Adaptive Temporal Difference Learning with
Linear Function Approximation

We first briefly review related works in both the areas of TD learning and adaptive stochastic gradient.

Temporal difference learning. The great empirical success of TD [2] motivated active studies on the theoretical foundation of TD. The first convergence analysis of TD was given by [6] using stochastic approximation techniques. In [4], the characterization of limit points in TD with linear function approximation has been studied, giving new intuition about the dynamics of TD learning. The ODE-based method (e.g., [7]) has dramatically inspired the subsequent development of research on asymptotic convergence of TD. Early convergence results of TD learning were mostly asymptotic, e.g., [8], because the TD update does not follow the (stochastic) gradient direction of any fixed objective function. Non-asymptotic analysis for the gradient TD — a variant of the original TD has been first studied by [9], in which the authors reformulate the original problem as new primal-dual saddle point optimization. The finite-time analysis of TD with linear function approximation under i.i.d observation has been studied in [10]; in particular, it is assumed that observations in each iteration of TD are independently drawn from the steady-state distribution. In a concurrent line of research, TD has been considered in the view of the stochastic linear system, whose improved results are given by [11]. Even without any fixed objective function to optimize, the proofs of [10, 11] still follow a Lyapunov analysis like the SGD due to the i.i.d sampling assumption and quadratic functions structure. Nevertheless, a more realistic assumption for the data sampling in TD is the Markov rather than the i.i.d process. The finite-time convergence analysis under Markov sampling is first presented in [12], whose results are based on the controls of the gradient basis [Lemma 9, [12]] and a coupling [Lemma 11, [12]]. The finite-time analysis for stochastic linear system under the Markov sampling is established by [13, 14], which is a general formulate of TD and applies potentially to other problems. The finite-time analysis of multi-agent TD is proved by [15]. However, all the aforementioned work leverages the original TD update. In [16], the authors proposed an improvement of Q-learning, which can be used to TD, but with only asymptotic analysis being provided. An adaptive variant of two time-scale stochastic approximation was introduced by [17], that can also be applied to TD. In [18], the authors introduce the momentum techniques for reinforcement learning and an extension on DQN. In [19], the authors propose an adaptive scaling mechanism for TRPO and show that it is the “RL version” of traditional trust-region methods from convex analysis.

Adaptive stochastic gradient descent. In machine learning areas different but related to RL, adaptive stochastic gradient descent methods have been actively studied. The first adaptive gradient (AdaGrad) is proposed by [20, 21], and the algorithm demonstrated impressive numerical results when the gradients are sparse. While the original AdaGrad has a performance guarantee only in the convex case, the nonconvex AdaGrad has been studied by [22]. Besides the convex results, sharp analysis for nonconvex AdaGrad has also been investigated in [23]. Variants of AdaGrad have been developed in [24, 25], which use alternative updating schemes (the exponential moving average schemes) rather than the average of the square of the past iterate. The momentum technique applied to the adaptive stochastic algorithms gives birth to Adam and Nadam [26, 27]. However, in [28], the authors demonstrate that Adam may diverge under certain circumstances, and provide a new convergent Adam algorithm called AMSGrad. Another method given by [29] is the use of decreasing factors for moving the average of the square of the past iterates. In [30], the convergence for generic Adam-type algorithms has been studied, which contains various adaptive methods.

Our analysis considers the Markov sampling setting and thus differs [10, 11]. It is worth mentioning that the stepsizes chosen in this paper are different from that of [17]: in [17], the “adaptive” means that the learning rate is reduced by multiplying a preset factor when transient error is dominated by the steady-state error; while in our paper, the “adaptive” inherits the notion from Ada training method, i.e., using the learning rate associated with the past gradients. Our analysis cannot directly follow the techniques given by [12, 13, 14, 16] since the learning rate in our algorithm is statistically dependent on past information, which breaks previous Lyapunov analysis.

