DIRECTION
方向相同Same direction
三球都沿当前负梯度。All follow the current negative gradient.
01 · 下楼梯的步幅01 · Your stride on stairs
明明每一步都朝下，为什么迈得太大仍会越过平台、摔到另一边？Every step points downward, so why can an oversized stride still leap across the landing?
02同曲面、同起点，三种步长并跑Same Surface and Start, Three Step Sizes
唯一变量：学习率 ηOne variable: learning rate η
青 · 过小Cyan · small
0.16
η × 0.26
橙 · 中间Amber · center
0.62
η
紫 · 过大Violet · large
5.89
η × 9.5
正在加载曲面；若提示持续显示，则需要 WebGL /
three.js。Loading surface; if this remains visible, WebGL or three.js is unavailable.
比较步 0 · 同起点comparison step 0 · same start
三条轨迹 · 拖动旋转three trails · drag to orbit
小：稳但慢small: stable but slow
中：稳定入谷center: settles well
大：越谷或爆跳large: overshoots or bounces
03学习率只做一件事：缩放更新Learning Rate Does One Job: Scale the Update
same gradient · different distanceSCALE
距离不同Different distance
η 直接放大或缩小更新量。η scales the update amount.
RECOMPUTE
落点不同Different landing
下一步从不同位置重算梯度。Next gradients come from different positions.
TRAJECTORY
轨迹分叉Paths diverge
小差异累积成慢、稳或震荡。Small differences become slow, stable, or oscillating paths.
04最小机制：η 乘在梯度前Minimal Mechanism: η Multiplies the Gradient
Δθ = −η∇LΔθ = −η∇L(θ)
η 太小：Δθ 很短；η 合适：持续下降；η
太大：一步跨过低谷，下一步梯度反向，再跨回来。Tiny η makes short updates; a suitable η keeps descending; excessive η crosses the
basin, flips the next gradient, and crosses back.
改变 ηChange η梯度方向不变direction unchanged
更新距离改变Update distance changes落在不同坡面land on a different slope
训练轨迹改变Training path changes慢、稳或发散slow, stable, or divergent
05边界实验：把学习率推到最大Boundary Test: Max Out the Learning Rate
主动制造失败Create a failure让紫球反复跨谷Make the violet ball cross the basin
方向计算没有错，失败来自每一步太长。观察紫色轨迹在谷两侧跳动，而青球仍慢慢前进。The direction is correct; failure comes from steps that are too long. Watch violet bounce while cyan still crawls.
尚未执行。Not run yet.
✗ 学习率越大，训练一定越快。A larger learning rate always trains faster.
→ ✓ 过大会震荡甚至发散。Too large can oscillate or diverge.
✗ 学习率改变下山方向。Learning rate changes the descent direction.
→ ✓ 它缩放梯度给出的更新。It scales the gradient update.
✗ 一个学习率适合所有曲面。One rate fits every surface.
→ ✓ 合适步长取决于地形与尺度。A suitable rate depends on geometry and scale.
06 · 一句话带走06 · One line to keep