Backpropagation | From a Dot to a Chatbot

The network after 171 updates

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0.010114total error E

0.1217, 0.9020outputs (want 0.01, 0.99)

1. Forward pass: compute the answer

net_h1 = w1·i1 + w2·i2 + b1 = 0.2105·0.05 + 0.3210·0.10 + 0.35 = 0.392629

h1 = σ(net_h1) = 0.596915 h2 = σ(0.407255) = 0.600429

net_o1 = w5·h1 + w6·h2 + b2 = −1.976306 o1 = 0.121713

net_o2 = w7·h1 + w8·h2 + b2 = 2.219383 o2 = 0.901977

E = ½(t1 − o1)² + ½(t2 − o2)² = 0.006240 + 0.003874 = 0.010113967

What to notice

Output weights are easy. w5's blame is just three numbers multiplied: how wrong o1 is, how steep o1's sigmoid is, and how big h1 was (a big input means the weight mattered more).

Hidden weights reuse that work. w1 can't see the error directly. Its blame arrives through both outputs: output 1's blame times w5, plus output 2's blame times w7. Then the same local slopes. That reuse is what makes backprop fast enough for billions of weights.

The check column agrees. "Nudge and measure" gives the same numbers without any calculus, but it needs two extra forward passes per weight. For an LLM that would be billions of passes per step; backprop needs one.

2. Backward pass: blame for every weight

Weight	Local slopes multiplied together (chain rule)	∂E/∂w	Check (nudge)	New value = w − 0.50 × ∂E/∂w
w5	(o₁ − t₁) 0.11171 × o₁(1 − o₁) 0.10690 × h₁ 0.59692	0.0071284	0.0071284	−2.16904 → −2.17260273
w6	(o₁ − t₁) 0.11171 × o₁(1 − o₁) 0.10690 × h₂ 0.60043	0.0071703	0.0071703	−2.13443 → −2.13801389
w7	(o₂ − t₂) −0.08802 × o₂(1 − o₂) 0.08841 × h₁ 0.59692	−0.0046455	−0.0046455	1.32494 → 1.32725789
w8	(o₂ − t₂) −0.08802 × o₂(1 − o₂) 0.08841 × h₂ 0.60043	−0.0046729	−0.0046729	1.37986 → 1.38219639
w1	δo₁·w5 + δo₂·w7 −0.03621 × h₁(1 − h₁) 0.24061 × i₁ 0.05000	−0.0004357	−0.0004357	0.21052 → 0.21073490
w2	δo₁·w5 + δo₂·w7 −0.03621 × h₁(1 − h₁) 0.24061 × i₂ 0.10000	−0.0008713	−0.0008713	0.32103 → 0.32146980
w3	δo₁·w6 + δo₂·w8 −0.03623 × h₂(1 − h₂) 0.23991 × i₁ 0.05000	−0.0004346	−0.0004346	0.30902 → 0.30923590
w4	δo₁·w6 + δo₂·w8 −0.03623 × h₂(1 − h₂) 0.23991 × i₂ 0.10000	−0.0008692	−0.0008692	0.41804 → 0.41847179

At the start with learning rate 0.5 you should see w5 → 0.35891648 and w1 → 0.14978072, the same as the original worked example. δo means (o − t)·o(1 − o), the output neuron's blame.

The code that just ran

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final class TinyNet
{
    public array $w = ['w1' => 0.15, 'w2' => 0.20, 'w3' => 0.25, 'w4' => 0.30,
                       'w5' => 0.40, 'w6' => 0.45, 'w7' => 0.50, 'w8' => 0.55,
                       'b1' => 0.35, 'b2' => 0.60];   // biases stay fixed, as in the original example
    public array $in = [0.05, 0.10];
    public array $target = [0.01, 0.99];

