We further show the mismatched sampling paradox: A learner who knows the rewards distributions and samples from the correct posterior distribution can perform exponentially worse than a learner who does not know the rewards and simply samples from a well-chosen Gaussian posterior.

We further show the mismatched sampling paradox: A learner who knows the rewards distributions and samples from the correct posterior distribution can perform exponentially worse than a learner who does not know the rewards and simply samples from a well-chosen Gaussian posterior.

In this paper we consider Thompson Sampling (TS) for combinatorial semi-bandits.

Thompson sampling (TS) has optimal regret and excellent empirical performance in multi-armed bandit problems. Yet, in Bayesian optimization, TS underperforms popular acquisition functions (e.g., EI, UCB). TS samples arms according to the probability that they are optimal. A recent algorithm, P-Star Sampler (PSS), performs such a sampling via Hit-and-Run. We present an improved version, Stagger Thompson Sampler (STS). STS more precisely locates the maximizer than does TS using less computation time. We demonstrate that STS outperforms TS, PSS, and other acquisition methods in numerical experiments of optimizations of several test functions across a broad range of dimension. Additionally, since PSS was originally presented not as a standalone acquisition method but as an input to a batching algorithm called Minimal Terminal Variance (MTV), we also demon-strate that STS matches PSS performance when used as the input to MTV.

We consider Thompson Sampling (TS) for linear combinatorial semi-bandits and subgaussian rewards. We propose the first known TS whose finite-time regret does not scale exponentially with the dimension of the problem. We further show the "mismatched sampling paradox": A learner who knows the rewards distributions and samples from the correct posterior distribution can perform exponentially worse than a learner who does not know the rewards and simply samples from a well-chosen Gaussian posterior.

Scientific experiments are usually expensive due to complex experimental preparation and processing. Experimental design is therefore involved with the task of finding the optimal experimental input that results in the desirable output by using as few experiments as possible. Experimenters can often acquire the knowledge about the location of the global optimum. However, they do not know how to exploit this knowledge to accelerate experimental design. In this paper, we adopt the technique of Bayesian optimization for experimental design since Bayesian optimization has established itself as an efficient tool for optimizing expensive black-box functions. Again, it is unknown how to incorporate the expert prior knowledge about the global optimum into Bayesian optimization process. To address it, we represent the expert knowledge about the global optimum via placing a prior distribution on it and we then derive its posterior distribution. An efficient Bayesian optimization approach has been proposed via posterior sampling on the posterior distribution of the global optimum. We theoretically analyze the convergence of the proposed algorithm and discuss the robustness of incorporating expert prior. We evaluate the efficiency of our algorithm by optimizing synthetic functions and tuning hyperparameters of classifiers along with a real-world experiment on the synthesis of short polymer fiber. The results clearly demonstrate the advantages of our proposed method.

We design and analyze TS-Cascade, a Thompson sampling algorithm for the cascading bandit problem. In TS-Cascade, Bayesian estimates of the click probability are constructed using a univariate Gaussian; this leads to a more efficient exploration procedure vis-\`a-vis existing UCB-based approaches. We also incorporate the empirical variance of each item's click probability into the Bayesian updates. These two novel features allow us to prove an expected regret bound of the form $\tilde{O}(\sqrt{KLT})$ where $L$ and $K$ are the number of ground items and the number of items in the chosen list respectively and $T\ge L$ is the number of Thompson sampling update steps. This matches the state-of-the-art regret bounds for UCB-based algorithms. More importantly, it is the first theoretical guarantee on a Thompson sampling algorithm for any stochastic combinatorial bandit problem model with partial feedback. Empirical experiments demonstrate superiority of TS-Cascade compared to existing UCB-based procedures in terms of the expected cumulative regret and the time complexity.