Learning rate schedule can significantly affect generalization performance in modern neural networks, but the reasons for this are not yet understood. Li et al. (2019) recently proved this behavior can exist in a simplified non-convex neural-network setting. In this work, we show that this phenomenon can exist even for convex learning problems -- in particular, linear regression in 2 dimensions. We give a toy convex problem where learning rate annealing (large initial learning rate, followed by small learning rate) can lead gradient descent to minima with provably better generalization than using a small learning rate throughout. In our case, this occurs due to a combination of the mismatch between the test and train loss landscapes, and early-stopping.