class: center, middle, inverse, title-slide .title[ # Temporal Difference (TD) methods for prediction ] .author[ ### Lars Relund Nielsen ] --- layout: true <div class="my-footer"> <span> <a href="https://bss-osca.github.io/rl/mod-td-pred.html" target="_blank">Notes</a> | <a href="https://bss-osca.github.io/rl/slides/07_td-pred-slides.html" target="_blank">Slides</a> | <a href="https://github.com/bss-osca/rl/blob/master/slides/07_td-pred-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 * Describe what Temporal Difference (TD) learning is. * Formulate the incremental update formula for TD learning. * Define the temporal-difference error. * Interpret the role of a fixed step-size. * Identify key advantages of TD methods over DP and MC methods. * Explain the TD(0) prediction algorithm. * Understand the benefits of learning online with TD compared to MC methods. --- ## What is TD learning? * TD learning is a combination of Monte Carlo (MC) and dynamic programming (DP) ideas * Like MC, TD can predict using a model-free environment and learn from experience. * Like DP, TD update estimates based on other learned estimates, without waiting for a final outcome (bootstrapping). * TD can learn on-line and do not need to wait until the whole sample-path is found. * Given a policy `\(\pi\)`, we want to estimate the state-value function: `$$v_\pi(s) = \mathbb{E}_\pi[G_t | S_t = s].$$` where the return is `$$G_t = R_{t+1} + \gamma R_{t+2} + \gamma^2 R_{t+3} + \cdots = \sum_{k=0}^{\infty} \gamma^k R_{t+k+1} = R_{t+1} + \gamma G_{t+1}$$` <!-- TD in general learn more efficiently than MC due to bootstrapping. In this module prediction using TD is considered. --> --- ## Estimating the state-value How do we find estimates `\(V\)` of `\(v_\pi\)` and the return `\(G_t\)`? * For MC we generated sample-paths and calculated returns `\(G_t\)` and updated the estimate by taking the average over all the realized returns. * For TD we use `\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] \\ &= \mathbb{E}_\pi[R_{t+1}| S_t = s] + \gamma \mathbb{E}_\pi[G_{t+1} | S_t = s] \\ &= \mathbb{E}_\pi[R_{t+1}| S_t = s] + \gamma \mathbb{E}_\pi[v_\pi(S_{t+1})]. \end{align}` * TD: Given a realized reward `\(R_{t+1}\)` and next state `\(S_{t+1}\)`, an estimate for the return `\(G_t\)` is `\(R_{t+1} + \gamma V(S_{t+1})\)` * TD update the estimate based on the estimate of the next state (bootstrapping). --- ## Incremental update .pull-left[ **Monte Carlo** * Generate sample-path and calculate `\(G_t\)` along the path. * Unbiased estimate. * Update the estimate using $$ V(S_t) \leftarrow V(S_t) + \alpha_n\left[G_t - V(S_t)\right]. $$ * Can update after the full sample-path known. ] -- .pull-right[ **Temporal Difference** * Replace `\(G_t\)` with the TD estimate `\(R_{t+1} + \gamma V(S_{t+1}).\)` * Update now becomes .midi[ `$$V(S_t) \leftarrow V(S_t) + \alpha_n\left[R_{t+1} + \gamma V(S_{t+1}) - V(S_t)\right].$$` ] * Can update when the next state `\(S_{t+1}\)` is observed. * As the estimate of `\(S_{t+1}\)` improve the estimate of `\(S_t\)` also improve. * The incremental update is called *TD(0)* or one-step TD because it use a one-step lookahead. ] --- ## A note about fixed step-size If the environment is non-stationary (e.g. transition probabilities change over time) then a fixed step-size `\(\alpha\in(0,1]\)` may be appropriate. A fixed step-size corresponds to a weighted average of the past observed returns and the initial estimate of `\(S_t\)`: $$ `\begin{align} V_{n+1} &= V_n +\alpha \left[G_n - V_n\right] \nonumber \\ &= \alpha G_n + (1 - \alpha)V_n \nonumber \\ &= \alpha G_n + (1 - \alpha)[\alpha G_{n-1} + (1 - \alpha)V_{n-1}] \nonumber \\ &= \alpha G_n + (1 - \alpha)\alpha G_{n-1} + (1 - \alpha)^2 V_{n-1} \nonumber \\ & \vdots \nonumber \\ &= (1-\alpha)^n V_1 + \sum_{i=1}^{n} \alpha (1 - \alpha)^{n-i} G_i \\ \end{align}` $$ That is, a larger weight is used for recent observations compared to old observations. Moreover, if `\(\alpha=1\)` then `\(V_{n+1} = G_t\)` and we use the realized return as the estimate. --- ## The TD error * The term `$$\delta_t = R_{t+1} + \gamma V(S_{t+1}) - V(S_t),$$` is denoted the *temporal difference error* (*TD error*). * Equals the difference between the current estimate `\(V(S_t)\)` and the updated estimate `\(R_{t+1} + \gamma V(S_{t+1})\)`. --- ## TD(0) prediction algorithm <img src="img/td0-pred.png" width="90%" style="display: block; margin: auto;" /> * Also works for continuing processes (infinite number of time-steps of inner loop). * No stopping criterion is given but could stop when small for state-value differences. --- ## TD prediction for action-values * Since model-free we need to estimate action-values instead so we can improve the policy. * Goal is to find `\(q_*\)` for which the optimal action is the greedy action. * To find `\(q_*\)`, we first need to predict action-values `\(q_\pi\)` for a policy `\(\pi\)`. * The incremental update equation 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