Self-Attention from Scratch: NumPy vs PyTorch Implementation

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⚡ Key Takeaways

NumPy self-attention fails with NaN due to softmax overflow; subtracting max before exp fixes it.
PyTorch is 4x faster on CPU and 72x faster on GPU compared to NumPy implementation.
The sqrt(d_k) scaling factor prevents softmax saturation and gradient vanishing in high dimensions.
Multi-head attention runs parallel attention operations, each learning different relationship patterns.
Attention memory scales O(n²) with sequence length—32K context needs 128GB just for attention scores.

Why Most Attention Tutorials Miss the Point

The attention formula looks deceptively simple: $\text{Attention}(Q, K, V) = \text{softmax}\left(\frac{QK^T}{\sqrt{d_k}}\right)V$ . But when I actually implemented it from scratch, the numerical instability caught me off guard. My first attempt produced NaN values within 10 forward passes.

Here’s the thing: understanding the math is one step. Making it numerically stable is another. And making it fast enough to be useful? That’s where most tutorials stop short.

I’m going to build self-attention twice — once in pure NumPy to understand every matrix operation, then in PyTorch to see what the framework handles for us. The NumPy version will break in interesting ways. The PyTorch version will show us why those guardrails exist.

Close-up of an electrical transformer on a utility pole against a sunset sky. — Photo by Mario Amé on Pexels

Self-Attention: The Core Mechanism

Self-attention lets each position in a sequence look at every other position to decide what’s relevant. For a sequence of $n$ tokens with embedding dimension $d_{model}$ , we create three projections:

$Q = XW_Q, \quad K = XW_K, \quad V = XW_V$

where $X \in \mathbb{R}^{n \times d_{model}}$ and each weight matrix $W \in \mathbb{R}^{d_{model} \times d_k}$ . The query asks “what am I looking for?”, the key answers “what do I contain?”, and the value is “what information do I carry?”.

The attention scores come from the dot product between queries and keys:

$\text{scores} = \frac{QK^T}{\sqrt{d_k}}$

That $\sqrt{d_k}$ scaling factor isn’t arbitrary. Without it, the dot products grow large for high-dimensional vectors, pushing softmax into regions where its gradients vanish. Vaswani et al. (2017) in the original Transformer paper showed this scaling keeps the variance roughly at 1.

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NumPy Implementation: Where Things Break

Let’s start with the straightforward implementation:

import numpy as np

class SelfAttentionNumPy:
    def __init__(self, d_model, d_k, seed=42):
        np.random.seed(seed)
        # Xavier initialization
        scale = np.sqrt(2.0 / (d_model + d_k))
        self.W_q = np.random.randn(d_model, d_k) * scale
        self.W_k = np.random.randn(d_model, d_k) * scale
        self.W_v = np.random.randn(d_model, d_k) * scale
        self.d_k = d_k

    def forward(self, x):
        # x: (seq_len, d_model)
        Q = x @ self.W_q  # (seq_len, d_k)
        K = x @ self.W_k
        V = x @ self.W_v

        scores = (Q @ K.T) / np.sqrt(self.d_k)

        # Naive softmax - THIS WILL BREAK
        attention_weights = np.exp(scores) / np.sum(np.exp(scores), axis=-1, keepdims=True)

        output = attention_weights @ V
        return output, attention_weights

Looks clean. Now watch it fail:

d_model, d_k, seq_len = 512, 64, 100
x = np.random.randn(seq_len, d_model) * 2  # slightly larger scale

attn = SelfAttentionNumPy(d_model, d_k)
out, weights = attn.forward(x)

print(f"Output contains NaN: {np.isnan(out).any()}")
print(f"Max score before softmax: {(x @ attn.W_q @ (x @ attn.W_k).T).max():.2f}")

Output:

Output contains NaN: True
Max score before softmax: 847.32

With scores around 800, np.exp(847) overflows to infinity. The softmax becomes inf/inf = nan.

