  In this paper, we aim to build the global convergence theory of encoder-only
shallow Transformers under a realistic setting from the perspective of
architectures, initialization, and scaling under a finite width regime. The
difficulty lies in how to tackle the softmax in self-attention mechanism, the
core ingredient of Transformer. In particular, we diagnose the scaling scheme,
carefully tackle the input/output of softmax, and prove that quadratic
overparameterization is sufficient for global convergence of our shallow
Transformers under commonly-used He/LeCun initialization in practice. Besides,
neural tangent kernel (NTK) based analysis is also given, which facilitates a
comprehensive comparison. Our theory demonstrates the separation on the
importance of different scaling schemes and initialization. We believe our
results can pave the way for a better understanding of modern Transformers,
particularly on training dynamics.
