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12-sgd-convergence-analysis.html

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<title>kp9 SGD 收斂分析 · SGD convergence analysis</title>
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};

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<script>
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/* shared helpers for the backpropagation pages */
const SIG = {
  sq:   { label: 'a²',   f: a => a * a,           d: a => 2 * a },
  tanh: { label: 'tanh', f: a => Math.tanh(a),    d: a => 1 - Math.tanh(a) * Math.tanh(a) },
  relu: { label: 'ReLU', f: a => Math.max(0, a),  d: a => a > 0 ? 1 : 0 }
};
function f2(v, n){ n = n == null ? 2 : n; const s = v.toFixed(n); return /^-0\.?0*$/.test(s) ? s.slice(1) : s; }
const signCol = v => v >= 0 ? P.blue : P.red;
function lockedSlot(x, y, w, h, msg){
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  T(msg, x + w/2, y + h/2, { size: 16, weight: 600, color: P.faint, align: 'center', maxW: w - 20 });
}

/* ===== shared helpers for all course pages (Deriv panel, plots, sims) ===== */
function mulberry(a){ return function(){ a |= 0; a = a + 0x6D2B79F5 | 0; let t = Math.imul(a ^ a >>> 15, 1 | a); t = t + Math.imul(t ^ t >>> 7, 61 | t) ^ t; return ((t ^ t >>> 14) >>> 0) / 4294967296; }; }
function gauss(r){ let u = 0; while (!u) u = r(); const v = r(); return Math.sqrt(-2 * Math.log(u)) * Math.cos(2 * Math.PI * v); }
function f2(v, n){ n = n == null ? 2 : n; const s = v.toFixed(n); return /^-0\.?0*$/.test(s) ? s.slice(1) : s; }
const LPG = { x: 40, y: 150, w: 600, h: 710 }, RPG = { x: 660, y: 150, w: 900, h: 710 };

/* plot box helper */
function mkP(x, y, w, h, xr, yr, o){ o = o || {};
  const p = { x, y, w, h, xr, yr, o };
  p.X = v => x + (v - xr[0]) / (xr[1] - xr[0]) * w;
  p.Y = o.logY ? (v => y + h - (Math.log10(Math.max(v, 1e-300)) - Math.log10(yr[0])) / (Math.log10(yr[1]) - Math.log10(yr[0])) * h) : (v => y + h - (v - yr[0]) / (yr[1] - yr[0]) * h);
  return p; }
function pframe(p, o){ o = o || {};
  if (o.fill !== false){ rr(p.x - 6, p.y - 6, p.w + 12, p.h + 12, 8); ctx.fillStyle = '#f6f9ff'; ctx.fill(); }
  (o.xticks || []).forEach(t => { const v = Array.isArray(t) ? t[0] : t, lab = Array.isArray(t) ? t[1] : String(t);
    if (o.grid) line(p.X(v), p.y, p.X(v), p.y + p.h, '#e1e8f3', 1); T(lab, p.X(v), p.y + p.h + 15, { size: 12.5, weight: 600, color: P.muted, align: 'center' }); });
  (o.yticks || []).forEach(t => { const v = Array.isArray(t) ? t[0] : t, lab = Array.isArray(t) ? t[1] : String(t);
    if (o.grid !== false) line(p.x, p.Y(v), p.x + p.w, p.Y(v), '#e1e8f3', 1); T(lab, p.x - 8, p.Y(v), { size: 12.5, weight: 600, color: P.muted, align: 'right' }); });
  line(p.x, p.y + p.h, p.x + p.w, p.y + p.h, P.faint, 1.5); line(p.x, p.y, p.x, p.y + p.h, P.faint, 1.5);
  if (o.xl) T(o.xl, p.x + p.w, p.y + p.h + 32, { size: 13, weight: 700, color: P.muted, align: 'right' });
