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On the Convergence of AdaGrad(Norm) on $\mathbb{R}^d$: Beyond Convexity, Non-Asymptotic Rate and Acceleration
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the self-bounding argument (bounding step size b_T via function gap) instead of bounded domain assumption for AdaGradNorm | AdaGradNorm, deterministic unconstrained · smooth convex / gamma-quasar convex | average function gap rate | Helped |
| Use standard (non-off-by-one) stepsize in stochastic AdaGradNorm via divide-by-(2b_t-b_0) technique | Stochastic AdaGradNorm · gamma-quasar convex, sub-Weibull noise | expected average function gap | Helped |
| Replace Assumption 2 (weak smoothness) with L-smoothness in deterministic AdaGradNorm analysis (ablation) | AdaGradNorm · gamma-quasar convex, deterministic | rate constant: improves rate by factor 1/gamma | Helped |
| Last-iterate variant with p_t=1/t and power 1/(2+Delta), Delta>0 (AdaGradNorm-Last) | AdaGradNorm-Last · gamma-quasar convex, smooth | last-iterate function gap rate | Helped |
| Accelerated variant with a_t=2/(t+1), q_t=2/t and power 1/(2+Delta) step size (AdaGradNorm-Acc) | AdaGradNorm-Acc · convex smooth, deterministic unconstrained | function gap rate: O(1/T^2) vs O(1/T) non-accelerated | Helped |
| Delta=0 (power 1/2, scale-invariant) variants of accelerated method (ablation) | AdaGradNorm-Acc with delta=1 · convex smooth, deterministic | rate: only O(1/T^2+1/T) non-asymptotic rate; asymptotic O(1/T^2) | Hurt |
| Scale-invariant step size b_t^delta b_{t-1}^{1-delta} with delta in [2/3,1) in place of standard power-1/2 update (ablation) | AdaGradNorm last-iterate variant · gamma-quasar convex, smooth | rate dependence on smoothness: rate depends exponentially on smoothness L | Hurt |
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