Javad Azimi, Ali Jalali, Xiaoli Fern Oregon State University University of Texas at Austin In NIPS 2011, Workshop in Bayesian optimization, experimental design and bandits: Theory and applications 1 Dynamic Batch Bayesian Optimization

Bayesian Optimization (BO) 2 Finding the Maximizer of an unknown function by requesting a small set of function evaluations (experiments) – experiments are costly BO assumes prior over – select next experiment based on posterior Current Experiments Gaussian Process Surface Select Single/Multiple Experiment Run Experiment(s)

Traditional Approaches 3 Sequential: Only one experiment is selected at each iteration Pros: Performance is optimized Cons: Can be very costly when running one experiment takes long time Batch: experiments are selected at each iteration Pros: times speed-up comparing to sequential approaches Cons: Can not performs as well as sequential algorithms

Batch Performance (Azimi et.al NIPS 2010) 4 Given a sequential policy, it chooses a batch of samples which are likely to be selected by the sequential policy. k=5 k=10

Motivation 5 Given a sequential policy, is it possible to simultaneously, select a batch of experiments approximately preserve the sequential policy performance. Size of the batch can change at each time step Dynamic batch size

Proposed Idea: Big Picture 6 Based on a given prior (blue circles) and an objective function (G), is selected To select the next experiment,, we need, which is not available The statistics of the samples inside the red circle are expected to change after observing at Set the G values for all samples inside the red circle as their upper bound value If the next selected experiment is outside of the red circle, we claim it is independent from. x1x1 x2x2 x3x3

Problems 7 Which samples statistics are changed after selecting an/a set of experiment? How can we upper bound the objective function G?

Gaussian Process (GP) 8 GP is used to model the posterior over the unobserved samples in BO Statistical prediction for each point by a normal random variable rather than deterministic prediction The posterior variance is independent from the observation

Definition 9 z z z Unobserved set of points Corresponding Outputs Any point :

GP Theorems 10

Expected Improvement (EI) 11 Our algorithm inputs a sequential policy to compete with. We choose Expected Improvement (EI) as criterion our approach extends to other policies EI simply computes the expected improvement after sampling at each point

Dynamic Batch 12 samples are asked at each iteration. if the selected samples are independent from each other. The first selected sample, is the same as sequential Choice of the second point depends on Setting (maximum possible value) EI of the next step is upper bounded The next sample is selected, if it is not inside the red circle (not significantly effected by )

Dynamic Batch: Algorithm 13

Experimental Results: Setting 14 GP with squared exponential kernel is used as the model We set n l = 20(total number of experiments), and n b =5 (maximum batch size) The average regret over 100 independent runs is reported where regret is: Speedup of each framework is reported which is the percentage of experiments asked in batch mode. ε=0.02 for 2-3 dimensional and 0.2 for higher dimensional frameworks An alternative and more realistic approach is to set M=(1+ α) y m which means (100* α)% improvement at each iteration.

Experimental Results: Results 15

Experimental Results: Speedup vs Budget 16

Conclusion and Future works 17 Conclusion The proposed dynamic batch approach selects variable number of experiments at each step The selected experiments are approximately independent from each other The proposed approach approximately preserves the sequential performance Future Works Theoretical analysis of the distance between selected samples in batch and sequential approach. The analysis of choose of epsilon in performance