  We develop a new method of online inference for a vector of parameters
estimated by the Polyak-Ruppert averaging procedure of stochastic gradient
descent (SGD) algorithms. We leverage insights from time series regression in
econometrics and construct asymptotically pivotal statistics via random
scaling. Our approach is fully operational with online data and is rigorously
underpinned by a functional central limit theorem. Our proposed inference
method has a couple of key advantages over the existing methods. First, the
test statistic is computed in an online fashion with only SGD iterates and the
critical values can be obtained without any resampling methods, thereby
allowing for efficient implementation suitable for massive online data. Second,
there is no need to estimate the asymptotic variance and our inference method
is shown to be robust to changes in the tuning parameters for SGD algorithms in
simulation experiments with synthetic data.
