# reinforce with baseline

While the learned baseline already gives a considerable improvement over simple REINFORCE, it can still unlearn an optimal policy. … We optimize hyperparameters for the different approaches by running a grid search over the learning rate and approach-specific hyperparameters. As our main objective is to compare the data efficiency of the different baselines estimates, we choose the parameter setting with a single beam as the best model. Note that as we only have to actions, it means in p/2% of the cases, we take a wrong action. Stochasticity seems to make the sampled beams too noisy to serve as a good baseline. Mark Saad in Reinforcement Learning with MATLAB 29 Nov • 6 min read. This would require 500*N samples which is extremely inefficient. V^(st,w)=wTst. This is a pretty significant difference, and this idea can be applied to our policy gradient algorithms to help reduce the variance by subtracting some baseline value from the returns. However, most of the methods proposed in thereinforcement learning community are not yet applicable to manyproblems such as robotics, motor control, etc. Another limitation of using the sampled baseline is that you need to be able to make multiple instances of the environment at the same (internal) state and many OpenAI environments do not allow this. Able is a place to discuss building things with software and technology. The capability of training machines to play games better than the best human players is indeed a landmark achievement. This can be even achieved with a single sampled rollout. In my next post, we will discuss how to update the policy without having to sample an entire trajectory first. A reward of +1 is provided for every time step that the pole remains upright. However, the unbiased estimate is to the detriment of the variance, which increases with the length of the trajectory. \end{aligned}∇w[21(Gt−V^(st,w))2]=−(Gt−V^(st,w))∇wV^(st,w)=−δ∇wV^(st,w). LMMâââNeural Network That Animates Video Game Characters, Building an artificially intelligent system to augment financial analysis, Neural Networks from Scratch with Python Code and Math in Detailâ I, A Short Story of Faster R-CNNâs Object detection, Hello World-Implementing Neural Networks With NumPy, number of update steps (1 iteration = 1 episode + gradient update step), number interactions (1 interaction = 1 action taken in the environment), The regular REINFORCE loss, with the learned value as a baseline, The mean squared error between the learned value and the observed discounted return. Likewise, we substract a lower baseline for states with lower returns. A not yet explored benefit of sampled baseline might be for partially observable environments. The source code for all our experiments can be found here: Rennie, S. J., Marcheret, E., Mroueh, Y., Ross, J., & Goel, V. (2017). With advancements in deep learning, these algorithms proved very successful using powerful networks as function approximators. We focus on the speed of learning not only in terms of number of iterations taken for successful learning but also the number of interactions done with the environment to account for the hidden cost in obtaining the baseline. It turns out that the answer is no, and below is the proof. Stochastic Beams and Where to Find Them: The Gumbel-Top-k Trick for Sampling Sequences Without Replacement. contrib. www is the weights parametrizing V^\hat{V}V^. episode length of 500). In. All together, this suggests that for a (mostly) deterministic environment, a sampled baseline reduces the variance of REINFORCE the best. At 10%, we experience that all methods achieve similar performance as with the deterministic setting, but with 40%, all our methods are not able to reach a stable performance of 500 steps. If you haven’t looked into the field of reinforcement learning, please first read the section “A (Long) Peek into Reinforcement Learning » Key Concepts”for the problem definition and key concepts. Several such baselines were proposed, each with its own set of advantages and disadvantages. Starting from the state, we could also make the agent greedy, by making it take only actions with maximum probability, and then use the resulting return as the baseline. For this implementation we use the average reward as our baseline. The following figure shows the result when we use 4 samples instead of 1 as before. We can update the parameters of V^\hat{V}V^ using stochastic gradient. However, more sophisticated baselines are possible. Self-critical sequence training for image captioning. Then the new set of numbers would be 100, 20, and 50, and the variance would be about 16,333. Applying this concept to CartPole, we