○ All acquisition functions calculate the mean μ(x) and variance σ2(x) based on the posterior distribution, designing exploration strategies through this

○ Type 1. EI (expected improvement): The second formula for EI(x) holds only when using a Gaussian Process

○ Note, Φ and 𝜙 are the CDF and PDF of the standard normal distribution, and Z standardizes x

○ Higher λ leads to more exploration

○ Type 2. MPI (maximum probability of improvement)

○Type 3. UCB

③ Gaussian process: Assuming normal distribution during Bayesian optimization

x <- c(-3, -1, 0, 2, 3)
f <- data.frame(x=x, y=cos(x))
x <- f$x
k_xx <- varCov(x,x)
k_xxs <- varCov(x, x_star)
k_xsx <- varCov(x_star, x)
k_xsxs <- varCov(x_star, x_star)

f_star_bar <- k_xsx %*% solve(k_xx)%*%f$y              # Mean
cov_f_star <- k_xsxs - k_xsx %*% solve(k_xx) %*% k_xxs # Var

plot_ly(type='scatter', mode="lines") %>%
  add_trace(x=x_star, y=f_star_bar, name="Function Estimate") %>%
  add_trace(x=x_star, y=f_star_bar-2*sqrt(diag(cov_f_star)), 
              name="Lower Confidence Band (Function Estimate)",
            line = list(width = 1, dash = 'dash')) %>%
  add_trace(x=x_star, y=f_star_bar+2*sqrt(diag(cov_f_star)), 
              name="Upper Confidence Band (Function Estimate)",
            line = list(width = 1, dash = 'dash')) %>%
  add_markers(x=f$x, y=f$y, name="Obs. Data", marker=list(size=20)) %>%
  add_trace(x=x_star, y=cos(x_star),line=list(width=5), name="True f()=cos(x)") %>%
  layout(title="Gaussian Process Simulation (5 observations)", legend = list(orientation='h'))

④ Hyperparameter Optimization

○ Explore hyperparameters that maximize marginal likelihood, the probability that data X and y occur, using Maximum Likelihood Estimation (MLE)

○ Marginal likelihood and Gaussian Process hyperparameters are defined as follows

○ Advantages: Improves underfitting and overfitting

⑷ Type 2. Stochastic Multi-armed Bandit

① Feature 1. Online Decision: Select the it-th arm among N arms at each time t = 1, ···, T

② Feature 2. Stochastic Feedback: The rewards provided by each arm follow a fixed but unknown distribution

③ Feature 3. Bandit Feedback: At each time step, only the reward for the chosen arm is visible

④ Objective: Maximize cumulative expected rewards over T steps

⑤ Maximizing rewards is equivalent to minimizing regret = μ* - μ_t. The regret over the entire K steps is defined as follows.

⑥ Bandit: Implies taking away when failed, like a robber

⑦ Example 1. Belief state update equation when $n$-th arm is chosen over N arms

⑧ Example 2. Thompson Sampling

2. ϵ-greedy Exploration

⑴ Procedure

① Step 1. At the beginning, play all arms.

② Step 2. With probability (1 - ϵ), choose the arm with the highest sample mean, and with probability ϵ, choose an arm at random.

③ Step 3. Update the sample mean for each arm and return to Step 2.

⑵ Characteristic

① Regret $R_K$ = Ο($K$) over the entire $K$ steps because the error is accumulated by ϵ for each step.

3. UCB

⑴ Overview

① Along with Thompson Sampling, one of the most widely used bandit algorithms

② UCB balances exploitation and exploration.

③ Provides optimism about the uncertainty.

⑵ Procedure

① Empirical mean for arm i during time t

○ Numerator: Total reward obtained from arm $i$

○ 𝕀{$i_s = i$}: Outputs 1 only when $i_s = i$, otherwise 0

○ $n_{i,t}$: Number of times i was chosen until time $t$

○ True mean is represented as $\mu_{i,t}$.

② Bonus term

○ It enables exploration.

○ It is linked to convergence as follows:

③ Upper Confidence Bound (UCB)

○ Optimism: Meaning to consider it higher than the actual expected value with high probability (over 99.9%)

④ Step 1. Play each arm once at the beginning.

⑤ Step 2. Choose the arm with the maximum $\text{UCB}_{i,t}$ at each step $i_t$.

⑥ Step 3. Update empirical mean and UCB at every time step: uncertainty represented by UCB decreases with each observation

⑶ Proof of Convergence

① Theorem

For all $N > 1$, if policy UCB is run on N arms having arbitrary reward distributions $P_1$, …, $P_N$ with support in [0,1], then its expected regret after any number of $T$ plays

is at most

where $\mu_1$, …, $\mu_N$ are the means of the distributions $P_1$, …, $P_N$.

