\(Z^{\pi_\text{old}}(s_t)\) is the partition function to normalize the distribution. Deterministic policy gradient (DPG) instead models the policy as a deterministic decision: \(a = \mu(s)\). The architecture of A3C versus A2C. 7): Fig. “Addressing Function Approximation Error in Actor-Critic Methods.” arXiv preprint arXiv:1802.09477 (2018). (Image source: Fujimoto et al., 2018). To reduce the high variance of the policy gradient \(\hat{g}\), ACER truncates the importance weights by a constant c, plus a correction term. Actually, in the DPG paper, the authors have shown that if the stochastic policy \(\pi_{\mu_\theta, \sigma}\) is re-parameterized by a deterministic policy \(\mu_\theta\) and a variation variable \(\sigma\), the stochastic policy is eventually equivalent to the deterministic case when \(\sigma=0\). Say, in the off-policy approach, the training trajectories are generated by a stochastic policy \(\beta(a \vert s)\) and thus the state distribution follows the corresponding discounted state density \(\rho^\beta\): Note that because the policy is deterministic, we only need \(Q^\mu(s, \mu_\theta(s))\) rather than \(\sum_a \pi(a \vert s) Q^\pi(s, a)\) as the estimated reward of a given state s. This policy gradient causes the parameters to move most in the direction that favors actions that has the highest return. When \(\alpha \rightarrow 0\), \(\theta\) is updated only according to the expected return \(J(\theta)\). As alluded to above, the goal of the policy is to maximize the total expected reward: Policy gradient methods have a number of benefits over other reinforcement learning methods. We can maximise the objective function J to maximises the return by adjusting the policy parameter θ to get the best policy. Hence, A3C is designed to work well for parallel training. The synchronized gradient update keeps the training more cohesive and potentially to make convergence faster. 9. When applying PPO on the network architecture with shared parameters for both policy (actor) and value (critic) functions, in addition to the clipped reward, the objective function is augmented with an error term on the value estimation (formula in red) and an entropy term (formula in blue) to encourage sufficient exploration. 2017. Return; or discounted future reward; \(G_t = \sum_{k=0}^{\infty} \gamma^k R_{t+k+1}\). Let’s use the state-value function as an example. One sentence summary is probably: “we first consider all combinations of parameters that result in a new network a constant KL divergence away from the old network. Out of all these possible combinations, we choose the one that minimizes our loss function.”. Many following algorithms were proposed to reduce the variance while keeping the bias unchanged. The state transition function involves all states, action and observation spaces \(\mathcal{T}: \mathcal{S} \times \mathcal{A}_1 \times \dots \mathcal{A}_N \mapsto \mathcal{S}\). This constant value can be viewed as the step size or learning rate. In either case, we can recover the following equation. To improve training stability, we should avoid parameter updates that change the policy too much at one step. The objective function of PPO takes the minimum one between the original value and the clipped version and therefore we lose the motivation for increasing the policy update to extremes for better rewards. We first start with the derivative of the state value function: This equation has a nice recursive form (see the red parts!) So we start the optimization from the last timestep \(T\): First, let us define the following functions: To solve the maximization optimization with inequality constraint, we can construct a Lagrangian expression with a Lagrange multiplier (also known as “dual variable”), \(\alpha_T\): Considering the case when we try to minimize \(L(\pi_T, \alpha_T)\) with respect to \(\alpha_T\) - given a particular value \(\pi_T\). Then plug in \(\pi_T^{*}\) and compute \(\alpha_T^{*}\) that minimizes \(L(\pi_T^{*}, \alpha_T)\). by Lilian Weng “Safe and efficient off-policy reinforcement learning” NIPS. Abstract: In this post, we are going to look deep into policy gradient, why it works, and many new policy gradient algorithms proposed in recent years: vanilla policy gradient, actor-critic, off-policy actor-critic, A3C, A2C, DPG, DDPG, D4PG, MADDPG, TRPO, PPO, ACER, ACTKR, SAC, TD3 & SVPG. When k = 0: \(\rho^\pi(s \to s, k=0) = 1\). Policy gradient is an approach to solve reinforcement learning problems. “Phasic Policy Gradient.” arXiv preprint arXiv:2009.04416 (2020). Fig. A3C builds up the foundation for ACER, but it is on policy; ACER is A3C’s off-policy counterpart. The twin-delayed deep deterministic policy gradient (TD3) algorithm is a model-free, online, off-policy reinforcement learning method. Policy Gradient Algorithms Ashwin Rao ICME, Stanford University Ashwin Rao (Stanford) Policy Gradient Algorithms 1/33. “Stein variational