have the following hyperparameters to tune: number of beams for estimating the state value (1, 2, and 4), the log basis of the sample interval (2, 3, and 4), and the learning rate (1e-4, 4e-4, 1e-3, 2e-3, 4e-3). Namely, there’s a high variance in … To reduce variance of the gradient, they subtract 'baseline' from sum of future rewards for all time steps. As maintainers of, and the first Ethereum client embracing Baseline, we are excited that the solutions delivered by Nethermind and Provide enable rapid adoption, allowing enterprises to reinforce their integrations with the unique notarization capabilities and liveness of the Ethereum mainnet. … New campaign to reinforce hygiene practices in dorms Programme aims to keep at bay fresh mass virus outbreaks among migrant workers. On the other hand, the learned baseline has not converged when the policy reaches the optimum because the value estimate is still behind. However, all these conclusions only hold for the deterministic case, which is often not the case. However, taking more rollouts leads to more stable learning. \end{aligned}E[t=0∑T∇θlogπθ(at∣st)b(st)]=E[∇θlogπθ(a0∣s0)b(s0)+∇θlogπθ(a1∣s1)b(s1)+⋯+∇θlogπθ(aT∣sT)b(sT)]=E[∇θlogπθ(a0∣s0)b(s0)]+E[∇θlogπθ(a1∣s1)b(s1)]+⋯+E[∇θlogπθ(aT∣sT)b(sT)], Because the probability of each action and state occurring under the current policy does change with time, all of the expectations are the same and we can reduce the expression to, E[∑t=0T∇θlogπθ(at∣st)b(st)]=(T+1)E[∇θlogπθ(a0∣s0)b(s0)]\mathbb{E} \left[\sum_{t = 0}^T \nabla_\theta \log \pi_\theta \left(a_t \vert s_t \right) b\left(s_t\right) \right] = \left(T + 1\right) \mathbb{E} \left[\nabla_\theta \log \pi_\theta \left(a_0 \vert s_0 \right) b\left(s_0\right)\right] Reinforcement Learning (RL) refers to both the learning problem and the sub-field of machine learning which has lately been in the news for great reasons. In the case of learned value functions, the state estimate for s=(a1,b) is the same as for s=(a2,b), and hence learns an average over the hidden dimensions. The goal is to keep the pendulum upright by applying a force of -1 or +1 (left or right) to the cart. REINFORCE with Baseline Policy Gradient Algorithm. The research community is seeing many more promising results. In my last post, I implemented REINFORCE which is a simple policy gradient algorithm. Actor Critic Algorithm (Detailed explanation can be found in Introduction to Actor Critic article) Actor Critic algorithm uses TD in order to compute value function used as a critic. But assuming no mistakes, we will continue. Reinforcement learning is probably the most general framework inwhich reward-related learning problems of animals, humans or machinecan be phrased. Implementation of REINFORCE with Baseline algorithm, recreation of figure 13.4 and demonstration on Corridor with switched actions environment. One of the restrictions is that the environment needs to be duplicated because we need to sample different trajectories starting from the same state. Now the estimated baseline is the average of the rollouts including the main trajectory (and excluding the jâth rollout). We will choose it to be V^(st,w)\hat{V}\left(s_t,w\right)V^(st,w) which is the estimate of the value function at the current state. We use same seeds for each gridsearch to ensure fair comparison. spaces import Discrete, Box: def get_traj (agent, env, max_episode_steps, render, deterministic_acts = False): ''' Runs agent-environment loop for one whole episdoe (trajectory). This approach, called self-critic, was first proposed in Rennie et al.Â¹ and also shown to give good results in Kool et al.Â² Another promising direction is to grant the agent some special powers - the ability to play till the end of the game from the current state, go back to the state and play more games following alternative decision paths. I do not think this is mandatory though. For example, assume we take a single beam. ∇wV^(st,w)=st, and we update the parameters according to, w=w+(Gt−wTst)stw = w + \left(G_t - w^T s_t\right) s_t In other words, as long as the baseline value we subtract from the return is independent of the action, it has no effect on the gradient estimate! The experiments of 20% have shown to be at a tipping point. This can be improved by subtracting a baseline value from the Q values. Kool, W., van Hoof, H., & Welling, M. (2018). Shop Baseline women's gym and activewear clothing, exclusively online. REINFORCE method and actor-critic methods are examples of this approach. In our case this usually means that in more than 75% of the cases, the episode length was optimal (500) but that there were a small set of cases where the episode length was sub-optimal. However, also note that by having more rollouts per iteration, we have many more interactions with the environment; and then we could conclude that more rollouts is not per se more efficient. 