  Increasing the size of a Transformer model does not always lead to enhanced
performance. This phenomenon cannot be explained by the empirical scaling laws.
Furthermore, improved generalization ability occurs as the model memorizes the
training samples. We present a theoretical framework that sheds light on the
memorization process and performance dynamics of transformer-based language
models. We model the behavior of Transformers with associative memories using
Hopfield networks, such that each transformer block effectively conducts an
approximate nearest-neighbor search. Based on this, we design an energy
function analogous to that in the modern continuous Hopfield network which
provides an insightful explanation for the attention mechanism. Using the
majorization-minimization technique, we construct a global energy function that
captures the layered architecture of the Transformer. Under specific
conditions, we show that the minimum achievable cross-entropy loss is bounded
from below by a constant approximately equal to 1. We substantiate our
theoretical results by conducting experiments with GPT-2 on various data sizes,
as well as training vanilla Transformers on a dataset of 2M tokens.
