# OFFO minimization algorithms for second-order optimality and their complexity

An Adagrad-inspired class of algorithms for smooth unconstrained optimization is presented in which the objective function is never evaluated and yet the gradient norms decrease at least as fast as O(1/\sqrt{k+1}) while second-order optimality measures converge to zero at least as fast as O(1/(k+1)^{1/3}). This latter rate of convergence is shown to be essentially sharp and is identical to that known for more standard algorithms (like trust-region or adaptive-regularization methods) using both function and derivatives' evaluations. A related divergent stepsize'' method is also described, whose essentially sharp rate of convergence is slighly inferior. It is finally discussed how to obtain weaker second-order optimality guarantees at a (much) reduced computional cost.