# 2.12 Machine learning lecture 12 course notes  (Page 2/8)

 Page 2 / 8

Or, when we are writing rewards as a function of the states only, this becomes

$R\left({s}_{0}\right)+\gamma R\left({s}_{1}\right)+{\gamma }^{2}R\left({s}_{2}\right)+\cdots .$

For most of our development, we will use the simpler state-rewards $R\left(s\right)$ , though the generalization to state-action rewards $R\left(s,a\right)$ offers no special difficulties.

Our goal in reinforcement learning is to choose actions over time so as to maximize the expected value of the total payoff:

$\mathrm{E}\left[R,\left({s}_{0}\right),+,\gamma ,R,\left({s}_{1}\right),+,{\gamma }^{2},R,\left({s}_{2}\right),+,\cdots \right]$

Note that the reward at timestep $t$ is discounted by a factor of ${\gamma }^{t}$ . Thus, to make this expectation large, we would like to accrue positive rewardsas soon as possible (and postpone negative rewards as long as possible). In economic applications where $R\left(·\right)$ is the amount of money made, $\gamma$ also has a natural interpretation in terms of the interest rate (where a dollar today isworth more than a dollar tomorrow).

A policy is any function $\pi :S↦A$ mapping from the states to the actions. We say that we are executing some policy $\pi$ if, whenever we are in state $s$ , we take action $a=\pi \left(s\right)$ . We also define the value function for a policy $\pi$ according to

${V}^{\pi }\left(s\right)=\mathrm{E}\left[R,\left({s}_{0}\right),+,\gamma ,R,\left({s}_{1}\right),+,{\gamma }^{2},R,\left({s}_{2}\right),+,\cdots \right|{s}_{0}=s,\pi \right].$

${V}^{\pi }\left(s\right)$ is simply the expected sum of discounted rewards upon starting in state $s$ , and taking actions according to $\pi$ . This notation in which we condition on $\pi$ isn't technically correct because $\pi$ isn't a random variable, but this is quite standard in the literature.

Given a fixed policy $\pi$ , its value function ${V}^{\pi }$ satisfies the Bellman equations :

${V}^{\pi }\left(s\right)=R\left(s\right)+\gamma \sum _{{s}^{\text{'}}\in S}{P}_{s\pi \left(s\right)}\left({s}^{\text{'}}\right){V}^{\pi }\left({s}^{\text{'}}\right).$

This says that the expected sum of discounted rewards ${V}^{\pi }\left(s\right)$ for starting in $s$ consists of two terms: First, the immediate reward $R\left(s\right)$ that we get rightaway simply for starting in state $s$ , and second, the expected sum of future discounted rewards. Examining the second termin more detail, we see that the summation term above can be rewritten ${\mathrm{E}}_{{s}^{\text{'}}\sim {P}_{s\pi \left(s\right)}}\left[{V}^{\pi }\left({s}^{\text{'}}\right)\right]$ . This is the expected sum of discounted rewards for starting in state ${s}^{\text{'}}$ , where ${s}^{\text{'}}$ is distributed according ${P}_{s\pi \left(s\right)}$ , which is the distribution over where we will end up after taking the first action $\pi \left(s\right)$ in the MDP from state $s$ . Thus, the second term above gives the expected sum of discounted rewardsobtained after the first step in the MDP.

Bellman's equations can be used to efficiently solve for ${V}^{\pi }$ . Specifically, in a finite-state MDP ( $|S|<\infty$ ), we can write down one such equation for ${V}^{\pi }\left(s\right)$ for every state $s$ . This gives us a set of $|S|$ linear equations in $|S|$ variables (the unknown ${V}^{\pi }\left(s\right)$ 's, one for each state), which can be efficiently solved for the ${V}^{\pi }\left(s\right)$ 's.

We also define the optimal value function according to

${V}^{*}\left(s\right)=\underset{\pi }{max}{V}^{\pi }\left(s\right).$

In other words, this is the best possible expected sum of discounted rewards that can be attained using any policy. There is also a version of Bellman'sequations for the optimal value function:

${V}^{*}\left(s\right)=R\left(s\right)+\underset{a\in A}{max}\phantom{\rule{1pt}{0ex}}\gamma \sum _{{s}^{\text{'}}\in S}{P}_{sa}\left({s}^{\text{'}}\right){V}^{*}\left({s}^{\text{'}}\right).$

The first term above is the immediate reward as before. The second term is the maximum over all actions $a$ of the expected future sum of discounted rewards we'll get upon after action $a$ . You should make sure you understand this equation and see why it makes sense.

We also define a policy ${\pi }^{*}:S↦A$ as follows:

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