Reinforcement Learning - Lec1

Policy-based Algorithm

Reinforcement Learning
RL / NTU / CS294

今年1 月的目標想複習三下時學的 RL，主要的參考教材為李宏毅老師的 DRL 8 講及 Sergey 在 Berkeley 開的 CS294。

先讓我們從 Policy Gradient 開始吧！

Terminology

這裡假設讀者熟悉 RL 的基本概念 (env/action/state 的交互作用及相關名詞)。

Policy

Policy $\pi$ 為 given state $s$，採取 action $a$ 的機率。而我們可以以一組參數 $\theta$ 去描述。

\[ \pi_\theta(a,s) = \mathbb{P}[a|s] \]

Given 一個固定的 policy 的 agent (或者又叫做 actor)，我們讓其去與環境互動，每玩一次(一個 episode) 會收集到一個序列 {$s_1,a_1,\cdots,s_T,a_T$}，稱為 trajectory $\tau$，並會得到一個 final reward $r(\tau)$

Value Function

Given policy, expected return from state $s$

\[ V^\pi(s) = \mathbb{E}_\pi[r(\tau) | s_t = s] \]

Note: $\pi$ 是指 Given policy，特定 trajectory 被 sample 出來的機率

Action-Value Function

Given policy, expected return when taking $a$ at $s$ ，之後稱其為 Q function

\[ Q^\pi(s,a) = \mathbb{E}_\pi[r(\tau)|s_t = s, a_t = a] \]

Goal

我們想找到一個最好的 policy 去 maximize 最後得到 reward 的期望值 ( optimize 的目標)

Optimal value function

\[ \begin{aligned} V_{\ast}(s) &= \max_\pi V_\pi(s) = \max_a Q_{\ast}(s,a)\\ Q_{\ast}(s,a) &= \max_\pi Q_\pi(s,a) = r(s,a) + \sum_{s^\prime} \mathbb{P}[s^\prime|a,s] V_{\ast}(s) \end{aligned} \]

Note: exploration 只在 training 的時候做，一旦摸清整個環境(i.e optimal policy is claimed)，在決定時便不用再加上隨機性了 (so that's why I directly write down $\max$)

Decoding Problem (how to construct such policy)

Only considered in non policy-based method, more discussion on $\rightarrow$ Lec3

Learning Problem

而後續討論的演算法，都是圍繞著如何找出這樣符合optimal value equation 的 policy。

  • DP-based Planning: value iteration, policy iteration
    • Discrete action/state and $|\mathcal{A}|,|\mathcal{S}|$ is small (i.e tabular): \
    • MDP is known
    • Contraction Mapping Theorem 可以 prove 最佳解的存在性) $\rightarrow$ see Lec3
  • Value-based Learning: Monte-Carlo RL, Temporal-Difference (bootstrapping) RL
    • In generate, discrete action (since we need to choose argmax from $\mathcal{A}$)
    • MDP is unknown
    • Tradeoff between biased or not and low/high variance
  • Policy-based Learning: REINFORCE
    • Continuous action/state (since we usually use NN as hypothesis set)
    • MDP is unknown
    • Variance reduction trick (see $\rightarrow$ Lec2)
    • Combined with value-based algorithm: Actor-Critic (see $\rightarrow$ Lec5)

How to update $\theta$ in Policy Gradient

Optimal policy 所對應到的 $\theta$ 滿足以下

\[ \theta^{\ast} \, = \, \arg \max_\theta \mathbb{E}_{\tau \sim \pi_\theta} r(\tau) \, = \, \arg \max_\theta \sum_t [\mathbb{E}_{(s_t,a_t) \sim \pi_\theta(s,a)} [r(s_t,a_t)]] \]

利用 Graient Descent。

$$ \begin{align} \nabla_\theta J(\theta) &= \int \color{blue}{\nabla_\theta[\pi_\theta(\tau)]} r(\tau) d\tau\\ &= \int \color{blue}{\pi_\theta \nabla_\theta[\log \pi_\theta(\tau)]}r(\tau) d\tau \\ &= \mathbb{E}_{\tau \; \sim \; \pi_\theta(\tau)} \nabla_\theta[\log\pi_\theta(\tau)]r(\tau) \\ \end{align} $$

直覺來說，相同於傳統 supervised learning 的 setting ，但更新的步伐大小，跟 reward 成線性關係\
(i.e try & error，哪個方向好就走多一點，weighted by reward)

但上述的 $\nabla_\theta J(\theta)$無法很直覺地用於 policy 的更新，所以我們做了以下改動

  • Policy Gradient Theorem 告訴我們

For any differentialble policy $\pi_{\theta}(a,s)$, for any of the policy objective functions $J = J_1,J \scriptstyle avR$…,\

\[ \nabla_\theta J(\theta) = \mathbb{E}_{\pi_\theta}[\nabla_\theta \log \pi_\theta(s,a)Q^{\pi_\theta}(s,a)] \]
  • 我們是對每一個 $(a,s)$ 做更新 (因 $\pi$ 能管的也只有 Given $s$，選擇採取 $a$ 的機率)
\[ \nabla_\theta [\log \pi_\theta (\tau)] = \sum_t \nabla_\theta [\log \pi_\theta (a_t , s_t)] \]
  • 在大多數情況，無法窮舉所有可能的 $\tau$ 去求期望值，只能用 sampling 去估計
\[ \boxed{ \nabla_\theta J(\theta) \approx \sum_n \sum_t \nabla_\theta[\log\pi_\theta(a_t,s_t)]\underbrace{\,r(\tau)\,}_{ Q^{\pi_\theta}(s,a)}} \]

Note: 在最基本的 policy gradient 中，我們用 $r(\tau)$ 來作為$Q^{\pi_\theta}(s,a)$ 的 unbiased estimation。

Drawback

  • Online policy，need to resample when $\theta$ is updated $\Rightarrow$ Data collection 費時
  • Sample is finite $\Rightarrow$ variance 大，難 train 。

Reference

 
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