# Vanilla Policy Gradient¶

For background on Deep RL, its core definitions and problem formulations refer to Deep RL Background

## Objective¶

The objective is to choose/learn a policy that will maximize a cumulative function of rewards received at each step, typically the discounted reward over a potential infinite horizon. We formulate this cumulative function as

$E\left[{\sum_{t=0}^{\infty}{\gamma^{t} r_{t}}}\right]$

where we choose the action $$a_{t} = \pi_{\theta}(s_{t})$$.

## Algorithm Details¶

### Collect Experience¶

To make our agent learn, we first need to collect some experience in an online fashion. For this we make use of the collect_rollouts method. This method is defined in the OnPolicyAgent Base Class.

For updation, we would need to compute advantages from this experience. So, we store our experience in a Rollout Buffer. Action Selection —————-

Note: We sample a stochastic action from the distribution on the action space by providing False as an argument to select_action.

For practical purposes we would assume that we are working with a finite horizon MDP.

### Update Equations¶

Let $$\pi_{\theta}$$ denote a policy with parameters $$\theta$$, and $$J(\pi_{\theta})$$ denote the expected finite-horizon undiscounted return of the policy.

At each update timestep, we get value and log probabilities:

Now, that we have the log probabilities we calculate the gradient of $$J(\pi_{\theta})$$ as:

$\nabla_{\theta} J(\pi_{\theta}) = E_{\tau \sim \pi_{\theta}}\left[{ \sum_{t=0}^{T} \nabla_{\theta} \log \pi_{\theta}(a_t|s_t) }\right],$

where $$\tau$$ is the trajectory.

We then update the policy parameters via stochastic gradient ascent:

$\theta_{k+1} = \theta_k + \alpha \nabla_{\theta} J(\pi_{\theta_k})$

The key idea underlying vanilla policy gradients is to push up the probabilities of actions that lead to higher return, and push down the probabilities of actions that lead to lower return, until you arrive at the optimal policy.

## Training through the API¶

import gym

from genrl.agents import VPG
from genrl.trainers import OnPolicyTrainer
from genrl.environments import VectorEnv

env = VectorEnv("CartPole-v0")
agent = VPG('mlp', env)
trainer = OnPolicyTrainer(agent, env, log_mode=['stdout'])
trainer.train()

timestep         Episode          loss             mean_reward
0                0                9.1853           22.3825
20480            10               24.5517          80.3137
40960            20               24.4992          117.7011
61440            30               22.578           121.543
81920            40               20.423           114.7339
102400           50               21.7225          128.4013
122880           60               21.0566          116.034
143360           70               21.628           115.0562
163840           80               23.1384          133.4202
184320           90               23.2824          133.4202
204800           100              26.3477          147.87
225280           110              26.7198          139.7952
245760           120              30.0402          184.5045
266240           130              30.293           178.8646
286720           140              29.4063          162.5397
307200           150              30.9759          183.6771
327680           160              30.6517          186.1818
348160           170              31.7742          184.5045
368640           180              30.4608          186.1818
389120           190              30.2635          186.1818