DREAM: Delay-Resistant Acceleration with Momentum

With exact, delay-free gradients, Nesterov's accelerated gradient (NAG) achieves optimal √κ dependence for L-smooth, $μ$-strongly convex objectives, where $κ=L/μ$; heavy-ball (HB) achieves the same scaling on strongly convex quadratics. In asynchronous and distributed settings, gradients can be inexact, with delays varying across workers and iterations. We isolate staleness using a fixed delay τ≥1, replacing ∇ f(yₖ) with ∇ f(yk-τ) at iteration k. Arjevani, Shamir, and Srebro establish a worst-case lower bound of Ω(τ√κlog(1/ε)) iterations to reach objective error at most ε as ε→0. We show that for any fixed positive delay, neither NAG nor HB retains √κ-acceleration on strongly convex quadratics when updating every iteration, for any stable constant step size and positive momentum. We introduce delay-resistant accelerated momentum (DREAM), which decouples gradient-query extrapolation from momentum and uses explicit delay-dependent parameters. DREAM attains a linear convergence rate 1-Θ(1/(τ√κ)) for all smooth strongly convex objectives as κ→∞, yielding complexity matching the delayed first-order lower bound up to logarithmic factors. Our proof combines characteristic-polynomial stability analysis on quadratics with an analytic Zames--Falb integral quadratic constraint certificate. We extend DREAM to smooth convex objectives, obtaining near-optimal iteration complexity up to logarithmic factors for sufficiently small target accuracy. Experiments on strongly convex quadratics and regularized logistic regression corroborate DREAM's accelerated condition-number dependence under delay, unlike NAG and HB, and suggest this scaling persists when parameters use a conservative bound on the true delay.