Thus, this paper uses a delayed expectation technique rather than bounding a coupling [Lemma 11, [12]]. Furthermore, our analysis needs to deal with several terms related to adaptive style learning rates, which are quite complicated but absent in [12]. Since the adaptive learning rate consists of past iterates instead of being preset, these terms do not enjoy simple explicit bounds. To this end, in this paper, we develop novel techniques (Lemmas 3 and 7). On the other hand, our analysis is also different from the adaptive SGD because TD update is not the stochastic gradient of any objective function; it uses biased samples generated from the Markov chain.

Complementary to existing theoretical RL efforts, we propose the first provably convergent adaptive projected variant of the TD learning algorithm with linear function approximation that has finite-time convergence guarantees. For completeness of our analytical results, we investigate both the TD(0) algorithm as well as the TD($\lambda$) algorithm. In a nutshell, our contributions are summarized in threefold:

c1) We develop the adaptive variants of the TD(0) and TD($\lambda$) algorithms with linear function approximation. The new algorithms AdaTD(0) and AdaTD($\lambda$) are simple to use.

c2) We establish the finite-time convergence guarantees of AdaTD(0) and AdaTD($\lambda$), and they are not worse than those of TD and TD($\lambda$) algorithms in the worst case.

c3) We test our AdaTD(0) and AdaTD($\lambda$) on several standard RL benchmarks and show how these compare favorably to existing alternatives like TD(0), TD($\lambda$), etc.

Consider a Markov decision process (MDP) described as a tuple $(\mathcal{S},\mathcal{A},\mathcal{P},\mathcal{R},\gamma$), where $\mathcal{S}$ denotes the state space, $\mathcal{A}$ denotes the action space, $\mathcal{P}$ represents the transition matrix, $\mathcal{R}$ is the reward function, and $0<\gamma<1$ is the discount factor. In this case, let $\mathcal{P}(s^{\prime}|s)$ denote the transition probability from state $s$ to state $s^{\prime}$. The corresponding transition reward is $\mathcal{R}(s,s^{\prime})$. We consider the finite-state case, i.e., $\mathcal{S}$ consists of $|\mathcal{S}|$ elements, and a stochastic policy $\mu:{\cal S}\rightarrow{\cal A}$ that specifies an action given the current state $s$. We use the following two assumptions on the stationary distribution and the reward.

Assumptions 1 and 2 are standard in MDP. For irreducible and aperiodic Markov chains, Assumption 2 can always hold [31]. The constant $\rho$ represents the Markov chain’s speed accessing the stationary distribution $\pi$. When the number of states is finite, the Markovian transition kernel is a matrix $\mathcal{P}$, and $\rho$ is identical to the second largest eigenvalue of $\mathcal{P}$. An important notion in the Markov chain is the mixing time, which measures the time that a Markov chain needs for its current state distribution roughly matches the stationary one $\pi$. Given an $\epsilon>0$, the mixing time is defined as

With Assumption 2, we can see $\tau(\epsilon)=O({\ln\frac{1}{\epsilon}}/{\ln\frac{1}{\rho}})$. That means if $\rho$ is small, the mixing time is short.

This paper considers the on policy setting, where both the target and behavior policies are $\mu$. For a given policy $\mu$, since the actions or the distribution of actions will be uniquely determined, we thus eliminate the dependence on the action in the rest of the paper. We denote the expected reward at a given state $s$ by

The value function $V_{\mu}:\mathcal{S}\to\mathbb{R}$ associated with a policy $\mu$ is the expected cumulative discounted reward from a given state $s$, that is

where the expectation is taken over the trajectory of states generated under $\mathcal{P}$ and $\mu$. The restriction on discount $0<\gamma<1$ can guarantee the boundedness of $V_{\mu}(s)$. The Markovian property of MDP yields the well-known Bellman equation

where the operator $\mathcal{T}_{\mu}$ on $V$ is defined as

Solving the (linear) Bellman equation allows us to find the value function $V_{\mu}$ induced by a given policy $\mu$. However, in practice, $|\mathcal{S}|$ is usually very large and computationally intractable. An alternative method is to leverage the linear [1] or non-linear function approximations (e.g., kernels and neural networks [32]). We focus on the linear case here, that is

