Provably learning a multi-head attention layer
(2402.04084)Abstract
The multi-head attention layer is one of the key components of the transformer architecture that sets it apart from traditional feed-forward models. Given a sequence length $k$, attention matrices $\mathbf{\Theta}1,\ldots,\mathbf{\Theta}m\in\mathbb{R}{d\times d}$, and projection matrices $\mathbf{W}1,\ldots,\mathbf{W}m\in\mathbb{R}{d\times d}$, the corresponding multi-head attention layer $F: \mathbb{R}{k\times d}\to \mathbb{R}{k\times d}$ transforms length-$k$ sequences of $d$-dimensional tokens $\mathbf{X}\in\mathbb{R}{k\times d}$ via $F(\mathbf{X}) \triangleq \summ_{i=1} \mathrm{softmax}(\mathbf{X}\mathbf{\Theta}i\mathbf{X}\top)\mathbf{X}\mathbf{W}i$. In this work, we initiate the study of provably learning a multi-head attention layer from random examples and give the first nontrivial upper and lower bounds for this problem: - Provided ${\mathbf{W}i, \mathbf{\Theta}i}$ satisfy certain non-degeneracy conditions, we give a $(dk){O(m3)}$-time algorithm that learns $F$ to small error given random labeled examples drawn uniformly from ${\pm 1}{k\times d}$. - We prove computational lower bounds showing that in the worst case, exponential dependence on $m$ is unavoidable. We focus on Boolean $\mathbf{X}$ to mimic the discrete nature of tokens in LLMs, though our techniques naturally extend to standard continuous settings, e.g. Gaussian. Our algorithm, which is centered around using examples to sculpt a convex body containing the unknown parameters, is a significant departure from existing provable algorithms for learning feedforward networks, which predominantly exploit algebraic and rotation invariance properties of the Gaussian distribution. In contrast, our analysis is more flexible as it primarily relies on various upper and lower tail bounds for the input distribution and "slices" thereof.
Overview
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The paper presents nontrivial learning guarantees for multi-head attention mechanisms in transformer architectures, focusing on an algorithm that learns these mechanisms under specific conditions.
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The algorithm involves a six-stage process that refines projection matrices using novel techniques, leveraging properties of convex functions and softmax functions.
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Two main theoretical contributions are provided: an algorithm for learning with high accuracy and computational lower bounds indicating potential exponential complexity.
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The research offers insights into the polynomial-time learnability of self-attention mechanisms and establishes theoretical limitations of efficient learnability based on cryptographic hypotheses.
Overview of Techniques
The study begins by establishing nontrivial learning guarantees for a multi-head attention mechanism using a sequence of novel algorithmic techniques and analytical methods. The multi-head attention mechanism is one of the essential components of transformer architectures, which are at the forefront of recent advances in AI, particularly in natural language processing tasks. This work concentrates on proving the existence of an algorithm to learn such mechanisms under specific conditions.
The primary approach revolves around an intricate algorithm that operates over six stages. These stages include creating a crude approximation of the sum of projection matrices, refining the enclosure around the attention matrices, and ultimately using this computed data to infer projection matrices with significant accuracy. Key to this refinement is observing examples that produce certain attention patterns, which then guide a least-squares problem towards estimating these projection matrices.
The paper also utilizes various properties of convex functions, concentration inequalities, and characteristics of softmax functions to model the self-attention mechanism's learning problem as a convex optimization task. Theoretical findings incorporate bounds on linear and quadratic forms, a Hanson-Wright inequality adapted for the cube-slice distribution, and an extension of the integro-local central limit theorem.
Main Results
The main theoretical contributions are twofold. First, the authors provide an algorithm that outputs, with high probability, an approximate multi-head attention layer that has small test loss. This result is achieved given access to labeled examples that are uniformly drawn and perfectly labeled by an unknown ground truth transformer, assuming that the examples and ground truth parameters satisfy certain non-degeneracy conditions.
Second, the authors establish a set of computational lower bounds that suggest, in the worst case, an exponential dependence on the number of attention heads is unavoidable. These results leverage hardness assumptions from learning with rounding and statistical query frameworks, demonstrating that under these models, learning even relatively simple transformers requires either exponential number of queries or a tolerance that is exponentially small.
Implications
The findings presented offer substantial value toward understanding how self-attention mechanisms could be learned in polynomial time under realistic settings. The insights gained through this work contribute to the broader quest of devising training algorithms for transformers and understanding their learning dynamics and sample complexity.
Furthermore, computational lower bounds present a firm theoretical basis for the inherent limitations of efficient learnability for transformers. The assumptions used align with contemporary cryptographic hypotheses, indicating that certain lower bound conjectures could be justified within real-life applications and practical considerations.
Conclusion
The work challenges the current understanding of the learnability of transformers by providing a mixture of upper and lower bounds on the complexity of the learning problem. The performance guarantee for the learning algorithm under certain assumptions juxtaposes intriguing computational lower bounds derived from plausible cryptographic conjectures, paving a way for subsequent explorations that could either harness the learnability under stricter conditions or prove more in-depth computational hardness results.
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