Query Complexity of Derivative-Free Optimization
This work addresses the challenge of optimization in scenarios where gradients are unavailable, such as with human comparisons, though it is incremental in improving DFO algorithms.
The paper tackles the problem of derivative-free optimization (DFO) with noisy evaluations, establishing lower bounds that reveal a performance gap compared to gradient-based methods, and proposes a near-optimal algorithm for strongly convex functions using only Boolean-valued comparisons, achieving the same convergence rate as with noisy evaluations.
This paper provides lower bounds on the convergence rate of Derivative Free Optimization (DFO) with noisy function evaluations, exposing a fundamental and unavoidable gap between the performance of algorithms with access to gradients and those with access to only function evaluations. However, there are situations in which DFO is unavoidable, and for such situations we propose a new DFO algorithm that is proved to be near optimal for the class of strongly convex objective functions. A distinctive feature of the algorithm is that it uses only Boolean-valued function comparisons, rather than function evaluations. This makes the algorithm useful in an even wider range of applications, such as optimization based on paired comparisons from human subjects, for example. We also show that regardless of whether DFO is based on noisy function evaluations or Boolean-valued function comparisons, the convergence rate is the same.