Answers.Body: <p>Your concern about the extra term seems due to a symbolic error. When you compute <span class="math-container">$\bf{z}_t$</span>, it already incorporates the contribution of the previous eligibility trace <span class="math-container">$\bf{z}_{t-1}$</span> due to the term <span class="math-container">$\gamma \lambda \bf{z}_{t-1}$</span> in your above definition which is (12.5) in S&amp;B book, and the contributions of all previous eligibility traces according to their recency with its obvious recursive nature which can be shown from the below corrected weights update formula:</p>
  <p><span class="math-container">\begin{align*}
  \bf{w}_{t+1} &amp;= \bf{w}_t + \alpha \delta_t (\gamma \lambda \bf{z}_{t-1} + \bf{x}(S_t)) \\
  &amp;=\bf{w}_t + \alpha \delta_t (\gamma \lambda (\gamma \lambda \bf{z}_{t-2} + \bf{x}(S_{t-1})) + \bf{x}(S_t)) \\
  &amp;= \bf{w}_t + \alpha \delta_t (\gamma^2 \lambda^2 \bf{z}_{t-2} + \gamma \lambda \bf{x}(S_{t-1}) + \bf{x}(S_t)) \\
  &amp;=...
  \end{align*}</span></p>
  <p>The eligibility trace serves as a <em>dynamic</em> memory of past state contributions automatically decaying their influence over time. Therefore the adjustment for <span class="math-container">$\bf{z}_{t-1}$</span> does not need to be included in the approximated value function's weight vector update at time step <span class="math-container">$t$</span> as you claimed, as it is already built into <span class="math-container">$\bf{z}_t$</span> as shown above corrected formula.</p>
  
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Answers.Text: Thank you for the explanation, however, I’m still puzzled why, in a purely symbolic derivation from the definitions provided, this extra term doesn't disappear explicitly. Intuitively, it seems that mathematically it should either cancel out or be shown redundant. Does this behavior suggest an approximation rather than an exact equality, or is there a way to show that $\bf z_t$ fully encapsulates all past influence in a way that aligns symbolically with equation 12.7?
  Thank you again for the help
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Answers.Text: @AJR I've updated per your request, seems you have a symbolic error in your question.
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Body: <p>I'm working through Sutton &amp; Barto's <em>Reinforcement Learning: An Introduction</em>, 2nd edition, and trying to understand the derivation of Equation 12.7 for TD(λ) weight updates in Chapter 12, specifically when using eligibility traces. Here’s the update equation in question:</p>
  <p><span class="math-container">$$\bf{w}_{t+1} = \bf{w}_t + \alpha \delta_t \bf{z}_t$$</span></p>
  <p>In this equation:</p>
  <ul>
  <li><span class="math-container">$\bf{w}_{t+1}$</span> is the updated weight vector.</li>
  <li><span class="math-container">$\alpha$</span> is the learning rate.</li>
  <li><span class="math-container">$\delta_t$</span> is the TD error defined as:
  <span class="math-container">$\delta_t = R_{t+1} + \gamma \hat{v}(S_{t+1}, \bf{w}_t) - \hat{v}(S_t, \bf{w}_t)$</span></li>
  <li><span class="math-container">$\bf{z}_t$</span> is the eligibility trace vector, which is recursively defined as:
  <span class="math-container">$\bf{z}_{t} = \gamma \lambda \bf{z}_{t-1} + \bf{x}(S_t)$</span></li>
  </ul>
  <h3>My Question:</h3>
  <p>I attempted to derive Equation 12.7 from first principles, but encountered an issue. Using the definitions provided, my derivation included an additional term:
  <span class="math-container">$$\bf{w}_{t+1} = \bf{w}_t + \alpha \delta_t (\bf{z}_{t} - \gamma \lambda \bf{z}_{t-1})$$</span></p>
  <p>Expanding this leads to:
  <span class="math-container">$$\bf{w}_{t+1} = \bf{w}_t + \alpha \delta_t \bf{z}_t - \alpha \delta_t \gamma \lambda \bf{z}_{t-1}$$</span></p>
  <p>This extra term <span class="math-container">$- \alpha \delta_t \gamma \lambda \bf{z}_{t-1}$</span> does not appear in Sutton &amp; Barto's formulation. I suspect that the recursion within <span class="math-container">$\bf{z}_t$</span> might implicitly account for this term, but I would like to confirm that understanding and clarify why it can be omitted.</p>
  <h3>Key Points I'm Looking For:</h3>
  <ol>
  <li>A step-by-step derivation from the definitions, starting from the TD error and eligibility trace definitions to show how the final weight update form is obtained without needing the extra term.</li>
  <li>An explanation, if possible, of how <span class="math-container">$\bf{z}_t$</span> itself encapsulates the historical contributions from prior time steps.</li>
  </ol>
  <p>Any references to additional literature that validate this would also be very helpful. Thank you!</p>
  