② Proof

Let $c_{t,s}$ := √(2log($t$)/$s$). Also define by the following $\bar{X}n^i = (1/n) \Sigma_t=1^n X_t$ where ${X_t^i}{t∈ℕ}$ is the random process of rewards for arm $i$ if it is played successively. For any arm $i$, we upper bound UCB_i($T$) on any sequence of plays. Let $I_t$ denote the arm played at time $t$, then we have with ℓ being an arbitrary positive integer that

Now we observe that X̄_s* + c_t,s ≤ X̄_{s_i}ⁱ + c_{t,s_i} implies that at least one of the following must hold: X̄_s* ≤ μ* - c_t,s, X̄_{s_i}ⁱ ≤ μ_i + c_{t,s_i}, μ* < μ_i + 2c_{t,s_i}. Otherwise, there is a contradiction: X̄_s* + c_t,s > μ* ≥ μ_i + 2c_{t,s_i} > X̄_{s_i}ⁱ + c_{t,s_i}. We can bound the probability of the first two events by using a version of the Azuma-Hoeffding inequality since both sX̄_s* - sμ* and s_iX̄_{s_i}ⁱ - s_iμ_i are margingales null at 0 with increments in [-μ, 1 - μ] and [-μ_i, 1 - μ_i], respectively. This then yields the following

Also note that we have

for s_i ≥ 8 log(T) / Δ_i² (since T ≥ t). Thus, if we take ℓ = ⌈8 log(T) / Δ_i²⌉, then we cannot have μ* < μ_i + 2c_{t,s_i} and only the first two inequalities can hold. Thus, we get

Therefore, from 𝔼[n_i,T] ≤ 𝔼[UCB_i(T)] ≤ 8log(T) / Δ_i² + C, we get Regret(T) ≤ 8 log(T) / Δ_i + C∑Δ_i.

③ Conclusion

○ Per time regret converges to 0: Meaning every choice becomes the best choice

○ Using only empirical mean does not guarantee optimal arm selection

○ May result in never attempting the optimal arm

⑷ Comparison: UCB vs ϵ-greedy Exploration

① Difference 1. The total regret of UCB is $O(\log T)$, whereas that of $\epsilon$-greedy exploration is $O(T)$.

② Difference 2. The average reward of UCB is higher than that of $\epsilon$-greedy exploration.

Figure 2. UCB vs ϵ-greedy exploration

⑸ Comparison: UCB vs Thompson Sampling

① Commonality 1. Total regret is O(log T): Per time regret converges to 0 as O(log T / T)

② Difference 1. Thompson Sampling often outperforms UCB: UCB is a slightly more conservative algorithm, converging slower

③ Difference 2. UCB is deterministic while Thompson Sampling is probabilistic (randomized)

4. Thompson Sampling

⑴ Overview

① Oldest bandit algorithm: Introduced by Thompson in 1933

② Natural and efficient heuristic algorithm

⑵ Process

① Maintain belief for parameters of each arm (e.g., mean reward)

○ Use prior distributions such as Beta distribution or Gaussian distribution

② Extract estimated reward from each prior distribution

○ Simple sampling, not extracting expected value

○ Higher variance for fewer observations allows easier extraction

③ Choose the arm with the maximum estimated reward and observe its posterior reward

○ Posterior distribution must always be maintained by the researcher

④ Update prior belief with the posterior using Bayesian approach

⑤ Works well even with two modes

⑶ Example: Thompson Sampling using Beta Distribution

① Start with Beta(1, 1) as the prior belief for all arms

② For each time t,

○ Prior: Beta(α, β)

○ Independently sample θi,t for each arm

○ Choose it = argmaxi θi,t

○ Beta(α+1, β): When 1 is observed

○ Beta(α, β+1): When 0 is observed

⑷ Example: Thompson Sampling using Gaussian Distribution

① Start with N(0, ν2) as the prior belief for all arms

② For each time t,

○ Prior: N(0, ν2)

○ Independently sample θi,t for each arm

○ Choose it = argmaxi θi,t

○ Update the empirical mean µˆ (posterior) for the chosen arm: Where n is the number of independent observations

⑸ Convergence

① After a certain number of steps, well-separation between two arms eventually creates the best arm

② Beta Distribution

③ Gaussian Distribution: Based on Azuma-Hoeffding inequality

⑹ Comparison of UCB and Thompson Sampling

① Commonality 1. Total regret is O(log T): Per time regret converges to 0 as O(log T / T)

② Difference 1. Thompson Sampling often outperforms UCB: UCB is a slightly more conservative algorithm, converging slower

③ Difference 2. UCB is deterministic while Thompson Sampling is probabilistic (randomized)

Input: 2021.12.10 00:52

Modified: 2026.03.13 16:09

Edited: 2024.11.21 15:30

2160

Chapter 11-1. Multi-armed bandits

1. Overview

2. ϵ-greedy Exploration

3. UCB

4. Thompson Sampling

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