methods learn a value function for a target policy given data due to a different behavior policy, they often exhibit greater variance in parameter updates.
When applied to problems involving function approximation, off-policy methods are slower to converge than onpolicy methods and may even diverge (Baird, 1995; Sutton & Barto, 2018).
Two general approaches have been investigated to address the challenge of developing stable and effective off-policy temporal-difference algorithms.
One approach is to use importance sampling methods to warp the update distribution back to the on-policy distribution (Precup et al., 2000;
Mahmood et al., 2014).
This approach is useful for decreasing the variance of parameter updates, but it does not address stability issues.
The second main approach to addressing the challenge of off-policy learning is to develop true gradient descent-based methods that are guaranteed to be stable regardless of the update distribution.
Sutton et al.
(2009a;b) proposed the ﬁrst off-policy gradientdescent-based temporal difference (GTD and GTD2, respectively) algorithms.
These algorithms are guaranteed to be stable, with computational complexity scaling linearly with the size of the function approximator.
Empirically, however, their convergence is much slower than conventional temporal difference (TD) learning, limiting their practical utility (Ghiassian et al., 2020; White & White, 2016).
Building on this work, extensions to the GTD family of algorithms (see Ghiassian et al.
(2018) for a review) have allowed for incorporating eligibility traces (Maei & Sutton, 2010; Geist & Scherrer, 2014), non-linear function approximation such as with a neural network (Maei, 2011), and reformulation of the optimization as a saddle point problem (Liu et al., 2015; Du et al., 2017).
However, due to their slow convergence, none of these stable off-policy methods are commonly used in practice.
In this work, we introduce a new gradient descent algorithm for temporal difference learning with linear value function approximation.
This algorithm, which we call gradient descent temporal difference-difference (Gradient-DD) learning, is an acceleration technique that employs secondorder differences in successive parameter updates.
The basic idea of Gradient-DD is to modify the error objective function by additionally considering the prediction error obtained in last time step, then to derive a gradient-descent algorithm based on this modiﬁed objective function.
In ad- dition to exploiting the Bellman equation to get the solution, this modiﬁed error objective function avoids drastic changes in the value function estimate by encouraging local search around the current estimate.
Algorithmically, the Gradient-DD approach only adds an additional term to the update rule of the GTD2 method, and the extra computational cost is negligible.
We show mathematically that applying this method signiﬁcantly improves the convergence rate relative to the GTD2 method for linear function approximation.
This result is supported by numerical experiments, which also show that Gradient-DD obtains better convergence in many cases than conventional TD learning. R ELATED W ORK There are studies (Liu et al., 2012; Ghiassian et al., 2020) that add some regularization term in the objective of the Gradient-TD algorithms. Liu et al.
(2012) learns sparse representations of value functions by the l 1 regularization on weights, and Ghiassian et al. (2020) uses the l 2 regularization on weights. Unlike these references, our approach modiﬁes the error objective function by regularizing the evaluation error obtained in last time step. From this modiﬁcation, our method provides a learning rule that contains second-order differences in successive parameter updates.
Ours is similar to trust region policy optimization (Peters & Schaal, 2008; Schulman et al., 2015) or relative entropy policy search (Peters et al., 2010) that penalize large changes to the weights being learned in policy learning, in which the constrained optimization is used to update the policy by considering the constraint on some measure between the new policy and the old policy.
Here, however, our aim here is to look for the optimal value function, and the regularization term uses the previous value function estimate to avoid drastic changes in the updating process.
G RADIENT DESCENT METHOD FOR OFF - POLICY TEMPORAL DIFFERENCE LEARNING 2. P ROBLEM DEFINITION AND BACKGROUND
In this section, we formalize the problem of learning the value function for a given policy under the Markov Decision Process (MDP) framework.
In this framework, the agent interacts with the environment over a sequence of discrete time steps, t = 1 , 2 , . . . .
At each time step the agent observes a partial summary of the state s t ∈ S and selects an action a t ∈ A .
In response, the environment emits a reward r t ∈ R and transitions the agent to its next state s t +1 ∈ S .
The state and action sets are ﬁnite.
State transitions are stochastic and dependent on the immediately preceding state and action.
Rewards are stochastic and dependent on the preceding state and action, as well as on the next state.
The process generating the agent’s actions is termed the behavior policy.
