Universal Tensor Methods for Monotone Variational Inequalities

We study monotone variational inequalities whose operators have H\"older continuous higher-order derivatives. For a fixed order $p\geq 2$, we assume that the $(p-1)$-th derivative of the monotone operator is H\"older continuous with parameter $\nu\in[0,1]$ on a bounded closed convex set. We develop regularized tensor extragradient methods that combine a high-order Taylor approximation of the operator with an extragradient correction step. When the H\"older parameter $\nu$ is known, our regularized tensor extragradient method finds an $\epsilon$-weak solution using $\mathcal{O}(\epsilon^{-2/(p+\nu)})$ tensor-oracle calls. When $\nu$ is unknown, we propose a universal tensor extragradient method whose tensor-oracle complexity is $\mathcal{O}(\epsilon^{-2p/((p+1)(p-1+\nu))})$.