Transformer large models are built on several mathematical foundations including linear algebra for matrix operations, probability theory for the softmax function, calculus for gradient descent during training, and statistics for normalization techniques. The architecture consists of an encoder and decoder, with the self-attention mechanism being the core innovation. This mechanism allows the model to weigh the importance of different parts of the input sequence when processing each element.
The self-attention mechanism is the core innovation of Transformer models. It computes the relationship between all tokens in a sequence using three matrices: Query, Key, and Value. The Query matrix represents what we're looking for, the Key matrix is what we match against, and the Value matrix contains the information we want to extract. The attention formula computes the dot product between Query and Key matrices, scales it by the square root of the dimension, applies a softmax function to get attention weights, and then multiplies these weights with the Value matrix. This allows each token to gather information from all other tokens in the sequence, weighted by their relevance.
Multi-head attention is a key enhancement to the self-attention mechanism. It performs the self-attention operation multiple times in parallel with different learned projections. This allows the model to jointly attend to information from different representation subspaces, enabling it to focus on different aspects of the input simultaneously. After the multi-head attention, each position is processed by a feed-forward network. This consists of two linear transformations with a ReLU activation function in between. The formula is FFN(x) equals max of zero and x times W1 plus b1, all multiplied by W2 plus b2. This combination of multi-head attention and feed-forward networks forms the building blocks of both the encoder and decoder in the Transformer architecture.
Transformer models incorporate two critical techniques to improve training stability and convergence. First, residual connections add the input of a sub-layer to its output, creating shortcuts for gradient flow during backpropagation. This helps mitigate the vanishing gradient problem in deep networks. The formula is simply x plus Sublayer of x. Second, layer normalization normalizes the activations across the features for each input example. This stabilizes the learning process and reduces training time. The formula normalizes each input by subtracting the mean and dividing by the standard deviation, then applies learnable scale and shift parameters gamma and beta. Together, these techniques enable training of very deep transformer networks that would otherwise be difficult to optimize.
Training transformer models involves optimizing millions of parameters using gradient-based methods. The primary loss function is typically cross-entropy for classification tasks like next-token prediction. The Adam optimizer is commonly used with a specialized learning rate schedule that includes a warmup phase followed by decay. This schedule is given by the formula shown, where the learning rate depends on the model dimension and training steps. Regularization techniques like dropout and weight decay help prevent overfitting. Gradient clipping is also essential to prevent exploding gradients during backpropagation. The training process can be visualized as gradient descent on a loss landscape, where the model parameters are iteratively updated to find the minimum loss. With these mathematical principles and optimization techniques, transformer models can effectively learn complex patterns in data, enabling their remarkable performance across various tasks.