On the Prior Sensitivity of Thompson Sampling

arXiv.org Machine Learning

The empirically successful Thompson Sampling algorithm for stochastic bandits has drawn much interest in understanding its theoretical properties. One important benefit of the algorithm is that it allows domain knowledge to be conveniently encoded as a prior distribution to balance exploration and exploitation more effectively. While it is generally believed that the algorithm's regret is low (high) when the prior is good (bad), little is known about the exact dependence. In this paper, we fully characterize the algorithm's worst-case dependence of regret on the choice of prior, focusing on a special yet representative case. These results also provide insights into the general sensitivity of the algorithm to the choice of priors. In particular, with $p$ being the prior probability mass of the true reward-generating model, we prove $O(\sqrt{T/p})$ and $O(\sqrt{(1-p)T})$ regret upper bounds for the bad- and good-prior cases, respectively, as well as \emph{matching} lower bounds. Our proofs rely on the discovery of a fundamental property of Thompson Sampling and make heavy use of martingale theory, both of which appear novel in the literature, to the best of our knowledge.

Bandit Learning Through Biased Maximum Likelihood Estimation

arXiv.org Machine Learning

We propose BMLE, a new family of bandit algorithms, that are formulated in a general way based on the Biased Maximum Likelihood Estimation method originally appearing in the adaptive control literature. We design the cost-bias term to tackle the exploration and exploitation tradeoff for stochastic bandit problems. We provide an explicit closed form expression for the index of an arm for Bernoulli bandits, which is trivial to compute. We also provide a general recipe for extending the BMLE algorithm to other families of reward distributions. We prove that for Bernoulli bandits, the BMLE algorithm achieves a logarithmic finite-time regret bound and hence attains order-optimality. Through extensive simulations, we demonstrate that the proposed algorithms achieve regret performance comparable to the best of several state-of-the-art baseline methods, while having a significant computational advantage in comparison to other best performing methods. The generality of the proposed approach makes it possible to address more complex models, including general adaptive control of Markovian systems.

Statistical Anomaly Detection for Train Fleets

AAAI Conferences

We have developed a method for statistical anomaly detection which has been deployed in a tool for condition monitoring of train fleets. The tool is currently used by several railway operators over the world to inspect and visualize the occurrence of event messages generated on the trains. The anomaly detection component helps the operators to quickly find significant deviations from normal behavior and to detect early indications for possible problems. The savings in maintenance costs comes mainly from avoiding costly breakdowns, and have been estimated to several million Euros per year for the tool. In the long run, it is expected that maintenance costs can be reduced with between 5 and 10 % by using the tool.

Bayesian Anomaly Detection Using Extreme Value Theory

arXiv.org Machine Learning

Data-driven anomaly detection methods typically build a model for the normal behavior of the target system, and score each data instance with respect to this model. A threshold is invariably needed to identify data instances with high (or low) scores as anomalies. This presents a practical limitation on the applicability of such methods, since most methods are sensitive to the choice of the threshold, and it is challenging to set optimal thresholds. We present a probabilistic framework to explicitly model the normal and anomalous behaviors and probabilistically reason about the data. An extreme value theory based formulation is proposed to model the anomalous behavior as the extremes of the normal behavior. As a specific instantiation, a joint non-parametric clustering and anomaly detection algorithm (INCAD) is proposed that models the normal behavior as a Dirichlet Process Mixture Model. A pseudo-Gibbs sampling based strategy is used for inference. Results on a variety of data sets show that the proposed method provides effective clustering and anomaly detection without requiring strong initialization and thresholding parameters.