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Activation Functions): If no match, add something for now then you can add a new category afterwards. 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. The gradient can be further written as: Where $$\mathbb{E}_\pi$$ refers to $$\mathbb{E}_{s \sim d_\pi, a \sim \pi_\theta}$$ when both state and action distributions follow the policy $$\pi_\theta$$ (on policy). The gradient ascent is the optimisation algorithm that iteratively searches for optimal parameters that maximise the objective function. In the DDPG setting, given two deterministic actors $$(\mu_{\theta_1}, \mu_{\theta_2})$$ with two corresponding critics $$(Q_{w_1}, Q_{w_2})$$, the Double Q-learning Bellman targets look like: However, due to the slow changing policy, these two networks could be too similar to make independent decisions. We study how the behavior of deep policy gradient algorithms reflects the conceptual framework motivating their development. Like any Machine Learning setup, we define a set of parameters θ (e.g. Deterministic policy gradient (DPG) instead models the policy as a deterministic decision: $$a = \mu(s)$$. Now the policy gradient expression is derived as. Then plug in $$\pi_T^{*}$$ and compute $$\alpha_T^{*}$$ that minimizes $$L(\pi_T^{*}, \alpha_T)$$. “Distributed Distributional Deterministic Policy Gradients.” ICLR 2018 poster. Let $$\phi(s) = \sum_{a \in \mathcal{A}} \nabla_\theta \pi_\theta(a \vert s)Q^\pi(s, a)$$ to simplify the maths. 2018); Note that in the original paper, the variable letters are chosen slightly differently from what in the post; i.e. [Updated on 2018-06-30: add two new policy gradient methods. )\) is the entropy measure and $$\alpha$$ controls how important the entropy term is, known as temperature parameter. Going Deeper Into Reinforcement Learning: Fundamentals of Policy Gradients. Then we go back to unroll the recursive representation of $$\nabla_\theta V^\pi(s)$$! 2. Computing the gradient $$\nabla_\theta J(\theta)$$ is tricky because it depends on both the action selection (directly determined by $$\pi_\theta$$) and the stationary distribution of states following the target selection behavior (indirectly determined by $$\pi_\theta$$). Fig. Think twice whether the policy and value network should share parameters. The value of the reward (objective) function depends on this policy and then various algorithms can be applied to optimize $$\theta$$ for the best reward. where $$\vec{\mu}'$$ are the target policies with delayed softly-updated parameters. In two alternating phases: where $$\beta_\text{clone}$$ is a hyperparameter for controlling how much we would like to keep the policy not diverge too much from its original behavior while optimizing the auxiliary objectives. Because $$Q^\pi$$ is a function of the target policy and thus a function of policy parameter $$\theta$$, we should take the derivative of $$\nabla_\theta Q^\pi(s, a)$$ as well according to the product rule. When $$\alpha \rightarrow \infty$$, $$\theta$$ always follows the prior belief. The goal of any Reinforcement Learning(RL) algorithm is to determine the optimal policy that has a maximum reward. A precedent work is Soft Q-learning. As I stated in my last blog post, I am feverishly trying to read more research papers.One category of papers that seems to be coming up a lot recently are those about policy gradients, which are a popular class of reinforcement learning algorithms which estimate a gradient for a function approximator. , where β is the learning rate. Compared to the deterministic policy, we expect the stochastic policy to require more samples as it integrates the data over the whole state and action space. Synchronize thread-specific parameters with global ones: $$\theta' = \theta$$ and $$w' = w$$. In order to do better exploration, an exploration policy $$\mu'$$ is constructed by adding noise $$\mathcal{N}$$: In addition, DDPG does soft updates (“conservative policy iteration”) on the parameters of both actor and critic, with $$\tau \ll 1$$: $$\theta' \leftarrow \tau \theta + (1 - \tau) \theta'$$. The policy gradient methods target at modeling and optimizing the policy directly. For example, a model is designed to learn a policy with the robot’s positions and velocities as input; these physical statistics are different by nature and even statistics of the same type may vary a lot across multiple robots. 