On the Complexity of Finding Small Subgradients in Nonsmooth Optimization
This work addresses fundamental complexity issues in optimization theory, providing insights into algorithm design for researchers in mathematical optimization and machine learning, though it is incremental in building on prior definitions and results.
The paper tackles the problem of finding small subgradients in nonsmooth optimization, showing that deterministic algorithms cannot achieve dimension-free rates for producing stationary points, but derandomization is possible for smooth functions with logarithmic dependence, and it establishes lower bounds and improved rates for convex functions.
We study the oracle complexity of producing $(δ,ε)$-stationary points of Lipschitz functions, in the sense proposed by Zhang et al. [2020]. While there exist dimension-free randomized algorithms for producing such points within $\widetilde{O}(1/δε^3)$ first-order oracle calls, we show that no dimension-free rate can be achieved by a deterministic algorithm. On the other hand, we point out that this rate can be derandomized for smooth functions with merely a logarithmic dependence on the smoothness parameter. Moreover, we establish several lower bounds for this task which hold for any randomized algorithm, with or without convexity. Finally, we show how the convergence rate of finding $(δ,ε)$-stationary points can be improved in case the function is convex, a setting which we motivate by proving that in general no finite time algorithm can produce points with small subgradients even for convex functions.