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Large Deviations Rates for Stochastic Gradient Descent with Strongly Convex Functions
| Authors | D. Bajovic, D. Jakovetic, and S. Kar |
| Title | Large Deviations Rates for Stochastic Gradient Descent with Strongly Convex Functions |
| Abstract | Recent works have shown that high probability metrics with stochastic gradient descent (SGD) exhibit informativeness and in some cases advantage over the commonly adopted mean-square error-based ones. In this work we provide a formal framework for the study of general high probability bounds with SGD, based on the theory of large deviations. The framework allows for a generic (not-necessarily bounded) gradient noise satisfying mild technical assumptions, allowing for the dependence of the noise distribution on the current iterate. Under the preceding assumptions, we find an upper large deviations bound for SGD with strongly convex functions. The corresponding rate function captures analytical dependence on the noise distribution and other problem parameters. This is in contrast with conventional mean-square error analysis that captures only the noise dependence through the variance and does not capture the effect of higher order moments nor interplay between the noise geometry and the shape of the cost function. We also derive exact large deviation rates for the case when the objective function is quadratic and show that the obtained function matches the one from the general upper bound hence showing the tightness of the general upper bound. Numerical examples illustrate and corroborate theoretical findings. |
| ISBN | TBA |
| Conference | Preprint |
| Date | TBA |
| Location | TBA |
| Year of Publication | 2022 |
| Url | https://zenodo.org/record/7537625 |
| DOI |  |
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Funding
This project has received funding from the European Union’s Horizon 2020 Research and Innovation program under grant agreement No 957337. The website reflects only the view of the author(s) and the Commission is not responsible for any use that may be made of the information it contains.
