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Beyond Scaling Laws: Understanding Transformer Performance with Associative Memory

Xueyan Niu, Bo Bai, Lei Deng, Wei Han

TL;DR

The paper tackles why scaling up Transformer models does not always yield better performance and posits memorization of training data as a central mechanism. It develops a theory that treats Transformer layers as associative memories via Hopfield networks, using a distance-based layer energy and a global energy across layers constructed through majorization-minimization. A key result is a lower bound on cross-entropy and a predicted quadratic trade-off between model size and data size, $N=O(D^2)$, for well-separated patterns, with empirical evidence from GPT-2, vanilla Transformers, and OpenELM variants. This energy-based perspective links attention to nearest-neighbor retrieval, clarifies when memorization dominates learning, and offers guidance for selecting model/data scales in pre-training, while connecting to Chinchilla scaling and Hopfield innovations in the literature.

Abstract

Increasing the size of a Transformer does not always lead to enhanced performance. This phenomenon cannot be explained by the empirical scaling laws. Furthermore, the model's enhanced performance is closely associated with its memorization of the training samples. We present a theoretical framework that sheds light on the memorization during pre-training 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. In particular, the energy function in modern continuous Hopfield networks serves as an explanation for the attention mechanism, which we approximate with a distance-based energy function. By observing that the softmax function corresponds to the gradient of the LogSumExp function in the energy, and employing the majorization-minimization technique, we construct a global energy function designed to capture the layered architecture. We demonstrate a dependency between the model size and the dataset size for the model to achieve optimal performance, and we show that the achievable cross-entropy loss is bounded from below.

Beyond Scaling Laws: Understanding Transformer Performance with Associative Memory

TL;DR

The paper tackles why scaling up Transformer models does not always yield better performance and posits memorization of training data as a central mechanism. It develops a theory that treats Transformer layers as associative memories via Hopfield networks, using a distance-based layer energy and a global energy across layers constructed through majorization-minimization. A key result is a lower bound on cross-entropy and a predicted quadratic trade-off between model size and data size, , for well-separated patterns, with empirical evidence from GPT-2, vanilla Transformers, and OpenELM variants. This energy-based perspective links attention to nearest-neighbor retrieval, clarifies when memorization dominates learning, and offers guidance for selecting model/data scales in pre-training, while connecting to Chinchilla scaling and Hopfield innovations in the literature.

Abstract

Increasing the size of a Transformer does not always lead to enhanced performance. This phenomenon cannot be explained by the empirical scaling laws. Furthermore, the model's enhanced performance is closely associated with its memorization of the training samples. We present a theoretical framework that sheds light on the memorization during pre-training 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. In particular, the energy function in modern continuous Hopfield networks serves as an explanation for the attention mechanism, which we approximate with a distance-based energy function. By observing that the softmax function corresponds to the gradient of the LogSumExp function in the energy, and employing the majorization-minimization technique, we construct a global energy function designed to capture the layered architecture. We demonstrate a dependency between the model size and the dataset size for the model to achieve optimal performance, and we show that the achievable cross-entropy loss is bounded from below.
Paper Structure (42 sections, 11 theorems, 45 equations, 4 figures, 3 tables)

This paper contains 42 sections, 11 theorems, 45 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Our Contribution:
  3. Related work
  4. Scaling laws.
  5. Energy-based models.
  6. Hopfield models.
  7. System model
  8. Associative memories
  9. Transformer blocks
  10. Attention mechanism.
  11. Feed-forward layers.
  12. A global energy function
  13. A layer-wise energy function
  14. The layered structure
  15. Cross-entropy loss
  16. ...and 27 more sections

Key Result

Proposition 1

Given $\mathcal{D}=\{\rho_1\ldots, \rho_d\},$ the layer-wise energy $E(x)$ satisfies

Figures (4)

  • Figure 1: Left:Energy landscapes for a set of 2-dimensional patterns $\mathcal{D}=\{(-2,-0.5), (0.2,-0.3), (1.5,1.5)\}.$ (a) The negative LogSumExp function with $\beta=1,$ as an extension of demircigil2017model. (b) The regularization terms $\frac{1}{2} x^Tx+\beta^{-1} \log d + \frac{\max_i \|\rho^i\|^2}{2}$ in the MCHN energy. (c) The MCHN energy $E^1_{\mathrm{MCHN}}(x).$ (d) The layer-wise energy \ref{['eq:energy-ours']} with squared Euclidean norm. Right:Energy landscapes for a set of 1-dimensional patterns $\mathcal{D} = \{-2, 0, 1\}.$ The orange curves correspond to the MCHN energy with $\beta = 1, 2.$
  • Figure 2: Top: Distribution of nearest neighbor distances for output activations utilizing $25\%, 50\%, 75\%,$ and $100\%$ of output data. The mean and median values of these distances consistently hover around 20, aligning with the magnitude $2\sqrt{n/2\pi e}$ as hypothesized. Bottom-left: Performance of vanilla Transformers with 6 layers (left) and 10 layers (middle), each trained on the 2M Question-Formation dataset. The models were configured according to the experimental setup detailed in murty2023grokking. The training losses for both models converge to a value of approximately 1, a finding that is consistent with Proposition \ref{['prop:loss-bound']}. Bottom-right: The pre-training loss (dots) and validation loss (squares) of an OpenELM model across five training runs. The minimal validation losses are displayed in dashed lines. Each run's performance is marked by distinct colors, with the minimum validation loss value for each run indicated along the y-axis.
  • Figure 3: The cross-entropy loss for one model configuration during pre-training (depicted with dots) and validation (depicted with squares) across five separate training runs. The minimal attainable validation loss is represented by dashed lines. Each individual run's performance is distinguished by a unique color, and the y-axis highlights the lowest validation loss for each respective run.
  • Figure 4: Cross-entropy losses of eight models employing the OpenELM architecture as presented in Table \ref{['tab:mse']}.

Theorems & Definitions (14)

  • Definition 1
  • Remark 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Remark 2
  • Lemma 1
  • Lemma 2
  • ...and 4 more