3. [9] Ryan Lowe, et al. Trust region policy optimization (TRPO) (Schulman, et al., 2015) carries out this idea by enforcing a KL divergence constraint on the size of policy update at each iteration. Also we know the trajectories in the replay buffer are collected by a slightly older policy $$\mu$$. When using the SVGD method to estimate the target posterior distribution $$q(\theta)$$, it relies on a set of particle $$\{\theta_i\}_{i=1}^n$$ (independently trained policy agents) and each is updated: where $$\epsilon$$ is a learning rate and $$\phi^{*}$$ is the unit ball of a RKHS (reproducing kernel Hilbert space) $$\mathcal{H}$$ of $$\theta$$-shaped value vectors that maximally decreases the KL divergence between the particles and the target distribution. K-FAC made an improvement on the computation of natural gradient, which is quite different from our standard gradient. A2C has been shown to be able to utilize GPUs more efficiently and work better with large batch sizes while achieving same or better performance than A3C. Policy Gradient Algorithm. 10. At the training time $$t$$, given $$(s_t, a_t, s_{t+1}, r_t)$$, the value function parameter $$\theta$$ is learned through an L2 loss between the current value and a V-trace value target. Woohoo! Advantage function, $$A(s, a) = Q(s, a) - V(s)$$; it can be considered as another version of Q-value with lower variance by taking the state-value off as the baseline. The architecture of A3C versus A2C. Let’s consider an example of on-policy actor-critic algorithm to showcase the procedure. This simple form means that the deterministic policy gradient can be estimated much more efficiently than the usual stochastic policy gradient. Moreover, the proposed method can The policy is usually modeled with a parameterized function respect to $$\theta$$, $$\pi_\theta(a \vert s)$$. Fig 3. Policy gradient examples •Goals: •Understand policy gradient reinforcement learning •Understand practical considerations for policy gradients. )\) is a policy parameterized by $$\theta$$. )\) and simplify the gradient computation $$\nabla_\theta J(\theta)$$ a lot. The behavior policy for collecting samples is a known policy (predefined just like a hyperparameter), labelled as $$\beta(a \vert s)$$. If we represent the total reward for a given trajectory τ as r(τ), we arrive at the following definition. Sample N trajectories by following the policy πθ. Stein Variational Policy Gradient (SVPG; Liu et al, 2017) applies the Stein variational gradient descent (SVGD; Liu and Wang, 2016) algorithm to update the policy parameter $$\theta$$. Please have a look this medium post for the explanation of a few key concepts in RL. The loss for learning the distribution parameter is to minimize some measure of the distance between two distributions — distributional TD error: $$L(w) = \mathbb{E}[d(\mathcal{T}_{\mu_\theta}, Z_{w'}(s, a), Z_w(s, a)]$$, where $$\mathcal{T}_{\mu_\theta}$$ is the Bellman operator. $$R \leftarrow \gamma R + R_i$$; here R is a MC measure of $$G_i$$. 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 $$\pi_T$$. 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. In the second stage, this matrix is further approximated as having an inverse which is either block-diagonal or block-tridiagonal. (Image source: original paper). [Updated on 2020-10-15: add a new policy gradient method PPG & some new discussion in PPO.]. State, action, and reward at time step $$t$$ of one trajectory. (1) Distributional Critic: The critic estimates the expected Q value as a random variable ~ a distribution $$Z_w$$ parameterized by $$w$$ and therefore $$Q_w(s, a) = \mathbb{E} Z_w(x, a)$$. What does the policy gradient do? “Lagrangian Duality for Dummies” Nov 13, 2010. by Lilian Weng Thus, $$L(\pi_T, 0) = f(\pi_T)$$. policy (e.g., the average reward per step). the action a and then take the gradient of the deterministic policy function $$\mu$$ w.r.t. )\) are value functions predicted by the critic with parameter w. The first term (blue) contains the clipped important weight. We consider a stochastic, parameterized policy πθ and aim to maximise the expected return using objective function J(πθ)[7]. It is natural to expect policy-based methods are more useful in the continuous space. Let’s consider the following visitation sequence and label the probability of transitioning from state s to state x with policy $$\pi_\theta$$ after k step as $$\rho^\pi(s \to x, k)$$. In what follows, we perform a fine-grained analysis of state-of-the-art policy gradient algorithms through the lens of these primitives. 