Perseus: A Simple High-Order Regularization Method for Variational Inequalities
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This paper settles an open and challenging question pertaining to the design of simple high-order regularization methods for solving smooth and monotone variational inequalities (VIs). A VI involves finding x^⋆∈𝒳 such that ⟨ F(x), x - x^⋆⟩≥ 0 for all x ∈𝒳 and we consider the setting where F: ℝ^d ↦ℝ^d is smooth with up to (p-1)^th-order derivatives. For the case of p = 2, <cit.> extended the cubic regularized Newton's method to VIs with a global rate of O(ϵ^-1). <cit.> proposed another second-order method which achieved an improved rate of O(ϵ^-2/3log(1/ϵ)), but this method required a nontrivial binary search procedure as an inner loop. High-order methods based on similar binary search procedures have been further developed and shown to achieve a rate of O(ϵ^-2/(p+1)log(1/ϵ)). However, such search procedure can be computationally prohibitive in practice and the problem of finding a simple high-order regularization methods remains as an open and challenging question in optimization theory. We propose a p^th-order method which does not require any binary search scheme and is guaranteed to converge to a weak solution with a global rate of O(ϵ^-2/(p+1)). A version with restarting attains a global linear and local superlinear convergence rate for smooth and strongly monotone VIs. Further, our method achieves a global rate of O(ϵ^-2/p) for solving smooth and non-monotone VIs satisfying the Minty condition; moreover, the restarted version again attains a global linear and local superlinear convergence rate if the strong Minty condition holds.
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