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10-sgd-single-sample.html
<!doctype html>
<html lang="zh-Hant">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>kp7 隨機梯度下降 · SGD</title>
<link rel="preconnect" href="https://fonts.googleapis.com">
<link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
<link href="https://fonts.googleapis.com/css2?family=Noto+Sans+TC:wght@400;500;600;700;800;900&display=swap" rel="stylesheet">
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<script defer src="https://cdn.jsdelivr.net/npm/katex@0.19.0/dist/katex.min.js"></script>
<style>
html,body{margin:0;height:100%;background:#0b1630;overflow:hidden}
body{display:flex;align-items:center;justify-content:center;font-family:"Noto Sans TC","Noto Sans CJK TC","PingFang TC","Microsoft JhengHei",sans-serif}
.stage{position:relative;width:min(100vw,177.7778vh);height:min(56.25vw,100vh);aspect-ratio:16/9;background:#f3f6fb;overflow:hidden}
canvas{position:absolute;inset:0;width:100%;height:100%;display:block;outline:none}
canvas:focus-visible{outline:3px solid #2459d6;outline-offset:-3px}
#math-layer{position:absolute;left:0;top:0;width:1600px;height:900px;transform-origin:top left;pointer-events:none}
.formula{position:absolute;white-space:nowrap;line-height:1.25}
.formula .katex-display{margin:0}
.formula .katex{font-size:1em}
.formula.left .katex-display,.formula.left .katex-display>.katex{text-align:left}
.formula.right .katex-display,.formula.right .katex-display>.katex{text-align:right}
.formula.center .katex-display,.formula.center .katex-display>.katex{text-align:center}
</style>
</head>
<body>
<div class="stage" id="stage">
<canvas id="canvas" width="1600" height="900" tabindex="0" role="img" aria-label="kp7 隨機梯度下降 · SGD"></canvas>
<div id="math-layer"></div>
</div>
<script>
'use strict';
/* ===== InfographicsCreation shared runtime (Canvas 1600x900 + KaTeX math layer) ===== */
const W = 1600, H = 900;
const P = { paper:'#f3f6fb', grid:'#dfe7f2', ink:'#0f2347', muted:'#475b7d', faint:'#8ea3c3', line:'#c7d4e8',
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amber:'#d9760a', amberS:'#fdebcd', violet:'#6a43dc', violetS:'#e7dffb', white:'#ffffff', navy:'#0f2347' };
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const stage = document.getElementById('stage'), mathLayer = document.getElementById('math-layer');
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}, 40);
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const size = o.size || 18, weight = o.weight || 500, align = o.align || 'left';
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ctx.fillStyle = o.color || P.ink; ctx.textAlign = align; ctx.textBaseline = 'middle';
ctx.fillText(s, x, y);
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function Tsup(s, x, y, o, size, weight, align){
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let cx = align === 'center' ? x - total/2 : align === 'right' ? x - total : x;
ctx.fillStyle = o.color || P.ink; ctx.textAlign = 'left'; ctx.textBaseline = 'middle';
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return total;
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} else if (ch === ' '){ buf += ch; out.push(buf); buf = ''; } else buf += ch;
}
if (buf) out.push(buf); return out;
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function wrapLines(s, maxW, size, weight){
ctx.font = weight + ' ' + size + 'px ' + FZ;
const lines = []; let cur = '';
for (const t of tokenize(s)){
const test = cur + t;
if (cur && ctx.measureText(test).width > maxW){ lines.push(cur.trimEnd()); cur = t.trimStart(); } else cur = test;
}
if (cur) lines.push(cur.trimEnd()); return lines;
}
/* wrap(text, x, y, maxWidth, lineHeight, fontSize, color, weight, align) -> number of lines.
