In this paper, we establish sharp lower bounds for Gaussian approximation of Polyak-Ruppert averaged stochastic gradient descent with polynomially decreasing step sizes. For every fixed exponent
γ∈(1/2,1), a single one-dimensional smooth, strongly convex example has approximation error with the limiting variance of order
n1/2−γ+nγ−1 in Kolmogorov and convex distances. This matches the existing upper bound throughout the range of
γ. The nonlinear bias contributes
n1/2−γ even when the Gaussian reference uses the finite linearized variance. We also derive a lower bound of order
n−(1−γ)/2 for the relative mean error of estimating the limiting variance from one SGD trajectory, uniformly over a scalar quadratic family. With the limiting variance known, achieving approximation error
ε on this example takes order
ε−4 iterations for the best choice
γ=3/4. Achieving both Gaussian approximation and relative mean variance-estimation errors at most
ε uniformly over the model class requires at least order
ε−6 iterations for every fixed
γ.