Lower bounds for finding stationary points I

Lower Bounds for Finding Stationary Points of Non-Convex, Smooth High-Dimensional Functions∗

We establish lower bounds on the complexity of finding -stationary points of smooth, non-convex, high-dimensional functions. For functions with Lipschitz continuous pth derivative, we show that all algorithms—even randomized algorithms observing arbitrarily high-order derivatives—have worst-case iteration count Ω( −(p+1)/p). Our results imply that the O( −2) convergence rate of gradient descent...

Iteration Bounds for Finding the -Stationary Points for Structured Nonconvex Optimization

In this paper we study proximal conditional-gradient (CG) and proximal gradient-projection type algorithms for a block-structured constrained nonconvex optimization model, which arises naturally from tensor data analysis. First, we introduce a new notion of -stationarity, which is suitable for the structured problem under consideration. We then propose two types of first-order algorithms for th...

Lower bounds for witness-finding algorithms

Can a procedure that decides whether a Boolean formula has a satisfying assignment help to find such an assignment? The naïve adaptive “search-todecision” reduction uses a linear number of (quite weak) queries. Is there a lower bound on the number of queries required for a nonadaptive search-to-decision reduction? We report on lower bounds for various classes of queries. Most interesting types ...

Some Lower Bounds for Estrada Index

Kernelization Lower Bounds for Finding Constant Size Subgraphs

Kernelization is an important tool in parameterized algorithmics. The goal is to reduce the input instance of a parameterized problem in polynomial time to an equivalent instance of the same problem such that the size of the reduced instance only depends on the parameter and not on the size of the original instance. In this paper, we provide a first conceptual study on limits of kernelization f...