where $\phi(s)\in\mathbb{R}^{d}$ is the feature vector for state $s$, and ${\bm{\theta}}\in\mathbb{R}^{d}$ is a parameter vector. To reduce difficulty caused by the dimension, $d$ is set smaller than $|\mathcal{S}|$. With the linear function approximator, the vector $V_{{\bm{\theta}}}\in\mathbb{R}^{|\mathcal{S}|}$ becomes

where the feature matrix is defined as

with $s_{n}$ being the $n$th state.

It is not hard to guarantee Assumption 3 since the feature map $\phi$ is chosen by the users and $|\mathcal{S}|>d$. With Assumptions 2 and 3, we can see that the matrix $\Phi^{\top}\textrm{Diag}(\pi)\Phi$ is positive define, and we denote its minimal eigenvalue as follows

With the linear approximation of value function, the task then is tantamount to finding ${\bm{\theta}}\in\mathbb{R}^{d}$ that obeys the Bellman equation given by

However, ${\bm{\theta}}$ that satisfies such an equation may not exist if $V_{\mu}\notin\{\Phi{\bm{\theta}}\mid{\bm{\theta}}\in\mathbb{R}^{d}\}$. Instead, there always exists a unique solution ${\bm{\theta}}^{*}$ for the projected Bellman equation [4], given by

where $\textbf{Proj}_{\Phi}$ is the projection onto the span of $\Phi$’s columns.

TD(0) algorithm starts with an initial parameter $\theta^{0}$. At iteration $k$, after sampling states $s_{k}$, $s_{k+1}$, and reward $r(s_{k+1},s_{k})$ from a Markov chain, we can compute the TD (temporal difference) error which is also called the Bellman error:

The TD error is subsequently used to compute the stochastic semi-gradient:

The traditional TD(0) with linear function approximation performs SGD-like update as

The update TD(0) makes sense because the direction $\overline{{\bf g}}({\bm{\theta}};s_{k},s_{k+1})$ is a good one since it is asymptotically close to the direction whose limit point is ${\bm{\theta}}^{*}$. Specifically, it has been established that [4]

where ${\bf g}({\bm{\theta}})$ is defined as

We term ${\bf g}({\bm{\theta}})$ as the limiting update direction, ensuring that ${\bf g}({\bm{\theta}}^{*})=\textbf{0}$. Note that while $\overline{{\bf g}}({\bm{\theta}};s_{k},s_{k+1})$ is an unbiased estimate under the stationary $\pi$, it is not for a finite $k$ due to the Markovian property of $s_{k}$. Therefore, the TD(0) update (6) is asymptotically akin to the stochastic approximation

where $s,s^{\prime}$ are independently drawn from the stationary distribution $\pi$.

Nevertheless, an important property of the limiting direction ${\bf g}({\bm{\theta}})$, found by [4], is that: for any ${\bm{\theta}}\in\mathbb{R}^{d}$, we have

An important observation follows from this inequality readily: only one ${\bm{\theta}}^{*}$ satisfies ${\bf g}({\bm{\theta}}^{*})=\textbf{0}$. We can show this by contradiction. If there exists another $\bar{{\bm{\theta}}}$ such that ${\bf g}(\bar{{\bm{\theta}}})=\textbf{0}$, we have $0=\langle{\bm{\theta}}^{*}-\bar{{\bm{\theta}}},{\bf g}(\bar{{\bm{\theta}}})\rangle\geq(1-\gamma)\omega\|{\bm{\theta}}^{*}-\bar{{\bm{\theta}}}\|^{2}$, which again means ${\bm{\theta}}^{*}=\bar{{\bm{\theta}}}$.

