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22-overparam-deep-prop5.html
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<title>kp15 過參數化收斂 (2/2) 深層與命題 5</title>
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'use strict';
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canvas.addEventListener('pointerdown', e => {
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requestDraw(); return;
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canvas.addEventListener('pointermove', e => {
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canvas.style.cursor = over ? 'pointer' : 'default';
});
canvas.addEventListener('pointerup', () => { UI.drag = null; });
canvas.addEventListener('keydown', e => {
const vis = UI.items.map((it, i) => i).filter(i => isVis(UI.items[i]));
if (!vis.length) return;
if (e.key === 'Tab'){
const cur = vis.indexOf(UI.focus); let nxt = cur + (e.shiftKey ? -1 : 1);
if (cur === -1) nxt = e.shiftKey ? vis.length - 1 : 0;
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UI.focus = vis[nxt]; e.preventDefault(); requestDraw(); return;
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if (it.type === 'slider' && (e.key === 'ArrowLeft' || e.key === 'ArrowRight' || e.key === 'ArrowDown' || e.key === 'ArrowUp')){
const dir = (e.key === 'ArrowRight' || e.key === 'ArrowUp') ? 1 : -1, t = sliderT(it);
const step = it.step ? (it.log ? 0 : it.step) : (it.max - it.min) / 50;
let v = it.log ? sliderVal(it, clamp(t + dir * 0.05, 0, 1)) : clamp(+(it.value + dir * step).toFixed(6), it.min, it.max);
it.value = v; it.onChange && it.onChange(v); e.preventDefault(); requestDraw();
} else if (it.type === 'seg' && (e.key === 'ArrowLeft' || e.key === 'ArrowRight')){
const k = it.options.findIndex(o => o.id === it.value), n = it.options.length;
it.value = it.options[(k + (e.key === 'ArrowRight' ? 1 : n - 1)) % n].id; it.onChange && it.onChange(it.value); e.preventDefault(); requestDraw();
} else if (it.type === 'button' && (e.key === 'Enter' || e.key === ' ')){ it.onClick && it.onClick(); e.preventDefault(); requestDraw(); }
});
canvas.addEventListener('blur', () => { UI.focus = -1; requestDraw(); });
/* ---------- render loop ---------- */
let drawFn = null, pending = false, animating = false;
function setAnimating(b){ animating = b; if (b) requestDraw(); }
function requestDraw(){ if (pending) return; pending = true; requestAnimationFrame(render); }
function fitCanvas(){
const r = stage.getBoundingClientRect(), dpr = Math.min(window.devicePixelRatio || 1, 2);
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();
}
function start(fn, ariaLabel){
drawFn = fn; if (ariaLabel) canvas.setAttribute('aria-label', ariaLabel);
syncMathLayer(); new ResizeObserver(() => { syncMathLayer(); requestDraw(); }).observe(stage);
window.addEventListener('resize', () => { syncMathLayer(); requestDraw(); });
requestDraw(); window.__ready = true;
}
/* ---------- audit hooks (used by the automated browser render audit) ---------- */
window.__ui = { set: uiSet, items: UI.items };
window.__audit = function(){
const issues = [], M = 4, boxes = AUD.text;
AUD.bad.forEach(b => issues.push('BAD: ' + b));
boxes.forEach(b => {
if (b.x0 < M || b.x1 > W - M || b.y0 < M || b.y1 > H - M) issues.push('OUT-OF-CANVAS: "' + b.s + '"');
if (b.maxW && (b.x1 - b.x0) > b.maxW + 2) issues.push('WIDER-THAN-MAXW: "' + b.s + '" (' + Math.round(b.x1 - b.x0) + '>' + Math.round(b.maxW) + ')');
});
