# Generalized Smoothness

The Burer–Monteiro factorized objective for low-rank matrix minimization is shown to be generalized smooth with exponent 2/3, and adaptive gradient descent and SafeAdam built on that property match or beat Dyn-NoLips on SymNMF text and image benchmarks without tuning relative-smoothness kernel parameters.

The Burer–Monteiro factorized objective for low-rank matrix minimization is shown to be generalized smooth with exponent 2/3, and adaptive gradient descent and SafeAdam built on that property match or beat Dyn-NoLips on SymNMF text and image benchmarks without tuning relative-smoothness kernel parameters.

In Bayesian LoRA fine-tuning of LLMs and vision-language models, minibatch gradient errors are heavy-tailed and grow with distance from the final iterate, and a robust estimator combining geometric-median centering with residual clipping, together with adaptive step sizes, retains non-asymptotic Fisher-information guarantees on non-log-concave targets.

In Bayesian LoRA fine-tuning of LLMs and vision-language models, minibatch gradient errors are heavy-tailed and grow with distance from the final iterate, and a robust estimator combining geometric-median centering with residual clipping, together with adaptive step sizes, retains non-asymptotic Fisher-information guarantees on non-log-concave targets.