Tags: <reinforcement-learning><temporal-difference-methods><stochastic-gradient-descent><eligibility-traces><td-lambda>
Title: Derivation of Sutton & Barto TD(λ) Weight Update Equation with Eligibility Traces
UserDisplayName: AJR
ThreadText: # Derivation of Sutton & Barto TD(λ) Weight Update Equation with Eligibility Traces
  
  (Asked by: username_0 on 2024-10-29)
  
  I'm working through Sutton & Barto's *Reinforcement Learning: An Introduction*, 2nd edition, and trying to understand the derivation of Equation 12.7 for TD(λ) weight updates in Chapter 12, specifically when using eligibility traces. Here’s the update equation in question:
  
  $$\bf{w}\_{t+1} = \bf{w}\_t + \alpha \delta\_t \bf{z}\_t$$
  
  In this equation:
  
  * $\bf{w}\_{t+1}$ is the updated weight vector.
  * $\alpha$ is the learning rate.
  * $\delta\_t$ is the TD error defined as:
    $\delta\_t = R\_{t+1} + \gamma \hat{v}(S\_{t+1}, \bf{w}\_t) - \hat{v}(S\_t, \bf{w}\_t)$
  * $\bf{z}\_t$ is the eligibility trace vector, which is recursively defined as:
    $\bf{z}\_{t} = \gamma \lambda \bf{z}\_{t-1} + \bf{x}(S\_t)$
  
  ### My Question:
  
  I attempted to derive Equation 12.7 from first principles, but encountered an issue. Using the definitions provided, my derivation included an additional term:
  $$\bf{w}\_{t+1} = \bf{w}\_t + \alpha \delta\_t (\bf{z}\_{t} - \gamma \lambda \bf{z}\_{t-1})$$
  
  Expanding this leads to:
  $$\bf{w}\_{t+1} = \bf{w}\_t + \alpha \delta\_t \bf{z}\_t - \alpha \delta\_t \gamma \lambda \bf{z}\_{t-1}$$
  
  This extra term $- \alpha \delta\_t \gamma \lambda \bf{z}\_{t-1}$ does not appear in Sutton & Barto's formulation. I suspect that the recursion within $\bf{z}\_t$ might implicitly account for this term, but I would like to confirm that understanding and clarify why it can be omitted.
  
  ### Key Points I'm Looking For:
  
  1. A step-by-step derivation from the definitions, starting from the TD error and eligibility trace definitions to show how the final weight update form is obtained without needing the extra term.
  2. An explanation, if possible, of how $\bf{z}\_t$ itself encapsulates the historical contributions from prior time steps.
  
  Any references to additional literature that validate this would also be very helpful. Thank you!
  
  **Tags:** reinforcement-learning, temporal-difference-methods, stochastic-gradient-descent, eligibility-traces, td-lambda
  
  ---
  
  ## 1 answers
  
  ### Answer by username_1 on 2024-10-30. Score: 2
  
  Your concern about the extra term seems due to a symbolic error. When you compute $\bf{z}\_t$, it already incorporates the contribution of the previous eligibility trace $\bf{z}\_{t-1}$ due to the term $\gamma \lambda \bf{z}\_{t-1}$ in your above definition which is (12.5) in S&B book, and the contributions of all previous eligibility traces according to their recency with its obvious recursive nature which can be shown from the below corrected weights update formula:
  
  \begin{align\*}
  \bf{w}\_{t+1} &= \bf{w}\_t + \alpha \delta\_t (\gamma \lambda \bf{z}\_{t-1} + \bf{x}(S\_t)) \\
  &=\bf{w}\_t + \alpha \delta\_t (\gamma \lambda (\gamma \lambda \bf{z}\_{t-2} + \bf{x}(S\_{t-1})) + \bf{x}(S\_t)) \\
  &= \bf{w}\_t + \alpha \delta\_t (\gamma^2 \lambda^2 \bf{z}\_{t-2} + \gamma \lambda \bf{x}(S\_{t-1}) + \bf{x}(S\_t)) \\
  &=...
  \end{align\*}
  
  The eligibility trace serves as a *dynamic* memory of past state contributions automatically decaying their influence over time. Therefore the adjustment for $\bf{z}\_{t-1}$ does not need to be included in the approximated value function's weight vector update at time step $t$ as you claimed, as it is already built into $\bf{z}\_t$ as shown above corrected formula.
  
  > **Comments:**
  > **username_0**: Thank you for the explanation, however, I’m still puzzled why, in a purely symbolic derivation from the definitions provided, this extra term doesn't disappear explicitly. Intuitively, it seems that mathematically it should either cancel out or be shown redundant. Does this behavior suggest an approximation rather than an exact equality, or is there a way to show that $\bf z_t$ fully encapsulates all past influence in a way that aligns symbolically with equation 12.7? Thank you again for the help  
  > **username_1**: @AJR I've updated per your request, seems you have a symbolic error in your question.  
  
  ---
  
  