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const a = boxes[i], b = boxes[j]; if (a.ok || b.ok) continue;
const ox = Math.min(a.x1, b.x1) - Math.max(a.x0, b.x0), oy = Math.min(a.y1, b.y1) - Math.max(a.y0, b.y0);
if (ox > 3 && oy > 5) issues.push('TEXT-OVERLAP: "' + a.s + '" x "' + b.s + '"');
}
const sr = stage.getBoundingClientRect(), sc = sr.width / W, fr = [];
Object.keys(MATH).forEach(id => {
const m = MATH[id]; if (m.el.style.visibility === 'hidden') return;
const h = m.el.querySelector('.katex-html'); if (!h){ issues.push('FORMULA-NOT-RENDERED: ' + id); return; }
const r = h.getBoundingClientRect(), box = { id, x0: (r.left - sr.left) / sc, x1: (r.right - sr.left) / sc, y0: (r.top - sr.top) / sc, y1: (r.bottom - sr.top) / sc };
fr.push(box);
if (box.x0 < M || box.x1 > W - M || box.y0 < M || box.y1 > H - M) issues.push('FORMULA-OUT-OF-CANVAS: ' + id);
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;
if (box.x0 < cx - 2 || box.x1 > cx + allow + 2) issues.push('FORMULA-OVERFLOW-WIDTH: ' + id + ' (' + Math.round(box.x1 - box.x0) + ' > ' + allow + ')');
if (/\\\\[A-Za-z]/.test(m.tex) && m.tex.indexOf('\\\\') !== -1) issues.push('SUSPECT-DOUBLE-BACKSLASH: ' + id);
const txt = h.textContent || '';
['hatp', 'frac', 'cdot', 'approx', 'theta', 'sigma', 'boldsymbol', 'nabla', 'partial'].forEach(w => { if (txt.indexOf(w) >= 0) issues.push('RAW-TEX-VISIBLE(' + w + '): ' + id); });
boxes.forEach(b => {
const ox = Math.min(box.x1, b.x1) - Math.max(box.x0, b.x0), oy = Math.min(box.y1, b.y1) - Math.max(box.y0, b.y0);
if (!b.ok && ox > 3 && oy > 5) issues.push('FORMULA-TEXT-OVERLAP: ' + id + ' x "' + b.s + '"');
});
});
for (let i = 0; i < fr.length; i++) for (let j = i + 1; j < fr.length; j++){
const a = fr[i], b = fr[j], ox = Math.min(a.x1, b.x1) - Math.max(a.x0, b.x0), oy = Math.min(a.y1, b.y1) - Math.max(a.y0, b.y0);
if (ox > 3 && oy > 3) issues.push('FORMULA-OVERLAP: ' + a.id + ' x ' + b.id);
}
return { issues, formulas: fr.map(f => ({ id: f.id, nx: +(f.x0 / W).toFixed(4), ny: +(f.y0 / H).toFixed(4), nw: +((f.x1 - f.x0) / W).toFixed(4) })), nText: boxes.length };
};
</script>
<script>
(function boot(){
if (!window.katex && !window.__katexWait){ window.__katexWait = 0; }
if (!window.katex && window.__katexWait++ < 60){ return setTimeout(boot, 100); }
if (!window.katex) console.warn('KaTeX failed to load');
/* 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*$/.test(s) ? s.slice(1) : s; }
const signCol = v => v >= 0 ? P.blue : P.red;
function lockedSlot(x, y, w, h, msg){
rr(x, y, w, h, 12); ctx.setLineDash([7, 5]); ctx.strokeStyle = P.faint; ctx.lineWidth = 1.6; ctx.stroke(); ctx.setLineDash([]);
T(msg, x + w/2, y + h/2, { size: 16, weight: 600, color: P.faint, align: 'center', maxW: w - 20 });
}
/* ===== shared helpers for all course pages (Deriv panel, plots, sims) ===== */
function mulberry(a){ return function(){ a |= 0; a = a + 0x6D2B79F5 | 0; let t = Math.imul(a ^ a >>> 15, 1 | a); t = t + Math.imul(t ^ t >>> 7, 61 | t) ^ t; return ((t ^ t >>> 14) >>> 0) / 4294967296; }; }
function gauss(r){ let u = 0; while (!u) u = r(); const v = r(); return Math.sqrt(-2 * Math.log(u)) * Math.cos(2 * Math.PI * v); }
function f2(v, n){ n = n == null ? 2 : n; const s = v.toFixed(n); return /^-0\.?0*$/.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); }
/* kp15 (2/2) — 深層網路(命題 5) */
const RP = RPG, st = { eps: 1e-2, wi: 4 }, WIDTHS = [4, 8, 16, 32, 64], NPT = 20, DIN = 4, STEPS = 600, LR = 0.1;
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 設定 Setup', formulas: [
