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21-overparam-shallow-prop4.html

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<title>kp15 過參數化收斂 (1/2) 淺層與命題 4</title>
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<div id="math-layer"></div>
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<script>
'use strict';
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}
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}
function sliderT(it){ return it.log ? (Math.log10(it.value) - Math.log10(it.min)) / (Math.log10(it.max) - Math.log10(it.min)) : (it.value - it.min) / (it.max - it.min); }
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function seg(o){ const it = Object.assign({ type:'seg', color:P.blue, h:42, visible: true }, o); UI.items.push(it); return it; }
function button(o){ const it = Object.assign({ type:'button', color:P.navy, h:42, visible: true }, o); UI.items.push(it); return it; }
function isVis(it){ return typeof it.visible === 'function' ? it.visible() : it.visible; }
function uiHit(it, px, py){
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}
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}
function uiSet(id, v){ const it = UI.items.find(i => i.id === id); if (!it) return; it.value = v; if (it.onChange) it.onChange(v); requestDraw(); }
function uiPoint(e){ const r = canvas.getBoundingClientRect(); return [(e.clientX - r.left) * W / r.width, (e.clientY - r.top) * H / r.height]; }
canvas.addEventListener('pointerdown', e => {
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});
canvas.addEventListener('pointermove', e => {
  const [px, py] = uiPoint(e);
  if (UI.drag){ const it = UI.drag, v = sliderVal(it, clamp((px - it.x) / it.w, 0, 1)); if (v !== it.value){ it.value = v; it.onChange && it.onChange(v); requestDraw(); } return; }
  let over = false; UI.items.forEach(it => { if (isVis(it) && uiHit(it, px, py)) over = true; });
  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);
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  const it = UI.items[UI.focus]; if (!it) return;
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  } 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 => {
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    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) + ')');
  });
  for (let i = 0; i < boxes.length; i++) for (let j = i + 1; j < boxes.length; j++){
    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; }
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};

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/* shared helpers for the backpropagation pages */
const SIG = {
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};
function f2(v, n){ n = n == null ? 2 : n; const s = v.toFixed(n); return /^-0\.?0*$/.test(s) ? s.slice(1) : s; }
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}

/* ===== 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 (1/2) — 過參數化網路的收斂保證:淺層(命題 4) */
const RP = RPG, st = { k: 2, target: 'indef', seed: 5 }, NPT = 8, RUNS = 40, STEPS = 1500, LR = 0.05;
const TARGET = { indef: [[1.0, 0.6], [0.6, -0.8]], pd: [[1.2, 0.3], [0.3, 0.8]] };
