Body: <p>This is comes from cs2852023Fall, hw3.
  I'm learning RL by myself and I cann't find answers related to this question.
  <strong>Althrough it's from a homework, I believe it would be beneficial to solve the question and get a deeper understanding for q-learning.</strong>
  The resource could be found in
  <a href="https://rail.eecs.berkeley.edu/deeprlcourse/" rel="nofollow noreferrer">https://rail.eecs.berkeley.edu/deeprlcourse/</a>.</p>
  <h2>Backgorund</h2>
  <p>Consider the N-step variant of Q-learning described.</p>
  <p><span class="math-container">\begin{align}
  y_{i,j} \gets (\sum_{t'=t}^{t+N-1 }\gamma^{t'-t}r_{j,t'} + \gamma^N \max_{a_{j, t+N}}Q_{\phi_{k}}(s_{j,t+N}, a_{j,t+N}) (1)\\
  \phi_{k+1} \gets \arg\min_{\phi\in\Phi}\sum_{j,t}(y_{j,t} - Q_{\phi}(s_{j,t}, a_{j,t}))^2 (2)
  \end{align}</span>
  In these equations, <span class="math-container">$j$</span> indicates an index in the replay buffer of trajectories <span class="math-container">$_{k}$</span>.  We first roll out a batch of B
  trajectories to update  <span class="math-container">$_{k}$</span> and compute the target values in (1). We then fit <span class="math-container">$Q_{\phi_{k+1}}$</span> to these target values with (2). After estimating <span class="math-container">$Q_{\phi_{k+1}}$</span>, we can then update the policy through an argmax:
  <span class="math-container">$$
  \pi_{k+1}(a_t | s_t) \gets 
  \begin{cases}
    &amp; 1\text{ if } a_{t} = \arg\max_{a_{t}}Q_{\phi_{k+1}}(s_{t}, a_{t})\\
    &amp; 0\text { otherwise }
  \end{cases}(3)
  $$</span></p>
  <p>We repeat the steps in eqs.(1) to (3) K times to improve the policy. In this question, you will analyze some properties of the algoriyhm, which is summarized in Algorithm 1.</p>
  <h1>Question</h1>
  <p>At each iteraction of the algorithm above after update from eq.(2), <span class="math-container">$Q_{\phi_{k}}$</span> can be viewed as an estimate of the true optimal <span class="math-container">$Q^\star$</span>. Consider the following statements:</p>
  <p><strong>I.</strong> <span class="math-container">$Q_{\phi_{k+1}}$</span> is an unbiased estimate of the <span class="math-container">$Q$</span> function of the last policy, <span class="math-container">$Q^{\pi_{k}}.$</span></p>
  <p><strong>II.</strong> As <span class="math-container">$k\to\infty$</span> for some fixed <span class="math-container">$B$</span> (replay_buffer), <span class="math-container">$Q^{\pi_{k}}$</span> is an unbiased estimate of <span class="math-container">$Q^{\star}$</span>, i.e., <span class="math-container">$\lim_{k\to\infty}\mathbb{E}[Q_{\phi_{k}}(s,q) - Q^{\star}(s,a)] = 0$</span></p>
  <p><strong>III.</strong> In the limit of infinite iterations and data we recover the optmimal <span class="math-container">$Q^{\star}$</span>, i.e.,  <span class="math-container">$\lim_{k,B\to\infty}\mathbb{E}[||Q_{\phi_{k}} - Q^{\star}||] = 0$</span></p>
  <p>We make the additional assumptions:</p>
  <ul>
  <li>The state and action spaces are finite.</li>
  <li>Every batch contains at least one experience for each action taken in each state.</li>
  <li>In the tabular setting, <span class="math-container">$Q_{\phi_{k}}$</span> can express any function, i.e., <span class="math-container">${Q_{\phi_{k}}:\phi\in\Phi}=\mathbb{R}^{S \times A}$</span></li>
  </ul>
  <p>When updating the buffer <span class="math-container">$_{k}$</span> with <span class="math-container">$B$</span>, we say:</p>
  <ul>
  <li>When learning on-policy, <span class="math-container">$_{k}$</span> is set to contain only the set of <span class="math-container">$B$</span> new rollouts of <span class="math-container">$\pi$</span> (so <span class="math-container">$|_{k} = B|$</span>). Thus,
  we only train on rollouts from the current policy</li>
  <li>When learning off-policy, we use a fixed dataset <span class="math-container">$_{k} = $</span> of <span class="math-container">$B$</span> trajectories from another policy <span class="math-container">$\pi'$</span>.</li>
  </ul>
  <p>Indicate which of the statements <strong>I-III</strong> always hold in the following cases.
  <span class="math-container">\begin{align}
  1. &amp;\text {N = 1 and...  }&amp;I\qquad &amp;II\qquad &amp;III\\
  &amp;a. on-policy in tablular setting. &amp;\Box\qquad &amp;\Box\qquad &amp;\Box \\
  &amp;b. off-policy in tabular setting.&amp;\Box\qquad &amp;\Box\qquad &amp;\Box \\
  2. &amp;\text { N &gt; 1 and...}\\
  &amp;a. on-policy in tablular setting.&amp;\Box\qquad &amp;\Box\qquad &amp;\Box \\
  &amp;b. off-policy in tabular setting.&amp;\Box\qquad &amp;\Box\qquad &amp;\Box \\
  3. &amp;\text {In the limite as $N\to\infty$ and...}\\
  &amp;a. on-policy in tablular setting.&amp;\Box\qquad &amp;\Box\qquad &amp;\Box \\
  &amp;b. off-policy in tabular setting.&amp;\Box\qquad &amp;\Box\qquad &amp;\Box \\
  \end{align}</span></p>
  <h1>My Understanding</h1>
  <ol>
  <li>When N = 1</li>
  </ol>
  <ul>
  <li>I: <span class="math-container">$
  Q_{\phi_{k+1}}(s, a)=r(s,a) + \gamma \max_{a}Q_{\phi_{k}}(s_{t+1}, a_{t+1})
  $</span>, which is the Q function of the last policy, so this is the unbiased estimate of last policy?</li>
  <li>II: Because of the contraction of Bellman operator, and the expression of Q is suffcient for Q value function, this is also right?</li>
  <li>III: Statement is also right because of the contraction of Bellman Operator.</li>
  </ul>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2024-04-02T14:35:07.870Z
Comments.Text: When you say "Don't know how to analysis", what exactly is unclear? Please, ask one question per post, even if they are related to the same homework. Also, don't provide screenshots but quote the relevant unclear part of the text
Comments.UserDisplayName: nbro
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2024-04-05T06:20:40.803Z
Comments.Text: Sorrry for not clarification. I've edited my question. Would your mind helping me?@nbro
Comments.UserDisplayName: yeebo xie
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2024-04-05T11:24:50.787Z
Comments.Text: I’m voting to close this question because its a homework
Comments.UserDisplayName: Kostya
Tags: <q-learning>
Title: Bias of multi-step Q Learning on/off policy
UserDisplayName: yeebo xie
ThreadText: # Bias of multi-step Q Learning on/off policy
  