{ tex: String.raw`\|\boldsymbol{x}_i\|_2=1,\qquad\|\boldsymbol{x}_i-\boldsymbol{x}_j\|_2\ge\delta\ \ (i\ne j)`, h: 56, size: 25 },
{ tex: String.raw`l\ \mathrm{layers},\ \ \mathrm{ReLU},\ \ \mathrm{MSE},\ \ \mathrm{random\ init}`, h: 46, size: 25 }],
note: ['l 層全連接 ReLU 網路、MSE 損失;資料已正規化,且任兩筆訓練樣本至少相距 δ,權重隨機初始化。', 'An l-layer fully-connected ReLU network with MSE; unit-norm inputs at least δ apart; random initialization.'] },
{ tab: '② 全批次 GD', formulas: [
{ tex: String.raw`m\ge\Omega\big(\mathrm{poly}(n,l,\delta^{-1})\,d\big),\quad\eta=\Theta\!\Big(\tfrac{d\,\delta}{\mathrm{poly}(n,l)\,m}\Big)`, h: 66, size: 22 },
{ tex: String.raw`T=\Theta\!\left(\frac{\mathrm{poly}(n,l)\,\log(\epsilon^{-1})}{\delta^{2}}\right) \tag{58}`, h: 84, size: 27 }],
note: ['寬度 m 夠大時,T 步後以高機率達到損失 < ε;T 隨 log(1/ε) 增長,並與 δ² 成反比。', 'For wide enough m, after T steps the loss is below ε with high probability; T grows with log(1/ε) and as 1/δ².'] },
{ tab: '③ Mini-batch SGD', formulas: [
{ tex: String.raw`m\ge\Omega\!\Big(\tfrac{\mathrm{poly}(n,l,\delta^{-1})\,d}{b}\Big)`, h: 66, size: 25 },
{ tex: String.raw`T=\Theta\!\left(\frac{\mathrm{poly}(n,l)\,\log(\epsilon^{-1})\,\log^{2}m}{\delta^{2}\,b}\right) \tag{59}`, h: 84, size: 25 }],
note: ['批次大小 b 的 SGD 同樣能在多項式時間內把訓練損失降到任意小,迭代數多一個 log²m/b 因子。', 'Mini-batch SGD also drives the training loss as low as desired in polynomial time, with an extra log²m / b factor.'] }
] });
slider({ id: 'eps', x: RP.x + 24, y: RP.y + 412, w: 380, min: 1e-5, max: 1e-1, log: true, value: st.eps, color: P.amber, label: '目標損失 Target ε', fmt: v => fmtSci(v), onChange: v => { st.eps = v; } });
slider({ id: 'wi', x: RP.x + 450, y: RP.y + 412, w: 280, min: 0, max: 4, step: 1, value: st.wi, color: P.blue, label: '強調寬度 Highlight m', fmt: v => 'm = ' + WIDTHS[v], onChange: v => { st.wi = v; } });
let DATA = null, X = [], Y = [], DELTA = 0, computing = false;
function build(){ const r = mulberry(3); X = []; Y = []; for (let i = 0; i < NPT; i++){ const x = []; let s = 0; for (let j = 0; j < DIN; j++){ const g = gauss(r); x.push(g); s += g * g; } s = Math.sqrt(s); X.push(x.map(v => v / s)); Y.push(r() < 0.5 ? -1 : 1); }
DELTA = 9; for (let i = 0; i < NPT; i++) for (let j = i + 1; j < NPT; j++) DELTA = Math.min(DELTA, Math.hypot(...X[i].map((v, k) => v - X[j][k]))); }
function train(m){ const rr = mulberry(m * 13 + 7), dims = [DIN, m, m, 1], L = 3, Ws = [];
for (let l = 0; l < L; l++){ const fi = dims[l], fo = dims[l + 1], W = new Float64Array(fi * fo), sc = Math.sqrt((l === L - 1 ? 1 : 2) / fi); for (let i = 0; i < W.length; i++) W[i] = sc * gauss(rr); Ws.push(W); }
const losses = [];
for (let s = 0; s <= STEPS; s++){ const G = Ws.map(W => new Float64Array(W.length)); let loss = 0;
for (let i = 0; i < NPT; i++){ const A = [X[i]], Z = [];
for (let l = 0; l < L; l++){ const fi = dims[l], fo = dims[l + 1], out = new Float64Array(fo); for (let o = 0; o < fo; o++){ let a = 0; for (let j = 0; j < fi; j++) a += Ws[l][o * fi + j] * A[l][j]; out[o] = a; } Z.push(out); A.push(l === L - 1 ? out : out.map(v => v > 0 ? v : 0)); }
const e = A[L][0] - Y[i]; loss += 0.5 * e * e / NPT; let delta = new Float64Array([e / NPT]);
for (let l = L - 1; l >= 0; l--){ const fi = dims[l], fo = dims[l + 1], nd = new Float64Array(fi); for (let o = 0; o < fo; o++) for (let j = 0; j < fi; j++){ G[l][o * fi + j] += delta[o] * A[l][j]; nd[j] += Ws[l][o * fi + j] * delta[o]; }
if (l > 0) for (let j = 0; j < fi; j++) if (Z[l - 1][j] <= 0) nd[j] = 0; delta = nd; } }