textAlign=left: x is the left edge; center: x is the visual centre; right: x is the right edge. */
function wrap(t, x, y, maxW, lh, size, color, weight, align){
t = String(t).replace(/\\n/g, '\n'); size = size || 18; weight = weight || 500; align = align || 'left';
let n = 0;
for (const para of t.split('\n')){
for (const l of wrapLines(para, maxW, size, weight)){ T(l, x, y + n*lh, { size, weight, color, align, maxW }); n++; }
}
return n;
}
function wrapCenter(t, cx, y, maxW, lh, size, color, weight){ return wrap(t, cx, y, maxW, lh, size, color, weight, 'center'); }
function textWidth(s, size, weight){ ctx.font = (weight||500) + ' ' + size + 'px ' + FZ; return ctx.measureText(String(s)).width; }
/* ---------- shapes ---------- */
function rr(x, y, w, h, r){
ctx.beginPath(); ctx.moveTo(x+r, y); ctx.arcTo(x+w, y, x+w, y+h, r); ctx.arcTo(x+w, y+h, x, y+h, r);
ctx.arcTo(x, y+h, x, y, r); ctx.arcTo(x, y, x+w, y, r); ctx.closePath();
}
function panel(x, y, w, h, o){
o = o || {}; rr(x, y, w, h, o.r == null ? 16 : o.r);
ctx.fillStyle = o.fill || '#fff'; ctx.fill();
if (o.lw !== 0){ ctx.strokeStyle = o.stroke || P.line; ctx.lineWidth = o.lw || 1.6; ctx.stroke(); }
}
function arrow(x1, y1, x2, y2, color, lw, hs){
lw = lw || 3; hs = hs || 11; const a = Math.atan2(y2-y1, x2-x1);
ctx.strokeStyle = color; ctx.fillStyle = color; ctx.lineWidth = lw; ctx.lineCap = 'round';
ctx.beginPath(); ctx.moveTo(x1, y1); ctx.lineTo(x2 - Math.cos(a)*hs*0.6, y2 - Math.sin(a)*hs*0.6); ctx.stroke();
ctx.beginPath(); ctx.moveTo(x2, y2);
ctx.lineTo(x2 - hs*Math.cos(a-0.45), y2 - hs*Math.sin(a-0.45));
ctx.lineTo(x2 - hs*Math.cos(a+0.45), y2 - hs*Math.sin(a+0.45)); ctx.closePath(); ctx.fill();
}
function dot(x, y, r, fill, stroke, lw){
ctx.beginPath(); ctx.arc(x, y, r, 0, Math.PI*2);
if (fill){ ctx.fillStyle = fill; ctx.fill(); }
if (stroke){ ctx.strokeStyle = stroke; ctx.lineWidth = lw || 1.5; ctx.stroke(); }
}
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ctx.beginPath(); ctx.moveTo(x1, y1); ctx.lineTo(x2, y2); ctx.strokeStyle = color; ctx.lineWidth = lw || 1.5;
ctx.setLineDash(dash || []); ctx.stroke(); ctx.setLineDash([]);
}
const clamp = (v, a, b) => Math.max(a, Math.min(b, v));
const lerp = (a, b, t) => a + (b - a) * t;
const ease = t => t < 0.5 ? 2*t*t : 1 - Math.pow(-2*t + 2, 2) / 2;
/* ---------- page chrome ---------- */
function chrome(m){
ctx.fillStyle = P.paper; ctx.fillRect(0, 0, W, H);
ctx.strokeStyle = P.grid; ctx.lineWidth = 1; ctx.beginPath();
for (let x = 0; x <= W; x += 40){ ctx.moveTo(x + .5, 0); ctx.lineTo(x + .5, H); }
for (let y = 0; y <= H; y += 40){ ctx.moveTo(0, y + .5); ctx.lineTo(W, y + .5); }
ctx.stroke();
const cw = textWidth(m.tag, 15.5, 700) + 30;
rr(40, 24, cw, 28, 14); ctx.strokeStyle = P.blue; ctx.lineWidth = 1.6; ctx.stroke();
T(m.tag, 40 + cw/2, 38.5, { size: 15.5, weight: 700, color: P.blue, align: 'center' });
T(m.zh, 40, 82, { size: 42, weight: 900, color: P.ink });
T(m.en, 42, 119, { size: 22, weight: 500, color: P.blue });
T('IDL 115 · Chapter 3 — Backpropagation & Optimization in Deep Neural Networks', 40, 886, { size: 13.5, weight: 400, color: P.muted });