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}\mapsto\boldsymbol{v}^{\top}\sigma(\boldsymbol{W}\boldsymbol{x}),\quad\boldsymbol{W}\in\mathbb{R}^{k\times d},\ \boldsymbol{v}\in\mathbb{R}^{k}`, h: 54, size: 24 },
        { tex: String.raw`\mathcal{L}(\boldsymbol{W})=\frac{1}{2n}\sum_{i=1}^{n}\Big(y_i-\boldsymbol{v}^{\top}\sigma(\boldsymbol{W}\boldsymbol{x}_i)\Big)^{2}`, h: 84, size: 25 }],
      note: ['一個隱藏層的網路,MSE 損失。Proposition 4 針對這個設定給出收斂保證。', 'A one-hidden-layer network with the MSE loss — the setting of Proposition 4.'] },
    { tab: '② 條件 Conditions', formulas: [
        { tex: String.raw`\sigma(z)=z^{2},\qquad k\ge 2d`, h: 58, size: 32 },
        { tex: String.raw`\#\{j:v_j>0\}\ge d,\qquad\#\{j:v_j<0\}\ge d`, h: 58, size: 28 }],
      note: ['激活函數為二次函數、隱藏層寬度 k ≥ 2d,且輸出權重 v 至少有 d 個正、d 個負。', 'A quadratic activation, hidden width k ≥ 2d, and an output vector v with at least d positive and d negative entries.'] },
    { tab: '③ 結論 Conclusions', formulas: [
        { tex: String.raw`\mathrm{local\ minimum}\ \Longrightarrow\ \mathrm{global\ minimum}`, h: 50, size: 27 },
        { tex: String.raw`\operatorname{vec}(\boldsymbol{U})^{\top}\,\nabla^{2}\mathcal{L}(\boldsymbol{W}_s)\,\operatorname{vec}(\boldsymbol{U})<0`, h: 54, size: 25 },
        { tex: String.raw`d\le n\le c\,d^{2}\ \Longrightarrow\ \min\mathcal{L}=0`, h: 54, size: 27 }],
      note: ['① 沒有假的局部極小;② 鞍點必有負曲率的逃逸方向;③ 資料量適中時可達零訓練損失。', '① No spurious local minima; ② every saddle has an escape direction of negative curvature; ③ zero training loss is attainable.'] }
  ] });
seg({ id: 'target', x: RP.x + 24, y: RP.y + 540, w: 380, h: 42, value: 'indef', fs: 14, color: P.violet, options: [{ id: 'indef', label: '目標:不定矩陣 indefinite' }, { id: 'pd', label: '目標:正定矩陣 PD' }], onChange: v => { st.target = v; } });
slider({ id: 'k', x: RP.x + 430, y: RP.y + 536, w: 300, min: 1, max: 8, step: 1, value: st.k, color: P.blue, label: '隱藏層寬度 Width k', fmt: v => 'k = ' + v, onChange: v => { st.k = v; } });
button({ id: 'inits', x: RP.x + 24, y: RP.y + 596, w: 260, h: 44, text: '換一批隨機初始值 New inits', color: P.navy, fs: 15, onClick: () => { st.seed += 11; } });
const cache = {};
function experiment(){ const key = st.k + st.target + st.seed; if (cache[key]) return cache[key];
  const M = TARGET[st.target], r0 = mulberry(5), X = [], Y = [];
  for (let i = 0; i < NPT; i++){ const x = [gauss(r0), gauss(r0)]; X.push(x); Y.push(M[0][0] * x[0] * x[0] + 2 * M[0][1] * x[0] * x[1] + M[1][1] * x[1] * x[1]); }
  const r = mulberry(st.seed + 999), k = st.k, finals = [], curves = [];
  for (let run = 0; run < RUNS; run++){
    const W = [], v = []; for (let j = 0; j < k; j++){ W.push([0.5 * gauss(r), 0.5 * gauss(r)]); v.push(j < Math.ceil(k / 2) ? 1 : -1); }
    const curve = []; let loss = 0;
    for (let s = 0; s <= STEPS; s++){ const g = W.map(() => [0, 0]); loss = 0;
      for (let i = 0; i < NPT; i++){ let yh = 0; const z = []; for (let j = 0; j < k; j++){ const a = W[j][0] * X[i][0] + W[j][1] * X[i][1]; z.push(a); yh += v[j] * a * a; }
        const e = yh - Y[i]; loss += 0.5 * e * e / NPT; for (let j = 0; j < k; j++){ const d = e * v[j] * 2 * z[j] / NPT; g[j][0] += d * X[i][0]; g[j][1] += d * X[i][1]; } }