  (Asked by: username_0 on 2024-03-31)
  
  This is comes from cs2852023Fall, hw3.
  I'm learning RL by myself and I cann't find answers related to this question.
  **Althrough it's from a homework, I believe it would be beneficial to solve the question and get a deeper understanding for q-learning.**
  The resource could be found in
  <https://rail.eecs.berkeley.edu/deeprlcourse/>.
  
  ## Backgorund
  
  Consider the N-step variant of Q-learning described.
  
  \begin{align}
  y\_{i,j} \gets (\sum\_{t'=t}^{t+N-1 }\gamma^{t'-t}r\_{j,t'} + \gamma^N \max\_{a\_{j, t+N}}Q\_{\phi\_{k}}(s\_{j,t+N}, a\_{j,t+N}) (1)\\
  \phi\_{k+1} \gets \arg\min\_{\phi\in\Phi}\sum\_{j,t}(y\_{j,t} - Q\_{\phi}(s\_{j,t}, a\_{j,t}))^2 (2)
  \end{align}
  In these equations, $j$ indicates an index in the replay buffer of trajectories $\_{k}$. We first roll out a batch of B
  trajectories to update $\_{k}$ and compute the target values in (1). We then fit $Q\_{\phi\_{k+1}}$ to these target values with (2). After estimating $Q\_{\phi\_{k+1}}$, we can then update the policy through an argmax:
  $$
  \pi\_{k+1}(a\_t | s\_t) \gets
  \begin{cases}
  & 1\text{ if } a\_{t} = \arg\max\_{a\_{t}}Q\_{\phi\_{k+1}}(s\_{t}, a\_{t})\\
  & 0\text { otherwise }
  \end{cases}(3)
  $$
  
