Year: 2020

Abstract:
Off-policy algorithms, in which a behavior policy differs from the target policy and is used to gain experience for learning, have proven to be of great practical value in reinforcement learning. 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 (GradientDD) 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 Boyan-chain prediction tasks, we find substantial improvement over GTD learning and, in several cases, better performance even than conventional TD learning.

heading: 1 INTRODUCTION
text: Off-policy algorithms for value function learning enable an agent to use a behavior policy that differs from the target policy in order to gain experience for learning. 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 on-policy 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 second-
order 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 addition 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.

heading: 1.1 RELATED WORK
text: In related approaches to ours, some previous studies have attempted to improve Gradient-TD algorithms by adding regularization terms to the objective function. Liu et al. (2012) have used l1 regularization on weights to learn sparse representations of value functions, and Ghiassian et al. (2020) has used l2 regularization on weights. Unlike these references, our approach modifies the error objective function by regularizing the evaluation error obtained in the most recent time step. With this modification, our method provides a learning rule that contains second-order differences in successive parameter updates.
Our approach is similar to trust region policy optimization (Peters & Schaal, 2008; Schulman et al., 2015) or relative entropy policy search (Peters et al., 2010), which penalize large changes being learned in policy learning. In these methods, 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.

heading: 2 GRADIENT DESCENT METHOD FOR OFF-POLICY TEMPORAL DIFFERENCE LEARNING

heading: 2.1 PROBLEM DEFINITION AND BACKGROUND
text: 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 st ∈ S and selects an action at ∈ A. In response, the environment emits a reward rt ∈ R and transitions the agent to its next state st+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:
V (s) = Eπ [ ∞∑ t=1 γt−1rt|s1 = s ] , (1)
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) ∈ Rp, with p ≤ |S|, such that the approximate value function is given by
Vw(s) := w >x(s). (2)
The goal is then to learn the parameters w such that Vw(s) ≈ V (s).

heading: 2.2 GRADIENT TEMPORAL DIFFERENCE LEARNING
text: 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:
V = R + γPV =: BV,
where R is the reward vector with components E(rn+1|sn = 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 Vw = Xw. We will also assume that there exists a limiting probability distribution such that ds = limn→∞ p(sn = s) (or, in the episodic case, ds is the proportion of time steps spent in state s). The projection operator is then given by
Π = X(X>DX)−1X>D,
where the matrix D is diagonal, with diagonal elements ds.
The natural measure of how closely the approximation Vw satisfies the Bellman equation is the mean-squared Bellman error:
MSBE(w) = ‖Vw −BVw‖2D, (3) where the norm is weighted by D, such that ‖V‖2D = V>DV. However, because the Bellman operator follows the underlying state dynamics of the Markov chain, irrespective of the structure of the linear function approximator, BVw will typically not be representable as Vw for any w. An alternative objective function, therefore, is the mean squared projected Bellman error (MSPBE), which we define as
J(w) = ‖Vw −ΠBVw‖2D. (4) 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∇wJ(w), where α > 0, and
−1 2 ∇wJ(w) =− E[(γxn+1 − xn)x>n ][E(xnx>n )]−1E(δnxn)
≈− E[(γxn+1 − xn)x>n ]η. (5) For notational simplicity, we have denoted the feature vector associated with sn as xn = x(sn). We have also introduced the temporal difference error δn = rn + (γxn+1 − xn)>wn, as well as η, a linear predictor to approximate [E(xnx>n )]
−1E(δnxn). Because the factors in Eqn. (5) can be directly sampled, the resulting updates in each step are
δn =rn + (γxn+1 − xn)>wn ηn+1 =ηn + βn(δn − x>n ηn)xn wn+1 =wn − αn(γxn+1 − xn)(x>n ηn). (6)
These updates define the GTD2 learning algorithm, which we will build upon in the following section.

