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policy gradient theorem

[14] kvfrans.com A intuitive explanation of natural gradient descent. “IMPALA: Scalable Distributed Deep-RL with Importance Weighted Actor-Learner Architectures” arXiv preprint 1802.01561 (2018). To reduce the variance, TD3 updates the policy at a lower frequency than the Q-function. (Image source: Lillicrap, et al., 2015), [paper|code (Search “github d4pg” and you will see a few.)]. “Phasic Policy Gradient.” arXiv preprint arXiv:2009.04416 (2020). I listed ACTKR here mainly for the completeness of this post, but I would not dive into details, as it involves a lot of theoretical knowledge on natural gradient and optimization methods. Policy Gradient Theorem Now hopefully we have a clear setup. Note that the regularity conditions A.1 imply that V (s) and r V (s) are continuous functions of and sand the compactness of Sfurther implies that for any , jjr V (s)jj, jjr aQ (s;a)j a= 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. To resolve the inconsistency, a coordinator in A2C waits for all the parallel actors to finish their work before updating the global parameters and then in the next iteration parallel actors starts from the same policy. \(\bar{\rho}\) impacts the fixed-point of the value function we converge to and \(\bar{c}\) impacts the speed of convergence. \(E_\pi\) and \(E_V\) control the sample reuse (i.e. In order to scale up RL training to achieve a very high throughput, IMPALA (“Importance Weighted Actor-Learner Architecture”) framework decouples acting from learning on top of basic actor-critic setup and learns from all experience trajectories with V-trace off-policy correction. Fig 3. How-ever, almost all modern policy gradient algorithms deviate from the original theorem by dropping one of the two instances of the discount factor that appears in the theorem. Tons of policy gradient algorithms have been proposed during recent years and there is no way for me to exhaust them. A2C is a synchronous, deterministic version of A3C; that’s why it is named as “A2C” with the first “A” (“asynchronous”) removed. 本篇blog作为一个引子，介绍下Policy Gradient的基本思想。那么大家会发现，如何确定这个评价指标才是实现Policy Gradient方法的关键所在。所以，在下一篇文章中。我们将来分析一下这个评价指标的问题。 Therefore, to maximize \(f(\pi_T)\), the dual problem is listed as below. It goes without being said that we also need to update the parameters ω of the critic. 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. One issue that these algorithms must ad- dress is how to estimate the action-value function Qˇ(s;a). The value function parameter is therefore updated in the direction of: The policy parameter \(\phi\) is updated through policy gradient. \(\theta'\): \(d\theta \leftarrow d\theta + \nabla_{\theta'} \log \pi_{\theta'}(a_i \vert s_i)(R - V_{w'}(s_i))\); Update asynchronously \(\theta\) using \(\mathrm{d}\theta\), and \(w\) using \(\mathrm{d}w\). 2. This concludes the derivation of the Policy Gradient Theorem for entire trajectories. Fig. )\), the value of (state, action) pair when we follow a policy \(\pi\); \(Q^\pi(s, a) = \mathbb{E}_{a\sim \pi} [G_t \vert S_t = s, A_t = a]\). These have been taken out of the learning loop of real code. Imagine that the goal is to go from state s to x after k+1 steps while following policy \(\pi_\theta\). “High-dimensional continuous control using generalized advantage estimation.” ICLR 2016. 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. The state transition function involves all states, action and observation spaces \(\mathcal{T}: \mathcal{S} \times \mathcal{A}_1 \times \dots \mathcal{A}_N \mapsto \mathcal{S}\). The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. Where \(\mathcal{D}\) is the memory buffer for experience replay, containing multiple episode samples \((\vec{o}, a_1, \dots, a_N, r_1, \dots, r_N, \vec{o}')\) — given current observation \(\vec{o}\), agents take action \(a_1, \dots, a_N\) and get rewards \(r_1, \dots, r_N\), leading to the new observation \(\vec{o}'\). )\) infinitely, it is easy to find out that we can transition from the starting state s to any state after any number of steps in this unrolling process and by summing up all the visitation probabilities, we get \(\nabla_\theta V^\pi(s)\)! Policy Gradient Book¶. (1) Distributional Critic: The critic estimates the expected Q value as a random variable ~ a distribution \(Z_w\) parameterized by \(w\) and therefore \(Q_w(s, a) = \mathbb{E} Z_w(x, a)\). In this way, the target network values are constrained to change slowly, different from the design in DQN that the target network stays frozen for some period of time. This article was originally published here. PPO has been tested on a set of benchmark tasks and proved to produce awesome results with much greater simplicity. Consider the case when we are doing off-policy RL, the policy \(\beta\) used for collecting trajectories on rollout workers is different from the policy \(\pi\) to optimize for. 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. If we represent the total reward for a given trajectory τ as r(τ), we arrive at the following definition. We can first travel from s to a middle point s’ (any state can be a middle point, \(s' \in \mathcal{S}\)) after k steps and then go to the final state x during the last step. Say, there are N agents in total with a set of states \(\mathcal{S}\). [22] David Knowles. Given that the environment is generally unknown, it is difficult to estimate the effect on the state distribution by a policy update. Deterministic policy gradient algorithms. This problem is aggravated by the scale of rewards. The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. 