losses.push(loss); if (s < STEPS) for (let l = 0; l < L; l++) for (let i = 0; i < Ws[l].length; i++) Ws[l][i] -= LR * G[l][i]; }
return losses; }
function ensure(){ if (DATA || computing) return; computing = true; setTimeout(() => { build(); DATA = WIDTHS.map(train); computing = false; window.__ready_data = true; requestDraw(); }, 30); }
const COLS = ['#8ea3c3', '#6a8fe0', '#2459d6', '#6a43dc', '#d3342a'];
function draw(){
chrome({ tag: 'kp15 · 過參數化收斂 Over-parameterized convergence (2/2)', zh: '過參數化 ②:深層網路與命題 5', en: 'Over-parameterization ②: deep networks and Proposition 5', foot: 'kp15 · 本知識點共 2 張 · page 2 of 2' });
D.draw(); ensure();
const { x, y, w, h } = RP; panel(x, y, w, h);
T('數值示意:寬度越大,擬合隨機標籤越快', x + 24, y + 30, { size: 23, weight: 800 }); T('Seeded experiment: 3-layer ReLU net, n = 20 unit vectors in ℝ⁴ with random ±1 labels, full-batch GD η = 0.1, ' + STEPS + ' steps', x + 24, y + 58, { size: 14, weight: 500, color: P.muted, maxW: 850 });
const p = mkP(x + 74, y + 100, 770, 290, [0, STEPS], [1e-6, 1], { logY: true }); pframe(p, { xticks: [0, 100, 200, 300, 400, 500, 600], yticks: [[1e-6, '10⁻⁶'], [1e-5, '10⁻⁵'], [1e-4, '10⁻⁴'], [1e-3, '10⁻³'], [1e-2, '10⁻²'], [1e-1, '10⁻¹'], [1, '1']], xl: 'step t', yl: '訓練損失 training loss(log)' });
if (!DATA){ T('計算中… computing the real training runs', p.x + p.w / 2, p.y + p.h / 2, { size: 18, weight: 800, color: P.muted, align: 'center' }); return; }
line(p.x, p.Y(st.eps), p.x + p.w, p.Y(st.eps), P.amber, 2, [7, 5]); T('ε = ' + fmtSci(st.eps), p.x + p.w - 4, p.Y(st.eps) - 12, { size: 13, weight: 800, color: P.amber, align: 'right' });
DATA.forEach((ls, i) => { const on = i === st.wi; pcurve(p, ls.map((v, t) => [t, clamp(v, 1e-6, 1)]).filter((_, t) => t % 3 === 0), COLS[i], on ? 4.2 : 2.2, on ? null : null); });
WIDTHS.forEach((m, i) => { badge(p.x + 12 + i * 112, p.y + 20, 'm = ' + m, i === st.wi ? COLS[i] : '#eef3fb', i === st.wi ? '#fff' : COLS[i], 12.5); });
/* table of T */
T('達到損失 < ε 所需步數 T · steps to reach loss < ε', x + 24, y + 486, { size: 16, weight: 800 });
WIDTHS.forEach((m, i) => { const t = DATA[i].findIndex(v => v < st.eps), cx = x + 24 + i * 172, on = i === st.wi; card(cx, y + 506, 160, 66, { fill: on ? '#eef4ff' : '#fff', stroke: on ? COLS[i] : P.line, lw: on ? 3 : 1.4 });
T('m = ' + m, cx + 14, y + 524, { size: 15, weight: 800, color: COLS[i] }); T(t < 0 ? '> ' + STEPS : String(t), cx + 14, y + 554, { size: 24, weight: 900, color: t < 0 ? P.red : P.ink, maxW: 140 }); });
T('資料的最小間距 min ‖xᵢ − xⱼ‖ = δ = ' + f2(DELTA, 3) + '(命題 5 需 δ > 0)', x + 24, y + 596, { size: 14.5, weight: 700, maxW: 850 });
wrap('寬度小的網路在這段步數內做不到;寬度夠大時,損失幾乎沿直線下降(log 軸),即 T ∝ log(1/ε)。\\nNarrow nets cannot do it within these steps; wide ones fall almost linearly on the log axis, i.e. T ∝ log(1/ε).', x + 24, y + 620, 850, 18, 13, P.muted, 600);
wrap('命題 5 保證的是訓練損失;擬合隨機標籤不代表會泛化。\\nProposition 5 guarantees training loss — fitting random labels says nothing about generalization.', x + 24, y + 664, 850, 18, 13, P.amber, 700);
}
window.__states = [{ name: 'default', set: {} }, { name: 'eps-max', set: { eps: 0.1, wi: 0 } }, { name: 'eps-min', set: { eps: 1e-5, wi: 4 } }, { name: 'wi-mid', set: { wi: 2, eps: 1e-3 } }, { name: 'step2', set: { step: 2 } }, { name: 'step3', set: { step: 3 } }];
window.__reset = () => { const U = window.__ui; U.set('step', 1); U.set('eps', 1e-2); U.set('wi', 4); };
window.__settle = () => {};
start(draw, 'kp15 深層過參數化網路(命題 5):不同寬度的 ReLU 網路以 GD 擬合隨機標籤的真實實驗,寬度越大越快。');
})();
</script>
</body>
</html>