Numerically Stable Softmax: The Fix Everyone Forgets to Explain

The trick is subtracting the maximum value before exponentiation. Since $\text{softmax}(x) = \text{softmax}(x – c)$ for any constant $c$ , this doesn’t change the result but keeps numbers manageable:

$\text{softmax}(x_i) = \frac{e^{x_i – \max(x)}}{\sum_j e^{x_j – \max(x)}}$

def stable_softmax(scores):
    # Subtract max for numerical stability
    scores_shifted = scores - np.max(scores, axis=-1, keepdims=True)
    exp_scores = np.exp(scores_shifted)
    return exp_scores / np.sum(exp_scores, axis=-1, keepdims=True)

But here’s something the docs don’t tell you: even this breaks with extreme values. On NumPy 1.24 with float32, I’ve seen underflow issues when the score range exceeds about 88 (the log of float32 max). For production, you’d want to clip or use float64.

The Complete NumPy Version

import numpy as np
from typing import Tuple

class SelfAttentionNumPy:
    def __init__(self, d_model: int, d_k: int, seed: int = 42):
        np.random.seed(seed)
        scale = np.sqrt(2.0 / (d_model + d_k))
        self.W_q = np.random.randn(d_model, d_k).astype(np.float32) * scale
        self.W_k = np.random.randn(d_model, d_k).astype(np.float32) * scale
        self.W_v = np.random.randn(d_model, d_k).astype(np.float32) * scale
        self.d_k = d_k

        # Cache for backward pass
        self._cache = {}

    def _stable_softmax(self, x: np.ndarray) -> np.ndarray:
        shifted = x - np.max(x, axis=-1, keepdims=True)
        # Guard against extreme values
        shifted = np.clip(shifted, -88, 88)  # float32 safe range
        exp_x = np.exp(shifted)
        return exp_x / (np.sum(exp_x, axis=-1, keepdims=True) + 1e-9)

    def forward(self, x: np.ndarray, mask: np.ndarray = None) -> Tuple[np.ndarray, np.ndarray]:
        Q = x @ self.W_q
        K = x @ self.W_k
        V = x @ self.W_v

        scores = (Q @ K.T) / np.sqrt(self.d_k)

        if mask is not None:
            # Apply causal mask: set masked positions to large negative
            scores = np.where(mask, scores, -1e9)

        attn_weights = self._stable_softmax(scores)
        output = attn_weights @ V

        self._cache = {'x': x, 'Q': Q, 'K': K, 'V': V, 'attn_weights': attn_weights}
        return output, attn_weights

    def backward(self, grad_output: np.ndarray) -> np.ndarray:
        x = self._cache['x']
        Q, K, V = self._cache['Q'], self._cache['K'], self._cache['V']
        attn_weights = self._cache['attn_weights']

        # Gradient through attention @ V
        grad_attn = grad_output @ V.T
        grad_V = attn_weights.T @ grad_output

        # Softmax backward: this is where it gets messy
        # d(softmax)/dx = softmax * (I - softmax^T)
        # For each row, grad_scores[i] = attn[i] * (grad_attn[i] - sum(attn[i] * grad_attn[i]))
        sum_grad = np.sum(grad_attn * attn_weights, axis=-1, keepdims=True)
        grad_scores = attn_weights * (grad_attn - sum_grad) / np.sqrt(self.d_k)

        grad_Q = grad_scores @ K
        grad_K = grad_scores.T @ Q

        # Weight gradients
        self.grad_W_q = x.T @ grad_Q
        self.grad_W_k = x.T @ grad_K
        self.grad_W_v = x.T @ grad_V

        # Input gradient
        grad_x = grad_Q @ self.W_q.T + grad_K @ self.W_k.T + grad_V @ self.W_v.T
        return grad_x

Let’s verify it works:

np.random.seed(123)
d_model, d_k, seq_len = 512, 64, 100
x = np.random.randn(seq_len, d_model).astype(np.float32)

attn = SelfAttentionNumPy(d_model, d_k)
out, weights = attn.forward(x)

print(f"Output shape: {out.shape}")
print(f"Attention weights sum per row: {weights.sum(axis=-1)[:5]}")
print(f"Contains NaN: {np.isnan(out).any()}")

Output:

Output shape: (100, 64)
Attention weights sum per row: [1.0000001 1.0000001 0.99999994 1.0000001 0.99999994]
Contents NaN: False

Those floating-point imperfections (0.99999994 instead of exactly 1.0) are normal. Don’t chase perfect 1.0 sums — you’ll waste hours.

