  Despite an extensive body of literature on deep learning optimization, our
current understanding of what makes an optimization algorithm effective is
fragmented. In particular, we do not understand well whether enhanced
optimization translates to improved generalizability. Current research
overlooks the inherent stochastic nature of stochastic gradient descent (SGD)
and its variants, resulting in a lack of comprehensive benchmarking and insight
into their statistical performance. This paper aims to address this gap by
adopting a novel approach. Rather than solely evaluating the endpoint of
individual optimization trajectories, we draw from an ensemble of trajectories
to estimate the stationary distribution of stochastic optimizers. Our
investigation encompasses a wide array of techniques, including SGD and its
variants, flat-minima optimizers, and new algorithms we propose under the Basin
Hopping framework. Through our evaluation, which encompasses synthetic
functions with known minima and real-world problems in computer vision and
natural language processing, we emphasize fair benchmarking under a statistical
framework, comparing stationary distributions and establishing statistical
significance. Our study uncovers several key findings regarding the
relationship between training loss and hold-out accuracy, as well as the
comparable performance of SGD, noise-enabled variants, and novel optimizers
utilizing the BH framework. Notably, these algorithms demonstrate performance
on par with flat-minima optimizers like SAM, albeit with half the gradient
evaluations. We anticipate that our work will catalyze further exploration in
deep learning optimization, encouraging a shift away from single-model
approaches towards methodologies that acknowledge and leverage the stochastic
nature of optimizers.
