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Frank said:
Do that. I want to actually watch one token go through the machinery with numbers, even if they are toy numbers.
GPT said:
Good. We will use tiny fake numbers so the math stays visible.
Real models might use:
- 4096 embedding dimensions
- 32 attention heads
- thousands of neurons in the feedforward layer
But here we will use:
- 3 tokens
- embedding dimension = 4
- 1 attention head
- small feedforward layer
That lets us see the whole thing without drowning in arithmetic.
Our tiny input sequence will be:
[The, cat, slept]
We are going to follow the token:
cat
through one transformer block.
1. The starting hidden states
GPT said:
Each token begins as a vector of 4 numbers. These are toy hidden states:
x_{\text{The}} = [1, 0, 1, 0]
x_{\text{cat}} = [0, 1, 1, 0]
x_{\text{slept}} = [0, 1, 0, 1]
Put together as a matrix:
X = \begin{bmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{bmatrix}
Each row is one token.
Each column is one latent feature dimension.
Frank said:
So “cat” is just [0,1,1,0] at this moment.
GPT said:
Exactly. That is the token’s current position in hidden space.
2. Projecting into Query, Key, and Value spaces
GPT said:
Now we apply three learned matrices:
W_Q,\; W_K,\; W_V
To keep things simple, each will map from 4 dimensions down to 2.
Let:
W_Q = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 1 & 0 \\ 0 & 1 \end{bmatrix}
W_K = \begin{bmatrix} 1 & 1 \\ 0 & 1 \\ 1 & 0 \\ 0 & 1 \end{bmatrix}
W_V = \begin{bmatrix} 1 & 0 \\ 1 & 0 \\ 0 & 1 \\ 0 & 1 \end{bmatrix}
Each token vector gets multiplied by these matrices.
Query for “cat”
x_{\text{cat}} = [0,1,1,0]
Compute:
q_{\text{cat}} = x_{\text{cat}} W_Q
That gives:
- first output dimension: 0\cdot1 + 1\cdot0 + 1\cdot1 + 0\cdot0 = 1
- second output dimension: 0\cdot0 + 1\cdot1 + 1\cdot0 + 0\cdot1 = 1
So:
q_{\text{cat}} = [1,1]
Keys for all tokens
For The:
k_{\text{The}} = [1,0,1,0]W_K = [2,1]
For cat:
k_{\text{cat}} = [0,1,1,0]W_K = [1,1]
For slept:
k_{\text{slept}} = [0,1,0,1]W_K = [0,2]
Values for all tokens
For The:
v_{\text{The}} = [1,0,1,0]W_V = [1,1]
For cat:
v_{\text{cat}} = [0,1,1,0]W_V = [1,1]
For slept:
v_{\text{slept}} = [0,1,0,1]W_V = [1,1]
Frank said:
Interesting. In this toy case the value vectors all ended up the same.
GPT said:
Yes. That makes the example less realistic, but still useful. The key thing is seeing the mechanism.
3. “cat” compares its query to every key
GPT said:
Now “cat” asks: which tokens matter to me?
It does this by taking the dot product of its query with each key.
We have:
q_{\text{cat}} = [1,1]
Now compare it to each key.
Score for “The”
[1,1]\cdot[2,1] = 2+1=3
Score for “cat”
[1,1]\cdot[1,1] = 1+1=2
Score for “slept”
[1,1]\cdot[0,2] = 0+2=2
So the raw attention scores are:
[3,2,2]
4. Scale the attention scores
GPT said:
Because the head dimension is 2, we divide by:
\sqrt{2}\approx1.414
So:
[3,2,2] / 1.414 \approx [2.12, 1.41, 1.41]
5. Softmax turns scores into attention weights
GPT said:
Now we apply softmax.
Approximate exponentials:
e^{2.12}\approx8.33
e^{1.41}\approx4.10
e^{1.41}\approx4.10
Total:
8.33+4.10+4.10=16.53
So the attention weights are approximately:
\alpha_{\text{cat}} = [0.50,\;0.25,\;0.25]
Frank said:
So “cat” is paying about half its attention to “The” and splitting the rest between itself and “slept.”
GPT said:
Exactly.
That means in this toy block, “cat” is leaning most on “The.”
6. Weighted sum of values
GPT said:
Now “cat” gathers information by taking the weighted sum of the value vectors:
z_{\text{cat}} = 0.50\,v_{\text{The}} + 0.25\,v_{\text{cat}} + 0.25\,v_{\text{slept}}
Since all three values were [1,1], this becomes:
z_{\text{cat}} = [1,1]
Frank said:
So the attention output for “cat” is now [1,1].
GPT said:
Right.
In a real model, the values would differ, and the result would be a richer blend. But this still shows the logic:
- query asks
- keys answer compatibility
- values provide content
- weighted mixing creates a new context-aware representation
7. Output projection back to model dimension
GPT said:
Now attention usually projects the result back up to the model dimension.