In off-policy learning, this behavior policy is in general different from the target policy π : S → A .
The objective is to learn an approximation to the state-value function under the target policy in a particular environment:
[Equation]
where γ ∈ [0 , 1) is the discount rate.
In problems for which the state space is large, it is practical to approximate the value function.
In this paper we consider linear function approximation, where states are mapped to feature vectors with fewer components than the number of states.
Speciﬁcally, for each state s ∈ S there is a corresponding feature vector x ( s ) ∈ R p , with p ≤ |S| , such that the approximate value function is given by
[Equation]
The goal is then to learn the parameters w such that V w ( s ) ≈ V ( s ) .
2. G RADIENT TEMPORAL DIFFERENCE LEARNING
A major breakthrough for the study of the convergence properties of MDP systems came with the introduction of the GTD and GTD2 learning algorithms (Sutton et al., 2009a;b).
We begin by brieﬂy recapitulating the GTD algorithms, which we will then extend in the following sections.
To begin, we introduce the Bellman operator B such that the true value function V ∈ R |S| satisﬁes the Bellman equation:
[Equation]
where R is the reward vector with components E ( r n +1 | s n = s ) , and P is a matrix of state transition probabilities.
In temporal difference methods, an appropriate objective function should minimize the difference between the approximate value function and the solution to the Bellman equation.
Having deﬁned the Bellman operator, we next introduce the projection operator Π , which takes any value function V and projects it to the nearest value function within the space of approximate value functions of the form (2).
Letting X be the matrix whose rows are x ( s ) , the approximate value function can be expressed as V w =
Xw .
We will also assume that there exists a limiting probability distribution such that d s = lim n →∞ p ( s n = s ) (or, in the episodic case, d s is the proportion of time steps spent in state s ).
The projection operator is then given by
[Equation]
where the matrix D is diagonal, with diagonal elements d s .
The natural measure of how closely the approximation V w satisﬁes the Bellman equation is the mean-squared Bellman error:
[Equation]
where the norm is weighted by D , such that (cid:107) V (cid:107) 2 D = V (cid:62) DV .
However, because the Bellman operator follows the underlying state dynamics of the Markov chain, irrespective of the structure of the linear function approximator, BV w will typically not be representable as V w for any w .
An alternative objective function, therefore, is the mean squared projected Bellman error (MSPBE), which we deﬁne as
[Equation]
Following (Sutton et al., 2009b), our objective is to minimize this error measure.
As usual in stochastic gradient descent, the weights at each time step are then updated by ∆ w = − α ∇ w J ( w ) , where α > 0 , and
[Equation]
For notational simplicity, we have denoted the feature vector associated with s n as x n = x ( s n ) .
We have also introduced the temporal difference error δ n = r n + ( γ x n +1 − x n ) (cid:62) w n , as well as η , a linear predictor to approximate [ E ( x n x (cid:62) n )] − 1 E ( δ n x n ) .
Because the factors in Eqn.
(5) can be directly sampled, the resulting updates in each step are
[Equation]
These updates deﬁne the GTD2 learning algorithm, which we will build upon in the following section.
G RADIENT DESCENT TEMPORAL DIFFERENCE - DIFFERENCE LEARNING
In order to improve the GTD2 algorithm described above, in this section we modify the objective function via additionally considering the approximation error V w − V w n − 1 given the previous time step n − 1 .
Speciﬁcally, we modify Eqn.
(4) as follows:
[Equation]
where κ ≥ 0 is a parameter of the regularization.
Minimizing Eqn.
(7) is equivalent to the following optimization [Equation]
straint (cid:107) V w
[Equation]
V w
[Equation] 1 (cid:107) [Equation] D ≤ µ where µ > 0 is a parameter which becomes large when κ is small, so that the MSPBE objective is recovered as µ → ∞ , equivalent to κ → 0 in Eqn.
We show in the Appendix that for any µ > 0 , there exist κ ≥ 0 such that the solution of Eqn.
(7) and that of Eqn.
(8) are the same.
Eqns.
(7) and (8) represent a tradeoff between minimizing the MSPBE error and preventing the estimated value function from changing too drastically.
Rather than simply minimizing the optimal prediction from the projected Bellman equation, the agent makes use of the most recent update to look for the solution.
Figure 1 gives a schematic view of the effect of the regularization.