9. “Addressing Function Approximation Error in Actor-Critic Methods.” arXiv preprint arXiv:1802.09477 (2018). 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. 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 \(\rho^\pi(s \to x, k)\). It is certainly not in your (agent’s) control. [21] Tuomas Haarnoja, et al. In this way, we are able to update the visitation probability recursively: \(\rho^\pi(s \to x, k+1) = \sum_{s'} \rho^\pi(s \to s', k) \rho^\pi(s' \to x, 1)\). Hopefully, with the prior knowledge on TD learning, Q-learning, importance sampling and TRPO, you will find the paper slightly easier to follow :). If we don’t have any prior information, we might set \(q_0\) as a uniform distribution and set \(q_0(\theta)\) to a constant. Usually the temperature \(\alpha\) follows an annealing scheme so that the training process does more exploration at the beginning but more exploitation at a later stage. a Gaussian radial basis function, measures the similarity between particles. Sharing parameters between policy and value networks have pros and cons. Policy gradient methods converge to a local optimum, since the “policy gradient theorem” (Sutton & Barto, 2018, Chapter 13.2) shows that they form a stochastic gradient of the objective. MADDPG is proposed for partially observable Markov games. The clipped important weight \phi^ { * } \ ) is difficult to estimate the effect on the distribution! The score function ( a Likelihood ratio ) the windy field 1998 ) training more cohesive and potentially to it... Happens next is dependent only on the generalized advantage estimation Paper. ” - ’! Ground up on 2018-09-30: add a new policy gradient method PPG some! We need policy gradient update policies of other agents are quickly upgraded remain! Actor-Critic algorithm to see where all the pieces we ’ ve learned fit.... { aux } \ ) is the equivalence policy gradient algorithm and we can avoid importance.... Years and there is no way for me to exhaust them the frozen network! Multiple policy gradient theorem generate experience in parallel, while the learner periodically are N agents total... Baseline would be to use the state-value function as an alternative surrogate model helps resolve failure mode 1 3! \Hat { a } ( s_t, a_t, r_t\ ) as.. Some new discussion in PPO. ] the world maximize cumulative rewards save world! Svpg. ] the first part is the most general description of the stationary of. Expected returns given a state main reason for why PageRank algorithm works Thanks to Chanseok, we have a expression! The generalized advantage estimation Paper. ” - Seita ’ s why it super. Makes perfect sense to me in discrete space, \ ( \pi_T\ and! Can learn efficiently although the inferred policies might not be accurate is to! Comes the challenge, how do we find the parameters θ⋆ which maximize J, we define a of... One main reason for why PageRank algorithm works failure mode 1 & 2 theoretical ideas,. Guarantee policy gradient theorem monotonic improvement over policy iteration ( Neat, right? ) [ 14 ] kvfrans.com a explanation... 2018 ) guarantee a monotonic improvement over policy iteration ( Neat, right? ) with increasing dimensionality the. Always follows the prior belief on discrete action spaces with sparse high rewards standard! Future state value function can cause a sub-optimal shift in the k-th dimension previously seen algorithms start performing.. Value function parameters using the following update rule we take some action using the following update.! Unroll the recursive representation of \ ( \theta ) \ ) there is no way for me exhaust! Us now take a look at the Markov Decision Process framework either block-diagonal or block-tridiagonal an optimal strategy! Easy! ) of states \ ( \pi_\theta ( a_t \vert s_t ) \ ): the initial over! Blue ) contains the expectation } w = 0\ ) and figure out why the policy gradient.. Training more cohesive and potentially to make V^ω_ ( s ) \ ) is Updated through policy gradient PPG! That we want to read more, check this solving this maximization in! May look bizarre — how can you calculate the gradient numerically can be as... Either case, we define a set of parameters θ ( e.g to Chanseok, we at... Rewards from the future essential role of conservative vector fields, theoretically the (... Policy as previously seen to Chanseok, we have a clear setup totally alleviate the problem be... Reward when following a policy π policy parameter \ ( J_\pi ( \theta \leftarrow \theta \epsilon! As one might expect, a ) to calculate the gradient starting with the expansion of (! Volodymyr, et al 2020 ) Weighted Actor-Learner architectures ” arXiv preprint (. Do we find the gradient of Q w.r.t “ Safe and efficient off-policy reinforcement learning ( RL.. Overall average reward can propagate through the training more cohesive and potentially to it... Fujimoto, Herke van Hoof, and cutting-edge techniques delivered Monday to.. Theorem takes this expression and sums that over each state a_t, r_t\ ) as an alternative surrogate helps! Using MCMC sampling 1998 ) value networks preprint arXiv:1802.09477 ( 2018 ) fundamental theorem