    public function forward(): array
    {
        $w = $this->w; [$i1, $i2] = $this->in; [$t1, $t2] = $this->target;
        $nh1 = $w['w1'] * $i1 + $w['w2'] * $i2 + $w['b1'];  $h1 = sigmoid($nh1);
        $nh2 = $w['w3'] * $i1 + $w['w4'] * $i2 + $w['b1'];  $h2 = sigmoid($nh2);
        $no1 = $w['w5'] * $h1 + $w['w6'] * $h2 + $w['b2'];  $o1 = sigmoid($no1);
        $no2 = $w['w7'] * $h1 + $w['w8'] * $h2 + $w['b2'];  $o2 = sigmoid($no2);
        $E1 = 0.5 * ($t1 - $o1) ** 2;
        $E2 = 0.5 * ($t2 - $o2) ** 2;
        return compact('i1', 'i2', 't1', 't2', 'nh1', 'h1', 'nh2', 'h2', 'no1', 'o1', 'no2', 'o2', 'E1', 'E2') + ['E' => $E1 + $E2];
    }

    public function loss(): float { return $this->forward()['E']; }

    // BACKPROPAGATION. Each gradient is a product of simple local slopes:
    //   (o − t)      how the error changes with the output
    //   o·(1 − o)    how the sigmoid output changes with its input
    //   h or i       how that input changes with the weight
    // Hidden weights get the SUM of the blame flowing back from both outputs.
    public function gradients(): array
    {
        $f = $this->forward(); $w = $this->w;
        $do1 = ($f['o1'] - $f['t1']) * $f['o1'] * (1 - $f['o1']);   // δ of output 1
        $do2 = ($f['o2'] - $f['t2']) * $f['o2'] * (1 - $f['o2']);   // δ of output 2
        $back1 = $do1 * $w['w5'] + $do2 * $w['w7'];                 // blame reaching h1
        $back2 = $do1 * $w['w6'] + $do2 * $w['w8'];                 // blame reaching h2
        $s1 = $f['h1'] * (1 - $f['h1']);
        $s2 = $f['h2'] * (1 - $f['h2']);

        $g = fn($factors) => ['factors' => $factors, 'value' => array_product(array_column($factors, 1))];
        return [
            'w5' => $g([['(o₁ − t₁)', $f['o1'] - $f['t1']], ['o₁(1 − o₁)', $f['o1'] * (1 - $f['o1'])], ['h₁', $f['h1']]]),
            'w6' => $g([['(o₁ − t₁)', $f['o1'] - $f['t1']], ['o₁(1 − o₁)', $f['o1'] * (1 - $f['o1'])], ['h₂', $f['h2']]]),
            'w7' => $g([['(o₂ − t₂)', $f['o2'] - $f['t2']], ['o₂(1 − o₂)', $f['o2'] * (1 - $f['o2'])], ['h₁', $f['h1']]]),
            'w8' => $g([['(o₂ − t₂)', $f['o2'] - $f['t2']], ['o₂(1 − o₂)', $f['o2'] * (1 - $f['o2'])], ['h₂', $f['h2']]]),
            'w1' => $g([['δo₁·w5 + δo₂·w7', $back1], ['h₁(1 − h₁)', $s1], ['i₁', $f['i1']]]),
            'w2' => $g([['δo₁·w5 + δo₂·w7', $back1], ['h₁(1 − h₁)', $s1], ['i₂', $f['i2']]]),
            'w3' => $g([['δo₁·w6 + δo₂·w8', $back2], ['h₂(1 − h₂)', $s2], ['i₁', $f['i1']]]),
            'w4' => $g([['δo₁·w6 + δo₂·w8', $back2], ['h₂(1 − h₂)', $s2], ['i₂', $f['i2']]]),
        ];
    }

    // Sanity check without any calculus: nudge the weight up and down a hair
    // and see how much the loss moves. Should match backprop almost exactly.
    public function numericalGradient(string $name, float $eps = 1e-5): float
    {
        $keep = $this->w[$name];
        $this->w[$name] = $keep + $eps; $up = $this->loss();
        $this->w[$name] = $keep - $eps; $down = $this->loss();
        $this->w[$name] = $keep;
        return ($up - $down) / (2 * $eps);
    }

    public function step(float $lr): void
    {
        foreach ($this->gradients() as $name => $g) $this->w[$name] -= $lr * $g['value'];
    }
}