Monte-Carlo return

$$\begin{align*} R_t &= r_t + \gamma r_{t+1} + \gamma^2 r_{t+2} + \ldots + \gamma^{T-t} r_{T} \\ &= \sum_{l=0}^{T-t} \gamma^l r_{t+l} \end{align*}$$

n-step return

$$\begin{align*} R^{(n)}_t &= r_t + \gamma r_{t+1} + \gamma^2 r_{t+2} + \ldots + \gamma^{n-1}r_{t+n-1} + \gamma^{n} V(s_{t+n}) \\ &= \sum_{l=0}^{n-1}\big[ \gamma^l r_{t+l} \big] + \gamma^n V(s_{t+n}) \end{align*}$$

Basically everything after n-th step is bootstrapped with the value function

:::message $n \rightarrow \infty$ recovers Monte-Carlo return. This assumes the rewards after horizon $T$ are all zeros:

$$\begin{align*} R^{(\infty)}_t &= \sum_{l=0}^{T-t} \big[ \gamma^l r_{t+l} \big] + \sum_{l=T - t + 1}^{\infty} \big[ \gamma^l \cdot 0 \big] + \gamma^\infty V(s_{t+\infty}) \\ &= \sum_{l=0}^{T-t}\big[ \gamma^l r_{t+l} \big] = R_t. \end{align*}$$

:::

$n$	what?	expression	biased?	variance?
1	1-step return	$r_t + \gamma V(s_{t+1})$	🙅‍♂️ biased	🙆‍♂️ low-variance
$\vdots$
$\infty$	Monte-Carlo return	$\sum_{l=0}^{T-t} \gamma^l r_{t+l}$	🙆‍♂️ unbiased	🙅‍♂️ high-variance

1-step return is common in Q-learning.

Notice that $n$ acts as a trade-off between bias and variance. Why don't we mix these? --> $\lambda$-return

λ-return (TD(λ))

Exponentially weighted average of n-step returns

$$R_t(\lambda) = (1 - \lambda) \sum_{n=1}^{\infty} \lambda^{n-1} R_t^{(n)}$$

Q. Why do we need $(1-\lambda)$ there?:

It's necessary for normalization, as you can see from

$$(1 - r) \sum_{k=1}^{\infty} ar^{k-1} = a$$

:::details Basics of Geometric progression

Geometric Progression

$$\sum_{k=1}^n ar^{k-1} = ar^0 + ar^1 + ar^2 + \cdots + ar^{n-1}$$

A trick: multiply $(1-r)$ on both sides

$$\begin{align*} (1-r)\sum_{k=1}^{n} ar^{k-1} &= (1-r)(ar^0 + ar^1 + ar^2 + \cdots + ar^{n-1}) \\ &= (1-r)(ar^0 + ar^1 + ar^2 + \cdots + ar^{n-1}) \\ &= (ar^0 + ar^1 + ar^2 + \cdots + ar^{n-1}) - (ar^1 + ar^2 + ar^3 + \cdots + ar^n) \\ &= a - ar^n \\ &= a(1-r^n) \\ \end{align*}$$

Thus,

$$\sum_{k=1}^{n}ar^{k-1} = \frac{a(1 - r^n)}{1 - r}$$

If $n \rightarrow \infty$,

$$\sum_{k=1}^{\infty} ar^{k-1} = \frac{a}{1 - r}$$

:::

• $\lambda=0 \Longrightarrow R^{(1)}_t$: 1-step return
• $\lambda=1 \Longrightarrow R^{(\infty)}_t (=R_t)$: Monte-Carlo return

Generalized Advantage Estimator: GAE(λ)

In one word: we simply repeat the same discussion for advantages (TD residuals), instead of returns.

