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Stochastic Bayesian Optimization with Unknown Continuous Context Distribution via Kernel Density Estimation
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Proposes SBO-KDE and DRBO-KDE, which estimate an unknown continuous context PDF online with kernel density estimation and optimize the expected or distributionally robust (total-variation) UCB acquisition, with sub-linear Bayesian cumulative regret bounds.
Problem
Bayesian optimization of expected black-box objectives over a continuous, unknown, and observable context distribution, where discretizing the context space is costly and inaccurate.
Approach
Replace the unknown context distribution with an online KDE estimate inside the UCB acquisition (SBO-KDE), and for robustness optimize a worst-case expectation over a total-variation ball around the KDE estimate, reformulated as a two-dimensional convex problem (DRBO-KDE).
Key results
- SBO-KDE and DRBO-KDE outperform DRBO-MMD, DRBO-MMD-MinimaxApprox, StableOpt and GP-UCB on the synthetic functions and the newsvendor and portfolio tasks (reported in Figure 1).
- DRBO-KDE beats SBO-KDE on the Complicated Hartmann function with a mixture context distribution, where KDE error is higher.
- Theory: SBO-KDE has BCR O(T^{(2+D_c)/(4+D_c)}) up to log factors, compared with O(T^{1/2}) for standard SBO; DRBO-KDE keeps the same order with delta_t = O(t^{-2/(4+D_c)}).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use SBO-KDE instead of the compared baselines | Synthetic functions · Synthetic functions (Ackley, Modified Branin, Hartmann, Complicated Hartmann) | cumulative regret or cumulative reward | Helped |
| Use SBO-KDE instead of DRBO-KDE | SBO-KDE · Ackley | cumulative regret | Helped |
| Use DRBO-KDE instead of SBO-KDE | DRBO-KDE · Complicated Hartmann | cumulative reward | Helped |
| Use SBO-KDE or DRBO-KDE instead of the compared baselines | SBO-KDE and DRBO-KDE · Modified Branin and Hartmann | cumulative regret | No clear effect |
| Use GP-UCB, which ignores the context variable | GP-UCB · Synthetic functions | cumulative regret or cumulative reward | Mixed |
| Use DRBO-MMD with discretized context | DRBO-MMD · Synthetic functions | cumulative regret or cumulative reward | Mixed |
| Use StableOpt's worst-case objective | StableOpt · Synthetic functions | cumulative regret or cumulative reward | Hurt |
| Use DRBO-MMD-MinimaxApprox | DRBO-MMD-MinimaxApprox · Synthetic functions | cumulative regret or cumulative reward | Hurt |
| Use SBO-KDE or DRBO-KDE instead of the compared baselines | SBO-KDE and DRBO-KDE · Newsvendor and portfolio optimization | cumulative regret or cumulative reward | Helped |
| Use SBO-KDE or DRBO-KDE instead of GP-UCB | SBO-KDE and DRBO-KDE · Newsvendor | cumulative regret or cumulative reward | Helped |
| Use DRBO-MMD instead of DRBO-KDE | DRBO-MMD · Portfolio optimization with normal context distribution | cumulative regret or cumulative reward | No clear effect |
Surprising findings
- GP-UCB, which ignores the context entirely, performs acceptably, possibly because it absorbs context effects into observation noise.
- DRBO-MMD is competitive with DRBO-KDE on portfolio optimization with a normal context distribution.
Limitations
- Assumes the context distribution is static over time.
- Theoretical bound for DRBO-KDE is looser than SBO-KDE because it goes through the estimated distribution.
- KDE estimation error grows for complicated context distributions.
Experimental setup
Datasets: Ackley Modified Branin Hartmann Complicated Hartmann Newsvendor (Burr XII demand) Portfolio optimization (3,000 GP samples from CVXPortfolio)
Benchmarks: Ackley Modified Branin Hartmann Complicated Hartmann Newsvendor Portfolio optimization
Metrics: cumulative regret cumulative reward
Baselines: DRBO-MMD DRBO-MMD-MinimaxApprox StableOpt GP-UCB
Builds on: GP-UCB DRBO-MMD StableOpt Sobol sequence initialization
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