heading: 3 GRADIENT DESCENT TEMPORAL DIFFERENCE-DIFFERENCE LEARNING
text: In order to improve the GTD2 algorithm described above, in this section we modify the objective function via additionally considering the approximation error Vw−Vwn−1 given the previous time step n− 1. Specifically, we modify Eqn. (4) as follows:
JGDD(w|wn−1) = J(w) + κ‖Vw −Vwn−1‖2D, (7)
Figure 1: Schematic diagram of Gradient-DD learning with w ∈ R2. Rather than updating w directly along the gradient of the MSPBE (arrow), the update rule selects wn that minimizes the MSPBE while satisfying the constraint ‖Vw −Vwn−1‖2D ≤ µ (shaded ellipse).
where κ ≥ 0 is a parameter of the regularization. Minimizing Eqn. (7) is equivalent to the following optimization
arg min w
J(w) s.t. ‖Vw −Vwn−1‖2D ≤ µ (8)
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. (7). 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 JGDD(w|wn−1) is
− 1 2 ∇wJGDD(w|wn−1)
=− E[(γxn+1 − xn)x>n ][E(xnx>n )]−1E(δnxn)− κE[(x>nwn − x>nwn−1)xn] ≈− E[(γxn+1 − xn)x>n ]ηn − κE[(x>nwn − x>nwn−1)xn]. (9)
Because the terms in Eqn. (9) can be directly sampled, the stochastic gradient descent updates are given by
δn =rn + (γxn+1 − xn)>wn ηn+1 =ηn + βn(δn − x>n ηn)xn wn+1 =wn − κn(x>nwn − x>nwn−1)xn − αn(γxn+1 − xn)(x>n ηn). (10)
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.

heading: 4 IMPROVED CONVERGENCE RATE
text: 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:
(I + κnxnx > n )(wn+1 −wn) =κnxnx>n (un+1 − un)− αn(γxn+1 − xn)(x>n ηn);
un+1 =wn+1 −wn. (11)
Let βn = ζαn, ζ > 0. We consider constant step sizes in the updates, i.e., κn = κ and αn = α. Denote Hn = [
0 0 0 xnx > n
] and Gn = [ √ ζxnx > n xn(xn − γxn+1)>
−(xn − γxn+1)x>n 0
] . We rewrite
the update rules of two iterations in Eqn. (11) as a single iteration in a combined parameter vector with 2n components, ρn = (η > n / √ ζ,w>n )
>, and a new reward-related vector with 2n components, gn+1 = (rnx > n ,0 >)>, as follows:
ρn+1 =ρn − κHn(ρn − ρn−1) + √ ζα(Gnρn + gn+1), (12)
Denoting ψn+1 = α −1(ρn+1 − ρn), Eqn. (12) is rewritten as[
ρn+1 − ρn ψn+1 −ψn
] =α [ I + κHn −καHn
I −αI ]−1 [ −√ζ(Gnρn − gn+1) ψn ] =α [ − √ ζGn −κHn
− √ ζα−1Gn −α−1(I + κHn) ] [ ρn ψn ] + α [ √ ζgn+1√ ζα−1gn+1 ] ,
(13) where the second step is from [
I + κHn −καHn I −αI
]−1 = [ I −κHn
α−1I −α−1(I + κHn)
] . De-
note Jn = [ − √ ζGn −κHn
− √ ζα−1Gn −α−1(I + κHn)
] . Eqn. (13) tells us that Jn is the update matrix of
the Gradient-DD algorithm. (Note that Gn 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)), we can analyze the improved convergence rate of Gradient-DD learning by comparing the eigenvalues of the matrices E(Gn) denoted by G, and
E(Jn) denoted by J (Atkinson et al., 2008). Obviously, J = [ − √ ζG −κH
− √ ζα−1G −α−1(I + κH)
] ,
where H = E(Hn). To simplify, we consider the case that the matrix E(xnx>n ) = I. Let λG be a real eigenvalue of the matrix √ ζG. (Note that G is defined here with opposite sign relative to G in Maei (2011).) From Maei (2011), the eigenvalues of the matrix −G are strictly negative. In other words, λG > 0. Let λ be an eigenvalue of the matrix J, i.e. a solution to the equation
|λI− J| =(λ+ λG)(λ+ α−1) + κα−1λ = λ2 + [α−1(1 + κ) + λG]λ+ α−1λG = 0. (14) The smaller eigenvalues λm of the pair solutions to Eqn. (14) are
λm < −λG, where details of the above derivations are given in the appendix. 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 constant step sizes by applying the ordinary differential equation method of stochastic approximation (Borkar & Meyn, 2000). Let the TD fixed point be w∗, such that Vw∗ = ΠBVw∗ . Under some conditions, we prove that, for any > 0, there exists b1 < ∞ such that lim sup
n→∞ P (‖wn − w∗‖ > ) ≤ b1α.
Details are provided in the appendix. For tapered step sizes, which would be necessary to obtain an even stronger convergence proof, the analysis framework in Borkar & Meyn (2000) does not apply into the Gradient-DD algorithm. Although theoretical investigation of the convergence under tapered step sizes 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.

