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06-momentum.html
<!doctype html>
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<title>kp5 動量 · Momentum</title>
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canvas.addEventListener('blur', () => { UI.focus = -1; requestDraw(); });
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let drawFn = null, pending = false, animating = false;
function setAnimating(b){ animating = b; if (b) requestDraw(); }
function requestDraw(){ if (pending) return; pending = true; requestAnimationFrame(render); }
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const s = clamp(r.width * dpr / W, 1, 2.4), w = Math.round(W * s), h = Math.round(H * s);
if (canvas.width !== w || canvas.height !== h){ canvas.width = w; canvas.height = h; }
return s;
}
function render(now){
pending = false; const s = fitCanvas(); ctx.setTransform(s, 0, 0, s, 0, 0);
AUD.text.length = 0; AUD.bad.length = 0;
ctx.clearRect(0, 0, W, H); drawFn(now || performance.now()); uiDraw();
if (animating) requestDraw();
}
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syncMathLayer(); new ResizeObserver(() => { syncMathLayer(); requestDraw(); }).observe(stage);
window.addEventListener('resize', () => { syncMathLayer(); requestDraw(); });
requestDraw(); window.__ready = true;
}
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window.__ui = { set: uiSet, items: UI.items };
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fr.push(box);
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const allow = m.o.width, cx = m.o.align === 'left' ? m.o.x : m.o.align === 'right' ? m.o.x - allow : m.o.x - allow/2;
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/* stepwise derivation panel: scenes = [{tab, formulas:[{tex,h,size}], note:[zh,en], draw?}] */
function Deriv(o){
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const top = o.y + 124, noteH = o.noteH || 92, noteTop = o.y + o.h - 24 - noteH, areaBottom = noteTop - 14;
seg({ id: o.id, x: o.x + 24, y: o.y + 62, w: o.w - 48, h: 44, value: 1, fs: n > 3 ? 14 : 15, color: P.blue,
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o.scenes.forEach((s, i) => {
const total = s.formulas.reduce((q, f) => q + f.h, 0) + Math.max(0, s.formulas.length - 1) * gap;
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sync();
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const k = wrap(s.note[0], o.x + 28, noteTop + 10, o.w - 56, 21, 15.5, P.ink, 700);
wrap(s.note[1], o.x + 28, noteTop + 10 + k * 21 + 6, o.w - 56, 18, 13.5, P.muted, 500);
};
D.set = v => uiSet(o.id, v);
return D;
}
/* generic contour lines by marching squares */
function contourLines(p, f, levels, color, lw, nx, ny){
nx = nx || 110; ny = ny || 70; const gx = [], gy = [], V = [];
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for (let j = 0; j <= ny; j++) gy.push(p.yr[0] + (p.yr[1] - p.yr[0]) * j / ny);
for (let i = 0; i <= nx; i++){ V.push([]); for (let j = 0; j <= ny; j++) V[i].push(f(gx[i], gy[j])); }
ctx.save(); ctx.beginPath(); ctx.rect(p.x, p.y, p.w, p.h); ctx.clip(); ctx.lineWidth = lw || 1.3; ctx.lineCap = 'round';
levels.forEach((lv, li) => {
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for (let i = 0; i < nx; i++) for (let j = 0; j < ny; j++){
const v = [V[i][j], V[i + 1][j], V[i + 1][j + 1], V[i][j + 1]], xs = [gx[i], gx[i + 1], gx[i + 1], gx[i]], ys = [gy[j], gy[j], gy[j + 1], gy[j + 1]], pts = [];