  We repeat the steps in eqs.(1) to (3) K times to improve the policy. In this question, you will analyze some properties of the algoriyhm, which is summarized in Algorithm 1.
  
  # Question
  
  At each iteraction of the algorithm above after update from eq.(2), $Q\_{\phi\_{k}}$ can be viewed as an estimate of the true optimal $Q^\star$. Consider the following statements:
  
  **I.** $Q\_{\phi\_{k+1}}$ is an unbiased estimate of the $Q$ function of the last policy, $Q^{\pi\_{k}}.$
  
  **II.** As $k\to\infty$ for some fixed $B$ (replay\_buffer), $Q^{\pi\_{k}}$ is an unbiased estimate of $Q^{\star}$, i.e., $\lim\_{k\to\infty}\mathbb{E}[Q\_{\phi\_{k}}(s,q) - Q^{\star}(s,a)] = 0$
  
  **III.** In the limit of infinite iterations and data we recover the optmimal $Q^{\star}$, i.e., $\lim\_{k,B\to\infty}\mathbb{E}[||Q\_{\phi\_{k}} - Q^{\star}||] = 0$
  
  We make the additional assumptions:
  
  * The state and action spaces are finite.
  * Every batch contains at least one experience for each action taken in each state.
  * In the tabular setting, $Q\_{\phi\_{k}}$ can express any function, i.e., ${Q\_{\phi\_{k}}:\phi\in\Phi}=\mathbb{R}^{S \times A}$
  
  When updating the buffer $\_{k}$ with $B$, we say:
  
  * When learning on-policy, $\_{k}$ is set to contain only the set of $B$ new rollouts of $\pi$ (so $|\_{k} = B|$). Thus,
    we only train on rollouts from the current policy
  * When learning off-policy, we use a fixed dataset $\_{k} = $ of $B$ trajectories from another policy $\pi'$.
  
  Indicate which of the statements **I-III** always hold in the following cases.
  \begin{align}
  1. &\text {N = 1 and... }&I\qquad &II\qquad &III\\
  &a. on-policy in tablular setting. &\Box\qquad &\Box\qquad &\Box \\
  &b. off-policy in tabular setting.&\Box\qquad &\Box\qquad &\Box \\
  2. &\text { N > 1 and...}\\
  &a. on-policy in tablular setting.&\Box\qquad &\Box\qquad &\Box \\
  &b. off-policy in tabular setting.&\Box\qquad &\Box\qquad &\Box \\
  3. &\text {In the limite as $N\to\infty$ and...}\\
  &a. on-policy in tablular setting.&\Box\qquad &\Box\qquad &\Box \\
  &b. off-policy in tabular setting.&\Box\qquad &\Box\qquad &\Box \\
  \end{align}
  
  # My Understanding
  
  1. When N = 1
  
  * I: $
    Q\_{\phi\_{k+1}}(s, a)=r(s,a) + \gamma \max\_{a}Q\_{\phi\_{k}}(s\_{t+1}, a\_{t+1})
    $, which is the Q function of the last policy, so this is the unbiased estimate of last policy?
  * II: Because of the contraction of Bellman operator, and the expression of Q is suffcient for Q value function, this is also right?
  * III: Statement is also right because of the contraction of Bellman Operator.
  
  **Tags:** q-learning
  > **Comments:**
  > **username_1**: When you say "Don't know how to analysis", what exactly is unclear? Please, ask one question per post, even if they are related to the same homework. Also, don't provide screenshots but quote the relevant unclear part of the text  
  > **username_0**: Sorrry for not clarification. I've edited my question. Would your mind helping me?@nbro  
  > **username_2**: I’m voting to close this question because its a homework  
  
  ---
  
  ## 0 answers
  
  