Rather than directly following the direction of the MSPBE gradient, the update chooses a w that minimizes the MSPBE while following the constraint that the estimated value function should not change too greatly.
In effect, the regularization term encourages searching around the estimate at previous time step, especially when the state space is large.
With these considerations in mind, the negative gradient of J GDD ( w | w n − 1 ) is
[Equation]
Because the terms in Eqn.
(9) can be directly sampled, the stochastic gradient descent updates are given by
[Equation]
These update equations deﬁne the Gradient-DD method, in which the GTD2 update equations (6) are generalized by including a second-order update term in the third update equation, where this term originates from the squared bias term in the objective (7).
In the following sections, we shall analytically and numerically investigate the convergence and performance of Gradient-DD learning.
I MPROVED CONVERGENCE RATE
In this section we analyze the convergence rate of Gradient-DD learning.
Note that the second-order update in the last line in Eqn.
(10) can be rewritten as a system of ﬁrst-order difference equations:
[Equation]
Let β n = ζα n , ζ > 0 .
We consider the constant step sizes in the updates, i.e., κ n = κ and α n = α .
Denote H n = [-(cid:20) 
-][Equation] n
[Equation] (cid:62) n (cid:21) and G n = (cid:20) [Equation]
x (cid:62) n (cid:21) .
We rewrite the update rules of two iterations in Eqn.
(11) as a single iteration in a combined parameter vector with 2 n components, ρ n = ( η (cid:62) n / √ ζ, w (cid:62) n ) (cid:62) , and a new reward-related vector with 2 n components, g n +1 =
( r n x (cid:62) n , [Equation] ) (cid:62) , as follows:
[Equation]
Denoting ψ n +1 =
α − 1 ( ρ n +1 − ρ n ) , Eqn.
(12) is rewritten as
[Equation]
where the second step is from (cid:20)
[Equation]
Denote J n =
[Equation]
and J =
[Equation]
where G = E ( G n ) and H = E ( H n ) .
Eqn.
(13) tells us that J n is the update matrix of the GradientDD algorithm.
Note: G n is the update matrix of the GTD2 algorithm.
Therefore, assuming the stochastic approximation in Eqn. (13) goes to the solution of an associated ordinary differential equation (ODE) under some regularity conditions (a convergence property is provided in the appendix by following Borkar & Meyn (2000)), solutions to system of linear ﬁrst-order ODEs (Atkinson et al., 2008) tells us that the improved convergence rate of Gradient-DD learning can be understood by considering the real eigenvalues of the matrices G and J . Let λ G be a real eigenvalue of the matrix √ ζ G .
Note:
By directly following the gradient, G deﬁned here is the negative one in Maei (2011)).
From Maei (2011), the eigenvalues of the matrix − G are strictly negative. In other words, λ G > 0 . Denote λ be an eigenvalue of the matrix J .
For simplifying, we consider the simple case that the matrix E ( x n x (cid:62) n ) = I . Besides of λ = λ G , other eigenvalues of J would be solutions to
[Equation]
Assuming 0 < κ < ( α − 1 / 2 − α 1 / 2 λ G ) 2 / 4 , the solutions to Eqn.
(14) are
[Equation]
See the details of derivations of Eqn.
(14) & (15) in the appendix.
The assumption on κ is to exclude the complex eigenvalues, so we just consider real eigenvalues here. the upper bound of requirement of κ is mild because ( α − 1 / 2 − α 1 / 2 λ G ) 2 / 4 is big when α is small. In practice, α is small even goes to 0 as n → ∞ . It follows that the effect of the second-order difference term might allow the second set of weights to learn better estimates of the expected TD error, which could account for the performance increase over GTD2.
This explains the enhanced speed of convergence in Gradient-DD learning.
We shall illustrate this enhanced speed of convergence in numerical experiments in Section 5.
Additionally, we also show a convergence property of Gradient-DD under the constant stepsizes by applying the ordinary differential equation method of stochastic approximation (Borkar & Meyn, 2000). Let the TD ﬁxed point be w ∗ , such that V w ∗ = ΠBV w ∗ . Under some conditions, we prove that, for any (cid:15) > 0 , there exists b 1 < ∞ such that lim sup n →∞ P ( (cid:107) w n − w ∗ (cid:107) > (cid:15) )
≤ b 1 α . See details in the appendix. For the tapered stepsizes, the analysis framework in Borkar & Meyn (2000) does not apply into the Gradient-DD algorithm. Although theoretical investigation on the convergence under the tapered stepsizes is a question to be studied, we ﬁnd empirically in numerical experiments that the algorithm does in fact converge with tapered step sizes and even obtains much better performance in this case than with ﬁxed step sizes.