Let the output matrix be:
W_O = \begin{bmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{bmatrix}
Then:
o_{\text{cat}} = z_{\text{cat}}W_O
With:
z_{\text{cat}}=[1,1]
we get:
- first output dimension: 1\cdot1 + 1\cdot0 = 1
- second: 1\cdot0 + 1\cdot1 = 1
- third: 1\cdot1 + 1\cdot0 = 1
- fourth: 1\cdot0 + 1\cdot1 = 1
So:
o_{\text{cat}} = [1,1,1,1]
8. Residual connection after attention
GPT said:
Now add this attention output back to the original token state:
Original “cat” state:
x_{\text{cat}} = [0,1,1,0]
Attention output:
o_{\text{cat}} = [1,1,1,1]
Add them:
x’_{\text{cat}} = [1,2,2,1]
Frank said:
So now “cat” has become a more context-loaded vector.
GPT said:
Exactly. It started as [0,1,1,0]. After attention, it became [1,2,2,1].
That is the token after one round of semantic negotiation with the other tokens.
9. Feedforward layer: expanding to neuron space
GPT said:
Now comes the MLP or feedforward network.
We will map from 4 dimensions up to 3 hidden neurons, then back down to 4.
Let:
W_1 = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 0 \end{bmatrix}
Take:
x’_{\text{cat}} = [1,2,2,1]
Now compute:
h = x’_{\text{cat}} W_1
Let’s do it.
Hidden neuron 1
1\cdot1 + 2\cdot0 + 2\cdot1 + 1\cdot0 = 3
Hidden neuron 2
1\cdot0 + 2\cdot1 + 2\cdot1 + 1\cdot1 = 5
Hidden neuron 3
1\cdot1 + 2\cdot1 + 2\cdot0 + 1\cdot0 = 3
So:
h = [3,5,3]
10. Nonlinearity
GPT said:
Now apply a nonlinearity. To keep it simple, we will use ReLU:
\text{ReLU}(x)=\max(0,x)
Since all values are positive:
h = [3,5,3]
stays the same.
Frank said:
So these hidden neurons are like pattern detectors lighting up.
GPT said:
Exactly.
Each hidden neuron is responding to a different learned combination of the token’s current features.
11. Project back down to model dimension
GPT said:
Now use a second matrix:
W_2 = \begin{bmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 1 & 0 & 0 \end{bmatrix}
Compute:
y = hW_2
with:
h=[3,5,3]
Now calculate each output coordinate.
Output dimension 1
3\cdot1 + 5\cdot0 + 3\cdot1 = 6
Output dimension 2
3\cdot0 + 5\cdot1 + 3\cdot1 = 8
Output dimension 3
3\cdot1 + 5\cdot0 + 3\cdot0 = 3
Output dimension 4
3\cdot0 + 5\cdot1 + 3\cdot0 = 5
So:
y = [6,8,3,5]
12. Final residual connection
GPT said:
Now add this feedforward output back to the post-attention state:
x’_{\text{cat}} = [1,2,2,1]
y = [6,8,3,5]
So the final token state leaving the block is:
x^{\text{next}}_{\text{cat}} = [7,10,5,6]
Frank said:
So “cat” entered the block as [0,1,1,0] and left as [7,10,5,6].
GPT said:
Yes.
Not because the model “decided” in words what cat means, but because:
- the token was projected into Q, K, V forms
- it compared itself to other tokens
- it absorbed weighted context
- it passed through neuron-based feature detectors
- it exited as a more richly transformed state
That is the whole basic transformer-block cycle.
What each step means conceptually
A. Embedding dimensions
Frank said:
Now interpret the numbers. What are the dimensions doing conceptually?
GPT said:
The original 4 dimensions are the token’s latent coordinates. They are not human-readable labels, but they act like distributed semantic ingredients.
So [0,1,1,0] is not “cat = furry + animal” in any neat way. It is just the token’s current internal state.
B. Weight matrices
GPT said:
The matrices are the learned ANN structure.
They define:
- how the token is viewed as a query
- how it is viewed as a key
- how it is viewed as a value
- how its updated representation is expanded into hidden neuron activations
- how those activations are folded back into token space
So the weights are not the token.
They are the rules for transforming the token.
C. Attention head
GPT said:
The attention head is the relational mechanism.
For “cat,” the head asked:
- how much should I attend to “The”?
- how much to myself?
- how much to “slept”?
Those weights determined how context got blended into the token.
So attention is where token-to-token interaction happens.
D. Feedforward neurons
GPT said:
The feedforward neurons are where the updated token is internally reinterpreted.
Attention says:
what other tokens matter to me?
The feedforward block says:
given that new context, what richer feature patterns should I now activate?
The deepest takeaway
Frank said:
So what is the simplest way to summarize this walkthrough?
GPT said:
This way:
A token starts as a vector of latent coordinates. Inside one transformer block, that vector is projected by learned weights into query, key, and value forms. Its query is compared to the keys of other tokens to decide where attention should go. The resulting weighted blend of values becomes a context-enriched update. That update is added back to the token, then passed through a feedforward neural layer that detects new feature combinations and reshapes the token again. The token exits the block not as the same vector it entered with, but as a more context-sensitive semantic state.
Final Frank-said / GPT-said closing
Frank said:
So the token is not just looked up and passed along. It is repeatedly transformed by the ANN structure, first through relational attention and then through neuron-based feature refinement.
GPT said:
Exactly.
The transformer block is a two-part engine:
- attention lets the token consult the rest of the sentence
- feedforward neurons let the token reinterpret itself after that consultation
The numbers change because the token is being mathematically re-expressed in light of context.
Frank said:
So one block is one full cycle of contextualization plus reinterpretation.
GPT said:
Perfectly said.
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