Kurdyka-\L ojasiewicz exponent via square transformation

TL;DR

This paper investigates how the square transformation affects the variational properties and KL exponent of composite objective functions, providing insights into reparameterization impacts in optimization.

Contribution

It offers a detailed analysis of the variational properties and KL exponent changes under square reparameterization for composite functions.

Findings

Characterizes the subdifferential of reparameterized functions.

Computes second subderivative on linear subspaces.

Links KL exponent of reparameterized and original functions.

Abstract

We consider one of the most common reparameterization techniques, the square transformation. Assuming the original objective function is the sum of a smooth function and a polyhedral function, we study the variational properties of the objective function after reparameterization. In particular, we first study the minimal norm of the subdifferential of the reparameterized objective function. Second, we compute the second subderivative of the reparameterized objective function on a linear subspace, which allows for fully characterizing the subclass of stationary points of the reparameterized objective function that correspond to stationary points of the original objective function. Finally, utilizing the representation of the minimal norm of the subdifferential, we show that the Kurdyka-\L ojasiewicz (KL) exponent of the reparameterized function can be deduced from that of the original function.