13.4 REINFORCE with Baseline. Thus,those systems need to be modeled as partially observableMarkov decision problems which o… By this, we prevent to punish the network for the last steps although it succeeded. Hyperparameter tuning leads to an optimal learning rates of Î±=2e-4 and Î²=2e-5 . W. Zaremba et al., "Reinforcement Learning Neural Turing Machines", arXiv, 2016. this baseline is chosen as expected future reward given previous states/actions. The following methods show two ways to estimate this expected return of the state under the current policy. Mark Saad in Reinforcement Learning with MATLAB 28 Nov • 7 min read. Eighty-three male and female patients aged from 13 to 73 years were randomized to either of the following two treatment groups in a 1:1 ratio: satralizumab (120 mg) or placebo added to baseline … Then, ∇wV^(st,w)=st\nabla_w \hat{V} \left(s_t,w\right) = s_t The variance of this set of numbers is about 50,833. The optimal learning rate found by gridsearch over 5 different rates is 1e-4. To implement this, we choose to use a log scale, meaning that we sample from the states at T-2, T-4, T-8, etc. A state that yields a higher return will also have a high value function estimate, so we subtract a higher baseline. The unfortunate thing with reinforcement learning is that, at least in my case, even when implemented incorrectly, the algorithm may seem to work, sometimes even better than when implemented correctly. This indicates that both methods provide a proper baseline for stable learning. What is interesting to note is that the mean is sometimes lower than the 25th percentile. Reinforce With Baseline in PyTorch. Latest commit b2d179a Jun 11, 2019 History. Interestingly, by sampling multiple rollouts, we could also update the parameters on the basis of the jâth rollout. However, this is not realistic because in real-world scenarios, external factors can lead to different next states or perturb the rewards. &= \sum_s \mu\left(s\right) b\left(s\right) \nabla_\theta \sum_a \pi_\theta \left(a \vert s \right) \\ We want to minimize this error, so we update the parameters using gradient descent: w=w+δ∇wV^(st,w)\begin{aligned} This way, the average episode length is lower than 500. layers as layers: from tqdm import trange: from gym. they applied REINFORCE algorithm to train RNN. &= 0 In all our experiments, we use the same neural network architecture, to ensure a fair comparison. Without any gradients, we will not be able to update our parameters before actually seeing a successful trial. However, when we look at the number of interactions with the environment, REINFORCE with a learned baseline and sampled baseline have similar performance. To reduce … The algorithm involved generating a complete episode and using the return (sum of rewards) obtained in calculating the gradient. However, in most environments such as CartPole, the last steps determine success or failure, and hence, the state values fluctuate most in these final stages. However, the method suffers from high variance in the gradients, which results in slow unstable learning and a lot of frustrationâ¦. This enables the gradients to be non-zero, and hence can push the policy out of the optimum which we can see in the plot above. We would like to have tested on more environments. This output is used as the baseline and represents the learned value. Of course, there is always room for improvement. The other methods suffer less from this issue because their gradients are mostly non-zero, and hence, this noise gives a better exploration for finding the goal. In the case of a stochastic environment, however, using a learned value function would probably be preferable. My intuition for this is that we want the value function to be learned faster than the policy so that the policy can be updated more accurately. Because Gt is a sample of the true value function for the current policy, this is a reasonable target. REINFORCE with a Baseline. where Ï(a|s, Î¸) denotes the policy parameterized by Î¸, q(s, a) denotes the true value of the state-action pair and Î¼(s) denotes the distribution over states. Consider the set of numbers 500, 50, and 250. Furthermore, in the environment with added stochasticity, we observed that the learned value function clearly outperformed the sampled baseline. \end{aligned}E[∇θlogπθ(a0∣s0)b(s0)]=s∑μ(s)a∑πθ(a∣s)∇θlogπθ(a∣s)b(s)=s∑μ(s)a∑πθ(a∣s)πθ(a∣s)∇θπθ(a∣s)b(s)=s∑μ(s)b(s)a∑∇θπθ(a∣s)=s∑μ(s)b(s)∇θa∑πθ(a∣s)=s∑μ(s)b(s)∇θ1=s∑μ(s)b(s)(0)=0. The