A Simplified Analysis of SGD for Linear Regression with Weight Averaging
This work makes the analysis more accessible for researchers studying optimization in overparameterized models, but it is incremental as it simplifies an existing theoretical framework without introducing new algorithms or results.
The authors tackled the theoretical analysis of stochastic gradient descent (SGD) for linear regression by providing a simplified proof that recovers existing sharp rates for bias and variance bounds, using basic linear algebra instead of complex operator manipulations.
Theoretically understanding stochastic gradient descent (SGD) in overparameterized models has led to the development of several optimization algorithms that are widely used in practice today. Recent work by~\citet{zou2021benign} provides sharp rates for SGD optimization in linear regression using constant learning rate, both with and without tail iterate averaging, based on a bias-variance decomposition of the risk. In our work, we provide a simplified analysis recovering the same bias and variance bounds provided in~\citep{zou2021benign} based on simple linear algebra tools, bypassing the requirement to manipulate operators on positive semi-definite (PSD) matrices. We believe our work makes the analysis of SGD on linear regression very accessible and will be helpful in further analyzing mini-batching and learning rate scheduling, leading to improvements in the training of realistic models.