E MPIRICAL S TUDY
In this section, we assess the practical utility of the Gradient-DD method in numerical experiments. To validate performance of Gradient-DD learning, we compare Gradient-DD learning with GTD  learning, TDC learning (TD with gradient correction (Sutton et al., 2009b)), TD learning, and Emphatic TD learning (Sutton & Mahmood, 2016) in tabular representation using a random-walk task and in linear representation using the Boyan-chain task.
For each method and each task, we performed a scan over the step sizes α n and the parameter κ so that the comprehensive performance of the different algorithms can be compared.
We considered two choices of step size sequence { α n } :
• (Case 1) α n is constant, i e., α n = α 0 .
• (Case 2)
The learning rate α n is tapered according to the schedule α n = α 0 (10 3 + 1) / (10 3 + n ) .
We set the κ = cα 0 where c
= 1 , 2 , 4 . Additionally, we also allow κ dependent on n and consider Case 3: α n is tapered as in Case 2, but κ n = cα n .
In order to simplify presentation, the results of Case 3 are reported in the Appendix.
Here we at ﬁrst simply set β n = α n , then later set β n = ζα n under ζ ∈ { 1 / 4 , 1 / 2 , 1 , 1 } for checking the effect of the two-timescale approach of the Gradientbased TD algorithms on Gradient-DD.
In all cases, we set γ = 1 .
5. R ANDOM WALK TASK
As a ﬁrst test of Gradient-DD learning, we conducted a simple random walk task (Sutton & Barto, 2018) with tabular representation of the value function.
The random walk task has a linear arrangement of m states plus an absorbing terminal state at each end.
Thus there are m +2 sequential states,
S 0 , S 1 , · · · , S m , S m +1 , where m = 20, 50, or 100.
Every walk begins in the center state.
At each step, the walk moves to a neighboring state, either to the right or to the left with equal probability.
If either edge state ( S 0 or S m +1 ) is entered, the walk terminates.
A walk’s outcome is deﬁned to be r = 0 at S 0 and r = 1 at S m +1 .
Our aim is to learn the value of each state V ( s ) , where the true values are (1 , · · · , m ) /
( m + 1) .
In all cases the approximate value function is initialized to the intermediate value V 0 ( s ) = 0 . For further checking the effect of V 0 ( s ) , we also initialize
V 0 ( s ) = 0 , and report the results in Figure 7 of the Appendix, where its performance is very similar as the initialization V 0 ( s ) = 0 .
We ﬁrst compare the methods by plotting the empirical RMS error from the ﬁnal episode during training as a function of step size α in Figure 2, where 5000 episodes are used. From the ﬁgure, we can make several observations. (1) Emphatic TD works well but is sensitive to α .
It prefers very small α even in the tapering case, and this preference becomes strong as the state space becomes large in size.
(2) Gradient-DD works well and is robust to α as the conventional TD learning does. (3) TDC works similarly as the GTD2 method, but requires a little larger α than GTD2. (4) GradientDD performs similarly as the conventional TD learning, while the GTD2 method works worse than conventional TD learning. This advantage is consistent in different settings.
(5) The domain of α for Gradient-DD is roughly similar as GTD2.
[Figure]
Next we look closely at the performance during training, which we show in Figure 3, where each method and parameter setting was run for 5000 episodes.
From the observation in Figure 2, in order to facilitate comparison of these methods, we set α 0 = 0 . 1 for 10 spaces, α 0 = 0 . 2 for 20 spaces, and α 0 = 0 . 5 for 50 spaces.
Because Emphatic TD requires the step size α to be especially small as shown in Figure 2, the plotted values of α 0 for Emphatic TD are tuned relative to the values used in the algorithm deﬁned in Sutton & Mahmood (2016), where the stepsizes of Emphatic TD α ( ETD )is chosen from { 0 . 01% } by the smallest area under the performance curve. Additionally we also tune α 0 for TDC because TDC requires α n larger a little than GTD as shown in Figure 2. The stepsizes for TDC is set as α ( TDC ) n = aα n , where a is chosen from { 1 , 1 . 5 , 2 , 3 } by the smallest area under the performance curve.