To ensure the boundedness of ${\bm{\theta}}^{k}$ and simplify the convergence analysis, projection is used in (7) (see e.g., [12]). In [12], it has been shown that if $R\geq 2B/\sqrt{\omega}(1-\gamma)^{\frac{3}{2}}$ ($B$ has appeared in Assumption 1), the projected TD(0) does not exclude all the limit points of the TD(0) (such a fact still holds for our proposed algorithms). The finite-time convergence of projected TD(0) is analyzed by [12]. The projection step is removed in [13], with almost the same results being proved. But in [13], the authors just studied the constant stepsize, while [12] shows more cases, including diminishing stepsize cases. In this paper, we consider a more complicated scheme that cannot be analyzed using techniques in [13]. Thus, the projection is still needed.

Motivated by the recent success of adaptive SGD methods such as [20, 24, 28], this paper aims to develop an adaptive version of TD with linear function approximation that we term AdaTD(0) (adaptive TD). Unlike TD(0) method, in which step size is often a constant, AdaTD(0) seeks to scale the step size depending on the norm of stochastic gradient. Additionally, instead of using a one-step gradient, AdaTD(0) utilizes momentum, which gives the algorithm a memory of history information.

As presented in the last section, $\overline{{\bf g}}({\bm{\theta}};s_{k},s_{k+1})$ is a stochastic estimate of ${\bf g}({\bm{\theta}})$. Based on this observation, we develop the adaptive scheme for TD(0). Different from projected TD(0), we use the update direction ${\bf m}^{k}$, which is the exponentially weighted average of stochastic gradients. It is further scaled by $v^{k}$, the moving average of the squared norm of stochastic semi-gradients. Intuitively, when the gradient is large, the algorithm will take smaller steps. To prevent a too large scaling, we use a positive hyper-parameter $\delta$ for numerical stability. The AdaTD(0) update is given by (9). The key difference between AdaTD(0) and the TD(0) method is that AdaTD(0) utilizes the history information in the update of both first moment estimate ${\bf m}^{k}$ and second moment estimate $v^{k}$. Unlike projected TD(0) whose asymptotically expected TD update, ${\bf g}({\bm{\theta}}^{k})$, is a good direction as is evident from (8), the update direction ${\bf m}^{k}$ in AdaTD(0) is an exponentially weighted version of $\overline{{\bf g}}({\bm{\theta}}^{k};s_{k},s_{k+1})$, which makes our analysis more challenging.

Although the variance term in AdaTD(0) is used as a sum form, it can be rewritten as an exponentially weighted moving average. If we denote $\hat{v}^{k}:=\frac{\sum_{i=1}^{k}\|{\bf g}^{i}\|^{2}}{k}=\frac{v^{k}}{k}$, the last two steps of Ada-TD(0) can be reformulated as

Note that in this form, weights are $\{1/k\}_{k\geq 1}$ and stepsizes are $\{\frac{\eta}{\sqrt{k}}\}_{k\geq 1}$, which obey the sufficient conditions to guarantee the convergence of Adam-type algorithms with exponentially weighted average variance term [29, 30].

Because the main results depend on constants related to the bounds, we present them in Lemma 1.

Lemma 1 follows readily. The bounds presented in Lemma 1 are critical for the subsequent analysis.

The convergence analysis of AdaTD(0) is more challenging than that of both TD and adaptive SGD. Compared with the analysis of adaptive SGD methods in e.g., [20, 24, 28], even under the i.i.d. sampling, the stochastic direction $\overline{{\bf g}}({\bm{\theta}}^{k};s_{k},s_{k+1})$ used in (9) fails to be the stochastic gradient of any objective function, aside from the fact that samples are drawn from a Markov chain. Compared with TD(0), the actual update of ${\bm{\theta}}^{k}$ in AdaTD(0) involves the history information of both the first and the second moments of $\overline{{\bf g}}({\bm{\theta}}^{k};s_{k},s_{k+1})$, which makes the analysis of TD in e.g., [12, 13] intractable.

