Bandit convex optimisation is a fundamental framework for studying zeroth-order convex optimisation. These notes cover the many tools used for this problem, including cutting plane methods, interior point methods, continuous exponential weights, gradient descent and online Newton step. The nuances between the many assumptions and setups are explained. Although there is not much truly new here, some existing tools are applied in novel ways to obtain new algorithms. A few bounds are improved in minor ways.
The paper discusses computational challenges in convex bandit and zeroth-order optimization, focusing on the manipulation of convex sets.
Various representations of convex sets and their computational implications are analyzed, including polytope, convex hull, separation oracle, and linear optimization oracle.
Methods for improving computational efficiency, such as projection algorithms, finding self-concordant barriers, and approximating MVEEs, are proposed.
Open problems in the field are identified, emphasizing the need for algorithms that balance efficiency and accuracy, adapt to different constraints, and scale with problem complexity.
Computational efficiency in convex bandit and zeroth-order optimization problems heavily depends on the representation and manipulation of convex sets. This paper discusses various computational challenges encountered when dealing with convex sets in optimization problems and proposes several methods to overcome these hurdles. The core of the discussion revolves around efficient ways to perform operations such as projections, finding self-concordant barriers, and approximating minimum volume enclosing ellipsoids (MVEEs). These operations are crucial in implementing algorithms for both convex bandit optimization and zeroth-order optimization.
Convex sets can be represented in several ways, each offering different computational advantages and challenges:
The computation in convex optimization, particularly in bandit and zeroth-order settings, presents various open problems:
Effective computation in convex bandit and zeroth-order optimization is contingent upon the precise representation of convex sets and the efficiency of operations such as projection, barrier identification, and MVEE approximation. Future research directions include improving algorithmic adaptability, enhancing computational efficiency, and solving open problems related to convex set manipulation.