From the ﬁgure we draw several observations. (1) For all conditions tested, Gradient-DD converges much more rapidly than GTD2 and TDC. The results indicate that Gradient-DD even converges faster than TD learning in some cases, though it is not as fast in the beginning episodes.
The advantage of Gradient-DD learning over other methods grows as the state space increases in size.
(3) Gradient-DD learning is robust to the choice of c , which controls the size κ of the second-order update, as long as c is not too large.
(Empirically c = 2 is a good choice.)
(4) Gradient-DD has consistent and good performance under both the constant step size setting and under the tapered step size setting.
In summary, compared with the GTD2 learning, Gradient-DD learning in this task leads to improved learning with good convergence.
[Figure]
Effect of β n .
The Gradient-based TD algorithms are kinds of two-timescale approaches ( α n and β n ), so thanks of one reviewer, one concern is whether the same effect can be achieved by a two timescale approach, where the value function weights are updated with a smaller step size than the second set of weights.
For investigating the effect of β n , we set β n = ζα n under ζ ∈ { 1 / 4 , 1 / 2 , 1 , 1 } , and report the results in Figure 8 of the Appendix.
The resulting results show that Gradient-DD works well consistently under various β n settings.
It supports that the secondorder difference term in our approach provides an improvement independent from the two time scale approach.
5. B OYAN - CHAIN TASK
We next investigate Gradient-DD learning on the Boyan-chain problem, which is a standard task for testing linear value-function approximation (Boyan, 2002).
In this task we allow for 4 p − 3 states, with p = 20 , each of which is represented by a p -dimensional feature vector.
The p -dimensional representation for every fourth state from the start is [1 , 0 , · · · , 0] for state s 1 , [0 , 1 , 0 , · · · , 0] for s 5 , · · · , and [0 , 0 , · · · , 0 , 1] for the terminal state s 4 p − 3 .
The representations for the remaining states are obtained by linearly interpolating between these.
The optimal coefﬁcients of the feature vector are ( − 4( p − 1) , − 4( p − 2) , · · · , 0) / 5 .
Simulations with p = 50 and 100 give similar results to those from the random walk task, and hence are not shown here.
In each state, except for the last one before the end, there are two possible actions: move forward one step or move forward two steps with equal probability 0.5.
Both actions lead to reward -0.3.
The last state before the end just has one action of moving forward to the terminal with reward -0.2.
As in the random-walk task, α 0 used in Emphatic TD is tuned from { 0 . 5% , 0 .
We report the results in Figure 4, which leads to conclusions similar to those already drawn from
Figure 3.
(1) Gradient-DD has much faster convergence than GTD2 and TDC, and generally converges to better values despite being somewhat slower than TD learning at the beginning episodes.
(2) Gradient-DD is competitive with Emphatic TD.
The improvement over other methods grows as the state space becomes larger.
(3) As κ increases, the performance of Gradient-DD improves.
Additionally, the performance of Gradient-DD is robust to changes in κ as long as κ is not very large.
Empirically a good choice is to set κ = α or 2 α .
(4) Comparing the performance with constant step size versus that with tapered step size, the Gradient-DD method performs better with tapered step size than it does with constant step size. [Figure] E m p i r i c a l R
M S
[Figure]
rr o [Figure] [Figure] E m p
i r i c a l R
M S e rr o r [Figure] 5. B AIRD ’ S COUNTEREXAMPLE We also verify the performance of Gradient-DD on Baird’s off-policy counterexample (Baird, 1995), for which TD learning famously diverges and TDC is not reported here due to its similarity to GTD2.
We consider three cases: one is 7-state, 100-state and 500-state.
We set α = 0 .
(but α = 10 − 5 for ETD), β = α and γ = 0 . We set κ = 0 . 2 for GDD1, κ = 0 .
4 for GDD2 and κ = 0 .
8 for GDD3. For the initial parameter values (1 , · · · , 1 , 10 , 1) (cid:62) .
