In the case of a stochastic environment, however, using a learned value function would probably be preferable. We want to learn a policy, meaning we need to learn a function that maps states to a probability distribution over actions. import gym: import tensorflow as tf: import numpy as np: import itertools: import tensorflow. The source code for all our experiments can be found here: Rennie, S. J., Marcheret, E., Mroueh, Y., Ross, J., & Goel, V. (2017). 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} The goal is to keep the pendulum upright by applying a force of -1 or +1 (left or right) to the cart. Technically, any baseline would be appropriate as long as it does not depend on the actions taken. Furthermore, in the environment with added stochasticity, we observed that the learned value function clearly outperformed the sampled baseline. Kool, W., van Hoof, H., & Welling, M. (2019). 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. In my last post, I implemented REINFORCE which is a simple policy gradient algorithm. While the learned baseline already gives a considerable improvement over simple REINFORCE, it can still unlearn an optimal policy. 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} With enough motivation, let us now take a look at the Reinforcement Learning problem. 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. What if we subtracted some value from each number, say 400, 30, and 200? 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. I included the 12\frac{1}{2}21​ just to keep the math clean. 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). 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) = … However, all these conclusions only hold for the deterministic case, which is often not the case. We compare the performance against: The number of iterations needed to learn is a standard measure to evaluate. But in terms of which training curve is actually better, I am not too sure. But most importantly, this baseline results in lower variance, hence better learning of the optimal policy. Instead, the model with the learned baseline performs best. The figure shows that in terms of the number of interactions, sampling one rollout is the most efficient in reaching the optimal policy. 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. We saw that while the agent did learn, the high variance in the rewards inhibited the learning. more info Size SIZE GUIDE. In this post, I will discuss a technique that will help improve this. We test this by adding stochasticity over the actions in the CartPole environment. 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} Baseline Reinforced Support 7/8 Tight Black. This can be even achieved with a single sampled rollout. In terms of number of iterations, the sampled baseline is only slightly better than regular REINFORCE. One of the restrictions is that the environment needs to be duplicated because we need to sample different trajectories starting from the same state. \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] \\ # - REINFORCE algorithm with baseline # - Policy/value function approximation # # ---# @author Yiren Lu # @email luyiren [at] seas [dot] upenn [dot] edu # # MIT License: import gym: import numpy as np: import random: import tensorflow as tf: import tensorflow. ∇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} Because Gt is a sample of the true value function for the current policy, this is a reasonable target. The algorithm does get better over time as seen by the longer episode lengths. 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. REINFORCE with baseline. Performing a gridsearch over these parameters, we found the optimal learning rate to be 2e-3. However, more sophisticated baselines are possible. A state that yields a higher return will also have a high value function estimate, so we subtract a higher baseline. The number of interactions is (usually) closely related to the actual time learning takes. REINFORCE with Baseline Algorithm Initialize the actor μ (S) with random parameter values θμ. 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. where μ(s)\mu\left(s\right)μ(s) is the probability of being in state sss. RL based systems have now beaten world champions of Go, helped operate datacenters better and mastered a wide variety of Atari games. Several such baselines were proposed, each with its own set of advantages and disadvantages. layers as layers: from tqdm import trange: from gym. &= \sum_s \mu\left(s\right) b\left(s\right) \sum_a \nabla_\theta \pi_\theta \left(a \vert s \right) \\ By this, we prevent to punish the network for the last steps although it succeeded. However, the time required for the sampled baseline will get infeasible for tuning hyperparameters. Consider the set of numbers 500, 50, and 250. Note that the plot shows the moving average (width 25). 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 θ of the network. As before, we also plotted the 25th and 75th percentile. In my next post, we will discuss how to update the policy without having to sample an entire trajectory first. It was soon discovered that subtracting a ‘baseline’ from the return led to reduction in variance and allowed faster learning. The REINFORCE algorithm with baseline is mostly the same as the one used in my last post with the addition of the value function estimation and baseline subtraction. 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 … 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. This method, which we call the self-critic with sampled rollout, was described in Kool et al.³ The greedy rollout is actually just a special case of the sampled rollout if you consider only one sample being taken by always choosing the greedy action.