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ctx.restore();
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function fmtSci(v){ if (v === 0) return '0'; const e = Math.floor(Math.log10(Math.abs(v))); return (v / Math.pow(10, e)).toFixed(0) + 'e' + e; }
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/* kp5 — 動量 */
const RP = RPG, st = { eta: 0.07, alpha: 0.5, K: 40, t0: 0 };
const D = Deriv({ id: 'step', x: LPG.x, y: LPG.y, w: LPG.w, h: LPG.h, title: ['逐步推導 Step by step'], gap: 16,
scenes: [
{ tab: '① 動量更新 Update rule', formulas: [
{ tex: String.raw`(\Delta\boldsymbol{w})^{(k)}:=\alpha\,(\Delta\boldsymbol{w})^{(k-1)}-\eta^{(k)}\,\nabla f\big(\boldsymbol{w}^{(k)}\big) \tag{8}`, h: 56, size: 24 },
{ tex: String.raw`\boldsymbol{w}^{(k+1)}:=\boldsymbol{w}^{(k)}+(\Delta\boldsymbol{w})^{(k)}`, h: 48, size: 28 }],
note: ['不只看當下的梯度,還保留上一步更新的 α 倍當「慣性」;α > 0 是動量係數。', 'Besides the current gradient, keep a fraction α of the previous update as inertia; α > 0 is the momentum coefficient.'] },
{ tab: '② 向量直覺 Vector sum', formulas: [], note: ['歷史 αΔw 與 −η∇f 相加後,側向分量互相抵消,沿山谷的分量被保留。', 'Adding the history αΔw and −η∇f cancels the sideways parts and keeps the along-valley part.'],
draw: b => { const ox = b.x + 150, oy = b.y + b.h / 2 + 60;
arrow(ox, oy, ox + 90, oy - 120, P.blue, 4, 13); T('αΔw 歷史慣性 history', ox + 100, oy - 150, { size: 15, weight: 800, color: P.blue, maxW: 230 });
arrow(ox, oy, ox + 90, oy + 120, P.red, 4, 13); T('−η∇f 當前梯度 current grad.', ox + 100, oy + 150, { size: 15, weight: 800, color: P.red, maxW: 260 });
arrow(ox, oy, ox + 220, oy, P.violet, 5, 15); T('Δw 合成 resultant', ox + 232, oy - 4, { size: 15, weight: 800, color: P.violet, maxW: 150 });
line(ox + 90, oy - 120, ox + 220, oy, P.faint, 1.6, [5, 4]); line(ox + 90, oy + 120, ox + 220, oy, P.faint, 1.6, [5, 4]);
T('側向抵消 sideways cancel', ox + 40, oy - 4 + 0, { size: 13.5, weight: 700, color: P.muted, ok: true }); } },
{ tab: '③ 何時有用 When', formulas: [], note: ['狹長山谷、或梯度方向步與步之間變化劇烈時最有用;後面的 RMSProp、Adam 也借用「累積歷史」的想法。', 'Most useful in narrow ravines or when gradients change wildly between steps; RMSProp and Adam reuse the idea of accumulating history.'],
draw: b => { ['狹長山谷 ravine:純 GD 左右撞壁 · plain GD bounces', '梯度方向劇烈變化 · gradient flips between steps', '動量把歷史平均進來 · momentum averages the history in'].forEach((t, i) => { card(b.x + 14, b.y + 24 + i * 100, b.w - 28, 82, { fill: '#fff' }); dot(b.x + 44, b.y + 65 + i * 100, 14, P.blueS); T(String(i + 1), b.x + 44, b.y + 66 + i * 100, { size: 14, weight: 800, color: P.blue, align: 'center' }); wrap(t, b.x + 70, b.y + 52 + i * 100, b.w - 100, 22, 16, P.ink, 700); }); } }
] });
const restart = () => { st.t0 = performance.now(); setAnimating(true); };
slider({ id: 'eta', x: RP.x + 24, y: RP.y + 566, w: 268, min: 0.01, max: 0.095, step: 0.005, value: st.eta, color: P.amber, label: '學習率 η', fmt: v => f2(v, 3), onChange: v => { st.eta = v; restart(); } });
slider({ id: 'alpha', x: RP.x + 316, y: RP.y + 566, w: 268, min: 0, max: 0.95, step: 0.05, value: st.alpha, color: P.green, label: '動量 α Momentum', fmt: v => f2(v, 2), onChange: v => { st.alpha = v; restart(); } });
slider({ id: 'K', x: RP.x + 608, y: RP.y + 566, w: 268, min: 10, max: 60, step: 5, value: st.K, color: P.violet, label: '步數 Steps', fmt: v => String(v), onChange: v => { st.K = v; restart(); } });