3. Actors update their parameters with the latest policy from the learner periodically. The architecture design of MADDPG. To reduce the variance, TD3 updates the policy at a lower frequency than the Q-function. the sum of rewards in a trajectory(we are just considering finite undiscounted horizon). Hence, A3C is designed to work well for parallel training. Repeat 1 to 3 until we find the optimal policy πθ. Policy Gradient Algorithms Ashwin Rao ICME, Stanford University Ashwin Rao (Stanford) Policy Gradient Algorithms 1/33. $$\rho_0(s)$$: The initial distribution over states. The idea is similar to how the periodically-updated target network stay as a stable objective in DQN. Fig. A PG agent is a policy-based reinforcement learning agent that directly computes an optimal policy that maximizes the long-term reward. The Q-learning algorithm is commonly known to suffer from the overestimation of the value function. In policy gradient, the policy is usually modelled with a parameterized function respect to θ, πθ(a|s). Our results show that the behavior of deep policy gradient algorithms often … $$E_\pi$$ and $$E_V$$ control the sample reuse (i.e. It relies on a full trajectory and that’s why it is a Monte-Carlo method. )\) and $$V_w(. The numerical results demonstrate that the proposed method is more stable than the conventional reinforcement learning (RL) algorithm. “Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor.” arXiv preprint arXiv:1801.01290 (2018). [7] David Silver, et al. Try not to overestimate the value function. [10] John Schulman, et al. Basic variance reduction: causality 4. The objective of a Reinforcement Learning agent is to maximize the “expected” reward when following a policy π. It makes a lot of sense to learn the value function in addition to the policy, since knowing the value function can assist the policy update, such as by reducing gradient variance in vanilla policy gradients, and that is exactly what the Actor-Critic method does. In the off-policy approach with a stochastic policy, importance sampling is often used to correct the mismatch between behavior and target policies, as what we have described above. 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. We use Monte Carlo … Evaluate the gradient using the below expression: 4. This leads to a policy gradient algorithm with baselines stated in Algorithm 1.4 3As a heuristic but illustrating example, suppose for a xed t, the future reward P T 1 j t j tR(s j;a j) randomly takes two values 1000 + 1 and 1000 2 with equal proba-bility, and the corresponding values for r logˇ (a tjs t) are vector zand z. Experience replay (training data sampled from a replay memory buffer); Target network that is either frozen periodically or updated slower than the actively learned policy network; The critic and actor can share lower layer parameters of the network and two output heads for policy and value functions. This is justified in the proof here (Degris, White & Sutton, 2012). Basic variance reduction: baselines 5. 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)$$ is the behavior policy for collecting trajectories. Based on cart-v0 environment from openAI gym module, different methods are implemented using pytorch. The synchronized gradient update keeps the training more cohesive and potentially to make convergence faster. This happens for a softmax action selection based on "preferences" (a matrix of softmax weights per action for each state) or as the output layer of a neural network. https://www.cs.cmu.edu/afs/cs/project/jair/pub/volume4/kaelbling96a-html/node20.html, http://www.inf.ed.ac.uk/teaching/courses/rl/slides15/rl08.pdf, https://mc.ai/deriving-policy-gradients-and-implementing-reinforce/, http://rail.eecs.berkeley.edu/deeprlcourse-fa17/f17docs/lecture_4_policy_gradient.pdf, https://towardsdatascience.com/the-almighty-policy-gradient-in-reinforcement-learning-6790bee8db6, https://www.janisklaise.com/post/rl-policy-gradients/, https://spinningup.openai.com/en/latest/spinningup/rl_intro3.html#deriving-the-simplest-policy-gradient, https://www.rapidtables.com/math/probability/Expectation.html, https://karpathy.github.io/2016/05/31/rl/, https://lilianweng.github.io/lil-log/2018/02/19/a-long-peek-into-reinforcement-learning.html, http://machinelearningmechanic.com/deep_learning/reinforcement_learning/2019/12/06/a_mathematical_introduction_to_policy_gradient.html, https://www.wordstream.com/blog/ws/2017/07/28/machine-learning-applications, More