# Ridge Regression from Poisson Resetting: A Renewal Perspective on Spectral Regularization

arXiv: https://arxiv.org/abs/2605.30059 · HTML: https://mlsanity.com/paper/2605.30059/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 46 references that are in the map.
- First posted on arXiv 28 May 2026.

## Builds on (most cited first, 16 of 16)

- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Stochastic Gradient Descent as Approximate Bayesian Inference](https://mlsanity.com/paper/1704.04289/) (2017)
- [The Primacy Bias in Deep Reinforcement Learning](https://mlsanity.com/paper/2205.07802/) (2022)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [Implicit Gradient Regularization](https://mlsanity.com/paper/2009.11162/) (2020)
- [Implicit regularization for deep neural networks driven by an Ornstein-Uhlenbeck like process](https://mlsanity.com/paper/1904.09080/) (2019)
- [A Continuous-Time View of Early Stopping for Least Squares](https://mlsanity.com/paper/1810.10082/) (2018)
- [The Implicit Regularization of Stochastic Gradient Flow for Least Squares](https://mlsanity.com/paper/2003.07802/) (2020)
- [The Benefits of Implicit Regularization from SGD in Least Squares Problems](https://mlsanity.com/paper/2108.04552/) (2021)
- [Implicit Regularization or Implicit Conditioning? Exact Risk Trajectories of SGD in High Dimensions](https://mlsanity.com/paper/2206.07252/) (2022)
- [Non-Stationary Learning of Neural Networks with Automatic Soft Parameter Reset](https://mlsanity.com/paper/2411.04034/) (2024)
- [On Regularization via Early Stopping for Least Squares Regression](https://mlsanity.com/paper/2406.04425/) (2024)
- [Comparing Classes of Estimators: When does Gradient Descent Beat Ridge Regression in Linear Models?](https://mlsanity.com/paper/2108.11872/) (2021)
- [Stochastic Resetting Mitigates Latent Gradient Bias of SGD from Label Noise](https://mlsanity.com/paper/2406.00396/) (2024)
- [First-Passage Approach to Optimizing Perturbations for Improved Training of Machine Learning Models](https://mlsanity.com/paper/2502.04121/) (2025)
- [On the Saturation Effects of Spectral Algorithms in Large Dimensions](https://mlsanity.com/paper/2503.00504/) (2025)

## Abstract (arXiv)

We connect stochastic resetting from non-equilibrium statistical physics with ridge regularization in statistical learning. For linear gradient flow, resetting to the origin at rate r produces stationary mean (X^top X+rI)-1X^top y, exactly the ridge estimator with penalty λ=r. This uses the known Laplace-transform relationship between ridge regression and exponential-time averaging of gradient flow, with the exponential time now interpreted as the stationary age associated with Poisson resetting. We then extend this identity to general renewal reset laws: the exponential reset time distribution is the unique renewal law whose stationary mean reproduces scalar ridge in every eigendirection as an exact filter identity for every positive curvature, while non-exponential renewal laws generate alternative spectral filters. At the fluctuation level, we study a separate additive Ornstein-Uhlenbeck extension with constant diffusion, interpreted as a stylized SGD approximation. In this setting, the equality holds only at the level of the mean, since the reset process has a nonzero stationary covariance from accumulated OU noise and reset-timing variance, whereas deterministic ridge is a fixed estimator with the same center. Stylized experiments compare the deterministic renewal-induced filters directly and illustrate when filters induced by non-exponential reset-time laws can differ predictively from ridge. The results for the stationary mean and the induced spectral filters are established for continuous-time gradient flow with isotropic resetting on quadratic objectives; the covariance and risk formulas additionally assume additive noise with state-independent covariance.