## Using Gradient Method on a Bernoulli Multi-Armed Bandit¶

For an introduction to Multi Armed Bandits, refer to Multi Armed Bandit Overview

This method is different compared to others. In other methods, we explicity attempt to estimate the ‘value’ of taking an action (its quality) whereas in this method we approach the problem in a different way. Here, instead of estimating how good an action is through its quality, we only care about its preference of being selected compared to other actions. We denote this preference by $$H_t(a)$$. The larger the preference of an action ‘a’, more are the chances of it being selected, but this preference has no interpretation in terms of the reward for that action. Only the relative preference is important.

The action probabilites are related to these action preferences $$H_t(a)$$ by a softmax function. The probability of taking action $$a_j$$ is given by:

$P(a_j) = \frac{e^{H_t(a_j)}}{\sum_{i=1}^A e^{H_t(a_i)}} = \pi_t(a_j)$

where, A is the total number of actions and $$\pi_t(a)$$ is the probability of taking action ‘a’ at timestep ‘t’.

We initialise the preferences for all the actions to be 0, meaning $$\pi_t(a) = \frac{1}{A}$$ for all actions.

After computing $$\pi_t(a)$$ for all actions at each timestep, the action is sampled using this probability. Then that action is performed and based on the reward we get, we update our preferences.

The update rule bacially performs stochastic gradient ascent:

$$H_{t+1}(a_t) = H_t(a_t) + \alpha (R_t - \bar{R_t})(1-\pi_t(a_t))$$, for $$a_t$$: action taken at time ‘t’

$$H_{t+1}(a) = H_t(a) - \alpha (R_t - \bar{R_t})(\pi_t(a))$$ for rest of the actions

where, $$\alpha$$ is the step size, $$R_t$$ is the reward obtained at time ‘t’ and $$\bar{R_t}$$ is the mean reward obtained upto time t. If current reward is larger than the mean reward, we increase our preference for that action taken at time ‘t’. If it is lower than the mean reward, we decrease our preference for that action. The preferences for the rest of the actions are updated in the opposite direction.

For a more detailed mathematical analysis and derivation of the update rule, refer to chapter 2 of Sutton & Barto.

Code to use the Gradient method on a Bernoulli Multi-Armed Bandit:

import gym
import numpy as np

from genrl.bandit import BernoulliMAB, GradientMABAgent, MABTrainer

bandits = 10
arms = 5

reward_probs = np.random.random(size=(bandits, arms))
bandit = BernoulliMAB(bandits, arms, reward_probs, context_type="int")