policy gradient.” arXiv preprint arXiv:1704.02399 (2017). Given that TRPO is relatively complicated and we still want to implement a similar constraint, proximal policy optimization (PPO) simplifies it by using a clipped surrogate objective while retaining similar performance. Policy gradient examples •Goals: •Understand policy gradient reinforcement learning •Understand practical considerations for policy gradients. The soft actor-critic algorithm. the coefficients of a complex polynomial or the weights and biases of units in a neural network) to parametrize this policy — π_θ​ (also written a π for brevity). In this paper we consider deterministic policy gradient algorithms for reinforcement learning with continuous actions. PG-PSOPE method. If you want to read more, check this. Sharing parameters between policy and value networks have pros and cons. In the first, the rows and columns of the Fisher are divided into groups, each of which corresponds to all the weights in a given layer, and this gives rise to a block-partitioning of the matrix. The objective function in an off-policy model measures the total advantage over the state visitation distribution and actions, while the mismatch between the training data distribution and the true policy state distribution is compensated by importance sampling estimator: where \(\theta_\text{old}\) is the policy parameters before the update and thus known to us; \(\rho^{\pi_{\theta_\text{old}}}\) is defined in the same way as above; \(\beta(a \vert s)\) is the behavior policy for collecting trajectories. A precedent work is Soft Q-learning. The value function parameter is therefore updated in the direction of: The policy parameter \(\phi\) is updated through policy gradient. If the above can be achieved, then 0 can usually be assured to converge to a locally optimal policy in the performance measure \(N_\pi\) is the number of policy update iterations in the policy phase. This week you will learn about these policy gradient methods, and their advantages over value-function based methods. By plugging it into the objective function \(J(\theta)\), we are getting the following: In the episodic case, the constant of proportionality (\(\sum_s \eta(s)\)) is the average length of an episode; in the continuing case, it is 1 (Sutton & Barto, 2017; Sec. It is usually intractable but does not contribute to the gradient. Discount factor; penalty to uncertainty of future rewards; \(0<\gamma \leq 1\). The algorithm of PPG. Recall how TD learning works for prediction: When the rollout is off policy, we need to apply importance sampling on the Q update: The product of importance weights looks pretty scary when we start imagining how it can cause super high variance and even explode. [Updated on 2019-06-26: Thanks to Chanseok, we have a version of this post in Korean]. Each agent’s stochastic policy only involves its own state and action: \(\pi_{\theta_i}: \mathcal{O}_i \times \mathcal{A}_i \mapsto [0, 1]\), a probability distribution over actions given its own observation, or a deterministic policy: \(\mu_{\theta_i}: \mathcal{O}_i \mapsto \mathcal{A}_i\). That means the RL agent sample from starting state to goal state directly from the environment, rather than bootstrapping compared to other methods such as Temporal Difference Learning and Dynamic programming. Because the policy \(\pi_t\) at time t has no effect on the policy at the earlier time step, \(\pi_{t-1}\), we can maximize the return at different steps backward in time — this is essentially DP. Each \(Q^\vec{\mu}_i\) is learned separately for \(i=1, \dots, N\) and therefore multiple agents can have arbitrary reward structures, including conflicting rewards in a competitive setting. Soft Actor-Critic (SAC) (Haarnoja et al. A widely used variation of REINFORCE is to subtract a baseline value from the return \(G_t\) to reduce the variance of gradient estimation while keeping the bias unchanged (Remember we always want to do this when possible). Reinforcement learning is probably the most general framework inwhich reward-related learning problems of animals, humans or machinecan be phrased. The \(n\)-step V-trace target is defined as: where the red part \(\delta_i V\) is a temporal difference for \(V\). This way, we can update the parameters θ in the direction of the gradient(Remember the gradient gives the direction of the maximum change, and the magnitude indicates the maximum rate of change ). Using gradient ascent, we can move \(\theta\) toward the direction suggested by the gradient \(\nabla_\theta J(\theta)\) to find the best \(\theta\) for \(\pi_\theta\) that produces the highest return. precisely PPO, to have separate training phases for policy and value functions. Policy Gradients. Then we go back to unroll the recursive representation of \(\nabla_\theta V^\pi(s)\)! To improve the convergence of the policy gradient algorithm… Markov Chain Monte Carlo Without all the