  In this paper, we study the numerical approximation of the one-point large
deviations rate functions of nonlinear stochastic differential equations (SDEs)
with small noise. We show that the stochastic $\theta$-method satisfies the
one-point large deviations principle with a discrete rate function for
sufficiently small step-size, and present a uniform error estimate between the
discrete rate function and the continuous one on bounded sets in terms of
step-size. It is proved that the convergence orders in the cases of
multiplicative noises and additive noises are $1/2$ and $1$ respectively. Based
on the above results, we obtain an effective approach to numerically
approximating the large deviations rate functions of nonlinear SDEs with small
time. To the best of our knowledge, this is the first result on the convergence
rate of discrete rate functions for approximating the one-point large
deviations rate functions associated with nonlinear SDEs with small noise.