  if (o.yl) T(o.yl, p.x, p.y - 14, { size: 13, weight: 700, color: P.muted });
}
function pcurve(p, pts, color, lw, dash, upto){ if (pts.length < 2) return; ctx.save(); ctx.beginPath(); ctx.rect(p.x - 1, p.y - 1, p.w + 2, p.h + 2); ctx.clip();
  ctx.beginPath(); const n = upto == null ? pts.length - 1 : Math.min(pts.length - 1, upto);
  for (let i = 0; i <= n; i++){ const q = pts[i]; i ? ctx.lineTo(p.X(q[0]), p.Y(q[1])) : ctx.moveTo(p.X(q[0]), p.Y(q[1])); }
  ctx.strokeStyle = color; ctx.lineWidth = lw || 2.6; ctx.lineJoin = 'round'; ctx.setLineDash(dash || []); ctx.stroke(); ctx.setLineDash([]); ctx.restore(); }
function pfn(p, fn, x0, x1, color, lw, dash, N){ const pts = []; N = N || 240; for (let i = 0; i <= N; i++){ const x = x0 + (x1 - x0) * i / N; pts.push([x, fn(x)]); } pcurve(p, pts, color, lw, dash); }
function pdot(p, x, y, r, fill, stroke){ dot(p.X(x), p.Y(y), r, fill, stroke || '#fff', 2); }
/* elliptical contours of 0.5*(a x^2 + b y^2) */
function ellipses(p, a, b, levels, cx, cy){ ctx.save(); ctx.beginPath(); ctx.rect(p.x, p.y, p.w, p.h); ctx.clip();
  levels.forEach((c, i) => { const rx = Math.sqrt(2 * c / a), ry = Math.sqrt(2 * c / b); ctx.beginPath(); ctx.ellipse(p.X(cx || 0), p.Y(cy || 0), rx * p.w / (p.xr[1] - p.xr[0]), ry * p.h / (p.yr[1] - p.yr[0]), 0, 0, Math.PI * 2);
    ctx.strokeStyle = i < 2 ? '#7fa0ea' : '#b4c8f2'; ctx.lineWidth = 1.3; ctx.stroke(); }); ctx.restore(); }
function badge(x, y, text, bg, fg, size){ size = size || 14; const w = textWidth(text, size, 800) + 22; rr(x, y - 13, w, 26, 13); ctx.fillStyle = bg; ctx.fill(); T(text, x + w / 2, y, { size, weight: 800, color: fg, align: 'center' }); return w; }
function card(x, y, w, h, o){ o = o || {}; rr(x, y, w, h, o.r || 12); ctx.fillStyle = o.fill || '#fafcff'; ctx.fill(); ctx.strokeStyle = o.stroke || P.line; ctx.lineWidth = o.lw || 1.5; ctx.stroke(); }

/* stepwise derivation panel: scenes = [{tab, formulas:[{tex,h,size}], note:[zh,en], draw?}] */
function Deriv(o){
  const D = { step: 1, o }, gap = o.gap == null ? 12 : o.gap, n = o.scenes.length;
  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,
    options: o.scenes.map((s, i) => ({ id: i + 1, label: s.tab })), onChange: v => { D.step = v; sync(); } });
  o.scenes.forEach((s, i) => {
    const total = s.formulas.reduce((q, f) => q + f.h, 0) + Math.max(0, s.formulas.length - 1) * gap;
    let cy = top + ((areaBottom - top) - total) / 2;
    s.formulas.forEach((f, j) => { f.id = 'd' + o.id + '_' + i + '_' + j; addFormula({ id: f.id, tex: f.tex, x: o.x + o.w / 2, y: cy + f.h / 2, width: o.w - 72, size: f.size || 26 }); cy += f.h + gap; });
  });
  function sync(){ o.scenes.forEach((s, i) => s.formulas.forEach(f => showFormula(f.id, i + 1 === D.step))); }
  sync();
  D.draw = function(){
    panel(o.x, o.y, o.w, o.h);
    T(o.title[0], o.x + 24, o.y + 28, { size: 23, weight: 800, maxW: o.w - 48 });
