Combinatorial Pure Exploration with Full-Bandit or Partial Linear Feedback

In this paper, we first study the problem of combinatorial pure exploration with full-bandit feedback (CPE-BL), where a learner is given action space X \subseteq {0,1}^d, and in each round pulls an x \in receives random reward expectation x^T \theta, \theta \R^d latent unknown environment vector. The objective to identify optimal highest expected reward, using as few samples possible. For CPE-BL, design polynomial-time adaptive algorithm, whose sample complexity matches lower bound (within logarithmic factor) for family instances has light dependence \Delta_min (the smallest gap between sub-optimal actions). Furthermore, propose novel generalization CPE-BL flexible structures, called partial linear (CPE-PL), which encompasses several families sub-problems including feedback, semi-bandit nonlinear functions. CPE-PL, pull reports vector M_x , R^{m_x \times d} transformation matrix x, gains (possibly nonlinear) related x. develop simultaneously addresses limited general function (e.g., matroids, matchings s-t paths), provide its analysis. Our empirical evaluation demonstrates that our algorithms run orders magnitude faster than existing ones, algorithm robust across different settings while CPE-PL one returning correct answers