With Theorem 1, to achieve $\epsilon$-accuracy for $\min_{1\leq k\leq K}\{\mathbb{E}(\|{\bm{\theta}}^{*}-{\bm{\theta}}^{k}\|^{2})\}$, we need

Since $C_{2}=O(1)$, with the first inequality of (12), $K=\tilde{O}(\frac{1}{\epsilon^{2}})$. Further, due to $C_{1}=O([\ln K/\ln\frac{1}{\rho}]^{2})=O([\ln\frac{1}{\epsilon}/\ln\frac{1}{\rho}]^{2})$, with the second inequality of (12), we have $C_{1}\ln K/\sqrt{K}=\tilde{O}(C_{1}/\sqrt{K})=O(\epsilon)$. Therefore, we obtain a solution whose square distance to ${\bm{\theta}}^{*}$ is $\epsilon$, the iteration needed is

When $\rho$ is much smaller than $1$, the term $\ln\frac{1}{\epsilon}/\ln\frac{1}{\rho}$ keeps at a relatively small level. Recall that the state-of-the-art convergence result of TD(0) given in [12] is $\tilde{O}(\ln^{2}\frac{1}{\epsilon}\big{/}(\epsilon^{2}\ln^{2}\frac{1}{\rho}))$, the rate of AdaTD(0) is close to TD(0) in the general case. We do not present a faster speed for technical reasons. In fact, this is also the case for the adaptive SGD [20, 24, 28]. Specifically, although numerical results demonstrate the advantage of adaptive methods, the worst-case convergence rate of adaptive methods is still similar to that of the stochastic gradient descent method. To reach a desired error as $\min_{1\leq k\leq K}\{\mathbb{E}(\|\nabla f({\bf x}^{k})\|^{2})\}\leq\epsilon$ in adaptive SGD, where ${\bf x}^{k}$ is the $k$th iterative and $\epsilon>0$, the iteration number $K$ needs to be set as $O(\frac{1}{\epsilon^{2}})$, which is identical to the SGD.

Sketch of the proofs: Now, we present the sketch of the proofs for the main result. Because AdaTD(0) does not have any objective function to optimize, we consider sequence the $(\|{\bm{\theta}}^{k}-{\bm{\theta}}^{*}\|^{2})_{k\geq 0}$.

Using the update (9), we have

Here, we split the term $\langle{\bf m}^{k},{\bm{\theta}}^{k}-{\bm{\theta}}^{*}\rangle/(v^{k}+\delta)^{\frac{1}{2}}$ as $I_{1}^{k}+I_{3}^{k}$ because both ${\bf m}^{k}$ and $v^{k}$ statistically depends on $\overline{{\bf g}}({\bm{\theta}}^{k};s_{k},s_{k+1})$, which makes it hard to calculate the expectation. Subtracting $-I_{1}^{k}$ to both sides of (3.3), we get

With division (15), the analysis can be divided into three parts: A) determine the lower bound of $-\mathbb{E}[I_{1}^{k}]$ (i.e, the upper bound of $\mathbb{E}[I_{1}^{k}]$); B) prove the summability of $(\mathbb{E}[I_{2}^{k}])_{k\geq 1}$, i.e., $\sum_{k=1}^{+\infty}I_{2}^{k}<+\infty$; C) prove the summability of $(\mathbb{E}[I_{3}^{k}])_{k\geq 1}$.

