Shop leggings, sports bras, shorts, gym tops and more. \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 technique, called whitening is often necessary for good optimization, especially in the deep learning setting. The major issue with REINFORCE is that it has high variance. The state is described by a vector of size 4, containing the position and velocity of the cart as well as the angle and velocity of the pole. Please correct me in the comments if you see any mistakes. This is why we were unfortunately only able to test our methods on the CartPole environment. &=\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'} - \sum_{t = 0}^T \nabla_\theta \log \pi_\theta \left(a_t \vert s_t \right) b\left(s_t\right) \right] \\ ∇θ​J(πθ​)=E[t=0∑T​∇θ​logπθ​(at​∣st​)t′=t∑T​γt′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∇θlog⁡πθ(at∣st)∑t′=tT(γt′rt′−b(st))]=E[∑t=0T∇θlog⁡πθ(at∣st)∑t′=tTγt′rt′−∑t=0T∇θlog⁡πθ(at∣st)b(st)]=E[∑t=0T∇θlog⁡πθ(at∣st)∑t′=tTγt′rt′]−E[∑t=0T∇θlog⁡πθ(at∣st)b(st)]\begin{aligned} The problem however is that the true value of a state can only be obtained by using an infinite number of samples. The environment consists of an upright pendulum joint to a cart. Ever since DeepMind published its work on AlphaGo, reinforcement learning has become one of the âcoolestâ domains in artificial intelligence. ##Performance of Reinforce trained on CartPole ##Average Performance of Reinforce for multiple runs ##Comparison of subtracting a learned baseline from the return vs. using return whitening … Nevertheless, this improvement comes with the cost of increased number of interactions with the environment. After hyperparameter tuning, we evaluate how fast each method learns a good policy. 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. 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. 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. Switch branch/tag. 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. In this way, if the obtained return is much better than the expected return, the gradients are stronger and vice-versa. REINFORCE with Baseline. Thus,those systems need to be modeled as partially observableMarkov decision problems which o… The number of interactions is (usually) closely related to the actual time learning takes. In a stochastic environment, the sampled baseline would thus be more noisy. 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. Achetez et téléchargez ebook Reinforced Carbon Carbon (RCC) oxidation resistant material samples - Baseline coated, and baseline coated with tetraethyl orthosilicate (TEOS) impregnation (English Edition): Boutique Kindle - Science : Amazon.fr The learned baseline apparently suffers less from the introduced stochasticity. High variance gradients leads to unstable learning updates, slow convergence and thus slow learning of the optimal policy. But what is b(st)b\left(s_t\right)b(st​)? 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 … However, the fact that we want to test the sampled baseline restricts our choice. That is, it is not used for bootstrapping (updating the value estimate for a state from the estimated values of subsequent states), but only as a baseline for the state whose … The critic is a state-value function. Also, it is a very classic example in reinforcement learning literature. 13.4 REINFORCE with Baseline. Implementation of One-Step Actor-Critic algorithm, we revisit Cliff Walking environment and show that Actor-Critic can learn the optimal … However, when we look at the number of interactions with the environment, REINFORCE with a learned baseline and sampled baseline have similar performance. &= \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] One slight difference here is versus my previous implementation is that I’m implementing REINFORCE with a baseline value and using the mean of the returns as my baseline. 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. w=w+(Gt​−wTst​)st​. 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 … In my last post, I implemented REINFORCE which is a simple policy gradient algorithm. &= \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] reinforcement-learning / PolicyGradient / CliffWalk REINFORCE with Baseline Solution.ipynb Go to file Go to file T; Go to line L; Copy path guotong1988 Update CliffWalk REINFORCE with Baseline Solution.ipynb. Several such baselines were proposed, each with its own set of advantages and disadvantages. This can be improved by subtracting a baseline value from the Q values. 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. \mathbb{E} \left[\nabla_\theta \log \pi_\theta \left(a_0 \vert s_0 \right) b\left(s_0\right) \right] &= \sum_s \mu\left(s\right) \sum_a \pi_\theta \left(a \vert s\right) \nabla_\theta \log \pi_\theta \left(a \vert s \right) b\left(s\right) \\ REINFORCE with Baseline Algorithm Initialize the actor μ (S) with random parameter values θμ. If the current policy cannot reach the goal, the rollouts will also not reach the goal. 89.95. Reinforce With Baseline in PyTorch. 