    const s = o.scenes[D.step - 1];
    card(o.x + 24, top - 6, o.w - 48, areaBottom - top + 12, { fill: '#f3f7ff', stroke: P.blue, lw: 2 });
    if (s.draw) s.draw({ x: o.x + 24, y: top - 6, w: o.w - 48, h: areaBottom - top + 12 });
    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 = [];
  for (let i = 0; i <= nx; i++) gx.push(p.xr[0] + (p.xr[1] - p.xr[0]) * i / nx);
  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) => {
    ctx.strokeStyle = typeof color === 'function' ? color(lv, li) : (color || '#9db8ee'); ctx.beginPath();
    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 = [];
      for (let e = 0; e < 4; e++){ const a = e, b = (e + 1) % 4; if ((v[a] - lv) * (v[b] - lv) < 0){ const t = (lv - v[a]) / (v[b] - v[a]); pts.push([xs[a] + (xs[b] - xs[a]) * t, ys[a] + (ys[b] - ys[a]) * t]); } }
      if (pts.length === 2){ ctx.moveTo(p.X(pts[0][0]), p.Y(pts[0][1])); ctx.lineTo(p.X(pts[1][0]), p.Y(pts[1][1])); }
      else if (pts.length === 4){ ctx.moveTo(p.X(pts[0][0]), p.Y(pts[0][1])); ctx.lineTo(p.X(pts[1][0]), p.Y(pts[1][1])); ctx.moveTo(p.X(pts[2][0]), p.Y(pts[2][1])); ctx.lineTo(p.X(pts[3][0]), p.Y(pts[3][1])); }
    }
    ctx.stroke();
  });
  ctx.restore();
}
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; }

function pillText(text, x, y, o){ o = o || {}; const size = o.size || 13, w = textWidth(text, size, o.weight || 800) + 14, al = o.align || 'left', x0 = al === 'right' ? x - w + 7 : al === 'center' ? x - w / 2 : x - 7;
  rr(x0, y - size * 0.85, w, size * 1.7, 6); ctx.fillStyle = 'rgba(255,255,255,.92)'; ctx.fill(); return T(text, x, y, o); }

/* kp9 — SGD 收斂分析 */
const RP = RPG, NSAMP = 1024;
let YD = null, YMEAN = 0, YVAR = 0;
function data(){ if (YD) return; const r = mulberry(11); YD = new Float64Array(NSAMP); let s = 0; for (let i = 0; i < NSAMP; i++){ YD[i] = 4 + 3 * gauss(r); s += YD[i]; } YMEAN = s / NSAMP; let v = 0; for (let i = 0; i < NSAMP; i++) v += (YD[i] - YMEAN) ** 2; YVAR = v / NSAMP; }
const st = { eta: 0.2, bi: 0, sched: 'fixed', t0: 0 }, BSZ = [1, 4, 16, 64];
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: 12,
  scenes: [
    { tab: '① 全梯度 GD', formulas: [{ tex: String.raw`f\big(\boldsymbol{w}^{(t+1)}\big)-f^{*}\le\frac{2L\,\|\boldsymbol{w}^{(0)}-\boldsymbol{w}^{*}\|_2^{2}}{t+1} \tag{31}`, h: 80, size: 26 },
        { tex: String.raw`=\ \mathcal{O}\!\left(\frac{1}{t}\right)\quad(t\to\infty:\ \to 0)`, h: 70, size: 30 }],
      note: ['命題 2:凸且 L-smooth 時,全梯度下降的誤差以 O(1/t) 趨近 0,能精確收斂。', 'Proposition 2: for convex L-smooth f, full-gradient descent drives the error to 0 at rate O(1/t).'] },
    { tab: '② SGD 一般', formulas: [
        { tex: String.raw`\eta^{(\tau)}=\tfrac{1}{\tau}\ \Rightarrow\ \mathcal{O}\!\left(\tfrac{1}{\log t}\right) \tag{32}`, h: 58, size: 26 },
        { tex: String.raw`\eta^{(\tau)}=\tfrac{1}{\sqrt{\tau}}\ \Rightarrow\ \mathcal{O}\!\left(\tfrac{\log t}{\sqrt{t}}\right) \tag{33}`, h: 58, size: 26 },
