Prepare for the Worst, Hope for the Best: Active Robust Learning On Distributions

In recent years, many learning systems have been developed for higher level forms of data, such as learning on distributions in which each example itself is a distribution. This article proposes active robust learning on distributions. In learning on distributions, there is no access to distributions themselves but rather access is through a sample drawn from a distribution. Therefore, similar to robust learning, any estimates of examples are inexact. In order to address these difficulties, we provide an upper bound on the risk of the classifier in the next stage of active learning, where the size of the labeled dataset increases. Based on this upper bound, we propose probabilistic minimax active learning (PMAL) as a general multiclass active learning method that is easy to use in many Bayesian settings, which provably selects an example with knowledge of its label minimizing the expected risk. We present an efficient approximation of the objective with a known error bound to deal with the intractability of the proposed method for active robust learning. Here, we face a nonconvex problem, which we solve by means of a related convex problem with a bound on the norm of the difference between their solutions. To utilize the information about the estimates of distributions, we propose active robust learning on the distributions method based on learning the kernel embedding of distributions by a recent Bayesian method. The experiments demonstrate the effectiveness of the resulting method on a set of synthetic and real-world distributional datasets.