Gradient Temporal Difference with Momentum: Stability and Convergence

In reinforcement learning (RL), the goal of the learner or the agent is to maximize its long term accumulated reward by interacting with the environment. One important task in most of RL algorithms is that of policy evaluation. It predicts the average accumulated reward an agent would receive from a state (called value function) if it follows the given policy. In model-free learning, the agent does not have access to the underlying dynamics of the environment and has to learn the value function from samples of the form (state, action, reward, next-state). Two very popular algorithms in the model-free setting are Monte-Carlo (MC) and temporal difference (TD) learning (see Sutton and Barto (2018), Sutton (1988)). It is a well known fact that TD learning diverges in the off-policy setting (see Baird (1995)). A class of algorithms called gradient temporal difference (Gradient TD) were introduced in (Sutton, Maei, and Szepesvári 2009) and (Sutton et al. 2009) which are convergent even in the off-policy setting. These algorithms fall under a larger class of algorithms called linear stochastic approximation (SA) algorithms.

A lot of literature is dedicated to studying the asymptotic behaviour of SA algorithms starting from the work of (Robbins and Monro 1951). In recent times, the ODE method to analyze asymptotic behaviour of SA (Ljung 1977; Kushner and Clark 1978; Borkar 2008b; Borkar and Meyn 2000) has become quite popular in the RL community. The Gradient TD methods were shown to be convergent using the ODE approach. A generic one-timescale (One-TS) SA iterate has the following form:

$$x_{n+1}=x_{n}+a(n)\left(h(x_{n})+M_{n+1}\right),$$

(1)

where $x\in\mathbb{R}^{d_{1}}$ are the iterates. The function $h:\mathbb{R}^{d_{1}}\rightarrow\mathbb{R}^{d_{1}}$ is assumed to be a Lipschitz continuous function. $M_{n+1}$ is a Martingale difference noise sequence and $a(n)$ is the step-size at time-step $n$. Under some mild assumptions, the iterate given by (1) converges (see Borkar 2008b; Borkar and Meyn 2000). When $h$ is a linear map of the form $b-Ax_{n}$, the matrix $A$ is often called the driving matrix. The three Gradient TD algorithms: GTD (Sutton, Maei, and Szepesvári 2009), GTD2 and TDC (Sutton et al. 2009) consist two iterates of the following form:

$$x_{n+1}=x_{n}+a(n)(h(x_{n},y_{n})+M_{n+1}^{(1)},$$

(2)

$$y_{n+1}=y_{n}+b(n)(g(x_{n},y_{n})+M_{n+1}^{(2)}),$$

(3)

where $x\in\mathbb{R}^{d_{1}}$, $y\in\mathbb{R}^{d_{2}}$. See section 2 for exact form of the iterates. The two iterates still form a One-TS SA scheme if $\lim_{n\rightarrow\infty}\frac{b(n)}{a(n)}=c$, where $c$ is a constant and a two-timescale (two-TS) scheme if $\lim_{n\rightarrow\infty}\frac{b(n)}{a(n)}=0$.

Separately, adding a momentum term to accelerate the convergence of iterates is a popular technique in stochastic gradient descent (SGD). The two most popular schemes are the Polyak’s Heavy ball method (Polyak 1964), and Nesterov’s accelerated gradient method (Nesterov 1983). A lot of literature is dedicated to studying momentum with SGD. Some recent works include (Ghadimi, Feyzmahdavian, and Johansson 2014; Loizou and Richtárik 2020; Gitman et al. 2019; Ma and Yarats 2019; Assran and Rabbat 2020). Momentum in the SA setting, which is the focus of the current work, has limited results. Very few works study the effect of momentum in the SA setting. A recent work by (Mou et al. 2020) studies SA with momentum briefly and shows an improvement of mixing rate. However, the setting considered is restricted to linear SA and the driving matrix is assumed to be symmetric. Further, the iterates involve an additional Polyak-Ruppert averaging (Polyak 1990). Here, in contrast, we analyze the asymptotic behaviour of the algorithm and make none of the above assumptions. A somewhat distant paper is by (Devraj, Bušíć, and Meyn 2019) that introduces Matrix momentum in SA and is not equivalent to heavy ball momentum.