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. Consider the set of numbers 500, 50, and 250. Then, ∇wV^(st,w)=st\nabla_w \hat{V} \left(s_t,w\right) = s_t The REINFORCE algorithm takes the Monte Carlo approach to estimate the above gradient elegantly. The algorithm involved generating a complete episode and using the return (sum of rewards) obtained in calculating the gradient. It turns out that the answer is no, and below is the proof. We optimize hyperparameters for the different approaches by running a grid search over the learning rate and approach-specific hyperparameters. There has never been a better time for enterprises to harness its power, nor has the … V^(st​,w)=wTst​. It learned the optimal policy with the least number of interactions, with the least variation between seeds. This output is used as the baseline and represents the learned value. 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 … Vanilla Policy Gradient (VPG) expands upon the REINFORCE algorithm and improves some of its major issues. Mark Saad in Reinforcement Learning with MATLAB 29 Nov • 6 min read. But assuming no mistakes, we will continue. This can be a big advantage as we still have unbiased estimates although parts of the state space is not observable. To tackle the problem of high variance in the vanilla REINFORCE algorithm, a baseline is subtracted from the obtained return while calculating the gradient. We output log probabilities of the actions by using the LogSoftmax as the final activation function. 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 have seen that using a baseline greatly increases the stability and speed of policy learning with REINFORCE. For example, for the LunarLander environment, a single run for the sampled baseline takes over 1 hour. This system is unstable, which causes the pendulum to fall over. 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. New campaign to reinforce hygiene practices in dorms Programme aims to keep at bay fresh mass virus outbreaks among migrant workers. It can be shown that introduction of the baseline still leads to an unbiased estimate (see for example this blog). \nabla_\theta J\left(\pi_\theta\right) &= \mathbb{E} \left[\sum_{t = 0}^T \nabla_\theta \log \pi_\theta \left(a_t \vert s_t \right) \sum_{t' = t}^T \left(\gamma^{t'} r_{t'} - b\left(s_t\right)\right) \right] \\ This is similar to adding randomness to the next state we end up in: we sometimes end up in another state than expected for a certain action. Our main new result is to show that the gradient can be written in a form suitable for estimation from experience aided by an approximate action-value or advantage function. … We again plot the average episode length over 32 seeds, compared to the number of iterations as well as the number of interactions. 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. 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. We do not use V in G. G is only the reward to go for every step in … 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. To reduce variance of the gradient, they subtract 'baseline' from sum of future rewards for all time steps. &= -\delta \nabla_w \hat{V} \left(s_t,w\right) But wouldn’t subtracting a random number from the returns result in incorrect, biased data? δ=Gt​−V^(st​,w), If we square this and calculate the gradient, we get, ∇w[12(Gt−V^(st,w))2]=−(Gt−V^(st,w))∇wV^(st,w)=−δ∇wV^(st,w)\begin{aligned} We can update the parameters of V^\hat{V}V^ using stochastic gradient. 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 … Latest commit b2d179a Jun 11, 2019 History. While most papers use these baselines in specific settings, we are interested in comparing their performance on the same task. The results on the CartPole environment are shown in the following figure. contrib. The REINFORCE with Baseline algorithm becomes. We saw that while the agent did learn, the high variance in the rewards inhibited the learning. However, all these conclusions only hold for the deterministic case, which is often not the case. Instead, the model with the learned baseline performs best. To conclude, in a simple, (relatively) deterministic environment we definitely expect the sampled baseline to be a good choice. E[t=0∑T​∇θ​logπθ​(at​∣st​)b(st​)]=(T+1)E[∇θ​logπθ​(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. more info Size SIZE GUIDE. Now the estimated baseline is the average of the rollouts including the main trajectory (and excluding the jâth rollout). Please let me know in the comments if you find any bugs. We compare the performance against: The number of iterations needed to learn is a standard measure to evaluate. 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 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 Although the REINFORCE-with-baseline method learns both a policy and a state-value function, we do not consider it to be an actor–critic method because its state-value function is used only as a baseline, not as a critic. The easy way to go is scaling the returns using the mean and standard deviation. But most importantly, this baseline results in lower variance, hence better learning of the optimal policy. Another problem is that the sampled baseline does not work for environments where we rarely reach a goal (for example the MountainCar problem). REINFORCE method and actor-critic methods are examples of this approach. Note that as we only have to actions, it means in p/2% of the cases, we take a wrong action. &= \sum_s \mu\left(s\right) b\left(s\right) \nabla_\theta \sum_a \pi_\theta \left(a \vert s \right) \\ Of course, there is always room for improvement. \end{aligned}∇w​[21​(Gt​−V^(st​,w))2]​=−(Gt​−V^(st​,w))∇w​V^(st​,w)=−δ∇w​V^(st​,w)​. As before, we also plotted the 25th and 75th percentile. layers as layers: from tqdm import trange: from gym. 