Gradient descent with a stepsize that makes the objective decrease at every iteration is considered well-behaved. The loss goes down monotonically. The iterates approach the optimum smoothly. The convergence rate is O(1/R), where R is the number of communication rounds in a local gradient descent setup.
Increasing the stepsize beyond the monotone threshold makes the loss oscillate. The objective goes up on some iterations, down on others. By classical criteria, the optimization is unstable. The well-behaved trajectory has been replaced by a jagged, non-monotonic path.
The jagged path converges faster.
For logistic regression with local gradient descent, any positive stepsize eta produces convergence at rate O(1/(eta K R)), where K is the number of local steps. This rate is strictly faster than the standard O(1/R) achieved by small stepsizes. The larger the stepsize — the more violent the oscillations — the faster the convergence, with no upper bound beyond numerical precision.
The mechanism: large stepsizes cause the iterate to overshoot the optimum, but the overshooting is not random. It carries the iterate across the loss landscape in a way that explores more of the relevant geometry per step. Each oscillation is a large displacement followed by a partial correction, and the corrections accumulate toward the optimum faster than the smooth, monotone trajectory reaches it.
Monotone decrease is not a feature of good optimization. It is a condition we impose because it makes convergence proofs tractable. The proofs that require monotonicity are correct — but they are proving convergence of a subset of possible trajectories. Removing the monotonicity requirement reveals faster trajectories that the restricted analysis cannot see.
This is a specific instance of a general pattern: controlled instability as a computational resource. Simulated annealing uses thermal fluctuations. Stochastic gradient descent uses minibatch noise. Here, stepsize-induced oscillation provides the acceleration. In each case, the “instability” is the mechanism, not the obstacle. Stability is the price of conservative analysis.