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There is a strong relationship between a value function and a return. Namely that a value function calculates the expected return from being in a certain state, or taking a specific action in a specific state. A value function is not a "return function", it is an "expected return function" and that is an important difference. A return is a measured value (...

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Advantage function: $A(s,a) = Q(s,a) - V(s)$ More interesting is the General Value Function (GVF), the expected sum of the (discounted) future values of some arbitrary signal, not necessarily reward. It is therefore a generalization of value function $V(s)$. The GVF is defined on page 459 of the 2nd edition of Sutton and Barto's RL book as $$v_{\pi,\gamma,C}(... 5 We are ultimately interested in getting an optimal policy, that is the optimal sequence of actions to reach the final goal. State values on its own don't provide that, they tell you expected return from specific state onward but they don't tell you which action to take. In order to derive an optimal action in a specific state you would have to simulate all ... 5 Not quite. You are missing the reward at time step t+1. The definition you are looking for is (leaving out the \pi subscripts for ease of notation)$$q(s,a) = \mathbb{E}[R_{t+1} + \gamma v(s') | S_t=s,A_t=a] = \sum_{r,s'}(r + v(s'))p(s',r|s,a)\;.$$Because q(s,a) relates to expected returns at time t, and returns are defined as G_t = \sum_{b = 0}... 4 When training a Deep Q network with experienced replay, you accumulate what is known as training experiences e_t = (s_t, a_t, r_t, s_{t+1}). You then sample a batch of such experiences and for each sample you do the following. Feed s_t into the network to get Q(s,a;\theta). Feed s_{t+1} into the network to get Q(s’,a’,\theta). Choose max_aQ(s’,a’,... 4 There needs to be an E_{\pi} over the infinite discounted return term because of two reasons- The policy could be stochastic in nature. That is, for any given state s_t at time t, the policy \pi(s_t) does not provide a deterministic action a, but rather, it provides us with a distribution over the possible next states, that is the action at time ... 4 There seem to be two different ideas in this question here: What's the impact / importance of our choice for reward values? What's the impact / importance of our choice for initial value estimates (how do we initialise our table of Q(s, a) values in the case of a simple, tabular RL algorithm like Sarsa or Q-learning)? The reward values are typically ... 4 In the Sutton and Barto book q(s,a) is used to denote the true expected value of taking action a in state s, whereas capital Q(s,a) is used to denote an estimate of q(s,a). However, there is likely to be a lot of inconsistency in the literature as each author has their own preference on how to denote things. I would encourage you to consider ... 3 I am wondering which definition is correct. The asterisk * in both the definitions stands for "optimal" in the sense of "value when following the optimal policy" So this one is correct: V^* actually assumes the optimal action in a given state, meaning V^* would be 50 in the above case However, you have got the definition of Q ... 3 The deep Q-learning (DQL) algorithm is really similar to the tabular Q-learning algorithm. I think that both algorithms are actually quite simple, at least, if you look at their pseudocode, which isn't longer than 10-20 lines. Here's a screenshot of the pseudocode of DQL (from the original paper) that highlights the Q target. Here's the screenshot of Q-... 3 What does the target Q-values represent? In a DQN, which uses off-policy learning, they represent a refined estimate for the expected future reward from taking an action a in state s, and from that point on following a target policy. The target policy in Q learning is based on always taking the maximising action in each state, according to current ... 3 These two definitions are not exactly the same, even though they have a very similar formulation. David Silver's notation is probably an abuse of notation. The first difference between those two definitions is that, in the case of David Silver's slides, the policy is parametrized by \theta (i.e. the policy could be represented e.g. by a neural network), ... 3 Can someone provide the reasoning behind why G_{t+1} is equal to v_*(S_{t+1})? The two things are not usually exactly equal, because G_{t+1} is a probability distribution over all possible future returns whilst v_*(S_{t+1}) is a probability distribution derived over all possible values of S_{t+1}. These will be different distributions much of the ... 3 Your calculations are correct, but you have misinterpreted the equations and the diagram. The index k in v_k for the diagram refers to the policy evaluation update iteration only, and is not related to the policy update step (which uses the notation \pi' and does not mention k). Policy improvement consists of multiple sweeps through states to fully ... 3 The value of a state depends on the policy that you use, so I'll make the assumption here that you're talking about value using the optimal policy. According to the optimal policy, the agent would choose to stay in the square (1,1) every time, but since it has a 0.8 probability of actually staying (and 0.2 probability of dying), we can compute the value of ... 3 Your understanding of the Bellman equation is not quite right. The state-action value function is defined as the expected (discounted) returns when taking action a in state s. Now, in your equation (2) you have conditioned on taking action a' in the inner expectation - this is not what happens in the state-action value function, you do not condition on ... 3 Removing the learning rate will likely yield poor convergence to the optimal policy and optimal Q-values. Note that the current policy is completely dependent on the Q-values, as we take the action with highest Q-value in a given state (with a few other considerations such as exploration, etc.). If we were to remove the learning rate, then we are making a ... 3 So, naturally, if you've observed something that contradicts the theoretical properties of Value Iteration, something's wrong, right? Well, the code you've linked, as it is, is fine. It works as intended when all the values are initialized to zero. HOWEVER, my guess is that you're the one introducing an (admittedly very subtle) error. I think you're changing ... 3 In general, \mathbb{E}_\pi[G_{t:t+n}|S_t = s] \neq v_\pi(s). v_\pi(s) is defined as \mathbb{E}_\pi[\sum_{k=0}^{\infty} \gamma^k R_{t+k+1} | S_t = s], so you should be able to see why the two are not equal when the LHS is an expectation of the nth step return. They would only be equal as n \rightarrow \infty. 3 I am using the convention of uppercase X for random variable and lowercase x for an individual observation. It is possible your source material did not do this, which might be causing your confusion. However, it is the convention used in Sutton & Barto's Reinforcement Learning: An Introduction. What I didn't understand what is 𝑋 here. i.e., what is ... 3 In Model Based Reinforcement learning, state and state-action values for all states can be calculated based on the bellman equations. The equations are taken from Andrew Ng's Algorithms for Inverse Reinforcement Learning$$V^{\pi}(s) = R(s) + \gamma \sum_{s'}P(s'|s,a)V^{\pi}(s') \\ Q^{\pi}(s,a) = R(s) + \gamma \sum_{s'}P(s'|s,a)V^{\pi}(s')$$In this setting, ... 3 However, from the blogs and texts I read, the equations are expressed in terms of V and NOT Q. Why is that? MC and TD are methods for associating value estimates to time step based on experienced gained in later time steps. It does not matter what kind of value estimate is being associated across time, because all value functions are expressing the same ... 3 Based on this and this resources, let me give an answer to my own question, but, essentially, I will just rewrite the contents of the first resource here, for reproducibility, with some minor changes to the notation (to be consistent with Sutton & Barto's book, 2nd edition). Note that I am not fully sure if this formulation is universal (i.e. maybe there ... 3 A Q table allows you to look up any state/action pair in it and find the associated action value. It is not itself a policy. However, in order to calculate the action values, you will have assumed something about the policy. The most common policy scenarios with Q learning are that it will converge on (learn) the values associated with a given target policy, ... 3 It seems that you are getting confused between the definition of a Q-value and the update rule used to obtain these Q-values. Remember that to simply obtain an optimal Q-value for a given state-action pair we can evaluate$$Q(s, a) = r + \gamma \max_{a'} Q(s', a)\;; where $s'$ is the state we transitioned into (note that this only holds when obtaining the ...