Bullshit, Reinforcement Learning: An Introduction; 2nd Edition, “High-dimensional continuous control using generalized advantage estimation.”, “Asynchronous methods for deep reinforcement learning.”, “Deterministic policy gradient algorithms.”, “Continuous control with deep reinforcement learning.”, “Multi-agent actor-critic for mixed cooperative-competitive environments.”, “Sample efficient actor-critic with experience replay.”, “Safe and efficient off-policy reinforcement learning”, “Scalable trust-region method for deep reinforcement learning using Kronecker-factored approximation.”, “Going Deeper Into Reinforcement Learning: Fundamentals of Policy Gradients.”, “Notes on the Generalized Advantage Estimation Paper.”, “Distributed Distributional Deterministic Policy Gradients.”, “Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor.”, “Addressing Function Approximation Error in Actor-Critic Methods.”, “Soft Actor-Critic Algorithms and Applications.”, “Stein variational gradient descent: A general purpose bayesian inference algorithm.”, “IMPALA: Scalable Distributed Deep-RL with Importance Weighted Actor-Learner Architectures”, “Revisiting Design Choices in Proximal Policy Optimization.”, ← A (Long) Peek into Reinforcement Learning, Implementing Deep Reinforcement Learning Models with Tensorflow + OpenAI Gym →. This overestimation can propagate through the training iterations and negatively affect the policy. We study how the behavior of deep policy gradient algorithms reflects the conceptual framework motivating their development. A basic policy gradient algorithm making use of the above gradient is known as the Reinforce algorithm, and here is how it works: A Basic Reinforce Algorithm: Start with a random vector θ and repeat the following 3 steps until convergence: 1. or learn it off-policy-ly by following a different stochastic behavior policy to collect samples. [Updated on 2018-06-30: add two new policy gradient methods. The deterministic policy gradient update becomes: (2) \(N\)-step returns: When calculating the TD error, D4PG computes \(N\)-step TD target rather than one-step to incorporate rewards in more future steps. Lecture 7: Policy Gradient Finite Di erence Policy Gradient Policy Gradient Let J( ) be any policy objective function Policy gradient algorithms search for a local maximum in J( ) by ascending the gradient of the policy, w.r.t. The original DQN works in discrete space, and DDPG extends it to continuous space with the actor-critic framework while learning a deterministic policy. First, let’s denote the probability ratio between old and new policies as: Then, the objective function of TRPO (on policy) becomes: Without a limitation on the distance between \(\theta_\text{old}\) and \(\theta\), to maximize \(J^\text{TRPO} (\theta)\) would lead to instability with extremely large parameter updates and big policy ratios. By repeating this process, we can learn the optimal temperature parameter in every step by minimizing the same objective function: The final algorithm is same as SAC except for learning \(\alpha\) explicitly with respect to the objective \(J(\alpha)\) (see Fig. Transition probability of getting to the next state \(s'\) from the current state \(s\) with action \(a\) and reward \(r\). Assuming we have one neural network for policy and one network for temperature parameter, the iterative update process is more aligned with how we update network parameters during training. \(\Delta \theta\) on the search distribution space, \(\Delta \theta\) on the kernel function space (edited). One detail in the paper that is particularly useful in robotics is on how to normalize the different physical units of low dimensional features. Centralized critic + decentralized actors; Actors are able to use estimated policies of other agents for learning; Policy ensembling is good for reducing variance. The policy gradient algorithm 2. Basic variance reduction: baselines 5. where \(r_t + \gamma v_{t+1}\) is the estimated Q value, from which a state-dependent baseline \(V_\theta(s_t)\) is subtracted. While (\(s_t\) != TERMINAL) and \(t - t_\text{start} \leq t_\text{max}\): Pick the action \(A_t \sim \pi_{\theta'}(A_t \vert S_t)\) and receive a new reward \(R_t\) and a new state \(s_{t+1}\). [2] Richard S. Sutton and Andrew G. Barto. In the setup of maximum entropy policy optimization, \(\theta\) is considered as a random variable \(\theta \sim q(\theta)\) and the model is expected to learn this distribution \(q(\theta)\). I use \(\mu(. (1) Distributional Critic: The critic estimates the expected Q value as a random variable ~ a distribution \(Z_w\) parameterized by \(w\) and therefore \(Q_w(s, a) = \mathbb{E} Z_w(x, a)\). Thus,those systems need to be modeled as partially observableMarkov decision problems which oftenresults in ex… 2018); Note that in the original paper, the variable letters are chosen slightly differently from what in the post; i.e.