A) Calculating the upper bound of $\mathbb{E}{[I_{1}^{k}]}$ is the most difficult part, which consists of two substeps. Since ${\bf m}^{k}$ is a convex combination of ${\bf m}^{k-1}$ and $\overline{{\bf g}}({\bm{\theta}}^{k};s_{k},s_{k+1})$. We recursively bound $\mathbb{E}[I_{1}^{k}]$ by analyzing $\mathbb{E}[\langle{\bm{\theta}}^{k}-{\bm{\theta}}^{*},\overline{{\bf g}}({\bm{\theta}}^{k};s_{k},s_{k+1})\rangle/(v^{k-1}+\delta)^{\frac{1}{2}}]$. Assume that the Markov chain has reached its stationary distribution, in which, $\mathbb{E}[\overline{{\bf g}}({\bm{\theta}}^{k};s_{k},s_{k+1})]={\bf g}$. But the stationary distribution is not assumed in our setting. Therefore, we need to bound the difference between $\mathbb{E}\overline{{\bf g}}({\bm{\theta}}^{k};s_{k},s_{k+1})$ and ${\bf g}$. Unlike i.i.d setting, directly bounding them is difficult in the Markovian setting since it will lead to some uncontrollable error in the final convergence result. To solve this technical issue, we consider $\mathbb{E}[\overline{{\bf g}}({\bm{\theta}}^{k-T};s_{k},s_{k+1})\mid\sigma^{k}]$ and ${\bf g}({\bm{\theta}}^{k-T})$. This is because although $s_{k}$ is biased, $\mathcal{P}(s_{k}|s_{k-T}=s)$ is very close to $\pi(s)$ when $T$ is large. Using this technique, we prove the following Lemma.

In the second substep, because we have got the biased bound caused by the Markovian stochastic process, it is possible to bound the difference $\mathbb{E}[\langle{\bm{\theta}}^{k}-{\bm{\theta}}^{*},\overline{{\bf g}}({\bm{\theta}}^{k};s_{k},s_{k+1})\rangle/(v^{k-1}+\delta)^{\frac{1}{2}}]$ between $\mathbb{E}[\langle{\bm{\theta}}^{k}-{\bm{\theta}}^{*},{\bf g}({\bm{\theta}}^{k})\rangle/(v^{k-1}+\delta)^{\frac{1}{2}}]$ with the following lemma.

B) The second part is to bound $\sum_{k}\mathbb{E}[\|{\bf m}^{k}\|^{2}/(v^{k}+\delta)]$. Because $v^{k}$ depends on $(\|{\bf g}^{k}\|^{2})_{k\geq 0}$ (${\bf g}^{k}:=\overline{{\bf g}}({\bm{\theta}}^{k};s_{k},s_{k+1})$), we expand $\|{\bf m}^{k}\|^{2}$ by $(\|{\bf g}^{j}\|^{2})_{j\geq 0}$, i.e., (18) in Lemma 4 in the Appendix. We then apply a provable result (Lemma 5 in the Appendix) to the right side of (18) and get (19).

C) While the third part is the easiest and obvious. The boundedness of the points indicates the uniform bound of $(|\langle{\bm{\theta}}^{k}-{\bm{\theta}}^{*},{\bf m}^{k}\rangle|)_{k\geq 0}$. The monotonicity of $(v^{k})_{k}$ then yields the summable bound.

Summing (15) from $k=1$ to $K$, with the proved bounds in these three parts, we then can bound $\sum_{k=1}^{K}\langle{\bm{\theta}}^{*}-{\bm{\theta}}^{k},{\bf g}({\bm{\theta}}^{k})\rangle/(v^{k}+\delta)^{\frac{1}{2}}$. Once with the descent property $\langle{\bm{\theta}}^{*}-{\bm{\theta}},{\bf g}({\bm{\theta}})\rangle\geq(1-\gamma)\omega\|{\bm{\theta}}^{*}-{\bm{\theta}}\|^{2}$, we can derive the main convergence result.

The acceleration of adaptive SGD is proved under extra assumptions like the sparsity of the stochastic gradients. In the following, we present the acceleration result of AdaTD(0) also with an extra assumption.

When $\epsilon\ll\ln\frac{1}{\rho}$ and $0<\nu<1$, the result in Theorem 2 significantly improves the speed of TD. When $\nu=1$, the complexity is the same as (13).