This simply estimates advantage with λ-return: $\hat{\mathcal{A}}_t = R_t(\lambda) - V(s_t).$

Basics: Value function and Q-function

\begin{align*} V^{\pi, \gamma}(s_t) &\coloneqq \mathbb{E_{ \substack{s_{t+1:\infty}, \\ {\color{blue} a_{t:\infty}}} }}[\sum_{l=0}^\infty \gamma^l r_{t+l}] \\ Q^{\pi, \gamma}(s_t, a_t) &\coloneqq \mathbb{E}_{ \substack{s_{t+1:\infty} \\ {\color{blue} a_{t+1:\infty}}} }[\sum_{l=0}^{\infty} \gamma^l r_{t+l}] \\ {/* &= r_t + \gamma \cdot \mathbb{E}_{\substack{s_{t+2:\infty} \\ a_{t+1:\infty}}}[\sum_{l=0}^{\infty} \gamma^l r_{t+l}] \\ */} {/* &= r_t + \gamma V^{\pi, \gamma}(s_{t+1}) */} \end{align*}

Advantage is defined as their difference:

$$A^{\pi, \gamma}(s_t, a_t) \coloneqq Q^{\pi, \gamma}(s_t, a_t) - V^{\pi, \gamma}(s_t)$$

TD residual

Now, we define TD residual $\delta^V_t$:

$$\delta_t^{V^{\pi, \gamma}} = r_t + \gamma V^{\pi, \gamma}(s_{t+1}) - V^{\pi, \gamma}(s_t)$$

This is in fact, an unbiased estimator of $A^{\pi, \gamma}(s_t, a_t)$:

$$\begin{align*} \mathbb{E}_{s_{t+1}} \big[ \delta_t^{V^{\pi, \gamma}} \big] &= \mathbb{E}_{s_{t+1}} [r_t + \gamma V^{\pi, \gamma}(s_{t+1}) - V^{\pi, \gamma}(s_t)] \\ &= \mathbb{E}_{s_{t+1}} [Q^{\pi, \gamma}(s_t, a_t) - V^{\pi, \gamma}(s_t)] \\ &= A^{\pi, \gamma}(s_t, a_t) \end{align*}$$

n-step TD residual

$$\begin{align*} \hat{A}_t^{(n)} &\coloneqq \sum_{l=0}^{n-1} \gamma^l \delta_{t+l}^{V} = \delta_t^V + \gamma \delta_{t+1}^V + \gamma^2 \delta_{t+2}^V + \cdots + \gamma^{n-1} \delta_{t+n-1}^{n-1} \\ &=^{*} r_t + \gamma r_{t+1} + ~\cdots~ + \gamma^{n-1}r_{t+n-1} +~\gamma^n V(s_{t+n}) - V(s_t) \\ &= \sum_{l=0}^{n-1}[\gamma^l r_{t+l}] + \gamma^n V(s_{t+n}) - V(s_t) \\ &= R^{(n)}_t - V(s_t) \end{align*}$$

* Notice that this is a telescoping sum. :::message $n\rightarrow\infty$ becomes: [empirical returns] - [value function baseline]

$$\hat{A}^{(\infty)}_t = \sum_{l=0}^{\infty} \gamma^l \delta_{t+l}^V = \sum_{l=0}^{\infty} [\gamma^l r_{t+l}] - V(s_t)$$

:::

GAE

Exponentially weighted average of n-step TD residuals

$$\begin{align*} \hat{A}_t^{\text{GAE}(\gamma, \lambda)} &\coloneqq (1-\lambda)\sum_{n=1}^{\infty}\lambda^{n-1}\hat{A}_t^{(n)} \\ &= (1-\lambda)\big(\delta_t^V + \lambda(\delta_t^V + \gamma \delta_{t+1}^V) + \lambda^2(\delta_t^V + \gamma \delta_{t+1}^V + \gamma^2 \delta_{t+2}^V) + \cdots \big) \\ &= (1-\lambda)\big(\delta_t^V(1 + \lambda + \lambda^2 + \cdots) + \gamma \delta_{t+1}^V(\lambda + \lambda^2 + \lambda^3 + \cdots) + \cdots \big) \\ &= (1-\lambda)\big(\delta_t^V(\frac{1}{1-\lambda}) + \gamma \delta_{t+1}^V(\frac{\lambda}{1 - \lambda}) + \gamma^2 \delta_{t+1}^V(\frac{\lambda^2}{1 - \lambda}) + \cdots \big) \\ &= \sum_{l=0}^{\infty} (\gamma \lambda)^l \delta_{t+l}^V \end{align*}$$

• $\lambda=0$ -> unbiased regardless of the accuracy of $V$, but high variance
• $\lambda=1$ -> unbiased only if $V = V^{\pi, \gamma}$, and low variance

Reference

• Appendix in DeepMimic paper
• GAE paper
• David Silver's lecture (TD(λ) slides on Lecture 4)