T(m.foot, W - 40, 886, { size: 13.5, weight: 600, color: P.muted, align: 'right' });
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const MATH = {};
function placeFormula(m){
const o = m.o, el = m.el;
el.style.fontSize = (o.size || 28) + 'px'; el.style.color = o.color || P.ink; el.style.width = o.width + 'px';
el.className = 'formula ' + (o.align || 'center');
const left = o.align === 'left' ? o.x : o.align === 'right' ? o.x - o.width : o.x - o.width/2;
el.style.left = left + 'px'; el.style.top = o.y + 'px';
el.style.transform = 'translate(' + (o.offsetX || 0) + 'px,' + ((o.offsetY || 0) - 0) + 'px) translateY(-50%)';
}
function addFormula(o){
const el = document.createElement('div'); mathLayer.appendChild(el);
const m = MATH[o.id] = { el, o: Object.assign({ align:'center', size:28, width:600 }, o), tex: null };
placeFormula(m); setFormula(o.id, o.tex); return m;
}
function setFormula(id, tex, patch){
const m = MATH[id]; if (!m) return;
if (patch){ Object.assign(m.o, patch); placeFormula(m); }
if (m.tex !== tex){
m.tex = tex;
if (window.katex) window.katex.render(tex, m.el, { displayMode: true, throwOnError: false });
else { m.el.textContent = ''; AUD.bad.push('KaTeX not loaded'); }
}
}
function showFormula(id, on){ const m = MATH[id]; if (m) m.el.style.visibility = on ? 'visible' : 'hidden'; }
function syncMathLayer(){
const r = stage.getBoundingClientRect();
mathLayer.style.transform = 'scale(' + (r.width / W) + ',' + (r.height / H) + ')';
}
/* ---------- canvas widgets (slider / segmented / button) ---------- */
const UI = { items: [], focus: -1, drag: null, hover: -1 };
function sliderVal(it, t){
let v = it.log ? Math.pow(10, lerp(Math.log10(it.min), Math.log10(it.max), t)) : lerp(it.min, it.max, t);
if (it.step) v = Math.round(v / it.step) * it.step;
return clamp(+v.toFixed(6), it.min, it.max);
}
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); }
function slider(o){ const it = Object.assign({ type:'slider', color:P.blue, fmt: v => String(v), value: o.min, visible: true }, o); UI.items.push(it); return it; }
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){
if (it.type === 'slider') return px >= it.x - 14 && px <= it.x + it.w + 14 && py >= it.y - 8 && py <= it.y + 46;
return px >= it.x && px <= it.x + it.w && py >= it.y && py <= it.y + it.h;
}
function uiDraw(){
UI.items.forEach((it, idx) => {
if (!isVis(it)) return;
const focus = UI.focus === idx;
if (it.type === 'slider'){
T(it.label, it.x, it.y + 6, { size: 16, weight: 700, color: P.ink });
T(it.fmt(it.value), it.x + it.w, it.y + 6, { size: 16, weight: 800, color: it.color, align: 'right' });
const ty = it.y + 34, t = sliderT(it);
rr(it.x, ty - 3, it.w, 6, 3); ctx.fillStyle = '#d5dfef'; ctx.fill();
rr(it.x, ty - 3, it.w * t, 6, 3); ctx.fillStyle = it.color; ctx.fill();
dot(it.x + it.w * t, ty, 11, '#fff', it.color, 3.2);
if (focus){ dot(it.x + it.w * t, ty, 17, null, P.ink, 2); }
} else if (it.type === 'seg'){
const n = it.options.length, bw = it.w / n;
panel(it.x, it.y, it.w, it.h, { r: 10, fill: '#eef3fb', stroke: P.line, lw: 1.4 });
it.options.forEach((op, i) => {
