class: center, middle, inverse, title-slide .title[ # Policies and value functions for MDPs ] .author[ ### Lars Relund Nielsen ] --- layout: true <div class="my-footer"> <span> <a href="https://bss-osca.github.io/rl/mod-mdp-2.html" target="_blank">Notes</a> | <a href="https://bss-osca.github.io/rl/slides/04_mdp-2-slides.html" target="_blank">Slides</a> | <a href="https://github.com/bss-osca/rl/blob/master/slides/04_mdp-2-slides.Rmd" target="_blank">Source</a> </span> </div> <!-- Templates --> <!-- .pull-left[] .pull-right[] --> <!-- knitr::include_graphics("img/bandit.png") --> <!-- .left-column-wide[] .right-column-small[] --> --- ## Learning outcomes * Identify a policy as a distribution over actions for each possible state. * Define value functions for a state and action. * Derive the Bellman equation for a value function. * Understand how Bellman equations relate current and future values. * Define an optimal policy. * Derive the Bellman optimality equation for a value function. --- ## Policies A *policy* `\(\pi\)` is a distribution over actions, given some state: `$$\pi(a | s) = \Pr(A_t = a | S_t = s).$$` * Since the MDP is stationary the policy is time-independent, i.e. given a state, we choose the same action no matter the time-step. * The policy is *deterministic* if `\(\pi(a | s) = 1\)` for a single state, i.e. an action is chosen with probability one always. * The policy is *stochastic* if `\(\pi(a | s) < 1\)` for some state. --- ## Value functions * The *state-value function* `\(v_\pi(s)\)` (expected return given state `\(s\)` and policy `\(\pi\)`): $$ `\begin{align} v_\pi(s) &= \mathbb{E}_\pi[G_t | S_t = s] \\ &= \mathbb{E}_\pi[R_{t+1} + \gamma G_{t+1} | S_t = s]. \end{align}` $$ Note the last equal sign comes from `\(G_t = R_{t+1} + \gamma G_{t+1}\)`. * The *action-value function* `\(q_\pi(s, a)\)` (expected return given `\(s\)`, action `\(a\)` and policy `\(\pi\)`): $$ `\begin{align} q_\pi(s, a) &= \mathbb{E}_\pi[G_t | S_t = s, A_t = a] \\ &= \mathbb{E}_\pi[R_{t+1} + \gamma G_{t+1} | S_t = s, A_t = a]. \end{align}` $$ * Combining above, the state-value function becomes an average over the q-values: `$$v_\pi(s) = \sum_{a \in \mathcal{A}(s)}\pi(a|s)q_\pi(s, a)$$` --- ## Bellman equations The value functions can be transformed into recursive equations (Bellman equations). Bellman equations (state-value): `$$v_\pi(s) = \sum_{a \in \mathcal{A}}\pi(a | s)\left( r(s,a) + \gamma\sum_{s' \in \mathcal{S}} p(s' | s, a) v_\pi(s')\right)$$` Bellman equations (action-value): `$$q_\pi(s, a) = r(s,a) + \gamma \sum_{s' \in \mathcal{S}} p(s' | s, a) \sum_{a' \in \mathcal{A(s')}} \pi(a'|s')q_\pi(s',a')$$` Let us have a closer look on the derivation blackboard. --- layout: true <div class="my-footer"> <span> <a href="https://bss-osca.github.io/rl/mod-mdp-2.html" target="_blank">Notes</a> | <a href="https://bss-osca.github.io/rl/slides/04_mdp-2-slides.html" target="_blank">Slides</a> | <a href="https://github.com/bss-osca/rl/blob/master/slides/04_mdp-2-slides.Rmd" target="_blank">Source</a> </span> </div> ## Visualization of Bellman equations --- `$$\phantom{v_\pi(s) = \sum_{a \in \mathcal{A}}\pi(a | s)}\phantom{(} r(s,a) + \gamma\sum_{s' \in \mathcal{S}} p(s' | s, a) v_\pi(s')\phantom{)}$$` .left-column-small[ * Policy `\(\pi\)` given. * Calc. `\(q_\pi(s, a)\)`. <!-- * Multiply with `\(\pi(a | s)\)`. --> ] .right-column-wide[ <img src="img/mdp-2-unnamed-chunk-6-1.png" width="100%" style="display: block; margin: auto;" /> ] --- `$$\phantom{v_\pi(s) = \sum_{a \in \mathcal{A}}\pi(a | s)}\phantom{(} r(s,a) + \gamma\sum_{s' \in \mathcal{S}} p(s' | s, a) v_\pi(s')\phantom{)}$$` .left-column-small[ * Policy `\(\pi\)` given. * Calc. `\(q_\pi(s, a)\)` for each `\(a\)`. <!-- * Multiply with `\(\pi(a | s)\)`. --> ] .right-column-wide[ <img src="img/mdp-2-unnamed-chunk-7-1.png" width="100%" style="display: block; margin: auto;" /> ] --- `$$v_\pi(s) = \sum_{a \in \mathcal{A}}\pi(a | s)( r(s,a) + \gamma\sum_{s' \in \mathcal{S}} p(s' | s, a) v_\pi(s'))$$` .left-column-small[ * Policy `\(\pi\)` given. * Calc. `\(q_\pi(s, a)\)` for each `\(a\)`. * Multiply with `\(\pi(a | s)\)`. ] .right-column-wide[ <img src="img/mdp-2-unnamed-chunk-8-1.png" width="100%" style="display: block; margin: auto;" /> ] --- layout: false <div class="my-footer"> <span> <a href="https://bss-osca.github.io/rl/mod-mdp-2.html" target="_blank">Notes</a> | <a href="https://bss-osca.github.io/rl/slides/04_mdp-2-slides.html" target="_blank">Slides</a> | <a href="https://github.com/bss-osca/rl/blob/master/slides/04_mdp-2-slides.Rmd" target="_blank">Source</a> </span> </div> ## Optimal policies and value functions * The objective function of an MDP is to find an optimal policy `\(\pi_*\)` with state-value function: `$$v_*(s) = \max_\pi v_\pi(s).