heading: 5 EMPIRICAL STUDY
text: 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 GTD2
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 αn and the parameter κ so that the comprehensive performance of the different algorithms can be compared. We considered two choices of step size sequence {αn}:
• (Case 1) αn is constant, i.e., αn = α0. • (Case 2) The learning rate αn is tapered according to the schedule αn = α0(103 +
1)/(103 + n).
We set the κ = cα0 where c = 1, 2, 4. Additionally, we also allow κ dependent on n and consider Case 3: αn is tapered as in Case 2, but κn = cαn. In order to simplify presentation, the results of Case 3 are reported in the Appendix. To begin, we set βn = αn, then later allow for βn = ζαn under ζ ∈ {1/4, 1/2, 1, 2} in order to investigate the effect of the two-timescale approach of the Gradient-based TD algorithms on Gradient-DD. In all cases, we set γ = 1.

heading: 5.1 RANDOM WALK TASK
text: 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 ofm states plus an absorbing terminal state at each end. Thus there arem+2 sequential states, S0, S1, · · · , Sm, Sm+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 (S0 or Sm+1) is entered, the walk terminates. A walk’s outcome is defined to be r = 0 at S0 and r = 1 at Sm+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 V0(s) = 0.5. In order to investigate the effect of the initialization V0(s), we also initialize V0(s) = 0, and report the results in Figure 7 of the Appendix, where its performance is very similar as the initialization V0(s) = 0.5.
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 is conventional TD learning. (3) TDC performs similarly to the GTD2 method, but requires slightly larger α than GTD2. (4) Gradient-DD performs similarly to conventional TD learning and better than the GTD2 method. This advantage is consistent in different settings. (5) The range of α leading to effective learning for Gradient-DD is roughly similar to that for GTD2.
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 observations in Figure 2, in order to facilitate comparison of these methods, we set α0 = 0.1 for 10 spaces, α0 = 0.2 for 20 spaces, and α0 = 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 α0 for Emphatic TD are tuned relative to the values used in the algorithm defined in Sutton & Mahmood (2016), where the step sizes of Emphatic TD α(ETD)0 are chosen from {0.5%, 0.1%, 0.05%, 0.01%} by the smallest area under the performance curve. Additionally we also tune α0 for TDC because TDC requires αn larger a little than GTD2 as shown in Figure 2. The step sizes for TDC are set as α(TDC)n = aαn, where a is chosen from {1, 1.5, 2, 3} by the smallest area under the performance curve.
From the results shown in Figure 3a, we draw several observations. (1) For all conditions tested, Gradient-DD converges much more rapidly than GTD2 and TDC. The results indicate that GradientDD even converges faster than TD learning in some cases, though it is not as fast in the beginning episodes. (2) 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 GTD2 learning and other methods, Gradient-DD learning in this task leads to improved learning with good convergence.
In addition to investigating the effects of the learning rate, size of the state space, and magnitude of the regularization parameter, we also investigated the effect of using distinct values for the two learning rates, αn and βn. To do this, we set βn = ζαn with ζ ∈ {1/4, 1/2, 1, 2} and report the results in Figure 8 of the appendix. The results show that comparably good performance of Gradient-DD is obtained under these various βn settings.

heading: 5.2 BOYAN-CHAIN TASK
text: 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 4p − 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 s1, [0, 1, 0, · · · , 0] for s5,
· · · , and [0, 0, · · · , 0, 1] for the terminal state s4p−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, α0 used in Emphatic TD is tuned from {0.5%, 0.2%, 0.1%, 0.05%}. 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 2α. (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.