const on = op.id === it.value, bx = it.x + i * bw;
if (on){ rr(bx + 3, it.y + 3, bw - 6, it.h - 6, 8); ctx.fillStyle = op.color || it.color; ctx.fill(); }
T(op.label, bx + bw/2, it.y + it.h/2, { size: it.fs || 16, weight: 800, color: on ? '#fff' : P.muted, align: 'center', maxW: bw - 8 });
});
if (focus){ rr(it.x - 3, it.y - 3, it.w + 6, it.h + 6, 12); ctx.strokeStyle = P.ink; ctx.lineWidth = 2; ctx.stroke(); }
} else if (it.type === 'button'){
const dis = typeof it.disabled === 'function' ? it.disabled() : it.disabled;
rr(it.x, it.y, it.w, it.h, 10);
ctx.fillStyle = dis ? '#dfe6f2' : (it.ghost ? '#fff' : it.color); ctx.fill();
ctx.strokeStyle = dis ? P.line : it.color; ctx.lineWidth = 1.8; ctx.stroke();
T(typeof it.text === 'function' ? it.text() : it.text, it.x + it.w/2, it.y + it.h/2, { size: it.fs || 16, weight: 800, align: 'center', color: dis ? P.faint : (it.ghost ? it.color : '#fff'), maxW: it.w - 8 });
if (focus){ rr(it.x - 3, it.y - 3, it.w + 6, it.h + 6, 12); ctx.strokeStyle = P.ink; ctx.lineWidth = 2; ctx.stroke(); }
}
});
}
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 => {
const [px, py] = uiPoint(e);
for (let i = UI.items.length - 1; i >= 0; i--){
const it = UI.items[i]; if (!isVis(it) || !uiHit(it, px, py)) continue;
UI.focus = i; canvas.focus();
if (it.type === 'slider'){ UI.drag = it; canvas.setPointerCapture(e.pointerId); const v = sliderVal(it, clamp((px - it.x) / it.w, 0, 1)); if (v !== it.value){ it.value = v; it.onChange && it.onChange(v); } }
else if (it.type === 'seg'){ const k = clamp(Math.floor((px - it.x) / (it.w / it.options.length)), 0, it.options.length - 1); const v = it.options[k].id; if (v !== it.value){ it.value = v; it.onChange && it.onChange(v); } }
else if (it.type === 'button'){ const dis = typeof it.disabled === 'function' ? it.disabled() : it.disabled; if (!dis && it.onClick) it.onClick(); }
requestDraw(); return;
}
});
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);
if (cur === -1) nxt = e.shiftKey ? vis.length - 1 : 0;
if (nxt < 0 || nxt >= vis.length){ UI.focus = -1; requestDraw(); return; }
UI.focus = vis[nxt]; e.preventDefault(); requestDraw(); return;
}
const it = UI.items[UI.focus]; if (!it) return;
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) + ')');
});
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; }
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); }
/* kp7 — SGD */
const RP = RPG, Y = [2, 1, 8, 5], YM = 4;
const st = { w: 0, k: 0, eta: 0.3, sched: 'const', pick: 1, trace: [0], last: null, rng: mulberry(7) };
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: 14,
scenes: [
{ tab: '① 問題設定 Setup', formulas: [
{ tex: String.raw`\underset{\boldsymbol{w}}{\operatorname{minimize}}\ \frac{1}{n}\sum_{i=1}^{n}f_i(\boldsymbol{w}) \tag{20}`, h: 74, size: 27 },
{ tex: String.raw`\nabla f(\boldsymbol{w})=\frac{1}{n}\sum_{i=1}^{n}\nabla f_i(\boldsymbol{w}) \tag{21}`, h: 74, size: 27 }],
note: ['總 loss 是 n 筆樣本 loss 的平均;全梯度每步要算完整求和,成本 O(n)。', 'The loss averages n per-sample losses; the full gradient needs the whole sum — O(n) per step.'] },
{ tab: '② SGD 更新 Update', formulas: [
