# How Far Can Sharpness and Complexity Jointly Explain Generalization?

arXiv: https://arxiv.org/abs/2606.29043 · HTML: https://mlsanity.com/paper/2606.29043/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 45 references that are in the map.
- First posted on arXiv 27 Jun 2026.

## Builds on (most cited first, 30 of 30)

- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Do ImageNet Classifiers Generalize to ImageNet?](https://mlsanity.com/paper/1902.10811/) (2019)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Deep Double Descent: Where Bigger Models and More Data Hurt](https://mlsanity.com/paper/1912.02292/) (2019)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Exploring Generalization in Deep Learning](https://mlsanity.com/paper/1706.08947/) (2017)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [Fantastic Generalization Measures and Where to Find Them](https://mlsanity.com/paper/1912.02178/) (2019)
- [Sharp Minima Can Generalize For Deep Nets](https://mlsanity.com/paper/1703.04933/) (2017)
- [In Search of the Real Inductive Bias: On the Role of Implicit Regularization in Deep Learning](https://mlsanity.com/paper/1412.6614/) (2014)
- [Gradient Descent Maximizes the Margin of Homogeneous Neural Networks](https://mlsanity.com/paper/1906.05890/) (2019)
- [A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks](https://mlsanity.com/paper/1707.09564/) (2017)
- [Norm-Based Capacity Control in Neural Networks](https://mlsanity.com/paper/1503.00036/) (2015)
- [ASAM: Adaptive Sharpness-Aware Minimization for Scale-Invariant Learning of Deep Neural Networks](https://mlsanity.com/paper/2102.11600/) (2021)
- [Implicit Bias of Gradient Descent on Linear Convolutional Networks](https://mlsanity.com/paper/1806.00468/) (2018)
- [User-friendly introduction to PAC-Bayes bounds](https://mlsanity.com/paper/2110.11216/) (2021)
- [On the Origin of Implicit Regularization in Stochastic Gradient Descent](https://mlsanity.com/paper/2101.12176/) (2021)
- [Towards Understanding Sharpness-Aware Minimization](https://mlsanity.com/paper/2206.06232/) (2022)
- [A Primer on PAC-Bayesian Learning](https://mlsanity.com/paper/1901.05353/) (2019)
- [On the properties of variational approximations of Gibbs posteriors](https://mlsanity.com/paper/1506.04091/) (2015)
- [Three Mechanisms of Weight Decay Regularization](https://mlsanity.com/paper/1810.12281/) (2018)
- [On the Maximum Hessian Eigenvalue and Generalization](https://mlsanity.com/paper/2206.10654/) (2022)
- [PAC-Bayes unleashed: generalisation bounds with unbounded losses](https://mlsanity.com/paper/2006.07279/) (2020)
- [More PAC-Bayes bounds: From bounded losses, to losses with general tail behaviors, to anytime validity](https://mlsanity.com/paper/2306.12214/) (2023)
- [PAC-Bayes-Chernoff bounds for unbounded losses](https://mlsanity.com/paper/2401.01148/) (2024)
- [Improving Generalization of Complex Models under Unbounded Loss Using PAC-Bayes Bounds](https://mlsanity.com/paper/2305.19243/) (2023)
- [Regularizing Deep Neural Networks with Stochastic Estimators of Hessian Trace](https://mlsanity.com/paper/2208.05924/) (2022)

## Abstract (arXiv)

Sharpness and complexity are two central factors in the generalization analysis of deep neural networks. Existing quantitative evaluations of generalization measures have largely focused on individual scalar measures, leaving the joint explanatory power of sharpness and complexity largely unexplored. This work studies how far sharpness and complexity can jointly explain generalization. We use linear regression and introduce a Pareto-based analysis to quantitatively evaluate the joint explanatory power of these two factors. Beyond the existing parameter-level definitions, we further propose realizations of sharpness and complexity that are closer to function space and less dependent on raw parameter representations. We find that function-oriented definitions of these two quantities expand the explanatory scope of the two-factor view beyond what is achieved by existing parameter-level metrics. Overall, our results support the sharpness-complexity perspective as an informative lens for understanding generalization across diverse settings. At the same time, the remaining failures indicate that whether this two-factor view can serve as a complete theory of generalization remains open.