[Paper Review] Online Learning with Multiple Operator-valued Kernels

Julien Audiffren, Hachem Kadri|arXiv (Cornell University)|Nov 1, 2013

Advanced Bandit Algorithms Research18 references3 citations

TL;DR

This paper proposes ONORMA and MONORMA, two online learning algorithms for vector-valued functions using operator-valued kernels, extending the NORMA algorithm to handle structured outputs. ONORMA provides theoretical cumulative error bounds for both regression and classification, while MONORMA learns a dynamic combination of kernels to adapt to unknown output structures, achieving performance close to batch methods with significantly reduced computational cost.

ABSTRACT

We consider the problem of learning a vector-valued function f in an online learning setting. The function f is assumed to lie in a reproducing Hilbert space of operator-valued kernels. We describe two online algorithms for learning f while taking into account the output structure. A first contribution is an algorithm, ONORMA, that extends the standard kernel-based online learning algorithm NORMA from scalar-valued to operator-valued setting. We report a cumulative error bound that holds both for classification and regression. We then define a second algorithm, MONORMA, which addresses the limitation of pre-defining the output structure in ONORMA by learning sequentially a linear combination of operator-valued kernels. Our experiments show that the proposed algorithms achieve good performance results with low computational cost.

Motivation & Objective

• To address the lack of efficient online algorithms for learning vector-valued functions in reproducing kernel Hilbert spaces with operator-valued kernels.
• To extend the NORMA algorithm to the operator-valued kernel setting, enabling online learning with structured outputs.
• To overcome the limitation of pre-defining output structure in existing methods by learning a linear combination of operator-valued kernels sequentially.
• To provide theoretical error bounds for online learning in both classification and regression settings using operator-valued kernels.
• To empirically validate the performance and efficiency of the proposed algorithms on synthetic and real-world multi-task datasets.

Proposed method

• ONORMA extends the NORMA algorithm to the operator-valued kernel framework, updating the function estimate incrementally using a kernel-induced Hilbert space structure.
• The algorithm uses a regularization term with a kernel matrix of size $ td \times td $, where $ t $ is the number of examples and $ d $ the output dimension.
• MONORMA introduces a sequential learning mechanism that combines multiple operator-valued kernels by adapting their weights during online updates.
• The method employs a learning rate $ \eta_t $ and regularization parameter $ \lambda $, ensuring $ \eta_t \lambda < 1 $ for stability.
• It uses two types of kernels: a separable Gaussian kernel $ K_\mu(x,x') = \exp(-\|x-x'\|_2^2/\mu) \mathbf{J} $, and a non-separable kernel $ K_\mu(x,x') = \mu\langle x,x'\rangle \mathbf{1} + (1-\mu)\langle x,x'\rangle^2 \mathbf{I} $.
• The algorithms are evaluated using Mean Squared Error (MSE) and compared against a batch algorithm on synthetic and real-world datasets with varying $ d $, $ t $, and kernel parameters.

Experimental results

Research questions

• RQ1 Can the NORMA algorithm be successfully extended to the operator-valued kernel setting for online learning of vector-valued functions?
• RQ2 Does the proposed ONORMA algorithm achieve theoretical cumulative error bounds for both classification and regression tasks?
• RQ3 Can MONORMA automatically learn the optimal combination of operator-valued kernels without prior knowledge of the output structure?
• RQ4 How does the performance of MONORMA compare to ONORMA and batch methods in terms of accuracy and computational efficiency?
• RQ5 What is the impact of kernel parameter selection on the generalization performance of the online algorithms?

Key findings

• ONORMA achieves a cumulative error bound that holds for both classification and regression, providing theoretical guarantees in the online setting.
• MONORMA outperforms ONORMA by learning a dynamic combination of kernels, adapting to unknown output structures and improving prediction accuracy.
• ONORMA and MONORMA achieve MSE levels close to the batch algorithm, demonstrating strong generalization performance despite online learning.
• The running time of ONORMA is significantly lower than the batch method—12s vs. 162s on the synthetic dataset—demonstrating efficiency.
• MONORMA reduces computational cost compared to the batch method (217s vs. 2514s on the CPU dataset), while maintaining high accuracy.
• The MSE of MONORMA is consistently lower than ONORMA across all datasets, especially when the true output structure is complex or unknown.