On Expressive Power of Looped Transformers: Theoretical Analysis and Enhancement via Timestep Encoding

Looped Transformers provide advantages in parameter efficiency, computational capabilities, and generalization for reasoning tasks. However, their expressive power regarding function approximation remains underexplored. In this paper, we establish the approximation rate of Looped Transformers by defining the modulus of continuity for sequence-to-sequence functions. This reveals a limitation specific to the looped architecture. That is, the analysis prompts the incorporation of scaling parameters for each loop, conditioned on timestep encodings. Experiments validate the theoretical results, showing that increasing the number of loops enhances performance, with further gains achieved through the timestep encoding. Code is available at https: //github.com/kevin671/tmlt .