  We study the implicit regularization of mini-batch stochastic gradient
descent, when applied to the fundamental problem of least squares regression.
We leverage a continuous-time stochastic differential equation having the same
moments as stochastic gradient descent, which we call stochastic gradient flow.
We give a bound on the excess risk of stochastic gradient flow at time $t$,
over ridge regression with tuning parameter $\lambda = 1/t$. The bound may be
computed from explicit constants (e.g., the mini-batch size, step size, number
of iterations), revealing precisely how these quantities drive the excess risk.
Numerical examples show the bound can be small, indicating a tight relationship
between the two estimators. We give a similar result relating the coefficients
of stochastic gradient flow and ridge. These results hold under no conditions
on the data matrix $X$, and across the entire optimization path (not just at
convergence).
