Multi-Armed Bandits 2: ε-Greedy and Non-Stationary Problems

Today we’re going to address some of the problems with an ε-first exploration approach for multi-armed bandits problems. In the last post we saw how ε-first can perform very well on stationary problems where the true value Q_* of each bandit arm (slot machine in our example) never changes. But in the real world we are often faced with problems where the true value of a choice changes over time. In these situations the ε-first exploration approach will not adapt to the changing environment, and will ignorantly keep selecting the same suboptimal action over and over. As with my previous post, you can follow along and run the code in this colab notebook. 📝

Non-stationary problems:

A non-stationary problem is one where the underlying true value (q_*) of each bandit arm can gradually change over the course of an episode. Using our slot machines analogy, we can imagine that the slot machines start the day with a random average payout value, and this average payout value can slowly increase of decrease throughout the day. We model this by adapting our BanditProblem class to allow the arm values to change gradually, this is done by simulating ‘random walks’ for each arm, by drawing a small random number from a normal distribution for each arm and adding these to the true value of each arm. So this means the true payouts can gradually go up or down for each arm.

class BanditProblem():
  def __init__(self, arms, seed, stationary=True):
    self.stationary = stationary

    self.bandit_arms_values = np.random.normal(loc=0.0, scale=1.0, size=arms)
    self.optimal_action_index = np.argmax(self.bandit_arms_values)

  def draw_from_arm(self, arm_index):
    chose_optimal = 1 if arm_index == self.optimal_action_index else 0
    reward = np.random.normal(loc=self.bandit_arms_values[arm_index], scale=1.0)
    return reward, chose_optimal

  def step_arm_values(self):
    Step to be called manually in episode loop.
    q_star_value_shift = np.random.normal(loc=0.0, scale=0.01, size=len(self.bandit_arms_values))
    self.bandit_arms_values += q_star_value_shift
    self.optimal_action_index = np.argmax(self.bandit_arms_values)

This new logic is handled in the step_arm_values function above, which makes small changes to the true arm values and is called after the bandits are done drawing from the arms.

Introducing ε-Greedy:

Our first attempt at tackling the non-stationary bandit problem uses the well-known ε-greedy approach. It is fairly simple and similar to ε-first but with a small difference; instead of exploring randomly for some fixed number of steps, ε-greedy explores randomly some % of the time throughout the entire episode. This means ε-greedy never stops exploring some small portion of the time, determined by the value ε. This could be a small number, like 0.01, meaning that the agent explores randomly approximately 1% of the time. It also means that ε-greedy can start exploiting the best found option right away – there is no long period of exploration time needed before exploitation. The incremental average update rule for the q-value estimate stays exactly the same as it was for ε-first, here it is as a reminder:

Q_{n+1} &=  Q_n + \frac{1}{n}[R_n - Q_n] 

Adding this one simple change – introducing some small chance of exploring randomly every step – is enough to allow the ε-greedy bandit to adapt to non-stationary problems, because it constantly updates its belief about the best choice by some small amount.
So is ε-greedy ‘better’ than ε-first? Well, it depends. On a stationary problem where the values of the slot machines never change, then ε-first is probably better (if you can afford the upfront exploration). On a non-stationary problem, ε-greedy will be better.

Here’s the code for an ε-greedy bandit:

class EpsilonGreedyAgent():
  def __init__(self, epsilon, bandit_problem, alpha=0.1, update_type="incremental"):
    self.epsilon = epsilon
    self.alpha = alpha
    self.problem = bandit_problem
    self.update_type = update_type

    self.arm_qs = np.zeros(len(bandit_problem.bandit_arms_values))
    self.arm_ns = np.zeros(len(bandit_problem.bandit_arms_values))

  def choose_action(self):
    if np.random.rand() > self.epsilon:
      # greedily pull best arm
      choice = np.argmax(self.arm_qs)
      # explore pull any random arm (still a chance to pull best arm too)
      choice = np.random.randint(0, len(self.arm_qs))

    self.arm_ns[choice] += 1

    reward, optimal = self.problem.draw_from_arm(choice)

    if self.update_type == "incremental":
      self.update_estimate_incremental(choice, reward)
    elif self.update_type == "weighted":
      self.update_estimate_weighted(choice, reward)

    return reward, optimal

  def update_estimate_incremental(self, choice_index, reward):
    self.arm_qs[choice_index] = self.arm_qs[choice_index] + (1/self.arm_ns[choice_index])*(reward - self.arm_qs[choice_index])

  def update_estimate_weighted(self, choice_index, reward):
    self.arm_qs[choice_index] = self.arm_qs[choice_index] + (self.alpha*(reward - self.arm_qs[choice_index]))

Once again most of the logic is handled in the choose_action function. Notice how at each step (choice) we draw a random real number between 0.0 and 1.0, and we exploit the best found arm if that random number is bigger than ε (epsilon). But, if the random number is less than ε, then we explore (choose any random arm). Then we take note of the reward/payout received and update our estimate. There are two update types here, but the weighted one can be ignored for now, I’ll explain it later.

Results: ε-First vs ε-Greedy

So how does ε-greedy fare against ε-first on a stationary problem? See the graphs below:

As expected, on our stationary 10-armed bandit problem the ε-first agent fares better than the ε-greedy agent. This is because the values of the bandit arms never change, so once the ε-first bandit has locked on to the best choice, it exploits that choice continually. In contrast, ε-greedy takes a while longer to find the optimal choice, and even when it does find it, it still explores the other sub-optimal options 10% of the time. The only upside to the ε-greedy approach here is that it starts gathering good rewards almost right away, whereas ε-first takes 1000 exploration steps before collecting high rewards.