episode ends when the pendulum falls over or when 500 time steps have passed. Also, the optimal policy is not unlearned in later iterations, which does regularly happen when using the learned value estimate as baseline. A simple baseline, that looks similar to a trick commonly used in optimization literature, is to normalize the returns of each step of the episode by subtracting the mean and dividing by the standard deviation of returns at all time steps within the episode. &= \sum_s \mu\left(s\right) \sum_a \pi_\theta \left(a \vert s\right) \frac{\nabla_\theta \pi_\theta \left(a \vert s \right)}{\pi_\theta \left(a \vert s\right)} b\left(s\right) \\ E[t=0∑T∇θlogπθ(at∣st)b(st)]=(T+1)E[∇θlogπθ(a0∣s0)b(s0)], I apologize in advance to all the researchers I may have disrespected with any blatantly wrong math up to this point. We saw that while the agent did learn, the high variance in the rewards inhibited the learning. Now, we will implement this to help make things more concrete. The results that we obtain with our best model are shown in the graphs below. We do not use V in G. G is only the reward to go for every step in … REINFORCE with Baseline Algorithm Initialize the actor μ (S) with random parameter values θμ. In terms of number of iterations, the sampled baseline is only slightly better than regular REINFORCE. Amongst all the approaches in reinforcement learning, policy gradient methods received a lot of attention as it is often easier to directly learn the policy without the overhead of learning value functions and then deriving a policy. Some states will yield higher returns, and others will yield lower returns, and the value function is a good choice of a baseline because it adjusts accordingly based on the state. This is what is done in state-of-the-art policy gradient methods like A3C. In the REINFORCE algorithm, Monte Carlo plays out the whole trajectory in an episode that is used to update the policy afterward. ∇θJ(πθ)=E[t=0∑T∇θlogπθ(at∣st)t′=t∑Tγt′rt′], Suppose we subtract some value, bbb, from the return that is a function of the current state, sts_tst, so that we now have, ∇θJ(πθ)=E[∑t=0T∇θlogπθ(at∣st)∑t′=tT(γt′rt′−b(st))]=E[∑t=0T∇θlogπθ(at∣st)∑t′=tTγt′rt′−∑t=0T∇θlogπθ(at∣st)b(st)]=E[∑t=0T∇θlogπθ(at∣st)∑t′=tTγt′rt′]−E[∑t=0T∇θlogπθ(at∣st)b(st)]\begin{aligned} I included the 12\frac{1}{2}21 just to keep the math clean. This is called whitening. In our case, analyzing both is important because the self-critic with sampled baseline uses more interactions (per iteration) than the other methods. Once we have sample a trajectory, we will know the true returns of each state, so we can calculate the error between the true return and the estimated value function as, δ=Gt−V^(st,w)\delta = G_t - \hat{V} \left(s_t,w\right) In contrast, the sample baseline takes the hidden parts of the state into account, as it will start from s=(a1,b). As a result, I have multiple gradient estimates of the value function which I average together before updating the value function parameters. By Phillip Lippe, Rick Halm, Nithin Holla and Lotta Meijerink. Ever since DeepMind published its work on AlphaGo, reinforcement learning has become one of the âcoolestâ domains in artificial intelligence. contrib. The REINFORCE with Baseline algorithm becomes. One of the earliest policy gradient methods for episodic tasks was REINFORCE, which presented an analytical expression for the gradient of the objective function and enabled learning with gradient-based optimization methods. However, it does not solve the game (reach an episode of length 500). past few years amazing results like learning to play Atari Games from raw pixels and Mastering the Game of Go have gotten a lot of attention While most papers use these baselines in specific settings, we are interested in comparing their performance on the same task. We do one gradient update with the weighted sum of both losses, where the weights correspond to the learning rates Î± and Î², which we tuned as hyperparameters. The results with different number of rollouts (beams) are shown in the next figure. … The environment consists of an upright pendulum joint to a cart. This shows that although we can get the sampled baseline stabilized for a stochastic environment, it gets less efficient than a learned baseline. Technically, any baseline would be appropriate as long as it does not depend on the actions taken. BUY 4 REINFORCE SAMPLES, GET A BASELINE FOR FREE! REINFORCE with sampled baseline: the average return over a few samples is taken to serve as the baseline. Here, Gt is the discounted cumulative reward at time step t. Writing the gradient as an expectation over the policy/trajectory allows us to update the parameter similar to stochastic gradient ascent: As with any Monte Carlo based approach, the gradients of the REINFORCE algorithm suffer from high variance as the returns exhibit high variability between episodes - some episodes can end well with high returns whereas some could be very bad with low returns. REINFORCE with baseline. Contrast this to vanilla policy gradient or Q-learning algorithms that continuously increment the Q-value, which leads to situations where a minor incremental update … \end{aligned}∇θJ(πθ)=E[t=0∑T∇θlogπθ(at∣st)t′=t∑T(γt′rt′−b(st))]=E[t=0∑T∇θlogπθ(at∣st)t′=t∑Tγt′rt′]. But what is b(st)b\left(s_t\right)b(st)? Kool, W., Van Hoof, H., & Welling, M. (2019). 