We explain a little about assumption (20). Note that the boundedness of the sequence $({\bm{\theta}}^{k})_{k\geq 0}$ together with Assumptions 1 and 3 directly yields $v^{k}=O(k)$ (i.e., $\nu=1$). In fact, assumption (20) is very standard for analyzing adaptive stochastic optimization [33, 20, 28, 34, 30, 35]. As far as we know, there is no very good explanation of the superiority of the adaptive SGD rather than assumption (20); it is widely used in previous works because many training tasks enjoy sparse stochastic gradients.

In our AdaTD algorithm, we have

which keeps the sparsity of $\phi(s_{k})$ because $r(s_{k+1},s_{k})+\gamma V_{{\bm{\theta}}^{k}}(s_{k+1})-V_{{\bm{\theta}}^{k}}(s_{k})\in\mathbb{R}$. Then if the features are sparse, the stochastic semi-gradients are also sparse. In RL, a class of methods, called as state-aggregation based approaches [36, 37, 38, 39, 40, 41], usually uses sparse features to approximate Bellman equation linearly. The main idea of state-aggregation is dividing the whole state apace into a few mutually disjoint clusters (i.e., $\mathcal{S}=\bigcup_{i}^{\tilde{d}}\hat{s}_{i}$ and $\hat{s}_{i}\bigcap\hat{s}_{j}=\emptyset$ if $i\neq j$) and regards each cluster as a meta-state. Compared with the vanilla MDP with linear approximation, the state-aggregation based one employs a structured feature matrix

where $\phi(s_{i})[j]=1$ if $s_{i}\in\hat{s}_{j}$ and $\phi(s_{i})[j]=0$ if $s_{i}\notin\hat{s}_{j}$. We can see that $\phi(s_{i})$ has only one non-zero element, which is very sparse. And then, the stochastic semi-gradients in the AdaTD used to the state-aggregation MDP are very sparse.

State aggregation is a degenerated form of linear representations. More general is the tile coding, which still retains the sparsity structure [42].

Using existing analysis of TD($\lambda$) [1, 4], if $V_{\mu}$ solves the Bellman equation (1), it also solves

where $\mathcal{T}_{\mu}^{(m)}$ denotes the $m$-step of $\mathcal{T}_{\mu}$. In this case, we can also represent $V_{\mu}$ as

Given $\lambda\in[0,1]$ and $V_{\mu}(s)=(1-\lambda)\sum_{m=0}^{\infty}\lambda^{m}V_{\mu}(s),$ $\forall s\in\mathcal{S}$, the $\lambda$-averaged Bellman operator is given by

Comparing operator $\mathcal{T}_{\mu}$ and the $\lambda$-averaged Bellman operator, it is clear that $\mathcal{T}_{\mu}^{0}=\mathcal{T}_{\mu}$.

By defining

the stochastic update in TD($\lambda$) can be presented as (6). Similar to TD(0), it has been established in [4] and [43] that

Like the limiting update direction ${\bf g}({\bm{\theta}})$ in TD(0), a critical property of the update direction in TD($\lambda$) is given by

where $\alpha:=\gamma(1-\lambda)/(1-\gamma\lambda)$ for any ${\bm{\theta}}\in\mathbb{R}^{d}$. By denoting ${\bf z}^{\infty}:=\sum_{t=0}^{\infty}(\gamma\lambda)^{t}\hat{s}_{t}$, where $(\hat{s}_{1},\hat{s}_{2},\ldots)$ is the stationary sequence. Then, it also holds

We present AdaTD($\lambda$) in Algorithm 3. AdaTD($\lambda$) and AdaTD(0) differ in a different update protocol. We directly have the following bounds for AdaTD($\lambda$)

where $\hat{G}:=2\hat{R}+B$.

The analysis of TD($\lambda$) is more complicated than TD due to the existence of ${\bf z}^{k}$. To this end, we need to bound the sequence $(\mathbb{E}{\bf z}^{k})_{k\geq 0}$ in Lemma 8. Similar to the analysis of AdaTD(0) in Sec. 3, we need to consider the “delayed” expectation. For a fixed $K_{0}$, we consider the following error decomposition