We measure the performance by the empirical RMS errors as function of sweep, and report the results in Figure 5. The ﬁgure demonstrates that
Gradient-DD works as well on this well-known counterexample as GTD2 does, and even works Sweep E m p i r i c a l R
M S e rr o r
{+TDETDGTD2TDCGDD1GDD2GDD 
+}
{+Sweep E m p i r i c a l R M S e rr o r 
+}
Sweep
E m p i r i c a l R M S e rr o r better than GTD2 for the 100-state case. We also observe that improvement of GDD increases as the state spaces increases.
Note: in this counterexample the RMS errors doe not go to zero, since the linear approximation leaves a gap in this counterexample due to the projection error.
Interestingly,
GDD can reduces the gap as the state space increases.
C ONCLUSION AND DISCUSSION
In this work, we have proposed Gradient-DD learning, a new gradient descent-based TD learning algorithm.
The algorithm is based on a modiﬁcation of the projected Bellman error objective function for value function approximation by introducing a second-order difference term.
The algorithm signiﬁcantly improves upon existing methods for gradient-based TD learning, obtaining better convergence performance than conventional linear TD learning.
Since GTD learning was originally proposed, the Gradient-TD family of algorithms has been extended for incorporating eligibility traces and learning optimal policies (Maei & Sutton, 2010; Geist & Scherrer, 2014), as well as for application to neural networks (Maei, 2011).
Additionally, many variants of the vanilla Gradient-TD methods have been proposed, including HTD (Hackman, 2012) and Proximal Gradient-TD (Liu et al., 2016).
Because Gradient-DD just modiﬁes the objective error of GTD2 by considering an additional squared-bias term, it may be extended and combined with these other methods, potentially broadening its utility for more complicated tasks.
In this work we have focused on value function prediction in the two simple cases of tabular representations and linear approximation.
An especially interesting direction for future study will be the application of Gradient-DD learning to tasks requiring more complex representations, including neural network implementations.
Such approaches are especially useful in cases where state spaces are large, and indeed we have found in our results that Gradient-DD seems to confer the greatest advantage over other methods in such cases.
Intuitively, we expect that this is because the difference between the optimal update direction and that chosen by gradient descent becomes greater in higher-dimensional spaces (cf.
Fig. 1).
This performance beneﬁt in large state spaces suggests that
Gradient-DD may be of practical use for these more challenging cases
A PPENDIX 6. O N THE EQUIVALENCE OF E QNS . (7) & (
[Equation]
The Karush-Kuhn-Tucker conditions of Eqn.
(8) are the following system of equations [Equation]
These equations are equivalent to
[Equation]
κ
[Equation] [Equation]
dd
[Equation]
and κ = 0 , if (cid:107) V w − V w n
[Equation]
1 (cid:107) 2 D < µ
[Equation]
Thus, for any µ > 0 , there exists a κ ≥
such that d d w J ( w ) + µ d d w (cid:107) V w − V w n − [Equation] 2 D = 0 . 6. E IGENVALUES OF J
Let λ be an eigenvalue of the matrix J .
We have that
[Equation]
From the assumption E ( x n x (cid:62) n ) =
I and the deﬁnition of H , some eigenvalues of the matrix J , λ , are solutions to
[Equation]
and other eigenvalues of the matrix J , λ , are solutions to
[Equation]
Noting λ G < 0 , the solutions to the equation above are
[Equation]
where the 4th and 5th steps are from λ G < 0 .
C ONVERGENCE UNDER THE CONSTANT STEPSIZES
At last we apply the ODE method of stochastic approximation to obtain the convergence performance.
Theorem 1 Consider the update rules (10) with constant step size sequences κ , α and β satisfying κ ≥ 0 , β = ζα , ζ > 0 , α ∈ (0 , 1) and β > 0 . Let the TD ﬁxed point be w ∗ , such that V w ∗ = ΠBV w ∗ . Suppose that (A1) ( x n , r n , x n +1 ) is an i.i.d.
sequence with uniformly bounded second moments, and (A2) E
[( x n − γ x n +1 )
x (cid:62) n ] and E ( x n x (cid:62) n ) are non-singular.
Then for any (cid:15) > 0 , there exists b 1 < ∞ such that
[Equation]
Proof From the constant step sizes in the conditions, we denote κ n = κ and α n = α .
Thus,
Eqn.
(12) equals
[Equation]
Denoting ψ n +1 =
α − 1 ( ρ n +1 − ρ n ) , Eqn.
