Landscape Complexity for the Empirical Risk of Generalized Linear Models
This provides theoretical insights into optimization landscapes for machine learning practitioners, though it is incremental as it builds on existing Kac-Rice methods.
The authors tackled the problem of analyzing the number of critical points in the empirical risk landscape for generalized linear models, extending the Kac-Rice method to high-dimensional non-Gaussian random functions and deriving explicit variational formulas for both annealed and quenched complexities.
We present a method to obtain the average and the typical value of the number of critical points of the empirical risk landscape for generalized linear estimation problems and variants. This represents a substantial extension of previous applications of the Kac-Rice method since it allows to analyze the critical points of high dimensional non-Gaussian random functions. Under a technical hypothesis, we obtain a rigorous explicit variational formula for the annealed complexity, which is the logarithm of the average number of critical points at fixed value of the empirical risk. This result is simplified, and extended, using the non-rigorous Kac-Rice replicated method from theoretical physics. In this way we find an explicit variational formula for the quenched complexity, which is generally different from its annealed counterpart, and allows to obtain the number of critical points for typical instances up to exponential accuracy.