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Why Momentum Really Works

distill.pub · 17,083 words · saved by 5 readers

We often think of optimization with momentum as a ball rolling down a hill. This isn't wrong, but there is much more to the story.

Why Momentum Really Works Distill About Prize Submit ⋆ \star ⋆ + + + − - − = = = α \alpha α λ \lambda λ β \beta β R R R α = \alpha= α = β = \beta= β = β = 0 \beta = 0 β = 0 β = 1 \beta=1 β = 1 α = 1 / λ i \alpha = 1/\lambda_i α = 1 / λ ​ i ​ ​ m o d e l \text{model} model 0 p 1 0 p_1 0 p ​ 1 ​ ​ 0 p ¯ 1 0 \bar{p}_1 0 ​ p ​ ¯ ​ ​ ​ 1 ​ ​ 2 β 2\sqrt{\beta} 2 √ ​ β ​ ​ ​ λ i \lambda_i λ ​ i ​ ​ λ i = 0 \lambda_i = 0 λ ​ i ​ ​ = 0 α > 1 / λ i \alpha > 1/\lambda_i α > 1 / λ ​ i ​ ​ max { ∣ σ 1 ∣ , ∣ σ 2 ∣ } > 1 \max\{|\sigma_1|,|\sigma_2|\} > 1 max { ∣ σ ​ 1 ​ ​ ∣ , ∣ σ ​ 2 ​ ​ ∣ } > 1 x i k − x i

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