# Convergence of Unregularized Online Learning Algorithms

arXiv: https://arxiv.org/abs/1708.02939 · HTML: https://mlsanity.com/paper/1708.02939/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Aug 2017.

## Built on by (most cited first, 3 of 3)

- [Generalization Analysis of Online Stochastic Gradient Descent for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/2407.07670/) (2024)
- [Decentralized Online Learning for Random Inverse Problems Over Graphs](https://mlsanity.com/paper/2303.11789/) (2023)
- [Convergence Rate for the Last Iterate of Stochastic Gradient Descent Schemes](https://mlsanity.com/paper/2507.07281/) (2025)

## Abstract (arXiv)

In this paper we study the convergence of online gradient descent algorithms in reproducing kernel Hilbert spaces (RKHSs) without regularization. We establish a sufficient condition and a necessary condition for the convergence of excess generalization errors in expectation. A sufficient condition for the almost sure convergence is also given. With high probability, we provide explicit convergence rates of the excess generalization errors for both averaged iterates and the last iterate, which in turn also imply convergence rates with probability one. To our best knowledge, this is the first high-probability convergence rate for the last iterate of online gradient descent algorithms without strong convexity. Without any boundedness assumptions on iterates, our results are derived by a novel use of two measures of the algorithm's one-step progress, respectively by generalization errors and by distances in RKHSs, where the variances of the involved martingales are cancelled out by the descent property of the algorithm.