A very recent work by (Avrachenkov, Patil, and Thoppe 2020) studied One-TS SA with heavy ball momentum in the univariate case (i.e., $d=1$ in iterate (1)) in the context of web-page crawling. The iterates took the following form:

$$x_{n+1}=x_{n}+a(n)\left(h(x_{n})+M_{n+1}\right)+\eta_{n}(x_{n}-x_{n-1}).$$

(4)

The momentum parameter $\eta_{n}$ was chosen to decompose the iterate into two recursions of the form given by (2) and (3). We use such a decomposition for Gradient TD methods with momentum. This leads to three separate iterates with three step-sizes. We analyze these three iterates and provide stability (iterates remain bounded throughout) and almost sure (a.s.) convergence guarantees.

In the standard RL setup, an agent interacts with the environment which is a Markov Decision Process (MDP). At each discrete time step $t$, the agent is in state $s_{t}\in\mathcal{S},$ takes an action $a_{t}\in\mathcal{A},$ receives a reward $r_{t+1}\equiv r(s_{t},a_{t},s_{t+1})\in\mathbb{R}$ and moves to another state $s_{t+1}\in\mathcal{S}$. Here $\mathcal{S}$ and $\mathcal{A}$ are finite sets of possible states and actions respectively. The transitions are governed by a kernel $\mathbb{P}$. A policy $\pi:\mathcal{S}\times\mathcal{A}\rightarrow[0,1]$ is a mapping that defines the probability of picking an action in a state. We let $P^{\pi}(s^{\prime}|s)$ be the transition probability matrix induced by $\pi$. Also, $\{d^{\pi}(s)\}_{s\in\mathcal{S}}$ represents the steady-state distribution for the Markov chain induced by $\pi$ and the matrix $D$ is a diagonal matrix of dimension $n\times n$ with the entries $d^{\pi}(s)$ on its diagonals.The state-value function associated with a policy $\pi$ for state $s$ is

$$V^{\pi}(s)=\mathbb{E}_{\pi}\left[\sum_{t=0}^{\infty}\gamma^{t}R_{t+1}|s_{0}=s\right],$$

where $\gamma$ $\in[0,1)$ is the discount factor.

In the linear architecture setting, policy evaluation deals with estimating $V^{\pi}(s)$ through a linear model $V_{\theta}(s)=\theta^{T}\phi(s)$, where $\phi(s)\equiv\phi_{s}$ is a feature associated with the state $s$ and $\theta$ is the parameter vector. We define the TD-error as $\delta_{t}=r_{t+1}+\gamma\theta_{t}^{T}\phi_{t+1}-\theta_{t}^{T}\phi_{t}$ and $\Phi$ as an $n\times d$ matrix where the $s^{th}$ row is $\phi(s)^{T}$. In the i.i.d setting it is assumed that the tuple $(\phi_{t},\phi_{t}^{\prime}$) (where $\phi_{t+1}\equiv\phi_{t}^{\prime}$ ) is drawn independently from the stationary distribution of the Markov chain induced by $\pi$. Let $\bar{A}=\mathbb{E}[\phi_{t}(\gamma\phi_{t}^{\prime}-\phi_{t})^{T}]$ and $\bar{b}=\mathbb{E}[r_{t+1}\phi_{t}]$, where the expectations are w.r.t. the stationary distribution of the induced chain. The matrix $\bar{A}$ is negative definite (see Maei (2011); Tsitsiklis and Van Roy (1997)). In the off-policy case, the importance weight is given by $\rho_{t}=\frac{\pi(a_{t}|s_{t})}{\mu(a_{t}|s_{t})}$, where $\pi$ and $\mu$ are the target and behaviour policies respectively. Introduced in (Sutton, Maei, and Szepesvári 2009), Gradient TD are a class of TD algorithms that are convergent even in the off-policy setting. Next, we present the iterates associated with the algorithms GTD (Sutton, Maei, and Szepesvári 2009), GTD2, TDC (Sutton et al. 2009).

