Constant-Step Adam Does Not Reach Stationarity: An Explicit Noise Floor and a Variance-versus-Standard-Deviation Separation from SGD
Adam's convergence theory is almost always stated for a vanishing step size eta_t = O(1/sqrt(T)); under that schedule it reaches an approximate stationary point. Practitioners instead run Adam with fixed hyperparameters. We show the distinction is decisive. For the one-dimensional strongly convex objective f(x)=x^2/2 with bounded, unbiased, i.i.d. stochastic-gradient noise of scale s, we prove that constant-step RMSProp (Adam with beta1=0), for ANY fixed eta>0, beta2 in (0,1) and eps>=0, never reaches stationarity: the time-averaged expected squared gradient is bounded below by an explicit positive constant, liminf (1/T) sum_t E[(grad f(x_t))^2] >= eta^2 s^2/(32 (G+eps)^2) > 0, where G is an almost-sure gradient bound we derive. We give the matching exact result for constant-step SGD, whose floor is eta s^2/(2-eta), and combine them into a provable separation on this function class: because Adam normalizes the gradient by a running root-mean-square of size ~ s, its stationarity floor scales with the noise standard deviation, whereas SGD's scales with the noise variance, so SGD's floor is smaller by a factor of order 1/s that diverges as the noise shrinks. A reproducible simulation (code included) confirms the scaling and shows it persists for full Adam (beta1=0.9) and for a non-convex objective. The message is clean: constant step size, not non-convexity or adversarial gradients, is by itself sufficient to prevent Adam from reaching a stationary point.