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# %load /Users/facai/Study/book_notes/
%matplotlib inline

import matplotlib.pyplot as plt
import seaborn as sns
#sns.set(font='SimHei', font_scale=2.5)
#plt.rcParams['axes.grid'] = False

import numpy as np

import pandas as pd
#pd.options.display.max_rows = 20

#import sklearn

#import itertools

#import logging
#logger = logging.getLogger()

from IPython.display import Image

Chapter 5 Monte Carlo Methods

Monte Carlo methods require only experience: sample sequences of states, actions, and rewards from actual or simulated interaction with an environment.

requirements: averaging sample returns => we define Monte Carlo methods only for episodic tasks.

5.1 Monte Carlo Prediction

the value of a state: expected return starting from that state.

An obvious way to estimate it from experience: simply to average the returns observed after visits to that state.

$s$ may be visited multiple times in the same episode:

  • first-visit MC method
  • every-visit MC method
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Monte Carlo methods do not bootstrap: the estimate for one state does not build upon the estimate of any other state.

The computational expense of estimating the value of a single state is independent of the number of states.
=> One can generate many sample episodes starting from the states of interest, averaging returns from only these states, ignoring all others.

5.2 Monte Carlo Estimation of Action Values

If a model is not available, useful to estimate action values $q_\pi(s, a)$ rather than state values.

maintain exploration problem: many state-action pairs may never be visited.

  • exploring starts: every pair has a nonzero probability of being selected as start point.
  • make policy stochasic with a nonzero probability of selecting all actions in each state.

5.3 Monte Carlo Control

$\pi(s) \doteq \operatorname{arg max}_a q(s, a)$

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5.4 Monte Carlo Control without Exploring Starts

ensure that all actions are selected infinitely often is for the agent to continue to select them:

  • on-policy: evaluate or imporve the policy that is used to make decisions.
  • off-policy: evaluate or improve a policy different from that used to generate the data.

policy is soft, meaning that $\pi(a \mid s) > 0$ for all $s \in \mathcal{S}$ and all $a \in \mathcal{A}(s)$, but gradually shifted closer and closer to a deterministic optimal policy.

$\epsilon$-soft policies: $\pi(a \mid s) \geq \frac{\epsilon}{|\mathcal{A}(s)|}$ for all states and actions, for some $\epsilon > 0$.

  • $\epsilon$-greedy policies: most of time greedy, sometimes random.
  • For any $\epsilon$-soft policy $\pi$, any $\epsilon$-greedy policy with respect to $q_\pi$ is guaranteed to be better than or equal to $\pi$.
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5.5 Off-policy Prediction via Importances Sampling

use two policies:

  • target policy $\pi$: the potimal policy that is learned about.
  • behavior policy $b$: more exploratory policy that is used to generate behavior.

assumption of coverage: $\pi(a \mid s) > 0$ implies $b(a \mid s) > 0$.

we wish to estimate $v_\pi$ or $q_\pi$, but all we have are episodes following another policy $b$, where $b \neq \pi$.
=> importance sampling: a general technique for estimating expected values under one distribution given samples from another.
=> importance-sampling ratio $\rho_{t:T-1}$:

Given a starting state $S_t$, we have: $\operatorname{Pr}\{A_t, S_{t+1}, A_{t+1}, \cdots, S_T \mid S_t, A_{t:T-1} \sim \pi\} = \prod_{K=t}^{T-1} \pi(A_k \mid S_k) p(S_{k+1} \mid S_k, A_k)$

\begin{equation} \rho_{t:T-1} \doteq \prod_{K=t}^{T-1} \frac{\pi(A_k \mid S_k)}{b(A_k \mid S_k)} \end{equation}

So we can have the right expected value by:

\begin{align} \mathbb{E}[G_t \mid S_t] &= v_b(S_t) \\ \mathbb{E}[\rho_{t:T-1} G_t \mid S_t] &= v_\pi(S_t) \end{align}

To estimate $v_\pi(s)$, we simply scale the returns by the ratios and average the results:

  • ordinary importance sampling: $V(s) \doteq \frac{\sum_{t \in \mathcal{J}(s)} \rho_{t:T(t)-1} G_t}{|\mathcal{J}(s)|}$: unbiased, but it can be extreme.
  • weighted importance sampling: $V(s) \doteq \frac{\sum_{t \in \mathcal{J}(s)} \rho_{t:T(t)-1} G_t}{\sum_{t \in \mathcal{J}(s)} \rho_{t:T(t)-1}}$: biased, but its variance is bounded.

5.6 Incremental Implementation

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5.7 Off-policy Monte Carlo Control

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Potential problem: this method learns only from the tails of episodes, when all of the remaining actions in the episode are greedy. If nongreedy actions are commom => greatly slow learning.

5.8 Disounting-aware Importance Sampling

5.9 Per-decision Importance Sampling