Papers › A fast and simple modification of Newton's method helping to avoid saddle points
A fast and simple modification of Newton's method helping to avoid saddle points
Tuyen Trung Truong, Tat Dat To, Tuan Hang Nguyen, Thu Hang Nguyen, Hoang Phuong Nguyen, Maged Helmy
We propose in this paper New Q-Newton's method. The update rule is very simple conceptually, for example xₙ₊₁=xₙ-wₙ where wₙ=pr_(Aₙ,+)(vₙ)-pr_(Aₙ,-)(vₙ), with Aₙ=∇²f(xₙ)+δₙ||∇f(xₙ)||².Id and vₙ=Aₙ⁻¹.∇f(xₙ). Here δₙ is an appropriate real number so that Aₙ is invertible, and pr_(Aₙ,±) are projections to the vector subspaces generated by eigenvectors of positive (correspondingly negative) eigenvalues of Aₙ. The main result of this paper roughly says that if f is C³ (can be unbounded from below) and a sequence {xₙ}, constructed by the New Q-Newton's method from a random initial point x₀, {\bf converges}, then the limit point is a critical point and is not a saddle point, and the convergence rate is the same as that of Newton's method. The first author has recently been successful incorporating Backtracking line search to New Q-Newton's method, thus resolving the convergence guarantee issue observed for some (non-smooth) cost functions. An application to quickly finding zeros of a univariate meromorphic function will be discussed. Various experiments are performed, against well known algorithms such as BFGS and Adaptive Cubic Regularization are presented.
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