What Makes Looped Transformers Perform Better Than Non-Recursive Ones (Provably)

While looped transformers (termed as Looped-Attn) often outperform standard transformers (termed as Single-Attn) on complex reasoning tasks, the theoretical basis for this advantage remains underexplored. In this paper, we explain this phenomenon through the lens of loss landscape geometry, inspired by empirical observations of their distinct dynamics at both sample and Hessian levels. To formalize this, we extend the River-Valley landscape model by distinguishing between U-shaped valleys (flat) and V-shaped valleys (steep). Based on empirical observations, we conjecture that the recursive architecture of Looped-Attn induces a landscape-level inductive bias towards River-V-Valley. Theoretical derivations based on this inductive bias guarantee a better loss convergence along the river due to valley hopping, and further encourage learning about complex patterns compared to the River-U-Valley induced by Single-Attn. Building on this insight, we propose SHIFT (Staged HIerarchical Framework for Progressive Training), a staged training framework that accelerates the training process of Looped-Attn while achieving comparable performances.