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Despite the impressive numerical performance of the quasi-Newton andAnderson/nonlinear acceleration methods, their global convergence rates haveremained elusive for over 50 years. This study addresses this long-standingissue by introducing a framework that derives novel, adaptive quasi-Newton andnonlinear/Anderson acceleration schemes. Under mild assumptions, the proposediterative methods exhibit explicit, non-asymptotic convergence rates that blendthose of the gradient descent and Cubic Regularized Newton's methods. Theproposed approach also includes an accelerated version for convex functions.Notably, these rates are achieved adaptively without prior knowledge of thefunction's parameters. The framework presented in this study is generic, andits special cases includes algorithms such as Newton's method with randomsubspaces, finite-differences, or lazy Hessian. Numerical experimentsdemonstrated the efficiency of the proposed framework, even compared to thel-BFGS algorithm with Wolfe line-search.

Adaptive Quasi-Newton and Anderson Acceleration Framework with Explicit Global (Accelerated) Convergence Rates