Backpropagation | From a Dot to a Chatbot

The network after 245 updates

Train it

Start 1 step 2 steps 10 steps 100 steps 1,000 steps 10,000 steps One more step

0.006468total error E

0.0969, 0.9166outputs (want 0.01, 0.99)

1. Forward pass: compute the answer

net_h1 = w1·i1 + w2·i2 + b1 = 0.2242·0.05 + 0.3485·0.10 + 0.35 = 0.396056

h1 = σ(net_h1) = 0.597740 h2 = σ(0.410677) = 0.601250

net_o1 = w5·h1 + w6·h2 + b2 = −2.232298 o1 = 0.096887

net_o2 = w7·h1 + w8·h2 + b2 = 2.397029 o2 = 0.916600

E = ½(t1 − o1)² + ½(t2 − o2)² = 0.003775 + 0.002694 = 0.006468450

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) 0.08689 × o₁(1 − o₁) 0.08750 × h₁ 0.59774	0.0045444	0.0045444	−2.37897 → −2.38124590
w6	(o₁ − t₁) 0.08689 × o₁(1 − o₁) 0.08750 × h₂ 0.60125	0.0045711	0.0045711	−2.34560 → −2.34788379
w7	(o₂ − t₂) −0.07340 × o₂(1 − o₂) 0.07644 × h₁ 0.59774	−0.0033539	−0.0033539	1.47081 → 1.47248991
w8	(o₂ − t₂) −0.07340 × o₂(1 − o₂) 0.07644 × h₂ 0.60125	−0.0033736	−0.0033736	1.52660 → 1.52828230
w1	δo₁·w5 + δo₂·w7 −0.02634 × h₁(1 − h₁) 0.24045 × i₁ 0.05000	−0.0003167	−0.0003167	0.22423 → 0.22438385
w2	δo₁·w5 + δo₂·w7 −0.02634 × h₁(1 − h₁) 0.24045 × i₂ 0.10000	−0.0006333	−0.0006333	0.34845 → 0.34876770
w3	δo₁·w6 + δo₂·w8 −0.02640 × h₂(1 − h₂) 0.23975 × i₁ 0.05000	−0.0003164	−0.0003164	0.32271 → 0.32286441
w4	δo₁·w6 + δo₂·w8 −0.02640 × h₂(1 − h₂) 0.23975 × i₂ 0.10000	−0.0006329	−0.0006329	0.44541 → 0.44572882

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

Everything on this page came from calling this code, which lives at the top of this same file.

Show or hide the source

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'];
    }
}