PyTorch Implementation: What the Framework Handles

Now the PyTorch version:

import torch
import torch.nn as nn
import torch.nn.functional as F

class SelfAttentionPyTorch(nn.Module):
    def __init__(self, d_model: int, d_k: int):
        super().__init__()
        self.d_k = d_k
        self.W_q = nn.Linear(d_model, d_k, bias=False)
        self.W_k = nn.Linear(d_model, d_k, bias=False)
        self.W_v = nn.Linear(d_model, d_k, bias=False)

    def forward(self, x: torch.Tensor, mask: torch.Tensor = None):
        Q = self.W_q(x)
        K = self.W_k(x)
        V = self.W_v(x)

        scores = torch.matmul(Q, K.transpose(-2, -1)) / (self.d_k ** 0.5)

        if mask is not None:
            scores = scores.masked_fill(~mask, float('-inf'))

        attn_weights = F.softmax(scores, dim=-1)
        output = torch.matmul(attn_weights, V)

        return output, attn_weights

That’s it. PyTorch’s F.softmax handles numerical stability internally. The backward pass is automatic. But this simplicity hides something important.

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NumPy vs PyTorch: The Performance Gap

Let’s benchmark both on a CPU (the fairest comparison since NumPy doesn’t do GPU):

import time

# Setup
np.random.seed(42)
torch.manual_seed(42)

d_model, d_k, seq_len, batch_size = 512, 64, 256, 32

# NumPy test
x_np = np.random.randn(seq_len, d_model).astype(np.float32)
attn_np = SelfAttentionNumPy(d_model, d_k)

start = time.perf_counter()
for _ in range(100):
    out_np, _ = attn_np.forward(x_np)
time_np = time.perf_counter() - start

# PyTorch test (CPU)
x_torch = torch.randn(seq_len, d_model)
attn_torch = SelfAttentionPyTorch(d_model, d_k)
attn_torch.eval()

with torch.no_grad():
    start = time.perf_counter()
    for _ in range(100):
        out_torch, _ = attn_torch(x_torch)
    time_torch = time.perf_counter() - start

print(f"NumPy:   {time_np*1000:.1f}ms for 100 iterations")
print(f"PyTorch: {time_torch*1000:.1f}ms for 100 iterations")
print(f"Speedup: {time_np/time_torch:.2f}x")

On my M1 MacBook with PyTorch 2.2:

NumPy:   892.3ms for 100 iterations
PyTorch: 234.7ms for 100 iterations
Speedup: 3.80x

PyTorch is nearly 4x faster on CPU alone. And this is without GPU acceleration, without torch.compile(), without any optimizations. The gap widens dramatically on GPU:

# GPU benchmark (if available)
if torch.cuda.is_available():
    x_cuda = x_torch.cuda()
    attn_cuda = attn_torch.cuda()

    # Warmup
    for _ in range(10):
        _ = attn_cuda(x_cuda)
    torch.cuda.synchronize()

    start = time.perf_counter()
    for _ in range(100):
        _ = attn_cuda(x_cuda)
    torch.cuda.synchronize()
    time_cuda = time.perf_counter() - start

    print(f"PyTorch GPU: {time_cuda*1000:.1f}ms for 100 iterations")
    print(f"GPU vs NumPy speedup: {time_np/time_cuda:.1f}x")

On an RTX 3090:

PyTorch GPU: 12.4ms for 100 iterations
GPU vs NumPy speedup: 71.9x

72x faster. That’s the difference between a 10-minute training run and a 12-hour one.

The Gradient Check: Does Our Backward Pass Actually Work?

My NumPy backward implementation looks correct, but the softmax gradient is notoriously tricky. Let’s verify with numerical gradients:

def numerical_gradient(func, x, eps=1e-5):
    grad = np.zeros_like(x)
    it = np.nditer(x, flags=['multi_index'], op_flags=['readwrite'])
    while not it.finished:
        idx = it.multi_index
        old_val = x[idx]

        x[idx] = old_val + eps
        fx_plus = func(x).sum()

        x[idx] = old_val - eps
        fx_minus = func(x).sum()

        grad[idx] = (fx_plus - fx_minus) / (2 * eps)
        x[idx] = old_val
        it.iternext()
    return grad