heading: 5.3 BAIRD’S COUNTEREXAMPLE
text: We also verify the performance of Gradient-DD on Baird’s off-policy counterexample (Baird, 1995), for which TD learning famously diverges. We consider three cases: 7-state, 100-state and 500-state. We set α = 0.02 (but α = 10−5 for ETD), β = α and γ = 0.99. 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)>. 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 better than GTD2 for the 100-state case. We also observe that the performance improvement of Gradient-DD increases as the state spaces increases. We also note that, because the linear approximation leaves a residual error in the value estimation due to the projection
error, the RMS errors in this task do not go to zero. Interestingly, Gradient-DD reduces this residual error as the size of the state space increases.

heading: 6 CONCLUSION AND DISCUSSION
text: 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.

heading: 6.1 ON THE EQUIVALENCE OF EQNS. (7) & (8)
text: The Karush-Kuhn-Tucker conditions of Eqn. (8) are the following system of equations d dwJ(w) + κ d dw (‖Vw −Vwn−1‖ 2 D − µ) = 0;
κ(‖Vw −Vwn−1‖2D − µ) = 0; ‖Vw −Vwn−1‖2D ≤ µ; κ ≥ 0.
These equations are equivalent to d dwJ(w) + κ d dw‖Vw −Vwn−1‖ 2 D = 0 and κ > 0,
if ‖Vw −Vwn−1‖2D = µ; d dwJ(w) = 0 and κ = 0, if ‖Vw −Vwn−1‖ 2 D < µ.
Thus, for any µ > 0, there exists a κ ≥ 0 such that ddwJ(w) + µ d dw‖Vw −Vwn−1‖ 2 D = 0.