button({ id: 'replay', x: RP.x + 700, y: RP.y + 646, w: 176, h: 44, text: '重播 Replay', color: P.navy, onClick: restart });
function ravine(){ const g = [[6, 1.2]], m = [[6, 1.2]]; for (let k = 0; k < 60; k++){ const [x, y] = g[k]; g.push([x - st.eta * x, y - st.eta * 25 * y]);
const pr = k ? m[k - 1] : m[0], [mx, my] = m[k]; m.push([mx - st.eta * mx + st.alpha * (mx - pr[0]), my - st.eta * 25 * my + st.alpha * (my - pr[1])]); } return { g, m }; }
function draw(now){
chrome({ tag: 'kp5 · 動量 Momentum', zh: '動量:加速收斂、減少震盪', en: 'Momentum: faster convergence, less zig-zag', foot: 'kp5 · 本知識點共 1 張 · 1 page' });
D.draw();
const { x, y, w, h } = RP; panel(x, y, w, h); const S = ravine(), prog = clamp((now - st.t0) / 3200, 0, 1); if (prog >= 1) setAnimating(false);
T('狹長山谷 f = ½(x² + 25y²):GD vs 動量', x + 24, y + 30, { size: 23, weight: 800 });
T('Same start (6, 1.2) and same η; α = 0 reduces momentum to plain GD', x + 24, y + 58, { size: 15, weight: 500, color: P.muted });
const p = mkP(x + 64, y + 96, 790, 232, [-7, 7], [-1.5, 1.5]); pframe(p, { xticks: [-6, -3, 0, 3, 6], yticks: [-1, 0, 1], xl: 'x', yl: 'y' });
ellipses(p, 1, 25, [0.5, 2, 5, 10, 18, 28, 40]);
const K = st.K, up = prog * K;
[[S.g, P.red], [S.m, P.green]].forEach(([arr, col]) => { const pts = arr.slice(0, K + 1); pcurve(p, pts, col, 2.4, null, Math.floor(up)); const n = Math.floor(up);
pts.slice(0, n + 1).forEach(q => { if (Math.abs(q[0]) < 7 && Math.abs(q[1]) < 1.5) dot(p.X(q[0]), p.Y(q[1]), 2.6, col); }); const q = pts[Math.min(n, pts.length - 1)]; if (Math.abs(q[0]) < 7 && Math.abs(q[1]) < 1.5) dot(p.X(q[0]), p.Y(q[1]), 6, col, '#fff', 2); });
dot(p.X(6), p.Y(1.2), 6, '#fff', P.ink, 2.4); dot(p.X(0), p.Y(0), 5, P.ink);
badge(p.x + 12, p.y + 20, '純 GD plain GD', P.redS, P.red, 13); badge(p.x + 12, p.y + 52, '動量 momentum', P.greenS, P.green, 13);
/* distance chart */
const q = mkP(x + 64, y + 380, 790, 110, [0, 60], [1e-3, 10], { logY: true }); pframe(q, { xticks: [0, 10, 20, 30, 40, 50, 60], yticks: [[1e-3, '10⁻³'], [1e-2, '10⁻²'], [1e-1, '10⁻¹'], [1, '1'], [10, '10']], xl: 'k', yl: '距谷底 distance to minimum' });
[[S.g, P.red], [S.m, P.green]].forEach(([arr, col]) => pcurve(q, arr.slice(0, K + 1).map((v, i) => [i, clamp(Math.hypot(v[0], v[1]), 1e-3, 10)]), col, 2.6, null, Math.floor(up)));
line(q.X(K), q.y, q.X(K), q.y + q.h, P.faint, 1.4, [4, 4]);
const dg = Math.hypot(...S.g[K]), dm = Math.hypot(...S.m[K]), fm = v => v > 50 ? '發散 diverged' : v < 1e-3 ? '< 10⁻³' : f2(v, 3);
T('第 ' + K + ' 步距谷底 · distance at step ' + K + ': GD = ' + fm(dg) + ' , 動量 = ' + fm(dm), x + 24, y + 548, { size: 15.5, weight: 800, maxW: 840 });
}
window.__states = [
{ name: 'default', set: {} }, { name: 'alpha0', set: { alpha: 0 } }, { name: 'alpha-max', set: { alpha: 0.95 } }, { name: 'eta-max', set: { eta: 0.095, alpha: 0.5 } },
{ name: 'eta-min-K60', set: { eta: 0.01, alpha: 0.9, K: 60 } }, { name: 'K-min', set: { K: 10 } }, { name: 'step2', set: { step: 2 } }, { name: 'step3', set: { step: 3 } }
];
window.__reset = () => { const U = window.__ui; U.set('step', 1); U.set('eta', 0.07); U.set('alpha', 0.5); U.set('K', 40); };
window.__settle = () => { st.t0 = performance.now() - 6000; };
start(draw, 'kp5 動量:式 (8) 的更新規則、向量直覺,以及在狹長山谷中比較純 GD 與動量的軌跡與距離。');
})();
</script>
</body>
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