• •

  GTD:

  	$$\displaystyle\theta_{t+1}=\theta_{t}+\alpha_{t}(\phi_{t}-\gamma\phi_{t}^{\prime})\phi_{t}^{T}u_{t},$$		(5)
  	$$\displaystyle u_{t+1}=u_{t}+\beta_{t}(\delta_{t}\phi_{t}-u_{t}).$$		(6)

• •

  GTD2:

  	$$\displaystyle\theta_{t+1}=\theta_{t}+\alpha_{t}(\phi_{t}-\gamma\phi_{t}^{\prime})\phi_{t}^{T}u_{t},$$		(7)
  	$$\displaystyle u_{t+1}=u_{t}+\beta_{t}(\delta_{t}-\phi_{t}^{T}u_{t})\phi_{t}.$$		(8)

• •

  TDC:

  	$$\displaystyle\theta_{t+1}=\theta_{t}+\alpha_{t}\delta_{t}\phi_{t}-\alpha_{t}\gamma\phi_{t}^{\prime}(\phi_{t}^{T}u_{t}),$$		(9)
  	$$\displaystyle u_{t+1}=u_{t}+\beta_{t}(\delta_{t}-\phi_{t}^{T}u_{t})\phi_{t}.$$		(10)

The objective function for GTD is Norm of Expected Error defined as $NEU(\theta)=\mathbb{E}[\delta\phi]$. The GTD algorithm is derived by expressing the gradient direction as $-\frac{1}{2}\nabla NEU(\theta)$ = $\mathbb{E}\left[(\phi-\gamma\phi^{\prime})\phi^{T}\right]\mathbb{E}[\delta(\theta)\phi]$. Here $\phi^{\prime}\equiv\phi(s^{\prime})$. If both the expectations are sampled together, then the term would be biased by their correlation. An estimate of the second expectation is maintained as a long-term quasi-stationary estimate (see (5)) and the first expectation is sampled (see (6)). For GTD2 and TDC, a similar approach is used on the objective function Mean Square Projected Bellman Error defined as $MSPBE(\theta)=||V_{\theta}-\Pi T^{\pi}V_{\theta}||_{D}$. Here, $\Pi$ is the projection operator that projects vectors to the subspace $\{\Phi\theta|\theta\in\mathbb{R}^{d}\}$ and $T^{\pi}$ is the Bellman operator defined as $T^{\pi}V=R^{\pi}+\gamma P^{\pi}V$. As originally presented, GTD and GTD2 are one-timescale algorithms ($\frac{\alpha_{t}}{\beta_{t}}$ is constant) while TDC is a two-timescale algorithm ($\frac{\alpha_{t}}{\beta_{t}}\rightarrow 0$). It was shown in all the three cases that $\theta_{n}\rightarrow\theta^{*}=-\bar{A}^{-1}\bar{b}$.

Although, Gradient TD starts with a gradient descent based approach, it ends up with two-TS SA recursions. Momentum methods are known to accelerate the convergence of SGD iterates. Motivated by this, we examine momentum in the SA setting, and ask if the SA recursions for Gradient TD with momentum even converge to the same TD solution. We probe the heavy ball extension of the three Gradient TD algorithms where, we keep an accumulation of the previous gradient values in $\zeta_{t}$. Then, at time step $t+1$ the new gradient value multiplied by the step size is added to the current accumulation vector $\zeta_{t}$ multiplied by the momentum parameter $\eta_{t}$ as below:

$$\zeta_{t+1}=\eta_{t}\zeta_{t}+\alpha_{t}(\phi_{t}-\gamma\phi_{t}^{\prime})\phi_{t}^{T}u_{t}.$$

The parameter $\theta$ is then updated in the negative of the direction $\zeta_{t+1}$, i.e., $\theta_{t+1}=\theta_{t}-\zeta_{t+1}.$ Since $u_{t+1}$ is computed as a long-term estimate of $\mathbb{E}[\delta(\theta)\phi]$, its update rule remains same. The momentum parameter $\eta_{t}$ is usually set to a constant in the stochastic gradient setting. An exception to this can however be found in (Gitman et al. 2019; Gadat, Panloup, and Saadane 2016), where $\eta_{t}\rightarrow 1$. Here, we consider the latter case. Substituting $\zeta_{t+1}$ into the iteration of $\theta_{t+1}$ and noting that $\zeta_{t}=\theta_{t}-\theta_{t-1}$, the iterates for GTD with Momentum (GTD-M) can be written as:

$$\theta_{t+1}=\theta_{t}+\alpha_{t}(\phi_{t}-\gamma\phi_{t}^{\prime})\phi_{t}^{T}u_{t}+\eta_{t}(\theta_{t}-\theta_{t-1}),$$

(11)

$$u_{t+1}=u_{t}+\beta_{t}(\delta_{t}\phi_{t}-u_{t}).$$

(12)

