  Stochastic gradient descent (SGD) or stochastic approximation has been widely
used in model training and stochastic optimization. While there is a huge
literature on analyzing its convergence, inference on the obtained solutions
from SGD has only been recently studied, yet is important due to the growing
need for uncertainty quantification. We investigate two computationally cheap
resampling-based methods to construct confidence intervals for SGD solutions.
One uses multiple, but few, SGDs in parallel via resampling with replacement
from the data, and another operates this in an online fashion. Our methods can
be regarded as enhancements of established bootstrap schemes to substantially
reduce the computation effort in terms of resampling requirements, while at the
same time bypassing the intricate mixing conditions in existing batching
methods. We achieve these via a recent so-called cheap bootstrap idea and
Berry-Esseen-type bound for SGD.