{ tex: String.raw`\boldsymbol{w}^{(k+1)}:=\boldsymbol{w}^{(k)}-\eta^{(k)}\,\nabla f_i\big(\boldsymbol{w}^{(k)}\big) \tag{22}`, h: 56, size: 28 },
{ tex: String.raw`\text{instead of}\quad\boldsymbol{w}^{(k+1)}:=\boldsymbol{w}^{(k)}-\eta\,\nabla f\big(\boldsymbol{w}^{(k)}\big)`.replace('\\text{instead of}\\quad', '\\mathrm{vs.}\\quad'), h: 52, size: 24 }],
note: ['每步隨機抽一筆 i,用它的梯度更新:有雜訊,但期望方向仍指向全梯度,且每步便宜得多。', 'Draw one random i per step: noisy, but the expected direction is still the full gradient, and each step is far cheaper.'] },
{ tab: '③ 學習率排程 Schedules', formulas: [
{ tex: String.raw`\eta^{(k)}:=\frac{1}{k}`, h: 62, size: 32 }, { tex: String.raw`\eta^{(k)}:=\frac{1}{\sqrt{k}}`, h: 62, size: 32 }, { tex: String.raw`\eta^{(k)}:=\eta\quad(\text{fixed})`.replace('(\\text{fixed})', '(\\mathrm{fixed})'), h: 50, size: 30 }],
note: ['離最優點遠時步子大、靠近時步子小;固定 η 實作最簡單,但會在最優點附近晃動。', 'Large steps far away, small steps close in; a fixed η is simplest but keeps jittering near the optimum.'] }
] });
const stepSize = () => st.sched === 'const' ? st.eta : st.sched === 'inv' ? st.eta / (st.k + 1) : st.eta / Math.sqrt(st.k + 1);
function doStep(i, full){ const g = full ? st.w - YM : st.w - Y[i - 1], e = stepSize(); st.last = { i: full ? 0 : i, from: st.w, e }; st.w = st.w - e * g; st.k++; st.trace.push(st.w); if (st.trace.length > 81) st.trace.shift(); }
const reset = () => { st.w = 0; st.k = 0; st.trace = [0]; st.last = null; st.rng = mulberry(7); };
seg({ id: 'pick', x: RP.x + 24, y: RP.y + 296, w: 430, h: 42, value: 1, fs: 14, color: P.red, options: Y.map((y, i) => ({ id: i + 1, label: 'i = ' + (i + 1) + ' (y = ' + y + ')' })), onChange: v => { st.pick = v; } });
seg({ id: 'sched', x: RP.x + 480, y: RP.y + 296, w: 396, h: 42, value: 'const', fs: 14, color: P.violet, options: [{ id: 'const', label: '固定 η' }, { id: 'inv', label: 'η / k' }, { id: 'sqrt', label: 'η / √k' }], onChange: v => { st.sched = v; reset(); } });
slider({ id: 'eta', x: RP.x + 24, y: RP.y + 356, w: 430, min: 0.05, max: 1, step: 0.05, value: st.eta, color: P.amber, label: '基礎學習率 Base η', fmt: v => f2(v, 2), onChange: v => { st.eta = v; reset(); } });
button({ id: 'sgd', x: RP.x + 24, y: RP.y + 430, w: 204, h: 44, text: 'SGD 一步(所選 i)', color: P.red, fs: 15, onClick: () => doStep(st.pick, false) });
button({ id: 'full', x: RP.x + 240, y: RP.y + 430, w: 204, h: 44, text: '全梯度一步 Full step', color: P.green, fs: 15, onClick: () => doStep(0, true) });
button({ id: 'rand', x: RP.x + 456, y: RP.y + 430, w: 204, h: 44, text: '隨機抽樣 ×20 random', color: P.blue, fs: 15, onClick: () => { for (let j = 0; j < 20; j++) doStep(1 + Math.floor(st.rng() * 4), false); } });
button({ id: 'reset', x: RP.x + 672, y: RP.y + 430, w: 204, h: 44, text: '重設 Reset', ghost: true, color: P.muted, fs: 15, onClick: reset });
function draw(){
chrome({ tag: 'kp7 · 隨機梯度下降 SGD', zh: '隨機梯度下降:用一筆帳單估全公司平均', en: 'Stochastic gradient descent: estimate the average from a single sample', foot: 'kp7 · 本知識點共 1 張 · 1 page' });
D.draw();
const { x, y, w, h } = RP; panel(x, y, w, h);
T('四筆資料:f_{i}(w) = ½(w − y_{i})²', x + 24, y + 30, { size: 23, weight: 800 });