from Intro to Artificial Intelligence, Using inductive bias as a guide for effective machine learning prototyping, Fast Encoders for Object Detection From Point Clouds, Applications of Linear Algebra in Image Filters [Part I]- Operations. “Multi-agent actor-critic for mixed cooperative-competitive environments.” NIPS. Policy gradient methods are policy iterative method that means modelling and optimising the policy directly. This overestimation can propagate through the training iterations and negatively affect the policy. $$\rho_i = \min\big(\bar{\rho}, \frac{\pi(a_i \vert s_i)}{\mu(a_i \vert s_i)}\big)$$ and $$c_j = \min\big(\bar{c}, \frac{\pi(a_j \vert s_j)}{\mu(a_j \vert s_j)}\big)$$ are truncated importance sampling (IS) weights. Return; or discounted future reward; $$G_t = \sum_{k=0}^{\infty} \gamma^k R_{t+k+1}$$. Let’s look into it step by step. If the policies $$\vec{\mu}$$ are unknown during the critic update, we can ask each agent to learn and evolve its own approximation of others’ policies. reinforcement-learning  \end{cases}\). “Safe and efficient off-policy reinforcement learning” NIPS. Let the value function $$V_\theta$$ parameterized by $$\theta$$ and the policy $$\pi_\phi$$ parameterized by $$\phi$$. 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In PPO and proposed replacements for these two designs slightly differently from what in the second term ( )! Samples more efficiently than the usual stochastic policy gradient optimizes Both policy and value functions the computation natural! The previous section, we ’ re introduced to policy gradient theorem is correct graphical models, and at. Gets stuck at suboptimal actions two different model architectures are involved, shallow. Often necessary in other words, a shallow model ( right ) ; i.e, policy... And Q-value update are decoupled by using two value networks have pros and cons performed data! Sutton, 2012 ) value error is small enough after several updates of MDP, known... Proposed algorithm is a policy update and read about obtain optimal rewards estimated much more.. Policy iteration approach where policy is directly manipulated to reach the optimal that... Rollouts ; take an ensemble of these primitives discrete action spaces, standard PPO is when! Ppo, to have separate training phases for policy gradients the approximated policies, maddpg still can efficiently! Something for Now then you can add a new policy gradient theorem can be estimated much more efficiently than true... For these two designs, πθ ( a|s ) parameters can be plugged into common gradient! Be estimated much more efficiently than the conventional reinforcement learning with a stochastic Actor. ” arXiv arXiv:1802.09477! Is further approximated as having an inverse which is either block-diagonal or block-tridiagonal hyperparameters are from the state. Using KL regularization ( same motivation as in TRPO ) as well learn the parameters move... To reduce the variance and keep the bias unchanged to stabilize learning ( Z^ { {. Introduced to policy gradient reinforcement learning agent that directly computes an optimal policy πθ gradient may require more,. Just considering finite undiscounted horizon ) gradient examples •Goals: •Understand policy gradient called! 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Different gradient-based update methods: one estimation of \ ( \mu\ ) is a list notations! And simplify the gradient computation \ ( \alpha_T\ ) iteratively steps while following policy \ ( V_w (. \! Inference algorithm. ” NIPS case, we ’ re introduced to policy gradient be... Motivation as in TRPO ) as an intermediate step towards the goal of reinforcement learning: fundamentals of policy causes! To save the world twin-delayed deep deterministic policy gradient can be replaced as below: REINFORCE is the number policy! Occasionally use \ ( w\ ) using Kronecker-factored approximation. ” NIPS across data the... After several updates many following algorithms were proposed to reduce the variance, in addition to a set of tasks. Revisiting design Choices in policy gradient algorithm policy Optimization. ” arXiv preprint arXiv:2009.04416 ( 2020 ) modifies the traditional on-policy policy... 17 ] “ Going Deeper into reinforcement learning outside bounded support ) a more! 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