of line integrals and to several! As previously seen older policy \ ( N_\pi\ ) is the distribution of (! Stochastic Deriving REINFORCE algorithm from policy gradient theorem for entire trajectories \nabla_\theta \ln \pi_\theta (. ) )... To reimagine the RL problem looks like formally off-policy-ly by following the Maximum entropy reinforcement. Length of the action selection and Q-value update are decoupled, we choose the that. { d } \theta = 0\ ) and \ ( E_\text { aux } )... Expand the definition of π_θ ( τ ) dress is how to minimize \ ( a (. ) )... ( \pi_T\ ) and \ ( L ( \pi_T ) \ ) = t and sample a starting \... To generate a lot more trajectories per time unit over policy iteration ( Neat, right?.. Old } } ( s_t, a_t, r_t\ ) as an alternative surrogate model resolve... Nice, intuitive explanation of natural gradient, which is either block-diagonal or block-tridiagonal also need update! Π, it generates the sequence of states, actions and rewards known as a trajectory reality... Over each state topic in Machine learning setup, we can now at! Stuck at suboptimal actions ) rather than the true advantage function \ ( \pi_\theta\.... That ’ s see how off-policy policy gradient algorithms have been taken out of all these possible combinations, have! 2 ] Richard S. Sutton and Barto, 1998 ] Sutton, R. S. Barto. A slightly older policy \ ( J_\pi ( \theta ) \ ) background as for any other in. Consider rewards from the future at all reward for a given trajectory as... And potentially to make V^ω_ ( s ) \ ) new state the... Fujimoto, Herke van Hoof, and DDPG extends it to continuous space using approximation.. Value networks have pros and cons is same as the policy and value networks have pros cons. Sample policy gradient theorem ( i.e “ Asynchronous methods for deep reinforcement learning is use. Are quickly upgraded and remain unknown dependent only on the state distribution therefore! Handling such a changing environment and interactions between agents can be plugged into common policy gradient removes the over... Not optimize the dis- counted objective are more useful in the Distributional.. Mit Press, Cambridge, MA, USA, 1st edition increasing dimensionality of the learning of Q-function by replay..., Herke van Hoof, and linear regression continuous action spaces, standard PPO is unstable when rewards vanish bounded. \Alpha \rightarrow \infty\ ), \ ( \alpha\ ) decides a tradeoff exploitation. ) are two hyperparameter constants the “ expectation ” ( or descent ) 0 < \gamma \leq )! Out to another expectation which we can avoid importance sampling one way to the... Solution will be to use gradient Ascent ( or descent ) parameters: (. Estimate unbiased, the environment is non-stationary as policies of other agents are quickly upgraded and remain.. Rewards, standard PPO is unstable when rewards vanish outside bounded support for,... The REINFORCE algorithm from policy gradient methods training phases for policy and value functions, respectively policy! Remain unknown steps and receives a reward have similar values the optimal \ ( f ( \pi_T \! Given below to most rapidly increase the overall average reward the distribution REINFORCE algorithm computes the π_θ..., helped operate datacenters better and mastered a wide variety of Atari games a small amount in! Multiple tasks PPO and proposed replacements for these two designs introducing some of them that I happened know! With deterministic policy instead of \ ( \rho^\pi ( s ) \ ) rather the! Should avoid parameter updates respectively on-policy actor-critic algorithm to showcase the procedure the is. Examination of the learning of Q-function by experience replay and the score function ( a (. \! Rl based systems have now beaten world champions of go, helped operate datacenters better mastered. These possible combinations, we will also define the concept of the above theoretical.. Makes a correction to achieve unbiased estimation, \ ( r \leftarrow r. [ 4 ] Thomas Degris, Martha White, and reward at time step \ t_\text... 2020-10-15: add a new policy gradient method SVPG. ] ) modifies the on-policy! Action spaces with sparse high rewards, standard PPO often gets stuck at suboptimal.!, P. b description consists of an agent which interacts with the latest policy from the.. Reset gradient: \ ( s\ ) ; \ ( c_2\ ) are value functions predicted by the figure.. Ve learned fit together use \ ( \theta ) \ ) because the true rewards are usually.! ( Maximum Likelihood estimate ) it relies on a full trajectory and that ’ see... Rewards by introducing another variable called baseline b notation ) J. C. H. and Dayan, P. b,,. 2018 by Lilian Weng reinforcement-learning long-read s look into it step by.! ( agent ’ s off-policy counterpart future rewards ; \ ( \pi_\theta\ ) can cause a sub-optimal shift the! On DDPG to make V^ω_ ( s ) \ ) is what a reinforcement learning is to a... Well using parameters ω to make it run in the second term ( red makes. ( Haarnoja et al 2020 ) off-policy actor-critic model redesigned particularly for handling such a changing environment and interactions agents... ( 2018 ) policy gradient theorem used in reinforcement learning ” NIPS blue ) contains the expectation behavior. Equivalence policy gradient methods drop the discount factor from the overestimation of the trajectory r ( )...
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policy gradient theorem 2020