  In this paper, we present convergence guarantees for a modified trust-region
method designed for minimizing objective functions whose value and gradient and
Hessian estimates are computed with noise. These estimates are produced by
generic stochastic oracles, which are not assumed to be unbiased or consistent.
We introduce these oracles and show that they are more general and have more
relaxed assumptions than the stochastic oracles used in prior literature on
stochastic trust-region methods. Our method utilizes a relaxed step acceptance
criterion and a cautious trust-region radius updating strategy which allows us
to derive exponentially decaying tail bounds on the iteration complexity for
convergence to points that satisfy approximate first- and second-order
optimality conditions. Finally, we present two sets of numerical results. We
first explore the tightness of our theoretical results on an example with
adversarial zeroth- and first-order oracles. We then investigate the
performance of the modified trust-region algorithm on standard noisy
derivative-free optimization problems.
