Momentum: Rolling Down the Valley
Plain SGD forgets everything after each step, so it zig-zags uselessly in narrow valleys. Momentum gives the optimizer memory: it keeps a running velocity that averages recent gradients, which cancels oscillation and multiplies steady progress by up to 10×. This topic covers the heavy-ball formulas, why β = 0.9 means "remember ~10 steps", and Nesterov's look-ahead upgrade.
01.The Problem: An Optimizer With No Memory
Plain gradient descent (Topic 6) does this every step:
- Feel the slope at your feet.
- Take a step downhill.
- Forget everything.
- Repeat.
Step 3 is the problem.
Picture a narrow valley (recall Topic 7): steep walls side-to-side, gentle slope along the floor toward the bottom.
At each step, SGD feels:
- a big slope sideways (the wall),
- a small slope forward (the floor).
So it steps hard sideways → now the other wall is under its feet → steps hard back → and so on.
Minutes of computation, and the model has bounced across the valley hundreds of times while crawling forward.
This is the zig-zag. Loss curves that look like a noisy staircase, learning that crawls in exactly the direction you care about.
The fix is embarrassingly simple:
Stop treating each gradient as news. Average the last few.
That average is momentum.
Oscillation Damping, Acceleration Amplifying
Oscillation Damping, Acceleration Amplifying
In an ill-conditioned valley, high-curvature directions oscillate and average out of v; low-curvature directions accumulate coherently — momentum acts as a spectral filter on gradient directions.
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