      if (run < 12 && s % 25 === 0) curve.push([s, Math.max(loss, 1e-12)]);
      if (s < STEPS) for (let j = 0; j < k; j++){ W[j][0] -= LR * g[j][0]; W[j][1] -= LR * g[j][1]; } }
    finals.push(loss); if (run < 12) curves.push(curve); }
  return (cache[key] = { finals, curves }); }
function draw(){
  chrome({ tag: 'kp15 · 過參數化收斂 Over-parameterized convergence (1/2)', zh: '過參數化 ①:淺層網路與命題 4', en: 'Over-parameterization ①: the shallow network and Proposition 4', foot: 'kp15 · 本知識點共 2 張 · page 1 of 2' });
  D.draw();
  const { x, y, w, h } = RP; panel(x, y, w, h); const E = experiment();
  T('數值示意:σ(z) = z² 的網路,40 組隨機初始值', x + 24, y + 30, { size: 23, weight: 800 }); T('Seeded toy experiment: d = 2, n = 8, full-batch GD (η = 0.05, 1500 steps); v has ⌈k/2⌉ positive and the rest negative entries', x + 24, y + 58, { size: 14, weight: 500, color: P.muted, maxW: 850 });
  const A = mkP(x + 74, y + 100, 520, 300, [0, STEPS], [1e-12, 10], { logY: true });
  pframe(A, { xticks: [0, 500, 1000, 1500], yticks: [[1e-12, '10⁻¹²'], [1e-9, '10⁻⁹'], [1e-6, '10⁻⁶'], [1e-3, '10⁻³'], [1, '1']], xl: 'step', yl: '訓練損失 training loss(前 12 組)' });
  E.curves.forEach(c => pcurve(A, c, 'rgba(36,89,214,.55)', 2));
  const B = mkP(x + 660, y + 100, 190, 300, [0, 1], [1e-12, 10], { logY: true }); pframe(B, { yticks: [[1e-12, '≈0'], [1e-6, '10⁻⁶'], [1, '1']], yl: '最終損失 final' });
  E.finals.forEach((f, i) => { const jx = 0.15 + 0.7 * ((i * 0.61803) % 1); dot(B.X(jx), B.Y(Math.max(f, 1e-12)), 5.5, f < 1e-3 ? P.green : P.red, '#fff', 1.4); });
  const zero = E.finals.filter(f => f < 1e-3).length, lo = Math.min(...E.finals), hi = Math.max(...E.finals);
  card(x + 24, y + 452, w - 48, 62, { fill: zero === RUNS ? P.greenS : '#fff6ec', stroke: zero === RUNS ? P.green : P.amber, lw: 2.2 });
  T('收斂到零損失 reach ≈ 0 loss: ' + zero + ' / ' + RUNS + '     最終損失範圍 range: ' + (hi < 1e-12 ? '≈ 0' : (lo < 1e-12 ? '≈ 0' : lo.toExponential(1)) + ' ~ ' + hi.toExponential(1)), x + 40, y + 472, { size: 16, weight: 800, maxW: 820 });
  T(zero === RUNS ? '全部相同 → 沒有比全域更差的假局部極小 · all runs agree: no worse local minimum' : (hi / lo < 1.01 ? '全部落在同一個正的損失水準:此寬度連零損失都表示不出來(仍無假極小)' : '各組結果不同'), x + 40, y + 498, { size: 14.5, weight: 700, color: P.muted, maxW: 820 });
  wrap('定理給的是充分條件(k ≥ 2d = 4);這個玩具例子在更小的寬度就成功,寬度不足時則全部卡在同一個正損失。\\nThe theorem gives sufficient conditions (k ≥ 2d = 4); this toy succeeds at smaller widths, and when too narrow every run ends at the same positive loss.', x + 24, y + 656, 850, 19, 13.5, P.muted, 600);
}
window.__states = [{ name: 'default', set: {} }, { name: 'k1', set: { k: 1 } }, { name: 'k8', set: { k: 8 } }, { name: 'pd-k2', set: { target: 'pd', k: 2 } }, { name: 'pd-k3', set: { target: 'pd', k: 3 } }, { name: 'new-inits', set: { k: 4 }, click: ['inits'] }, { name: 'step2', set: { step: 2 } }, { name: 'step3', set: { step: 3 } }];
window.__reset = () => { const U = window.__ui; U.set('step', 1); U.set('target', 'indef'); U.set('k', 2); };
window.__settle = () => {};
start(draw, 'kp15 過參數化淺層網路(命題 4):σ=z² 網路在多組隨機初始值下的真實 GD 實驗,顯示無假的局部極小。');

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