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Regret Analysis for Randomized Gaussian Process Upper Confidence Bound

Shion Takeno, Yu Inatsu, Masayuki Karasuyama

TL;DR

Improved randomized GP-UCB is analyzed, which uses the confidence parameter generated from the shifted exponential distribution, and it is shown that randomization plays a key role in avoiding an increase in confidence parameter by showing that GP-UCB using a constant confidence parameter can incur linearly growing expected cumulative regret.

Abstract

Gaussian process upper confidence bound (GP-UCB) is a theoretically established algorithm for Bayesian optimization (BO), where we assume the objective function $f$ follows a GP. One notable drawback of GP-UCB is that the theoretical confidence parameter $β$ increases along with the iterations and is too large. To alleviate this drawback, this paper analyzes the randomized variant of GP-UCB called improved randomized GP-UCB (IRGP-UCB), which uses the confidence parameter generated from the shifted exponential distribution. We analyze the expected regret and conditional expected regret, where the expectation and the probability are taken respectively with $f$ and noise and with the randomness of the BO algorithm. In both regret analyses, IRGP-UCB achieves a sub-linear regret upper bound without increasing the confidence parameter if the input domain is finite. Furthermore, we show that randomization plays a key role in avoiding an increase in confidence parameter by showing that GP-UCB using a constant confidence parameter can incur linearly growing expected cumulative regret. Finally, we show numerical experiments using synthetic and benchmark functions and real-world emulators.

Regret Analysis for Randomized Gaussian Process Upper Confidence Bound

TL;DR

Improved randomized GP-UCB is analyzed, which uses the confidence parameter generated from the shifted exponential distribution, and it is shown that randomization plays a key role in avoiding an increase in confidence parameter by showing that GP-UCB using a constant confidence parameter can incur linearly growing expected cumulative regret.

Abstract

Gaussian process upper confidence bound (GP-UCB) is a theoretically established algorithm for Bayesian optimization (BO), where we assume the objective function follows a GP. One notable drawback of GP-UCB is that the theoretical confidence parameter increases along with the iterations and is too large. To alleviate this drawback, this paper analyzes the randomized variant of GP-UCB called improved randomized GP-UCB (IRGP-UCB), which uses the confidence parameter generated from the shifted exponential distribution. We analyze the expected regret and conditional expected regret, where the expectation and the probability are taken respectively with and noise and with the randomness of the BO algorithm. In both regret analyses, IRGP-UCB achieves a sub-linear regret upper bound without increasing the confidence parameter if the input domain is finite. Furthermore, we show that randomization plays a key role in avoiding an increase in confidence parameter by showing that GP-UCB using a constant confidence parameter can incur linearly growing expected cumulative regret. Finally, we show numerical experiments using synthetic and benchmark functions and real-world emulators.
Paper Structure (30 sections, 20 theorems, 108 equations, 4 figures, 1 algorithm)

This paper contains 30 sections, 20 theorems, 108 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background
  4. Bayesian Optimization
  5. Regularity Assumptions
  6. Regret
  7. Maximum information gain
  8. GP-UCB
  9. Algorithm
  10. Regret Analysis
  11. Expected Regret Bounds
  12. Conditional Expected Regret Bounds
  13. High-Probability Regret Bounds
  14. Expected Regret of GP-UCB with Constant Confidence Parameter
  15. Experiments
  16. ...and 15 more sections

Key Result

Lemma 4.1

Suppose that $f$ is a sample path from a GP with zero mean and a positive semidefinite kernel $k$, and ${\cal X}$ is finite. Pick $\delta \in (0, 1)$ and $t \geq 1$. Then, for any given ${\cal D}_{t-1}$, where $\beta_{\delta} = 2 \log (|{\cal X}| /( 2 \delta))$.

Figures (4)

  • Figure 1: This figure shows the confidence parameters of GP-UCB-based methods. For GP-UCB, the solid line represents $\beta_t$. For RGP-UCB and IRGP-UCB, the solid lines and shaded areas represent the expectations $\mathbb{E}[\zeta_t]$ and $95 \%$ credible intervals, respectively.
  • Figure 2: This figure shows the average and standard errors of simple regrets.
  • Figure 3: Average and standard errors of simple regret on benchmark functions.
  • Figure 4: Average and standard errors of simple regret on real-world datasets.

Theorems & Definitions (39)

  • Definition 2.1: Maximum information gain
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Theorem 4.1: BCR bound for finite domain
  • Theorem 4.2: BCR bound for infinite domain
  • Remark 4.1
  • Theorem 4.3
  • proof
  • Theorem 4.4
  • ...and 29 more