But what about a non-stationary problem – when the values of the bandit arms change? Can you predict what will happen in this case? Which approach will fare better?

Aha! Now the tables have turned. While ε-first does for a short moment often find a better choice than ε-greedy right after it is done with 1000 exploration steps, this choice quickly becomes stale as the values of the arms change. ε-greedy fare much better overall, and continues to increase its average score over the course of the episode! However, ε-greedy seems to exhibit the same staleness problem (though not as bad) as ε-first: as the episode goes on it chooses the optimal action less and less. Can you think of a reason for this? Hint: think about the q-value update rule shown above (we discussed this in my last post too).

The reason is because the update rule puts n in the denominator of the fraction that behaves as the step size, so this step size gets smaller and smaller as the episode goes on. Eventually, this step size will be so small that the q-value estimates will barely change on each update, and their rank (the order of each bandit arm according to our estimates) will almost never change after many episodes. So, eventually, ε-greedy almost becomes exactly like ε-first and gets stuck pulling the same arm when exploiting (but still pulls other arms randomly some % of the time). This explains why ε-greedy slowly makes less and less optimal choices – its arm value estimates are not keeping up with the changes to each arms’ true value q_* the longer the episode goes on!

Recency Weighted Q-Update:

To truly solve the stale choice issue for non-stationary bandit problems we need to be a bit more intelligent with our q-value estimate update rule. We saw previously how ε-greedy with a sample average update rule has some problems: the step size gradually gets smaller and smaller. The fix is simple: keep this step size constant! The reason why this works is subtle.

Theory warning! The following section will be a bit math heavy. But if you find maths a little dense (I can relate) then I also include a plain English description and accompanying code below. Hopefully that helps! 😊

So we want to change our update rule to include a fixed step size \alpha. That’s easy enough – see line (1) below. Line (1) below is all we need to implement this in code, the rest are just there for understanding. The reason why this works – and the key to understanding why it works – as a recency-weighted update (new updates are given more weight) is due to the recursive way \alpha is applied to the existing estimate Q_n. Realise that Q_n at any time step is the result of previous updates being applied one after another. Lines (2) to (6) demonstrate this below, where we are essentially unrolling Q_n as a sum of updates from all previous time steps.

Q_{n+1} &= Q_n + \alpha[R_n - Q_n] \\
&=  \alpha R_n + (1- \alpha)Q_n \\
&= \alpha R_n + (1- \alpha)[\alpha R_{n-1} + (1- \alpha)Q_{n-1}] \\
&= \alpha R_n + (1 - \alpha) \alpha R_{n-1} + (1- \alpha)^2 Q_{n-1} \\
&= \alpha R_n + (1 - \alpha)\alpha R_{n-1} + (1 - \alpha)^2 \alpha R_{n-2} + \\
& \qquad \qquad \qquad \cdots + (1- \alpha)^{n-1}\alpha R_{1} + (1- \alpha)^nQ_1\\
&= (1-\alpha)^nQ_1 + \sum_{i=1}^{n} (1-\alpha)^{n-i}\alpha R_i

On line (7) we show this idea in a single line where we express Q_{n+1} as the sum of all previous rewards weighted by alpha raised to the power of n-i (the number of steps ago this update occurred). So as we update Q, the contribution of old updates to our current estimate gets exponentially smaller and smaller.

It’s this attribute of our new recency-weighted average that allows our estimates to stay fresh as the ε-greedy continually explores and the true values of the bandit arms change over time. This update rule reacts much faster to changes in the values of the bandit arms and can do so for as long as the episode continues. Although, if it experiences an unlucky streak of exploration it may be temporarily misled into believing that the optimal action is not actually the best. Even so, it will usually fix this mistake quite quickly when the unlucky streak breaks.

You can find the code for this update rule in the update_estimate_weighted function in the code snippet above.

Results: ε-Fixed vs ε-Greedy vs Recency-Weighted ε-Greedy

Naturally, we wouldn’t expect the recency-weighted update ε-greedy to be any better when it comes to stationary problems, and this is true based on our results in the graphs below.

It’s clear that recency-weighted averages hurt performance in the stationary setting. Whereas ε-greedy eventually approaches choosing the best action 90% of the time (the remaining steps are exploration), the recency weighted ε-greedy chooses the optimal action 80% of the time and does not seems to be improving. In this situation the recency-weighted ε-greedy is limited by the \alpha value, which determines the ‘forgetfulness’ of the update rule, so this could be tuned a bit to improve performance for the stationary problem setting. But where recency-weighted ε-greedy really shines is in the non-stationary setting:

Much better! Adding the recency-weighted update rule allows the ε-greedy agent to outperform both prior approaches on non-stationary problems. The recency-weighted update gradually ‘forgets’ about old updates so that it can quickly switch to the new optimal choice as the episode progresses and the values of the bandit arms gradually change.

Discussion and future work:

So that’s it, right? We’ve solved the exploration/exploitation problem? Uhm… actually no. In the last two posts we’ve seen some really simple (and quite effective) methods to balance exploration/exploitation, but there’s one more method I want to cover. This method addresses a problem with ε-greedy: when it explores it does so totally at random, but couldn’t there be a way to focus exploration on choices that seem most promising? This is the topic of our next post when I’ll cover Upper Confidence Bound (UCB) action selection, and also some neat tricks to make ε-greedy exploration more effective too! 🤖

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