  Stochastic structured prediction under bandit feedback follows a learning
protocol where on each of a sequence of iterations, the learner receives an
input, predicts an output structure, and receives partial feedback in form of a
task loss evaluation of the predicted structure. We present applications of
this learning scenario to convex and non-convex objectives for structured
prediction and analyze them as stochastic first-order methods. We present an
experimental evaluation on problems of natural language processing over
exponential output spaces, and compare convergence speed across different
objectives under the practical criterion of optimal task performance on
development data and the optimization-theoretic criterion of minimal squared
gradient norm. Best results under both criteria are obtained for a non-convex
objective for pairwise preference learning under bandit feedback.
