Alternating minimization and alternating descent over nonconvex sets
This work addresses optimization challenges in high-dimensional statistics and signal processing, but it is incremental as it builds on existing notions like local concavity coefficients.
The paper analyzes alternating minimization for optimizing loss functions over two variables with potentially nonconvex constraints, a common setting in high-dimensional statistics and signal processing, and demonstrates the framework on examples like low rank + sparse decomposition and multitask regression with numerical validation.
We analyze the performance of alternating minimization for loss functions optimized over two variables, where each variable may be restricted to lie in some potentially nonconvex constraint set. This type of setting arises naturally in high-dimensional statistics and signal processing, where the variables often reflect different structures or components within the signals being considered. Our analysis relies on the notion of local concavity coefficients, which has been proposed in Barber and Ha to measure and quantify the concavity of a general nonconvex set. Our results further reveal important distinctions between alternating and non-alternating methods. Since computing the alternating minimization steps may not be tractable for some problems, we also consider an inexact version of the algorithm and provide a set of sufficient conditions to ensure fast convergence of the inexact algorithms. We demonstrate our framework on several examples, including low rank + sparse decomposition and multitask regression, and provide numerical experiments to validate our theoretical results.