在REINFORCE算法中，训练的目标函数是最小化reward期望值的负值，即 . In a stochastic environment, the sampled baseline would thus be more noisy. We could circumvent this problem and reproduce the same state by rerunning with the same seed from start. Therefore, we expect that the performance gets worse when we increase the stochasticity. Using the learned value as baseline, and Gt as target for the value function, leads us to two loss terms: Taking the gradients of these losses results in the following update rules for the policy parameters Î¸ and the value function parameters w, where Î± and Î² are the two learning rates: Implementation-wise, we simply added one more output value to our existing network. &= -\delta \nabla_w \hat{V} \left(s_t,w\right) Shop online today! Code: REINFORCE with Baseline. or make 4 interest-free payments of $22.48 AUD fortnightly with. I am just a lowly mechanical engineer (on paper, not sure what I am in practice). An implementation of Reinforce Algorithm with a parameterized baseline, with a detailed comparison against whitening. The division by stepCt could be absorbed into the learning rate. This effect is due to the stochasticity of the policy. As mentioned before, the optimal baseline is the value function of the current policy. For example, for the LunarLander environment, a single run for the sampled baseline takes over 1 hour. As maintainers of, and the first Ethereum client embracing Baseline, we are excited that the solutions delivered by Nethermind and Provide enable rapid adoption, allowing enterprises to reinforce … In my implementation, I used a linear function approximation so that, V^(st,w)=wTst\hat{V} \left(s_t,w\right) = w^T s_t To find out when the stochasticity makes a difference, we test choosing random actions with 10%, 20% and 40% chance. Then we will show results for all different baselines on the deterministic environment. But in terms of which training curve is actually better, I am not too sure. where www and sts_tst are 4×14 \times 14×1 column vectors. &= \mathbb{E} \left[\sum_{t = 0}^T \nabla_\theta \log \pi_\theta \left(a_t \vert s_t \right) \sum_{t' = t}^T \gamma^{t'} r_{t'} \right] - \mathbb{E} \left[\sum_{t = 0}^T \nabla_\theta \log \pi_\theta \left(a_t \vert s_t \right) b\left(s_t\right) \right] Thus, the learned baseline is only indirectly affected by the stochasticity, whereas a single sampled baseline will always be noisy. Using samples from trajectories, generated according the current parameterized policy, we can estimate the true gradient. The issue of the learned value function is that it is following a moving target, meaning that as soon as we change the policy the slightest, the value function is outdated, and hence, biased. Baseline Reinforced Support 7/8 Tight Black. The learned baseline apparently suffers less from the introduced stochasticity. Note that the plot shows the moving average (width 25). Why does Java have support for time zone offsets with seconds precision? We saw that while the agent did learn, the high variance in the rewards inhibited the learning. \nabla_w \left[ \frac{1}{2} \left(G_t - \hat{V} \left(s_t,w\right) \right)^2\right] &= -\left(G_t - \hat{V} \left(s_t,w\right) \right) \nabla_w \hat{V} \left(s_t,w\right) \\ Note that I update both the policy and value function parameters once per trajectory. In this way, if the obtained return is much better than the expected return, the gradients are stronger and vice-versa. In the case of the sampled baseline, all rollouts reach 500 steps so that our baseline matches the value of the current trajectory, resulting in zero gradients (no learning) and hence, staying stable at the optimum. Policy gradient is an approach to solve reinforcement learning problems. This helps to stabilize the learning, particularly in cases such as this one where all the rewards are positive because the gradients change more with negative or below-average rewards than they would if … We use ELU activation and layer normalization between the hidden