Learned Parameters to Predictions Conversion

[GasimV]

Let's walk through the steps of how a GPT model generates text, using a simple mathematical approach.

Step-by-Step Illustration

1. Initialization and Loading the Model

• Model Architecture: Assume a simple GPT-like model with an embedding layer, multiple Transformer blocks, and a final linear layer followed by softmax.
• Learned Parameters:
  • Embedding matrix (E) of size (V \times d), where (V) is the vocabulary size and (d) is the embedding dimension.
  • Weights and biases for each Transformer block (e.g., self-attention weights (W_Q, W_K, W_V) and feed-forward layers (W_1, W_2)).

2. Input Preprocessing

• Tokenization: Convert input text (e.g., "Hello") to token IDs. Let's say "Hello" corresponds to token ID 42.
• Embedding Lookup: Convert token ID to an embedding vector. [ \mathbf{x}_0 = E[42] ]
• Assume (E[42]) is a vector (\mathbf{e}_0 \in \mathbb{R}^d).

3. Forward Pass (Inference)

• Input Embedding: The initial input embedding vector is (\mathbf{x}_0 = \mathbf{e}_0).

  Transformer Block Operations:

• Self-Attention Mechanism:

  • Compute Query, Key, and Value vectors: [ \mathbf{Q} = \mathbf{x}_0 W_Q, \quad \mathbf{K} = \mathbf{x}_0 W_K, \quad \mathbf{V} = \mathbf{x}_0 W_V ]
  • Assume (W_Q, W_K, W_V \in \mathbb{R}^{d \times d}) are weight matrices for the query, key, and value projections.
  • Compute attention scores: [ \text{scores} = \frac{\mathbf{Q} \mathbf{K}^T}{\sqrt{d}} ]
  • Apply softmax to obtain attention weights: [ \text{attention_weights} = \text{softmax}\left(\frac{\mathbf{Q} \mathbf{K}^T}{\sqrt{d}}\right) ]
  • Compute the attention output: [ \mathbf{z} = \text{attention_weights} \cdot \mathbf{V} ]

• Feed-Forward Network (FFN):

  • Apply a linear transformation followed by a non-linear activation and another linear transformation: [ \mathbf{h}_1 = \text{ReLU}(\mathbf{z} W_1 + b_1) ] [ \mathbf{h}_2 = \mathbf{h}_1 W_2 + b_2 ]
  • Assume (W1 \in \mathbb{R}^{d \times d{\text{ff}}}), (W2 \in \mathbb{R}^{d{\text{ff}} \times d}), and (d_{\text{ff}}) is the dimension of the feed-forward layer.

• Repeat the self-attention and FFN steps for each Transformer block.

4. Output Generation

• Final Linear Transformation:

  • The output of the last Transformer block (\mathbf{h}_L) is transformed into logits for each token in the vocabulary: [ \mathbf{logits} = \mathbf{h}L W{\text{out}} + b_{\text{out}} ]
  • Assume (W{\text{out}} \in \mathbb{R}^{d \times V}) and (b{\text{out}} \in \mathbb{R}^V).

• Softmax to Obtain Probabilities: [ \mathbf{p} = \text{softmax}(\mathbf{logits}) ]

  • The softmax function converts logits into a probability distribution over the vocabulary.

5. Post-Processing

• Select Next Token:
  • The next token is selected based on the highest probability in (\mathbf{p}): [ \text{next_token_id} = \arg\max(\mathbf{p}) ]
  • Assume the token with the highest probability corresponds to token ID 23 (which might correspond to "world").
• Convert Token ID to Text:
  • The token ID 23 is converted back to its corresponding word, resulting in the word "world".

Summary

To summarize, given the input "Hello":

1. Token ID 42 is converted to its embedding (\mathbf{e}_0).
2. The embedding is processed through multiple Transformer blocks using self-attention and feed-forward networks.
3. The final output is transformed to logits and passed through softmax to obtain a probability distribution.
4. The most probable next token ID is selected and converted back to text, resulting in "world".

The process can then repeat for generating subsequent tokens.