$$` * A policy `\(\pi'\)` is better than policy `\(\pi\)` if its expected return is greater than or equal for all states and is strictly greater for at least one state. * But the objective is not a scalar, but if the agent start in state `\(s_0\)`: `$$v_*(s_0) = \max_\pi \mathbb{E}_\pi[G_0 | S_0 = s_0] = \max_\pi v_\pi(s_0)$$` That is, maximize the expected return given starting state `\(s_0\)`. * If the MDP has the right properties, there exists an optimal deterministic policy `\(\pi_*\)` which is better than or just as good as all other policies. --- ## Bellman optimality equations * The Bellman equations define the recursive equations. * Bellman optimality equations define how to find the optimal value functions. * Bellman optimality action-value function `\(q_*\)`: `\begin{align} q_*(s, a) &= \max_\pi q_\pi(s, a) \\ &= r(s,a) + \gamma\sum_{s' \in \mathcal{S}} p(s' | s, a) \max_{a'} q_*(s', a'). \end{align}` * *Bellman optimality equation* for `\(v_*\)`: `\begin{align} v_*(s) &= \max_\pi v_\pi(s) = \max_a q_*(s, a) \\ &= \max_a \left( r(s,a) + \gamma\sum_{s' \in \mathcal{S}} p(s' | s, a) v_*(s') \right) \end{align}` * Let us have a look at the derivations on the blackboard. --- layout: true <div class="my-footer"> <span> <a href="https://bss-osca.github.io/rl/mod-mdp-2.html" target="_blank">Notes</a> | <a href="https://bss-osca.github.io/rl/slides/04_mdp-2-slides.html" target="_blank">Slides</a> | <a href="https://github.com/bss-osca/rl/blob/master/slides/04_mdp-2-slides.Rmd" target="_blank">Source</a> </span> </div> --- ## Optimality vs approximation * Using the Bellman optimality equations optimal policies and value functions can be found. * It may be expensive to solve the equations if the number of states is huge. * *Curse of dimensionality:* Consider a state `\(s = (x_1,\ldots,x_n)\)` with state variables `\(x_i\)` each taking two possible values, then the number of states is `\(|\mathcal{S}| = 2^n\)`. That is, the state space grows exponentially with the number of state variables. * Large state or action spaces may happen in practice; moreover, they may also be continuous. * In such cases we approximate the value functions instead. * In RL we approximate the value functions if state stace is high or parameters unknown (e.g. transition probabilities). * In RL we often focus on states with high encountering probability while allowing the agent to make sub-optimal decisions in states that have a low probability. --- ## Semi-MDPs (non-fixed time length) * Finite MDPs consider a fixed length between each time-step. * Semi-MDPs consider non-fixed time-lengths. * Let `\(l(s'|s,a)\)` denote the length of a time-step given that the system is in state `\(s\)`, action `\(a\)` is chosen and makes a transition to state `\(s'\)`. * The discount rate over a time-step with length `\(l(s'|s,a)\)` is `$$\gamma(s'|s,a) = \gamma^{l(s'|s,a)},$$` * The Bellman optimality equations becomes: `$$v_*(s) = \max_a \left( r(s,a) + \sum_{s' \in \mathcal{S}} p(s' | s, a) \gamma(s'|s,a) v_*(s') \right)$$` `$$q_*(s, a) = r(s,a) + \sum_{s' \in \mathcal{S}} p(s' | s, a) \gamma(s'|s,a) \max_{a'} q_*(s', a')$$` <!-- # References --> <!-- ```{r, results='asis', echo=FALSE} --> <!-- PrintBibliography(bib) --> <!-- ``` --> [BSS]: https://bss.au.dk/en/ [bi-programme]: https://kandidat.au.dk/en/businessintelligence/ [course-help]: https://github.com/bss-osca/rl/issues [cran]: https://cloud.r-project.org [cheatsheet-readr]: https://rawgit.com/rstudio/cheatsheets/master/data-import.pdf [course-welcome-to-the-tidyverse]: https://github.com/rstudio-education/welcome-to-the-tidyverse [DataCamp]: https://www.datacamp.com/ [datacamp-signup]: https://www.datacamp.com/groups/shared_links/cbaee6c73e7d78549a9e32a900793b2d5491ace1824efc1760a6729735948215 [datacamp-r-intro]: https://learn.datacamp.com/courses/free-introduction-to-r [datacamp-r-rmarkdown]: https://campus.datacamp.com/courses/reporting-with-rmarkdown 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