layers. Wouter Kool University of Amsterdam ORTEC w.w.m.kool@uva.nl Herke van Hoof University of Amsterdam h.c.vanhoof@uva.nl Max Welling University of Amsterdam CIFAR m.welling@uva.nl ABSTRACT REINFORCE can be used to train models in structured prediction settings to di-rectly optimize the test-time objective. frames before the terminating state T. Using these value estimates as baselines, the parameters of the model are updated as shown in the following equation. We again plot the average episode length over 32 seeds, compared to the number of iterations as well as the number of interactions. Developing the REINFORCE algorithm with baseline. The outline of the blog is as follows: we first describe the environment and the shared model architecture. w=w+(Gt−wTst)st. Besides, the log basis did not seem to have a strong impact, but the most stable results were achieved with log 2. This is why we were unfortunately only able to test our methods on the CartPole environment. Vanilla Policy Gradient (VPG) expands upon the REINFORCE algorithm and improves some of its major issues. Reinforcement Learning is the mos… To tackle the problem of high variance in the vanilla REINFORCE algorithm, a baseline is subtracted from the obtained return while calculating the gradient. Policy Gradient Theorem 1. By contrast, Pigeon DRO8 showed clear evidence of symmetry: Its comparison-response rates were considerably higher on probe trials that reversed the symbolic baseline relations on which comparison responding was reinforced (positive trials) than on probe trials that reversed the symbolic baseline relations on which not-responding was reinforced (negative trials), F (1, 62) = … \end{aligned}∇θJ(πθ)=E[t=0∑T∇θlogπθ(at∣st)t′=t∑T(γt′rt′−b(st))]=E[t=0∑T∇θlogπθ(at∣st)t′=t∑Tγt′rt′−t=0∑T∇θlogπθ(at∣st)b(st)]=E[t=0∑T∇θlogπθ(at∣st)t′=t∑Tγt′rt′]−E[t=0∑T∇θlogπθ(at∣st)b(st)], We can also expand the second expectation term as, E[∑t=0T∇θlogπθ(at∣st)b(st)]=E[∇θlogπθ(a0∣s0)b(s0)+∇θlogπθ(a1∣s1)b(s1)+⋯+∇θlogπθ(aT∣sT)b(sT)]=E[∇θlogπθ(a0∣s0)b(s0)]+E[∇θlogπθ(a1∣s1)b(s1)]+⋯+E[∇θlogπθ(aT∣sT)b(sT)]\begin{aligned} Also, while most comparative studies focus on deterministic environments, we go one step further and analyze the relative strengths of the methods as we add stochasticity to our environment. The problem however is that the true value of a state can only be obtained by using an infinite number of samples. For comparison, here are the results without subtracting the baseline: We can see that there is definitely an improvement in the variance when subtracting a baseline. The network takes the state representation as input and has 3 hidden layers, all of them with a size of 128 neurons. Why? For an episodic problem, the Policy Gradient Theorem provides an analytical expression for the gradient of the objective function that needs to be optimized with respect to the parameters Î¸ of the network. more info Size SIZE GUIDE. &= \sum_s \mu\left(s\right) b\left(s\right) \nabla_\theta 1 \\ &= \sum_s \mu\left(s\right) b\left(s\right) \left(0\right) \\ It was soon discovered that subtracting a âbaselineâ from the return led to reduction in variance and allowed faster learning. \mathbb{E} \left[\sum_{t = 0}^T \nabla_\theta \log \pi_\theta \left(a_t \vert s_t \right) b\left(s_t\right) \right] &= \mathbb{E} \left[\nabla_\theta \log \pi_\theta \left(a_0 \vert s_0 \right) b\left(s_0\right) + \nabla_\theta \log \pi_\theta \left(a_1 \vert s_1 \right) b\left(s_1\right) + \cdots + \nabla_\theta \log \pi_\theta \left(a_T \vert s_T \right) b\left(s_T\right)\right] \\ This is also applied on all other plots of this blog. The results were slightly worse than for the sampled one which suggests that exploration is crucial in this environment. In terms of number of interactions, they are equally bad. Download source code. In my last post, I implemented REINFORCE which is a simple policy gradient algorithm. As before, we also plotted the 25th and 75th percentile. As in my previous posts, I will test the algorithm on the discrete-cart pole environment. Nevertheless, this improvement comes with the cost of increased number of interactions with the environment. E[t=0∑T∇θlogπθ(at∣st)b(st)]=0, ∇θJ(πθ)=E[∑t=0T∇θlogπθ(at∣st)∑t′=tT(γt′rt′−b(st))]=E[∑t=0T∇θlogπθ(at∣st)∑t′=tTγt′rt′]\begin{aligned} Comparing all baseline methods together we see a strong preference for REINFORCE with the sampled baseline as it already learns the optimal policy before 200 iterations. We have implemented the simplest case of learning a value function with weights w. A common way to do it is to use the observed return Gt as a âtargetâ of the learned value function.

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