[Paper Review] Mapping of attention mechanisms to a generalized Potts model

Riccardo Rende, Federica Gerace+2 more|arXiv (Cornell University)|Apr 14, 2023

Speech and dialogue systems51 references4 citations

TL;DR

This paper establishes an exact analytical mapping between a single-layer self-attention mechanism in transformers and the inverse problem of learning a generalized Potts model with interactions between word positions and token types. It shows that masked language modeling training is equivalent to solving the inverse Potts problem via pseudo-likelihood, and analytically computes the generalization error using the replica method, revealing the statistical limits of self-attention in learning structured distributions.

ABSTRACT

Transformers are neural networks that revolutionized natural language processing and machine learning. They process sequences of inputs, like words, using a mechanism called self-attention, which is trained via masked language modeling (MLM). In MLM, a word is randomly masked in an input sequence, and the network is trained to predict the missing word. Despite the practical success of transformers, it remains unclear what type of data distribution self-attention can learn efficiently. Here, we show analytically that if one decouples the treatment of word positions and embeddings, a single layer of self-attention learns the conditionals of a generalized Potts model with interactions between sites and Potts colors. Moreover, we show that training this neural network is exactly equivalent to solving the inverse Potts problem by the so-called pseudo-likelihood method, well known in statistical physics. Using this mapping, we compute the generalization error of self-attention in a model scenario analytically using the replica method.

Motivation & Objective

• To understand the statistical structures that self-attention learns during masked language modeling (MLM) training.
• To determine the family of conditional probability distributions that a single-layer self-attention mechanism can learn efficiently.
• To quantify the sample complexity required for self-attention to generalize well in learning structured data distributions.
• To establish a formal connection between self-attention and statistical physics models, particularly the Potts model.

Proposed method

• Models sequences of words as spins in a generalized Potts model with interactions between positions and word types.
• Uses factored self-attention, where attention weights depend only on positions and values only on embeddings, enabling analytical tractability.
• Maps the self-attention mechanism to the inverse Potts problem, with attention matrix corresponding to interaction couplings and value matrix to color similarity.
• Applies the replica method to compute the typical generalization error in the zero-temperature limit with Gaussian priors and square loss.
• Derives saddle-point equations for the order parameters (q, r, l, etc.) that characterize the learning performance.
• Uses the resulting free-energy density to analytically evaluate the generalization error of the self-attention model.

Experimental results

Research questions

• RQ1 What type of data distribution can a single-layer self-attention mechanism learn efficiently under masked language modeling?
• RQ2 How does the training of self-attention relate to statistical physics models such as the Potts model?
• RQ3 What is the generalization error of self-attention in learning structured, high-dimensional distributions?
• RQ4 Can the learning process of self-attention be exactly mapped to a known inverse problem in statistical mechanics?
• RQ5 What is the sample complexity required for self-attention to achieve good generalization in learning conditional distributions?

Key findings

• A single layer of factored self-attention learns the conditional distributions of a generalized Potts model exactly in the large training set limit.
• The training of self-attention via masked language modeling is mathematically equivalent to solving the inverse Potts problem using the pseudo-likelihood method.
• The generalization error of self-attention can be computed analytically using the replica method, yielding a closed-form expression in the zero-temperature limit.
• The solution of the saddle-point equations reveals that label noise in the learning task arises directly from input noise, not independently tunable.
• The derived free-energy density matches the form of supervised learning with noisy labels and structured data, but with noise intrinsically linked to input variability.
• The system exhibits a phase transition in learning performance, with the order parameters (q, r, l) capturing the trade-off between generalization and overfitting.