(A.1) is rewritten as
[Equation]
where the second step is from
[Equation]
Denoting G
= E ( G n ) , g = E ( g n ) and H
=
E ( H n ) , then the TD ﬁxed point of Eqn.
(A.1) is given by
[Equation]
We apply the ordinary differential equation approach of the stochastic approximation in Theorem 1 (Theorem 2.3 of (Borkar & Meyn, 2000)) into Eqn.
(A.2).
Note that (Sutton et al., 2009a) and (Sutton et al., 2009b) also applied Theorem 2.3 of (Borkar & Meyn, 2000) in using the gradientdescent method for temporal-difference learning to obtain their convergence results.
For simplifying notation, denote
[Equation]
ζα
[Equation]
and h =
[Equation]
Eqn.
[Equation]
A.2) is rewritten as
[Equation]
where f ( y n ) =
Jy n and M n +1 =
( J n − J )
y n + h n − h .
Now we verify the conditions (c1-c4) of Lemma 1.
Firstly, Condition (c1) is satisﬁed under the assumption of constant step sizes.
Secondly, f ( y ) is Lipschitz and f ∞ ( y )
=
Gy .
Following Sutton et al.
(2009a)
, the Assumption A2 implies the real parts of all the eigenvalues of G are positive.
Therefore, Condition (c2) is satisﬁed.
Because E ( M n +1 |F n ) = 0 and E ( (cid:107) M n +1 (cid:107) 2 |F n ) ≤ c 0 (1+ (cid:107) y n (cid:107) 2 ) , where F n = σ ( y i , M i , i ≤ n ) , is a martingale difference sequence, we have that
[Equation]
A.
[Equation]
From the assumption A1, Eqn.
(A.5) follows that there are constants c j and c h such that
[Equation]
Thus, Condition (c3) is satisﬁed.
Finally, Condition (c4) is satisﬁed by noting that y ∗ =
G − 1 g is the unique globally asymptotically stable equilibrium.
Theorem 1 bounds the estimation error of w in probability.
Note that the convergence of GradientDD learning provided in Theorem 1 is a somewhat weaker result than the statement that w n → w ∗ with probability 1 as n → ∞ .
The technical reason for this is the condition on step sizes.
In
Theorem 1, we consider the case of constant step sizes, with α n = α and κ n = κ .
This restriction is imposed so that Eqn.
(12) can be written as a system of ﬁrst-order difference equations, which cannot be done rigorously when step sizes are tapered as in (Sutton et al., 2009b).
As shown below, however, we ﬁnd empirically in numerical experiments that the algorithm does in fact converge with tapered step sizes and even obtains much better performance in this case than with ﬁxed step sizes.
A N ODE RESULT ON STOCHASTIC APPROXIMATION
We introduce an ODE result on stochastic approximation in the following lemma, then prove Theorem 1 by applying this result.
Lemma 1 (Theorem 2.3 of Borkar & Meyn (2000)) Consider the stochastic approximation algorithm described by the d -dimensional recursion
[Equation]
Suppose the following conditions hold: (c1) The sequence { α n } satisﬁes for some constant 0 < α < ¯ α < 1 , α < α n < ¯ α ; (c2)
The function f is Lipschitz, and there exists a function f ∞ such that lim r →∞ f r ( y ) =
f ∞ ( y ) , where the scaled function f r : R d → R d is given by f r ( y ) =
f ( r y ) /r .
Furthermore, the ODE ˙ y = f ∞ ( y ) has the origin as a globally asymptotically stable equilibrium;
(c3)
The sequence { M n , F n } , with F n = σ ( y i , M i , i ≤ n ) , is a martingale difference sequence
Moreover, for some c 0 < ∞ and any initial condition y 0 , E ( (cid:107) M n +1 (cid:107) 2 |F n ) ≤ c 0 (1 + (cid:107) y n (cid:107) 2 ) .
(c4)
The ODE ˙ y ( t ) =
f ( y ( t )) has a unique globally asymptotically stable equilibrium y ∗ .
Then for any (cid:15) > 0 , there exists b 1 < ∞ such that lim sup n →∞ P ( (cid:107) y n − y ∗ (cid:107) > (cid:15)
[Equation]
≤ b 1 ¯ α . 6. A DDITIONAL EMPIRICAL RESULTS
[Figure] [Figure] [Figure]
[Figure]