We study Nesterov's accelerated gradient method with constant step-size and
momentum parameters in the stochastic approximation setting (unbiased gradients
with bounded variance) and the finite-sum setting (where randomness is due to
sampling mini-batches). To build better insight into the behavior of Nesterov's
method in stochastic settings, we focus throughout on objectives that are
smooth, strongly-convex, and twice continuously differentiable. In the
stochastic approximation setting, Nesterov's method converges to a neighborhood
of the optimal point at the same accelerated rate as in the deterministic
setting. Perhaps surprisingly, in the finite-sum setting, we prove that
Nesterov's method may diverge with the usual choice of step-size and momentum,
unless additional conditions on the problem related to conditioning and data
coherence are satisfied. Our results shed light as to why Nesterov's method may
fail to converge or achieve acceleration in the finite-sum setting.

研究了 Nesterov 加速梯度方法在随机逼近和有限和设置下的表现，发现使用通常的步长和动量参数，该方法在后者可能发散，进而阐明了这种方法在此情况下可能失败的原因。