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. The division by stepCt could be absorbed into the learning rate. And if none of the rollouts reach the goal, this means that all returns will be the same, and thus the gradient will be zero. So I am not sure if the above results are accurate, or if there is some subtle mistake that I made. In this post, I will discuss a technique that will help improve this. Stochastic Beams and Where to Find Them: The Gumbel-Top-k Trick for Sampling Sequences Without Replacement. However, the method suffers from high variance in the gradients, which results in slow unstable learning and a lot of frustrationâ¦. The following methods show two ways to estimate this expected return of the state under the current policy. Hyperparameter tuning leads to an optimal learning rates of Î±=2e-4 and Î²=2e-5 . Attention, Learn to Solve Routing Problems!. This will allow us to update the policy during the episode as opposed to after which should allow for faster training. Policy Gradient Theorem 1. For this implementation we use the average reward as our baseline. 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. Mark Saad in Reinforcement Learning with MATLAB 28 Nov • 7 min read. 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. Therefore, E[∑t=0T∇θlog⁡πθ(at∣st)b(st)]=0\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] = 0 reinforce_with_baseline.py import gym: import tensorflow as tf: import numpy as np: import itertools: import tensorflow. All together, this suggests that for a (mostly) deterministic environment, a sampled baseline reduces the variance of REINFORCE the best. Why does Java have support for time zone offsets with seconds precision? Reinforcement Learning is the mos… 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. The outline of the blog is as follows: we first describe the environment and the shared model architecture. 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. &= \sum_s \mu\left(s\right) b\left(s\right) \sum_a \nabla_\theta \pi_\theta \left(a \vert s \right) \\ LMMâââNeural 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-CNNâs 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. 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. Implementation of REINFORCE with Baseline algorithm, recreation of figure 13.4 and demonstration on Corridor with switched actions environment. 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. Interestingly, by sampling multiple rollouts, we could also update the parameters on the basis of the jâth rollout. Reinforcement learning is probably the most general framework inwhich reward-related learning problems of animals, humans or machinecan be phrased. Stochasticity seems to make the sampled beams too noisy to serve as a good baseline. they applied REINFORCE algorithm to train RNN. Note that the plot shows the moving average (width 25). Furthermore, in the environment with added stochasticity, we observed that the learned value function clearly outperformed the sampled baseline. RL based systems have now beaten world champions of Go, helped operate datacenters better and mastered a wide variety of Atari games. To find out when the stochasticity makes a difference, we test choosing random actions with 10%, 20% and 40% chance. Nevertheless, there is a subtle difference between the two methods when the optimum has been reached (i.e. Policy gradient is an approach to solve reinforcement learning problems. The optimal learning rate found by gridsearch over 5 different rates is 1e-4. However, taking more rollouts leads to more stable learning. Self-critical sequence training for image captioning. The results with different number of rollouts (beams) are shown in the next figure. 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. In. Now, we will implement this to help make things more concrete. Then we will show results for all different baselines on the deterministic environment. Find file Select Archive Format. With advancements in deep learning, these algorithms proved very successful using powerful networks as function approximators. BUY 4 REINFORCE SAMPLES, GET A BASELINE FOR FREE! But we also need a way to approximate V^\hat{V}V^. So far, we have tested our different baselines on a deterministic environment: if we do some action in some state, we always end up in the same next state. The average of returns from these plays could serve as a baseline. Code: REINFORCE with Baseline. REINFORCE with a Baseline. A not yet explored benefit of sampled baseline might be for partially observable environments. This is called whitening. Kool, W., van Hoof, H., & Welling, M. (2018). E[t=0∑T​∇θ​logπθ​(at​∣st​)b(st​)]=0, ∇θJ(πθ)=E[∑t=0T∇θlog⁡πθ(at∣st)∑t′=tT(γt′rt′−b(st))]=E[∑t=0T∇θlog⁡πθ(at∣st)∑t′=tTγt′rt′]\begin{aligned} 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! If we learn a value function that (approximately) maps a state to its value, it can be used as a baseline. \nabla_\theta J\left(\pi_\theta\right) &= \mathbb{E} \left[\sum_{t = 0}^T \nabla_\theta \log \pi_\theta \left(a_t \vert s_t \right) \sum_{t' = t}^T \left(\gamma^{t'} r_{t'} - b\left(s_t\right)\right) \right] \\ If we are learning a policy, why not learn a value function simultaneously? This means that cumulative reward of the last step is the reward plus the discounted, estimated value of the final state, similarly to what is done in A3C. This indicates that both methods provide a proper baseline for stable learning. Note that I update both the policy and value function parameters once per trajectory. 