Augmented Newton Method for Optimization: Global Linear Rate and Momentum Interpretation
This work provides incremental improvements to optimization algorithms for researchers and practitioners in machine learning and numerical analysis.
The authors tackled the problem of improving Newton's method for unconstrained optimization by proposing two variants, the Penalty Newton and Augmented Newton methods, which achieve global linear convergence under mild assumptions and can be interpreted as incorporating adaptive heavy ball momentum.
We propose two variants of Newton method for solving unconstrained minimization problem. Our method leverages optimization techniques such as penalty and augmented Lagrangian method to generate novel variants of the Newton method namely the Penalty Newton method and the Augmented Newton method. In doing so, we recover several well-known existing Newton method variants such as Damped Newton, Levenberg, and Levenberg-Marquardt methods as special cases. Moreover, the proposed Augmented Newton method can be interpreted as Newton method with adaptive heavy ball momentum. We provide global convergence results for the proposed methods under mild assumptions that hold for a wide variety of problems. The proposed methods can be sought as the penalty and augmented extensions of the results obtained by Karimireddy et. al [24].