# Parsimonious Online Learning with Kernels via sparse projections in function space

@article{Koppel2017ParsimoniousOL, title={Parsimonious Online Learning with Kernels via sparse projections in function space}, author={Alec Koppel and Garrett Warnell and Ethan Stump and Alejandro Ribeiro}, journal={2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)}, year={2017}, pages={4671-4675} }

We consider stochastic nonparametric regression problems in a reproducing kernel Hilbert space (RKHS), an extension of expected risk minimization to nonlinear function estimation. Popular perception is that kernel methods are inapplicable to online settings, since the generalization of stochastic methods to kernelized function spaces require memory storage that is cubic in the iteration index (“the curse of kernelization”). We alleviate this intractability in two ways: (1) we consider the use…

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## References

SHOWING 1-10 OF 76 REFERENCES

### Scalable Kernel Methods via Doubly Stochastic Gradients

- Computer ScienceNIPS
- 2014

An approach that scales up kernel methods using a novel concept called "doubly stochastic functional gradients" based on the fact that many kernel methods can be expressed as convex optimization problems, which can readily scale kernel methods up to the regimes which are dominated by neural nets.

### Breaking the curse of kernelization: budgeted stochastic gradient descent for large-scale SVM training

- Computer ScienceJ. Mach. Learn. Res.
- 2012

Comprehensive empirical results show that BSGD achieves higher accuracy than the state-of-the-art budgeted online algorithms and comparable to non-budget algorithms, while achieving impressive computational efficiency both in time and space during training and prediction.

### Online learning with kernels

- Computer ScienceIEEE Transactions on Signal Processing
- 2004

This paper considers online learning in a reproducing kernel Hilbert space, and allows the exploitation of the kernel trick in an online setting, and examines the value of large margins for classification in the online setting with a drifting target.

### Online Kernel Learning with a Near Optimal Sparsity Bound

- Computer ScienceICML
- 2013

This work focuses on Online Sparse Kernel Learning that aims to online learn a kernel classifier with a bounded number of support vectors and shows promising performance of the proposed algorithm compared to the state-of-the-art algorithms for online sparse kernel learning.

### Dual Space Gradient Descent for Online Learning

- Computer ScienceNIPS
- 2016

The Dual Space Gradient Descent (DualSGD) is presented, a novel framework that utilizes random features as an auxiliary space to maintain information from data points removed during budget maintenance while simultaneously mitigating the impact of the dimensionality issue on learning performance.

### Nonparametric Budgeted Stochastic Gradient Descent

- Computer ScienceAISTATS
- 2016

This paper proposes the Nonparametric Budgeted Stochastic Gradient Descent that allows the model size to automatically grow with data in a principled way and provides theoretical analysis to show that this framework is guaranteed to converge for a large collection of loss functions.

### Non-parametric Stochastic Approximation with Large Step sizes

- Computer Science, Mathematics
- 2014

In a stochastic approximation framework, it is shown that the averaged unregularized least-mean-square algorithm, given a sufficient large step-size, attains optimal rates of convergence for a variety of regimes for the smoothnesses of the optimal prediction function and the functions in $\mathcal{H}$.

### Large Scale Online Kernel Learning

- Computer ScienceJ. Mach. Learn. Res.
- 2016

A new framework for large scale online kernel learning, making kernel methods efficient and scalable for large-scale online learning applications, and presents two different online kernel machine learning algorithms that apply the random Fourier features for approximating kernel functions.

### Online Prediction of Time Series Data With Kernels

- Computer ScienceIEEE Transactions on Signal Processing
- 2009

This paper investigates a new model reduction criterion that makes computationally demanding sparsification procedures unnecessary and incorporates the coherence criterion into a new kernel-based affine projection algorithm for time series prediction.

### Error analysis for online gradient descent algorithms in reproducing kernel Hilbert spaces

- Computer Science
- 2006

This work considers online gradient descent algorithms with general convex loss functions in reproducing kernel Hilbert spaces (RKHS) and provides general conditions ensuring convergence of the algorithm in the RKHS norm.