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Importance sampling is typically used when the distribution of interest is difficult to sample from - e.g. it could be computationally expensive to draw samples from the distribution - or when the distribution is only known up to a multiplicative constant, such as in Bayesian statistics where it is intractable to calculate the marginal likelihood; that is $$...


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I'm using OpenAI's cartpole environment. First of all, is this environment not Markov? The OpenAI Gym CartPole environment is Markov. Whether or not you know the transition probabilities does not affect whether the state has the Markov property. All that matters is that knowing the current state is enough to be determine the next state and reward in ...


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Recall that our goal is to be able to accurately estimate the true value of each state by computing a sample average over returns starting from that state: $$v_{q}(s) \doteq \mathbb{E}_{q}\left[G_{t} | S_{t}=s\right] \approx \frac{1}{n} \sum_{i=1}^{n} Return_i $$ where $Return_i$ is the return obtained from the $i^{th}$ trajectory. The problem is that the $\...


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This post contains many answers that describe the difference between on-policy vs. off-policy. Your book may be referring to how the current (DQN-based) state-of-the-art (SOTA) algorithms, such as Ape-X, R2D2, Agent57 are technically "off-policy", since they use a (very large!) replay buffer, often filled in a distributed manner. This has a number ...


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We estimate a value using sampling on whole episodes, and we take this values to construct the target policy. The crucial bit that you are missing is that there is no single value of $V(s)$ (or $Q(s,a)$) of a state (or a state action pair). These value functions are always defined with respect to some policy $\pi(a|s)$ and is given the notation of $V^{\pi}(...


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What I want to know is whether I can add expert data to the replay buffer, given that DDPG is an off-policy algorithm? You certainly can, that is indeed one of the advantages of off-policy learning algorithms; they're still "correct", regardless of which policy generated the data that you're learning from (and a human expert providing the ...


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In the book, the phrase "generate the data" refers to the data from observations about states, actions, next states and rewards, that then get used to make value estimate updates. In both the SARSA and Q learning pseudocode from the book, there is a behaviour policy that selects the next action to take. Other than the initial start state, this ...


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In the application of importance sampling to RL, is the expectation of the function $f$ equivalent to the value of the trajectories, which is represented by the trajectories $x$? I believe what you are asking here is if when using importance sampling in the off-policy RL setting that we set $f(x)$ from the general importance sampling formula to be our ...


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First, some preliminary questions: in this case, what is the optimal policy? It is the policy that maximises return from any given time step $G_t$. You need to be careful with your definition of return with continuing environments. The simple expected sum of future rewards is likely to be positive or negative infinity. There are three basic approaches: ...


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Let's fix some notation: we're collecting data from behavior policy $\pi_0$ and we want to evaluate a policy $\pi$. Of course, if we had plenty of data from policy $\pi$ that would be the best way to evaluate $\pi$ as we just take the empirical average (without any importance sampling) and CLT gives us confidence intervals that shrink at $\frac{1}{\sqrt n}$ ...


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The pseudocode you have copied looks incorrect to me, and I think it is from the first edition. The main issue is at the end of the loop. Where the book has $\qquad W \leftarrow W \frac{1}{\mu(A_t|S_t)}$ $\qquad \text{If } W = 0 \text{ then ExitForLoop}$ It should have either $\qquad W \leftarrow W \frac{1}{\mu(A_t|S_t)}$ $\qquad \text{If } \pi(S_t) \neq ...


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DDPG is an off-policy algorithm simply because of the objective taking expectation with respect to some other distribution that we are not learning about, i.e. the deterministic policy gradient can be expressed as $$\nabla _{\theta^\mu} J \approx \mathbb{E}_{s_t \sim \rho^\beta} \left[ \nabla _{\theta^\mu} Q(s,a|\theta^Q) | s=s_t, a=\mu(s_t ; \theta ^\mu) \...


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DQN is famous for doing over-approximation on Q function. However, having over approximated Q does not imply that it does not perform well in the environment. (unless it looks ridiculously high) From my experience, high learning rate usually cause over approximated Q, or mistakes made in the code. Best way to check is to see plot of Q function when running ...


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As mentioned in the comments your assumption about independence is wrong. Here's why. To prove independence we need to show the following holds: $$P(X=x, Y=y) = P(X=x)P(Y=y)$$ in the case of RL this becomes: $$P(X=a, X=a') = P(X=a)P(Y=a')$$ The left hand side has the value: $$P(X=a, Y=a') = b(A_t = a| S_t = s) p(s'|a,s) b(A_{t+1} = a'|, S_{t+1} = s')$$ while ...


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According to my understanding, you don't use just the current behavior policy for sampling. The importance sampling ratio is calculated as the product of the probability ratios for both the target and behaviour policy throughout the trajectory. See the calculation below, where the product is happening for all the probabilities throughout the trajectories. (...


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What you're describing is off-policy learning. A classic example is $Q$-learning, where you follow some policy $\pi$ whilst learning about the greedy policy. If you're interested in actor-critic methods then a popular off-policy method is the Deep Deterministic Policy Gradient.


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You can simply train a policy from the inputs to predict the actions in your dataset. You can use the cross entropy loss for this, i.e. maximize the the log probability that the policy assigns to the actions in the data set when given the corresponding inputs. This is called behavioral cloning. The result is an approximation of the behavioral policy that ...


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In the DRL nanodegree in Udacity, the instructor says it is possible to combine on- and off-policy learning and suggests the following paper where this has been done: Q-Prop: Sample-Efficient Policy Gradient with An Off-Policy Critic (ICLR 2017). Citing the paper: The core idea is to use the first-order Taylor expansion of the critic as a control ...


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