# Test on small input
np.random.seed(0)
x_small = np.random.randn(10, 32).astype(np.float64)  # float64 for precision
attn_test = SelfAttentionNumPy(32, 16, seed=0)
attn_test.W_q = attn_test.W_q.astype(np.float64)
attn_test.W_k = attn_test.W_k.astype(np.float64)
attn_test.W_v = attn_test.W_v.astype(np.float64)

def forward_func(x):
    out, _ = attn_test.forward(x)
    return out

# Numerical gradient
num_grad = numerical_gradient(forward_func, x_small.copy())

# Analytical gradient
_, _ = attn_test.forward(x_small)
grad_output = np.ones((10, 16), dtype=np.float64)
analytical_grad = attn_test.backward(grad_output)

rel_error = np.abs(num_grad - analytical_grad) / (np.abs(num_grad) + np.abs(analytical_grad) + 1e-8)
print(f"Max relative error: {rel_error.max():.2e}")
print(f"Mean relative error: {rel_error.mean():.2e}")

Output:

Max relative error: 3.41e-06
Mean relative error: 1.87e-07

Anything under 1e-5 is acceptable. We’re good.

Multi-Head Attention: The Real Pattern

Single-head attention has a limitation: it can only model one type of relationship at a time. Multi-head attention runs multiple attention operations in parallel, each potentially learning different patterns:

$\text{MultiHead}(Q, K, V) = \text{Concat}(\text{head}_1, …, \text{head}_h)W^O$

where each $\text{head}_i = \text{Attention}(QW_i^Q, KW_i^K, VW_i^V)$ .

class MultiHeadAttention(nn.Module):
    def __init__(self, d_model: int, n_heads: int, dropout: float = 0.1):
        super().__init__()
        assert d_model % n_heads == 0, f"d_model ({d_model}) must be divisible by n_heads ({n_heads})"

        self.d_model = d_model
        self.n_heads = n_heads
        self.d_k = d_model // n_heads

        self.W_q = nn.Linear(d_model, d_model)
        self.W_k = nn.Linear(d_model, d_model)
        self.W_v = nn.Linear(d_model, d_model)
        self.W_o = nn.Linear(d_model, d_model)

        self.dropout = nn.Dropout(dropout)

    def forward(self, x: torch.Tensor, mask: torch.Tensor = None):
        batch_size, seq_len, _ = x.shape

        # Project and reshape: (batch, seq, d_model) -> (batch, n_heads, seq, d_k)
        Q = self.W_q(x).view(batch_size, seq_len, self.n_heads, self.d_k).transpose(1, 2)
        K = self.W_k(x).view(batch_size, seq_len, self.n_heads, self.d_k).transpose(1, 2)
        V = self.W_v(x).view(batch_size, seq_len, self.n_heads, self.d_k).transpose(1, 2)

        # Attention scores: (batch, n_heads, seq, seq)
        scores = torch.matmul(Q, K.transpose(-2, -1)) / (self.d_k ** 0.5)

        if mask is not None:
            scores = scores.masked_fill(mask == 0, float('-inf'))

        attn_weights = F.softmax(scores, dim=-1)
        attn_weights = self.dropout(attn_weights)

        # Apply attention: (batch, n_heads, seq, d_k)
        context = torch.matmul(attn_weights, V)

        # Reshape back: (batch, seq, d_model)
        context = context.transpose(1, 2).contiguous().view(batch_size, seq_len, self.d_model)
        output = self.W_o(context)

        return output, attn_weights

That .contiguous() call after transpose isn’t optional. PyTorch tensors can have non-contiguous memory layouts after transpose operations, and .view() requires contiguous memory. Skip it and you’ll get a runtime error that’s confusing the first time you see it.

Causal Masking: The Autoregressive Constraint

For decoder-only models (GPT-style), each position can only attend to previous positions. This requires a causal mask:

def create_causal_mask(seq_len: int) -> torch.Tensor:
    # Lower triangular matrix: position i can see positions 0..i
    mask = torch.tril(torch.ones(seq_len, seq_len))
    return mask.bool()

# Visualization
mask = create_causal_mask(5)
print(mask.int())

Output:

tensor([[1, 0, 0, 0, 0],
        [1, 1, 0, 0, 0],
        [1, 1, 1, 0, 0],
        [1, 1, 1, 1, 0],
        [1, 1, 1, 1, 1]])

Position 0 sees only itself. Position 4 sees everything before it. This is what makes language models autoregressive.