heading: 6.2 EIGENVALUES OF J
text: Let λ be an eigenvalue of the matrix J. We have that |λI− J| = ∣∣∣∣ λI +√ζG κH√ζα−1G λI + α−1(I + κH) ∣∣∣∣ = ∣∣∣∣ λI +√ζG κH−λα−1I λI + α−1I ∣∣∣∣
= ∣∣∣∣ λI +√ζG κH0 λI + α−1I + κα−1λ(λI +√ζG)−1H ∣∣∣∣
=|(λI + √ ζG)(λI + α−1I) + κα−1λH|.
From the assumption E(xnx>n ) = I and the definition of H, some eigenvalues of the matrix J, λ, are solutions to |λI− J| =(λ+ λG)(λ+ α−1) = 0; and other eigenvalues of the matrix J, λ, are solutions to
|λI− J| =(λ+ λG)(λ+ α−1) + κα−1λ =λ2 + [α−1(1 + κ) + λG]λ+ α −1λG = 0.
Note λG > 0. the pair solutions to the equation above are
λ =− 1 2 [α−1(1 + κ) + λG]± 1 2
√ [α−1(1 + κ) + λG]2 − 4α−1λG
=− 1 2 [α−1(1 + κ) + λG]± 1 2
√ [α−1(1 + κ)− λG]2 + 4α−1λGκ.
Thus, the smaller eigenvalues of the pairs are
λm =− 1
2 [α−1(1 + κ) + λG]−
1
2
√ [α−1(1 + κ)− λG]2 + 4α−1λGκ
<− 1 2 [α−1(1 + κ) + λG]− 1 2
√ [α−1(1 + κ)− λG]2,
where the inequality is from λG > 0. When α−1(1 + κ)− λG > 0, then
λm <− 1
2 [α−1(1 + κ) + λG]−
1 2 (α−1(1 + κ)− λG)
=− α−1(1 + κ) <− λG,
When α−1(1 + κ)− λG ≤ 0, then
λm <− 1
2 [α−1(1 + κ) + λG] +
1 2 (α−1(1 + κ)− λG)
=− λG,
CONVERGENCE WITH CONSTANT STEP SIZES
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 Vw∗ = ΠBVw∗ . Suppose that (A1) (xn, rn,xn+1) is an i.i.d. sequence with uniformly bounded second moments, and (A2) E[(xn − γxn+1)x>n ] and E(xnx>n ) are non-singular. Then for any > 0, there exists b1 <∞ such that
lim sup n→∞
P (‖wn −w∗‖ > ) ≤ b1α.
Proof From the constant step sizes in the conditions, we denote κn = κ and αn = α. Thus, Eqn. (12) equals
(I + κHn)(ρn+1 − ρn)− κHn(ρn+1 − 2ρn + ρn−1) =− √ ζα(Gnρn − gn+1). (A.1)
Denoting ψn+1 = α −1(ρn+1 − ρn), Eqn. (A.1) is rewritten as[
ρn+1 − ρn ψn+1 −ψn ] =α [ I + κHn −καHn
I −αI ]−1 [ −√ζ(Gnρn − gn+1) ψn ] =α [ − √ ζGn −κHn
− √ ζα−1Gn −α−1(I + κHn) ] [ ρn ψn ] + α [ √ ζgn+1√
ζα−1gn+1
] , (A.2)
where the second step is from[ I + κHn −καHn
I −αI
]−1 = [ I −κHn
α−1I −α−1(I + κHn)
] .
Denoting G = E(Gn), g = E(gn) and H = E(Hn), then the TD fixed point of Eqn. (A.1) is given by
−Gρ+ g = 0 (A.3)
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
Jn =
[ − √ ζGn −κHn
− √ ζα−1Gn −α−1(I + κHn)
] ,
J =
[ − √ ζG −κH
− √ ζα−1G −α−1(I + κH)
] ,
yn = [ ρn ψn ] , hn = [ √ ζgn+1√ ζα−1gn+1 ] , and h = [ √ ζg√ ζα−1g ] . Eqn. (A.2) is rewritten as
yn+1 = yn + α(f(yn) + h + Mn+1), (A.4)
where f(yn) = Jyn and Mn+1 = (Jn − J)yn + hn − 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.
BecauseE(Mn+1|Fn) = 0 and E(‖Mn+1‖2|Fn) ≤ c0(1+‖yn‖2), whereFn = σ(yi,Mi, i ≤ n), is a martingale difference sequence, we have that
‖Mn+1‖2 ≤ 2(‖Jn − J‖2‖yn‖2 + ‖hn − h‖2). (A.5)
From the assumption A1, Eqn. (A.5) follows that there are constants cj and ch such that
E(‖Jn − J‖2|Fn) ≤ cj ; E(‖hn+1 − h‖2) ≤ ch.
Thus, Condition (c3) is satisfied.
Finally, Condition (c4) is satisfied by noting that y∗ = G−1g 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 wn → 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 αn = α and κn = 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.

heading: AN ODE RESULT ON STOCHASTIC APPROXIMATION
text: 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
yn+1 = yn + an[f(yn) + Mn+1].
Suppose the following conditions hold: (c1) The sequence {αn} satisfies for some constant 0 < α < ᾱ < 1, α < αn < ᾱ; (c2) The function f is Lipschitz, and there exists a function f∞ such that limr→∞ fr(y) = f∞(y), where the scaled function fr : Rd → Rd is given by fr(y) = f(ry)/r. Furthermore, the ODE ẏ = f∞(y) has the origin as a globally asymptotically stable equilibrium; (c3) The sequence {Mn,Fn}, with Fn = σ(yi,Mi, i ≤ n), is a martingale difference sequence. Moreover, for some c0 <∞ and any initial condition y0, E(‖Mn+1‖2|Fn) ≤ c0(1 + ‖yn‖2). (c4) The ODE
ẏ(t) = f(y(t))
has a unique globally asymptotically stable equilibrium y∗. Then for any > 0, there exists b1 <∞ such that lim sup
n→∞ P (‖yn − y∗‖ > ) ≤ b1ᾱ.
6.3 ADDITIONAL EMPIRICAL RESULTS

Review summaries:
This paper proposes a variant of the GTD2 algorithm by adding an additional regularization term to the objective function, and the new algorithm is named as Gradient-DD (GDD). The regularization ensures that the value function does not change drastically between consecutive iterations. The authors show that the update rule of GDD can be written as a difference equation and aim to further show the convergence via Lyapunov based analysis. An simulation study is provided to compare the proposed GDD algorithm with TD, ETD, and GTD.