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Note that for a general policy $\pi$ we have that $q_{\pi}(s,a) = \mathbb{E}_{\pi}[G_t | S_t = s, A_t = a]$, where in state $S_t$ we take action $a$ and thereafter following policy $\pi$. Note that the expectation is taken with respect to the reward transition distribution $\mathbb{P}(R_{t+1} = r, S_{t+1} = s' | A_t = a, S_t = s)$ which I will denote as $p(s'... 2 By definition of$V_{n+1}$, we have:$V_{n+1} = \frac{\sum_{k=1}^{n} W_{k} G_{k}}{\sum_{k=1}^{n} W_{k}} \; \tag{1}$Then, taking the$n^{th}$term out of the sum in the numerator, we have:$V_{n+1} = \frac{W_{n}G_{n} \; + \; \sum_{k=1}^{n-1} W_{k} G_{k}}{\sum_{k=1}^{n} W_{k}} \; \tag{2}$Then, from the definition of$V_n$,$V_{n} = \frac{\sum_{k=1}^{n-1} ...

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Do policy independent state and action values exist in reinforcement learning? No. They do not exist, because in order to progress in any MDP and receive any reward - i.e. to get any measure of value - you must take an action. Any consistent means of selecting actions is a policy, and the nature of that policy impacts which transitions and rewards you ...

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In addition to this answer, I would like to note that, if the future trajectories were fixed (i.e. the environment and the policies were deterministic, and the agent always starts from the same state), the expectation of the sum (of the fixed rewards) would simply correspond to the actual sum, because the sum is a constant (i.e. the expectation of a constant ...

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I guess I'm having difficulty grasping the concept that the goodness of a state changes depending on how an agent got there It doesn't. The value of a state changes depending on what the agent will do next. That is where the dependency on the policy comes in, not in past behaviour, but expectations of future behaviour. The future behaviour depends on the ...

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