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On Lower Bounds for Standard and Robust Gaussian Process Bandit Optimization
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Apply the alternative KL-based lower-bound proof | Standard noisy GP bandit optimization with the SE kernel · Standard noisy GP bandits, SE kernel | cumulative regret lower bound: \Omega\Big(\sqrt{T(\log T)^{d/2}}\Big) -> \Omega^{*}\Big(\sqrt{T(\log T)^{d/2}\log\frac{1}{\delta}}\Big) | Helped |
| Apply the alternative KL-based lower-bound proof | Standard noisy GP bandit optimization with the Matérn-ν kernel · Standard noisy GP bandits, Matérn-ν kernel | cumulative regret lower bound: \Omega\Big(T^{\frac{\nu+d}{2\nu+d}}\Big) -> \Omega\Big(T^{\frac{\nu+d}{2\nu+d}}\big(\log\frac{1}{\delta}\big)^{\frac{\nu | Helped |
| Account for joint dependence on corruption level and time horizon | Deterministic GP bandit algorithms with corrupted samples · Corrupted samples, SE and Matérn-ν kernels | cumulative regret lower bound: \Omega\Big(\underline{R}_T^{\rm std} + C\Big) -> \Omega\Big(\underline{R}_T^{\rm std} + C(\log T)^{d/2}\Big) (SE); \Omeg | Helped |
| Allow the corrupted-final-point lower bound to apply at arbitrary success probabilities above 2/3 | Robust GP bandit optimization with a corrupted final point · Corrupted final point, SE and Matérn-ν kernels | time to ε-optimality: \Omega\Big(\frac{1}{\epsilon^{2}}\big(\log\frac{1}{\epsilon}\big)^{\frac{d}{2}}\Big) -> \Omega\Big(\frac{1}{\epsilon^{2} | Helped |
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