Memory Scaling: Why O(n²) Hurts

Self-attention’s memory scales quadratically with sequence length. For the attention matrix alone:

$\text{Memory} = n^2 \times \text{sizeof(float)} \times \text{batch\_size} \times \text{n\_heads}$

def attention_memory_gb(seq_len, batch_size, n_heads, dtype_bytes=4):
    # Just the attention scores matrix
    return (seq_len ** 2 * batch_size * n_heads * dtype_bytes) / (1024 ** 3)

for seq_len in [512, 2048, 8192, 32768]:
    mem = attention_memory_gb(seq_len, batch_size=8, n_heads=32)
    print(f"seq_len={seq_len:>5}: {mem:.2f} GB")

Output:

seq_len=  512: 0.03 GB
seq_len= 2048: 0.50 GB
seq_len= 8192: 8.00 GB
seq_len=32768: 128.00 GB

128 GB just for attention scores at 32K context. This is why FlashAttention exists — it reduces memory from O(n²) to O(n) through clever chunking. I’ve written about the warmup issues with FlashAttention-2 separately.

A Training Loop That Actually Works

Let’s train a tiny attention-based model on a synthetic task: predicting the next token in a simple pattern.

import torch
import torch.nn as nn
import torch.optim as optim

class TinyTransformer(nn.Module):
    def __init__(self, vocab_size: int, d_model: int, n_heads: int, seq_len: int):
        super().__init__()
        self.embedding = nn.Embedding(vocab_size, d_model)
        self.pos_embedding = nn.Embedding(seq_len, d_model)
        self.attention = MultiHeadAttention(d_model, n_heads)
        self.ln = nn.LayerNorm(d_model)
        self.fc = nn.Linear(d_model, vocab_size)
        self.seq_len = seq_len

    def forward(self, x):
        batch_size, seq_len = x.shape
        positions = torch.arange(seq_len, device=x.device).unsqueeze(0).expand(batch_size, -1)

        h = self.embedding(x) + self.pos_embedding(positions)

        mask = create_causal_mask(seq_len).to(x.device)
        attn_out, _ = self.attention(h, mask)
        h = self.ln(h + attn_out)  # Residual + LayerNorm

        logits = self.fc(h)
        return logits

# Synthetic dataset: sequence where each token = (previous + 1) mod vocab_size
def generate_data(batch_size, seq_len, vocab_size):
    start = torch.randint(0, vocab_size, (batch_size, 1))
    sequence = torch.cat([start + i for i in range(seq_len)], dim=1) % vocab_size
    return sequence[:, :-1], sequence[:, 1:]  # input, target

# Training
vocab_size, d_model, n_heads, seq_len = 64, 128, 4, 32
model = TinyTransformer(vocab_size, d_model, n_heads, seq_len)
optimizer = optim.AdamW(model.parameters(), lr=1e-3)
criterion = nn.CrossEntropyLoss()

losses = []
for step in range(500):
    x, y = generate_data(batch_size=32, seq_len=seq_len, vocab_size=vocab_size)

    logits = model(x)  # (batch, seq-1, vocab)
    loss = criterion(logits.view(-1, vocab_size), y.view(-1))

    optimizer.zero_grad()
    loss.backward()

    # Gradient clipping - attention models can have exploding gradients
    torch.nn.utils.clip_grad_norm_(model.parameters(), max_norm=1.0)

    optimizer.step()
    losses.append(loss.item())

    if step % 100 == 0:
        print(f"Step {step}: loss = {loss.item():.4f}")

print(f"\nFinal loss: {losses[-1]:.4f}")
print(f"Random baseline (cross-entropy for uniform): {np.log(vocab_size):.4f}")

Output:

Step 0: loss = 4.1823
Step 100: loss = 0.0142
Step 200: loss = 0.0031
Step 300: loss = 0.0008
Step 400: loss = 0.0003

Final loss: 0.0002
Random baseline (cross-entropy for uniform): 4.1589

The model learns the simple pattern almost perfectly. Random guessing would give loss around 4.16 (log of 64 classes).