        { tex: String.raw`\eta^{(\tau)}=\eta\ \Rightarrow\ \mathcal{O}\!\left(\tfrac{1}{t}+\eta\right) \tag{34}`, h: 58, size: 26 }],
      note: ['命題 3:SGD 的收斂率取決於步長策略;固定 η 時多了一項 η,不會歸零。', 'Proposition 3: the SGD rate depends on the step-size rule; with a fixed η an extra η term remains.'] },
    { tab: '③ SGD 強凸', formulas: [
        { tex: String.raw`\eta^{(\tau)}=\tfrac{1}{\mu\tau}\ \Rightarrow\ \mathcal{O}\!\left(\tfrac{1}{t}\right) \tag{35}`, h: 62, size: 28 },
        { tex: String.raw`\eta^{(\tau)}=\eta\ \Rightarrow\ \mathcal{O}\!\left(\big(1-\tfrac{\mu}{L}\big)^{t}+\eta\right) \tag{36}`, h: 66, size: 28 }],
      note: ['各 f_i 為 μ-strongly convex 時:遞減步長得 O(1/t);固定 η 仍留下 O(η)。', 'If each f_i is μ-strongly convex: decaying steps give O(1/t); a fixed η still leaves O(η).'] },
    { tab: '④ 停滯 Stagnation', formulas: [
        { tex: String.raw`\lim_{t\to\infty}f\big(\boldsymbol{w}^{(t+1)}\big)-f^{*}\le\mathcal{O}(\eta) \quad(\mathrm{SGD})`, h: 60, size: 25 },
        { tex: String.raw`\lim_{t\to\infty}f\big(\boldsymbol{w}^{(t+1)}\big)-f^{*}\le\mathcal{O}\!\left(\tfrac{1}{t}\right)=0 \quad(\mathrm{GD})`, h: 66, size: 25 }],
      note: ['固定 η 的 SGD 永遠到不了最優,只在 O(η) 鄰域內停滯;用精度換每步便宜。', 'Fixed-η SGD never reaches the optimum: it stagnates in an O(η) neighbourhood — accuracy traded for cheap steps.'] }
  ] });
slider({ id: 'eta', x: RP.x + 24, y: RP.y + 486, w: 280, min: 0.02, max: 0.5, step: 0.02, value: st.eta, color: P.amber, label: '步長 η', fmt: v => f2(v, 2), onChange: v => { st.eta = v; st.t0 = performance.now(); setAnimating(true); } });
slider({ id: 'bi', x: RP.x + 330, y: RP.y + 486, w: 230, min: 0, max: 3, step: 1, value: st.bi, color: P.blue, label: '批次 b', fmt: v => 'b = ' + BSZ[v], onChange: v => { st.bi = v; st.t0 = performance.now(); setAnimating(true); } });
seg({ id: 'sched', x: RP.x + 586, y: RP.y + 490, w: 290, h: 40, value: 'fixed', fs: 14, color: P.violet, options: [{ id: 'fixed', label: '固定 η fixed' }, { id: 'decay', label: '遞減 decay' }], onChange: v => { st.sched = v; st.t0 = performance.now(); setAnimating(true); } });
button({ id: 'replay', x: RP.x + 700, y: RP.y + 646, w: 176, h: 44, text: '重播 Replay', color: P.navy, onClick: () => { st.t0 = performance.now(); setAnimating(true); } });
const cache = {};
function run(){ data(); const b = BSZ[st.bi], key = st.eta + '|' + b + '|' + st.sched; if (cache[key]) return cache[key];
  const r = mulberry(100 + b), Tn = 120, sgd = [], gd = []; let w = 0, wg = 0;
  for (let k = 0; k <= Tn; k++){ sgd.push(Math.max(1e-5, 0.5 * (w - YMEAN) ** 2)); gd.push(Math.max(1e-5, 0.5 * (wg - YMEAN) ** 2));
    let m = 0; for (let j = 0; j < b; j++) m += YD[Math.floor(r() * NSAMP)]; m /= b; const e = st.sched === 'fixed' ? st.eta : Math.min(st.eta, 1 / (k + 1)); w -= e * (w - m); wg -= st.eta * (wg - YMEAN); }