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). where www and sts_tst​ are 4×14 \times 14×1 column vectors. … or make 4 interest-free payments of22.48 AUD fortnightly with. Thus, we want to sample more frequently the closer we get to the end. This is considerably higher than for the previous two methods, suggesting that the sampled baseline give a much lower variance for the CartPole environment. We test this by adding stochasticity over the actions in the CartPole environment. However, more sophisticated baselines are possible. In my last post, I implemented REINFORCE which is a simple policy gradient algorithm. The network takes the state representation as input and has 3 hidden layers, all of them with a size of 128 neurons. 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. Using the definition of expectation, we can rewrite the expectation term on the RHS as, E[∇θlog⁡πθ(a0∣s0)b(s0)]=∑sμ(s)∑aπθ(a∣s)∇θlog⁡πθ(a∣s)b(s)=∑sμ(s)∑aπθ(a∣s)∇θπθ(a∣s)πθ(a∣s)b(s)=∑sμ(s)b(s)∑a∇θπθ(a∣s)=∑sμ(s)b(s)∇θ∑aπθ(a∣s)=∑sμ(s)b(s)∇θ1=∑sμ(s)b(s)(0)=0\begin{aligned} The experiments of 20% have shown to be at a tipping point. Likewise, we substract a lower baseline for states with lower returns. 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. \end{aligned}∇θ​J(πθ​)​=E[t=0∑T​∇θ​logπθ​(at​∣st​)t′=t∑T​(γt′rt′​−b(st​))]=E[t=0∑T​∇θ​logπθ​(at​∣st​)t′=t∑T​γt′rt′​−t=0∑T​∇θ​logπθ​(at​∣st​)b(st​)]=E[t=0∑T​∇θ​logπθ​(at​∣st​)t′=t∑T​γt′rt′​]−E[t=0∑T​∇θ​logπθ​(at​∣st​)b(st​)]​, We can also expand the second expectation term as, E[∑t=0T∇θlog⁡πθ(at∣st)b(st)]=E[∇θlog⁡πθ(a0∣s0)b(s0)+∇θlog⁡πθ(a1∣s1)b(s1)+⋯+∇θlog⁡πθ(aT∣sT)b(sT)]=E[∇θlog⁡πθ(a0∣s0)b(s0)]+E[∇θlog⁡πθ(a1∣s1)b(s1)]+⋯+E[∇θlog⁡πθ(aT∣sT)b(sT)]\begin{aligned} episode length of 500). We saw that while the agent did learn, the high variance in the rewards inhibited the learning. In the deterministic CartPole environment, using a sampled self-critic baseline gives good results, even using only one sample. While the learned baseline already gives a considerable improvement over simple REINFORCE, it can still unlearn an optimal policy. By this, we prevent to punish the network for the last steps although it succeeded. We would like to have tested on more environments. Buy 4 REINFORCE Samples, Get a Baseline for Free! 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. 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. Sign in with GitHub … We work with this particular environment because it is easy to manipulate, analyze and fast to train. &= \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) \\ By executing a full trajectory, you would know its true reward. where μ(s)\mu\left(s\right)μ(s) is the probability of being in state sss. \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) \\ On the other hand, the learned baseline has not converged when the policy reaches the optimum because the value estimate is still behind. By Phillip Lippe, Rick Halm, Nithin Holla and Lotta Meijerink. I am just a lowly mechanical engineer (on paper, not sure what I am in practice). In the REINFORCE algorithm, Monte Carlo plays out the whole trajectory in an episode that is used to update the policy afterward. However, the most suitable baseline is the true value of a state for the current policy. -REINFORCE with baseline → we use (G-mean (G))/std (G) or (G-V) as gradient rescaler. We see that the learned baseline reduces the variance by a great deal, and the optimal policy is learned much faster. Because Gt is a sample of the true value function for the current policy, this is a reasonable target. However, the stochastic policy may take different actions at the same state in different episodes. As in my previous posts, I will test the algorithm on the discrete-cart pole environment. REINFORCE with Baseline Policy Gradient Algorithm. This can be even achieved with a single sampled rollout. reinforce-with-baseline. Developing the REINFORCE algorithm with baseline. This method more efficiently uses the information obtained from the interactions with the environmentâ´. This is what we will do in this blog by experimenting with the following baselines for REINFORCE: We will go into detail for each of these methods later in the blog, but here is already a sneak peek of our models we test out. However, the policy gradient estimate requires every time step of the trajectory to be calculated, while the value function gradient estimate requires only one time step to be calculated. 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. We use ELU activation and layer normalization between the hidden layers. This is what is done in state-of-the-art policy gradient methods like A3C.

## reinforce with baseline

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