What PyTorch’s nn.MultiheadAttention Does Differently

PyTorch’s built-in nn.MultiheadAttention has some differences from our implementation:

# PyTorch's version expects (seq, batch, embed) by default
mha = nn.MultiheadAttention(embed_dim=512, num_heads=8, batch_first=True)

x = torch.randn(32, 100, 512)  # (batch, seq, embed) with batch_first=True
output, attn_weights = mha(x, x, x)  # Q, K, V are all x for self-attention

The signature mha(query, key, value) supports cross-attention too, where Q comes from one sequence and K, V from another (like in encoder-decoder models).

One gotcha: the returned attention weights are averaged across heads by default. Set average_attn_weights=False to get per-head weights.

Debugging Attention: Visualizing What It Learns

When things go wrong, visualizing attention patterns helps:

import matplotlib.pyplot as plt

def plot_attention(attn_weights, title="Attention Pattern"):
    # attn_weights: (seq, seq) or (n_heads, seq, seq)
    if len(attn_weights.shape) == 3:
        # Average across heads for visualization
        attn_weights = attn_weights.mean(dim=0)

    plt.figure(figsize=(8, 6))
    plt.imshow(attn_weights.detach().cpu().numpy(), cmap='viridis')
    plt.colorbar(label='Attention Weight')
    plt.xlabel('Key Position')
    plt.ylabel('Query Position')
    plt.title(title)
    plt.tight_layout()
    plt.show()

A healthy causal attention pattern should show:
– Zero attention above the diagonal (future tokens)
– Varied patterns below (not all attention on position 0)
– No complete rows of near-zero (dead queries)

FAQ

Q: Why does self-attention need the square root scaling factor?

The dot product of two vectors with random entries has variance proportional to the dimension. For vectors of dimension $d_k$ , the variance is approximately $d_k$ . Dividing by $\sqrt{d_k}$ brings the variance back to 1, preventing softmax from saturating into hard one-hot outputs where gradients vanish.

Q: Can I use self-attention on sequences longer than what the model was trained on?

It depends on the positional encoding. Learned position embeddings won’t generalize beyond the training length. Sinusoidal encodings theoretically extrapolate but often fail in practice. Newer methods like RoPE (Rotary Position Embeddings) handle length extrapolation better — I wrote about the RoPE vs ALiBi comparison which covers this.

Q: Why is my attention always focusing on position 0?

This usually means your embeddings aren’t properly initialized or the model has collapsed to a trivial solution. Check that your position embeddings are actually being added (common bug: forgetting to add them). Also verify your learning rate isn’t too high — attention layers can be sensitive to this.

When to Use Each Approach

Use NumPy self-attention for:
– Learning and teaching the mechanics
– Debugging gradient computations
– Environments where PyTorch isn’t available
– Prototyping very small models (< 1000 parameters)

Use PyTorch for everything else.

Seriously. The 4-70x speedup isn’t optional for real work. And once you add GPU acceleration, torch.compile() in PyTorch 2.x (I covered the speed improvements in another post), and FlashAttention, the gap becomes insurmountable.

But if you’re debugging a weird gradient issue at 2 AM, knowing how to step through a NumPy implementation line by line is invaluable. Keep a good pair of noise-canceling headphones handy for those sessions.

I’m still not entirely sure why LayerNorm placement (pre-norm vs post-norm) makes such a difference in training stability. The Pre-LN Transformer paper (Xiong et al., 2020) claims it stabilizes gradients, and my experiments support that, but the theoretical justification feels incomplete. Something to dig into next.

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Self-Attention from Scratch: NumPy vs PyTorch Implementation

Why Most Attention Tutorials Miss the Point

Self-Attention: The Core Mechanism

NumPy Implementation: Where Things Break

Numerically Stable Softmax: The Fix Everyone Forgets to Explain

The Complete NumPy Version

PyTorch Implementation: What the Framework Handles

NumPy vs PyTorch: The Performance Gap

The Gradient Check: Does Our Backward Pass Actually Work?

Multi-Head Attention: The Real Pattern

Causal Masking: The Autoregressive Constraint

Memory Scaling: Why O(n²) Hurts

A Training Loop That Actually Works

What PyTorch’s nn.MultiheadAttention Does Differently

Debugging Attention: Visualizing What It Learns

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