On the Expressive Power of Self-Attention Matrices

• URL: http://arxiv.org/abs/2106.03764v2
• Date: Tue, 8 Jun 2021 10:02:26 GMT
• Title: On the Expressive Power of Self-Attention Matrices
• Authors: Valerii Likhosherstov, Krzysztof Choromanski, Adrian Weller
• Abstract summary: We show that a fixed self-attention module can approximate arbitrary sparse patterns depending on the input. We propose an algorithm for finding adaptive inputs and fixed self-attention parameters in order to approximate a given matrix.
• Score: 41.72598286888797
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Transformer networks are able to capture patterns in data coming from many domains (text, images, videos, proteins, etc.) with little or no change to architecture components. We perform a theoretical analysis of the core component responsible for signal propagation between elements, i.e. the self-attention matrix. In practice, this matrix typically exhibits two properties: (1) it is sparse, meaning that each token only attends to a small subset of other tokens; and (2) it changes dynamically depending on the input to the module. With these considerations in mind, we ask the following question: Can a fixed self-attention module approximate arbitrary sparse patterns depending on the input? How small is the hidden size $d$ required for such approximation? We make progress in answering this question and show that the self-attention matrix can provably approximate sparse matrices, where sparsity is in terms of a bounded number of nonzero elements in each row and column. While the parameters of self-attention are fixed, various sparse matrices can be approximated by only modifying the inputs. Our proof is based on the random projection technique and uses the seminal Johnson-Lindenstrauss lemma. Our proof is constructive, enabling us to propose an algorithm for finding adaptive inputs and fixed self-attention parameters in order to approximate a given matrix. In particular, we show that, in order to approximate any sparse matrix up to a given precision defined in terms of preserving matrix element ratios, $d$ grows only logarithmically with the sequence length $L$ (i.e. $d = O(\log L)$).

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