A Proof of Learning Rate Transfer under $μ$P
This provides theoretical foundations for parameterization choices in neural network training, which is important for researchers developing optimization methods.
The authors tackled the problem of understanding learning rate behavior in wide neural networks by proving that under μP parameterization, the optimal learning rate converges to a non-zero constant as width increases, providing theoretical justification for learning rate transfer, while showing this fails under SP and NTP parameterizations.
We provide the first proof of learning rate transfer with width in a linear multi-layer perceptron (MLP) parametrized with $μ$P, a neural network parameterization designed to ``maximize'' feature learning in the infinite-width limit. We show that under $μP$, the optimal learning rate converges to a \emph{non-zero constant} as width goes to infinity, providing a theoretical explanation to learning rate transfer. In contrast, we show that this property fails to hold under alternative parametrizations such as Standard Parametrization (SP) and Neural Tangent Parametrization (NTP). We provide intuitive proofs and support the theoretical findings with extensive empirical results.