Provably learning a multi-head attention layer

التفاصيل البيبلوغرافية
العنوان: Provably learning a multi-head attention layer
المؤلفون: Chen, Sitan, Li, Yuanzhi
سنة النشر: 2024
المجموعة: Computer Science
Statistics
مصطلحات موضوعية: Computer Science - Machine Learning, Computer Science - Data Structures and Algorithms, Statistics - Machine Learning
الوصف: 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 $F(\mathbf{X}) \triangleq \sum^m_{i=1} \mathrm{softmax}(\mathbf{X}\mathbf{\Theta}_i\mathbf{X}^\top)\mathbf{X}\mathbf{W}_i$. 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 $(dk)^{O(m^3)}$-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 large language models, 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.
Comment: 105 pages, comments welcome
نوع الوثيقة: Working Paper
URL الوصول: http://arxiv.org/abs/2402.04084
رقم الأكسشن: edsarx.2402.04084
قاعدة البيانات: arXiv