T('y = (2, 1, 8, 5), so ∇f_{i} = w − y_{i} and the full gradient is w − ȳ with ȳ = 4', x + 24, y + 58, { size: 15, weight: 500, color: P.muted });
const p = mkP(x + 60, y + 150, 780, 4, [-1, 10], [0, 1]); const X = p.X;
card(x + 24, y + 84, w - 48, 190, { fill: '#f6f9ff', stroke: '#dbe5f3' });
line(X(-1), y + 200, X(10), y + 200, P.faint, 2); for (let t = 0; t <= 9; t++){ line(X(t), y + 195, X(t), y + 205, P.faint, 1.5); T(String(t), X(t), y + 218, { size: 12.5, weight: 600, color: P.muted, align: 'center' }); }
Y.forEach((v, i) => { const on = st.pick === i + 1; dot(X(v), y + 200, on ? 11 : 8, on ? P.red : P.redS, P.red, 2.4); T('y_{' + (i + 1) + '} = ' + v, X(v), y + 172 - (i % 2) * 0, { size: 13.5, weight: 800, color: P.red, align: 'center' }); });
const tri = (cx, col) => { ctx.beginPath(); ctx.moveTo(cx, y + 206); ctx.lineTo(cx - 8, y + 222); ctx.lineTo(cx + 8, y + 222); ctx.closePath(); ctx.fillStyle = col; ctx.fill(); };
tri(X(YM), P.green); T('ȳ = 4(全梯度的目標 full-gradient target)', X(YM), y + 240, { size: 13.5, weight: 800, color: P.green, align: 'center', maxW: 300 });
const e = stepSize();
Y.forEach((v, i) => { const tx = st.w + e * (v - st.w); arrow(X(st.w), y + 200 - 4 - i * 0, X(tx) + 0.01, y + 200 - 4, i + 1 === st.pick ? 'rgba(211,52,42,.9)' : 'rgba(211,52,42,.25)', i + 1 === st.pick ? 4 : 2, i + 1 === st.pick ? 12 : 8); });
arrow(X(st.w), y + 188, X(st.w + e * (YM - st.w)) + 0.01, y + 188, P.green, 3.4, 11);
dot(X(st.w), y + 200, 9, P.ink, '#fff', 2.4); T('w', X(st.w), y + 258, { size: 14, weight: 800, color: P.ink, align: 'center' });
T('k = ' + st.k + ' w = ' + f2(st.w, 3) + ' η_{k} = ' + f2(e, 3), x + 500, y + 372, { size: 16, weight: 800, maxW: 376 });
T('|w − ȳ| = ' + f2(Math.abs(st.w - YM), 3), x + 500, y + 398, { size: 15, weight: 700, color: P.muted, maxW: 376 });
const q = mkP(x + 64, y + 530, 790, 112, [0, 80], [-1, 9]); pframe(q, { xticks: [0, 20, 40, 60, 80], yticks: [0, 4, 8], xl: 'step k', yl: 'w' });
line(q.x, q.Y(YM), q.x + q.w, q.Y(YM), P.green, 2, [6, 4]);
const k0 = st.k - (st.trace.length - 1); pcurve(q, st.trace.map((v, i) => [k0 + i - k0, v]), P.red, 2.4);
T('目標 target ȳ = 4', q.x + q.w - 4, q.Y(YM) - 12, { size: 13, weight: 800, color: P.green, align: 'right' });
T('w 隨步數的軌跡(最近 80 步)· trace of w over the last 80 steps', x + 24, y + 500, { size: 14.5, weight: 700 });
}
window.__states = [
{ name: 'start', set: {} }, { name: 'sgd-i3', set: { pick: 3 }, click: ['sgd'] }, { name: 'full', set: {}, click: ['full', 'full'] }, { name: 'rand-const', set: { eta: 0.5 }, click: ['rand', 'rand'] },
{ name: 'rand-inv', set: { sched: 'inv', eta: 1 }, click: ['rand', 'rand', 'rand'] }, { name: 'eta-min', set: { eta: 0.05, sched: 'sqrt' }, click: ['rand'] }, { name: 'step2', set: { step: 2 } }, { name: 'step3', set: { step: 3 } }
];
window.__reset = () => { const U = window.__ui; U.set('step', 1); U.set('pick', 1); U.set('sched', 'const'); U.set('eta', 0.3); U.items.find(i => i.id === 'reset').onClick(); };
window.__settle = () => {};
start(draw, 'kp7 隨機梯度下降:每步只抽一筆樣本估計全梯度;右側以四筆資料的平方損失互動比較 SGD 與全梯度。');
})();
</script>
</body>
</html>