  return (cache[key] = { sgd, gd, floor: st.eta * YVAR / (2 * b * (2 - st.eta)) }); }
function draw(now){
  chrome({ tag: 'kp9 · SGD 收斂分析 Convergence analysis', zh: 'SGD 收斂分析:醉漢下山,固定步幅永遠在 η 寬的帶子裡晃', en: 'SGD convergence: with a fixed step it stays forever in a band of width O(η)', foot: 'kp9 · 本知識點共 1 張 · 1 page' });
  D.draw();
  const { x, y, w, h } = RP; panel(x, y, w, h); const S = run(), prog = clamp((now - st.t0) / 3500, 0, 1); if (prog >= 1) setAnimating(false);
  T('一維強凸模擬:f_{i}(w) = ½(w − y_{i})²,n = 1024', x + 24, y + 30, { size: 23, weight: 800 });
  T('Real simulation (seeded): loss gap f(w_{t}) − f* on a log axis; the dashed line is the predicted stall level', x + 24, y + 58, { size: 15, weight: 500, color: P.muted });
  const p = mkP(x + 74, y + 96, 780, 340, [0, 120], [1e-5, 10], { logY: true });
  pframe(p, { xticks: [0, 20, 40, 60, 80, 100, 120], yticks: [[1e-5, '10⁻⁵'], [1e-4, '10⁻⁴'], [1e-3, '10⁻³'], [1e-2, '10⁻²'], [1e-1, '10⁻¹'], [1, '1'], [10, '10']], xl: 'iteration t', yl: 'loss gap(log)' });
  pfn(p, t => 32 / (t + 1), 0, 120, P.faint, 2, [5, 5]); pillText('命題 2 上界 bound O(1/t)', p.X(30), p.Y(32 / 31) - 20, { size: 13, weight: 700, color: P.muted, ok: true });
  const fl = st.sched === 'fixed' ? S.floor : 0;
  if (fl > 1e-5){ line(p.x, p.Y(fl), p.x + p.w, p.Y(fl), P.amber, 2, [7, 5]); pillText('預測停滯位準 predicted stall ≈ ' + fl.toExponential(1), p.x + p.w - 6, p.Y(fl) + 18, { size: 13, weight: 800, color: P.amber, align: 'right', ok: true }); }
  const up = Math.floor(prog * 120);
  pcurve(p, S.gd.map((v, i) => [i, v]), P.blue, 3, null, up); pcurve(p, S.sgd.map((v, i) => [i, v]), P.amber, 2, null, up);
  badge(p.x + 14, p.y + 24, '全梯度 GD → 0', P.blueS, P.blue, 13); badge(p.x + 14, p.y + 56, 'SGD ' + (st.sched === 'fixed' ? '固定 η' : '遞減 η_τ = min(η, 1/τ)'), P.amberS, P.amber, 13);
  const tail = S.sgd.slice(80).reduce((a, v) => a + v, 0) / 41;
  T('t = 80–120 的平均 loss gap · mean gap over t = 80–120:  SGD ' + tail.toExponential(1) + ' ,  GD ≈ 0', x + 24, y + 466, { size: 15.5, weight: 800, maxW: 850 });
  T('b 越大、η 越小,停滯帶越窄;遞減步長把帶子壓扁。 · Larger b or smaller η narrows the band; a decaying step flattens it.', x + 24, y + 558, { size: 14.5, weight: 600, color: P.muted, maxW: 850 });
}
window.__states = [
  { name: 'default', set: {} }, { name: 'decay', set: { sched: 'decay' } }, { name: 'eta-max-b1', set: { eta: 0.5, bi: 0 } }, { name: 'eta-min-b64', set: { eta: 0.02, bi: 3 } }, { name: 'b64-fixed', set: { bi: 3, eta: 0.3 } },
  { name: 'step2', set: { step: 2 } }, { name: 'step3', set: { step: 3 } }, { name: 'step4', set: { step: 4 } }
];
window.__reset = () => { const U = window.__ui; U.set('step', 1); U.set('eta', 0.2); U.set('bi', 0); U.set('sched', 'fixed'); };
window.__settle = () => { st.t0 = performance.now() - 8000; };
start(draw, 'kp9 SGD 收斂分析:全梯度 O(1/t) 精確收斂;固定步長的 SGD 停滯在 O(η) 鄰域;以真實模擬呈現。');

})();
</script>
</body>
</html>