Convergence analysis of gradient descent for eigenvector computation

We present a novel, simple and systematic convergence analysis of gradient descent for eigenvector computation. As a popular, practical, and provable approach to numerous machine learning problems, gradient descent has found successful applications to eigenvector computation as well. However, surprisingly, it lacks a thorough theoretical analysis for the underlying geodesically non-convex problem. In this work, the convergence of the gradient descent solver for the leading eigenvector computation is shown to be at a global rate O(min{(_∆^λ1_p )² log¹¹ }), where ∆_p = λ_p − λ_p+1 > 0 represents the generalized positive eigengap and always exists without loss of generality with λ_i being the i-th largest eigenvalue of the given real symmetric matrix and p being the multiplicity of λ₁. The rate is linear at O((_∆^λ1_p )² log¹ ) if (_∆^λ1_p )² = O(1), otherwise sub-linear at O(¹ ). We also show that the convergence only logarithmically instead of quadratically depends on the initial iterate. Particularly, this is the first time the linear convergence for the case that the conventionally considered eigengap ∆₁ = λ₁ − λ₂ = 0 but the generalized eigengap ∆_p satisfies (_∆^λ1_p )² = O(1), as well as the logarithmic dependence on the initial iterate are established for the gradient descent solver. We are also the first to leverage for analysis the log principal angle between the iterate and the space of globally optimal solutions. Theoretical properties are verified in experiments.