Limits to Depth Efficiencies of Self-Attention

Self-attention architectures, which are rapidly pushing the frontier in natural language processing, demonstrate a surprising depth-inefficient behavior: Empirical signals indicate that increasing the internal representation (network width) is just as useful as increasing the number of self-attention layers (network depth). In this paper, we theoretically study the interplay between depth and width in self-attention, and shed light on the root of the above phenomenon. We invalidate the seemingly plausible hypothesis by which widening is as effective as deepening for self-attention, and show that in fact stacking self-attention layers is so effective that it quickly saturates a capacity of the network width. Specifically, we pinpoint a "depth threshold" that is logarithmic in $d_x$, the network width: $L_{\textrm{th}}=\log_{3}(d_x)$. For networks of depth that is below the threshold, we establish a double-exponential depth-efficiency of the self-attention operation, while for depths over the threshold we show that depth-inefficiency kicks in. Our predictions strongly accord with extensive empirical ablations in Kaplan et al. (2020), accounting for the different behaviors in the two depth-(in)efficiency regimes. By identifying network width as a limiting factor, our analysis indicates that solutions for dramatically increasing the width can facilitate the next leap in self-attention expressivity.