Similarly the iterates for GTD2-M are given by:

$$\theta_{t+1}=\theta_{t}+\alpha_{t}(\phi_{t}-\gamma\phi_{t}^{\prime})\phi_{t}^{T}u_{t}+\eta_{t}(\theta_{t}-\theta_{t-1}),$$

(13)

$$u_{t+1}=u_{t}+\beta_{t}(\delta_{t}-\phi_{t}^{T}u_{t})\phi_{t}.$$

(14)

Finally, the iterates for TDC-M are given by:

$$\begin{split}\theta_{t+1}=\theta_{t}+\alpha_{t}(\delta_{t}\phi_{t}-\gamma\phi_{t}^{\prime}(\phi_{t}^{T}u_{t}))+\eta_{t}(\theta_{t}-\theta_{t-1}),\end{split}$$

(15)

$$u_{t+1}=u_{t}+\beta_{t}(\delta_{t}-\phi_{t}^{T}u_{t})\phi_{t}.$$

(16)

We choose the momentum parameter $\eta_{t}$ as in (Avrachenkov, Patil, and Thoppe 2020) as follows: $\eta_{t}=\frac{\varrho_{t}-w\alpha_{t}}{\varrho_{t-1}}$, where $\{\varrho_{t}\}$ is a positive sequence s.t. $\varrho_{t}\rightarrow 0$ as $t\rightarrow\infty$ and $w\in\mathbb{R}$ is a constant. Note that $\eta_{t}\rightarrow 1$ as $t\rightarrow\infty$. We later provide conditions on $\varrho_{t}$ and $w$ to ensure a.s. convergence. As we would see in section 4, the condition on $w$ in the One-TS setting is restrictive. Specifically, it depends on the norm of the driving matrix $\bar{A}$. This motivates us to look at the Three-TS setting and then the corresponding condition on $w$ is less restrictive. Using the momentum parameter as above,

$$\begin{split}\theta_{t+1}&=\theta_{t}+\alpha_{t}(\phi_{t}-\gamma\phi_{t}^{\prime})\phi_{t}^{T}u_{t}+\frac{\varrho_{t}-w\alpha_{t}}{\varrho_{t-1}}(\theta_{t}-\theta_{t-1})\end{split}$$

Rearranging the terms and dividing by $\rho_{t}$, we get:

$$\begin{split}\frac{\theta_{t+1}-\theta_{t}}{\varrho_{t}}&=\frac{\theta_{t}-\theta_{t-1}}{\varrho_{t-1}}\\ &+\frac{\alpha_{t}}{\varrho_{t}}\Bigg{(}(\phi_{t}-\gamma\phi_{t}^{\prime})\phi_{t}^{T}u_{t}-w\left(\frac{\theta_{t}-\theta_{t-1}}{\varrho_{t-1}}\right)\Bigg{)}.\end{split}$$

We let

$$\frac{\theta_{t+1}-\theta_{t}}{\varrho_{t}}=v_{t+1},\xi_{t}=\frac{\alpha_{t}}{\varrho_{t}}\mbox{ and }\varepsilon_{t}=v_{t+1}-v_{t}.$$

Then, the GTD-M iterates in (11) and (12) can be re-written with the following three iterates:

	$$\displaystyle v_{t+1}=v_{t}+\xi_{t}\left((\phi_{t}-\gamma\phi_{t}^{\prime})\phi_{t}^{T}u_{t}-wv_{t}\right)$$		(17)
	$$\displaystyle u_{t+1}=u_{t}+\beta_{t}(\delta_{t}\phi_{t}-u_{t})$$		(18)
	$$\displaystyle\theta_{t+1}=\theta_{t}+\varrho_{t}(v_{t}+\varepsilon_{t})$$		(19)

A similar decomposition can be done for the GTD2-M and TDC-M iterates.

In this section we analyze the asymptotic behaviour of the GTD-M iterates given by (17), (18) and (19). Throughout the section, we consider $v_{t},u_{t},\theta_{t}\in\mathbb{R}^{d}$. We first consider the One-TS case when $\beta_{t}=c_{1}\xi_{t}$ and $\varrho_{t}=c_{2}\xi_{t}$ $\forall t$, for some real constants $c_{1},c_{2}>0$. Subsequently, we consider the Three-TS setting where $\frac{{\beta_{t}}}{\xi_{t}}\rightarrow 0$ and $\frac{{\varrho_{t}}}{\beta_{t}}\rightarrow 0$ as $t\rightarrow\infty$.