Closing the Bias-Correction Gap in Discounted Regret Analysis of Adam

Recent discounted-to-nonconvex analyses of Adam establish convergence guarantees by bounding the discounted regret of an associated online learner. However, these analyses typically remove Adam's original bias-correction factors by absorbing them into the learning-rate schedule or clipping radius, and therefore prove guarantees for a reparameterized update. To accurately understand its convergence behavior, we analyze clipped Adam while retaining the original bias correction in this paper. The key difficulty is that bias correction introduces nonmonotone time-dependent coefficients into the regret analysis: the relevant coefficient sequence first decreases and then increases, forming a single-valley structure. Under standard stochastic cost-vector assumptions and the mild parameter coupling, we prove that the effective step sizes remain automatically monotone and develop a summation argument showing that the transient valley cost is controlled by sufficiently many post-valley terms. The resulting discounted-regret guarantees match the known $1/\sqrt{1-{\beta }_1}$ dependence of analyses without retained bias correction, and hence preserve Adam's original bias-corrected parameterization without worsening the regret order.