[Paper Review] Totally Corrective Boosting for Regularized Risk Minimization

Chunhua Shen, Hanxi Li+1 more|arXiv (Cornell University)|Aug 30, 2010

Sparse and Compressive Sensing Techniques32 references3 citations

TL;DR

This paper introduces CGBoost, a novel boosting framework that formulates regularized risk minimization through totally-corrective primal optimization, enabling efficient training without complex convex solvers. It unifies various loss functions and regularizers, demonstrating that even non-totally-corrective algorithms like AdaBoost can be interpreted within this framework, with empirical performance matching existing methods on UCI datasets.

ABSTRACT

Consideration of the primal and dual problems together leads to important new insights into the characteristics of boosting algorithms. In this work, we propose a general framework that can be used to design new boosting algorithms. A wide variety of machine learning problems essentially minimize a regularized risk functional. We show that the proposed boosting framework, termed CGBoost, can accommodate various loss functions and different regularizers in a totally-corrective optimization fashion. We show that, by solving the primal rather than the dual, a large body of totally-corrective boosting algorithms can actually be efficiently solved and no sophisticated convex optimization solvers are needed. We also demonstrate that some boosting algorithms like AdaBoost can be interpreted in our framework--even their optimization is not totally corrective. We empirically show that various boosting algorithms based on the proposed framework perform similarly on the UCIrvine machine learning datasets [1] that we have used in the experiments.

Motivation & Objective

• To develop a unified framework for designing boosting algorithms that optimize the primal problem in a totally-corrective manner.
• To enable efficient training of boosting models without requiring sophisticated convex optimization solvers.
• To unify existing boosting algorithms, including AdaBoost, under a common theoretical framework based on regularized risk minimization.
• To demonstrate that primal-based optimization can achieve performance comparable to dual-based approaches while maintaining computational efficiency.
• To provide theoretical insights into the duality between primal and dual formulations in boosting, leading to new algorithmic design principles.

Proposed method

• The framework formulates boosting as a regularized risk minimization problem in the primal space, directly optimizing the objective function over weak learners.
• It employs a totally-corrective strategy where all weak learner weights are re-optimized at each iteration, ensuring convergence to a globally optimal solution at each step.
• The method supports arbitrary differentiable loss functions and various regularizers, enabling flexibility across diverse machine learning tasks.
• By focusing on primal optimization, the approach avoids the need for complex dual decomposition or interior-point methods, enabling simpler and faster implementations.
• The framework allows reinterpretation of existing algorithms such as AdaBoost as approximations within a totally-corrective setting, even when they do not strictly enforce total correction.
• Empirical evaluation is conducted on standard UCIrvine datasets to compare performance across different loss and regularization choices.

Experimental results

Research questions

• RQ1 Can a totally-corrective boosting framework be designed that operates efficiently in the primal space without requiring advanced convex optimization tools?
• RQ2 How can various loss functions and regularizers be systematically incorporated into a unified boosting framework?
• RQ3 To what extent can existing non-totally-corrective algorithms like AdaBoost be reinterpreted within a totally-corrective primal optimization framework?
• RQ4 Does primal-based totally-corrective boosting achieve performance comparable to existing state-of-the-art boosting algorithms on standard benchmarks?
• RQ5 What insights into the duality between primal and dual formulations in boosting can be derived from this new framework?

Key findings

• The proposed CGBoost framework enables efficient training of totally-corrective boosting algorithms by solving the primal problem directly, eliminating the need for complex convex optimization solvers.
• The framework successfully accommodates a wide range of loss functions and regularizers, demonstrating broad applicability across different machine learning problems.
• AdaBoost and similar algorithms can be interpreted within the CGBoost framework, even though their optimization is not strictly totally corrective, revealing deeper theoretical connections.
• Empirical results on UCIrvine datasets show that CGBoost-based algorithms achieve performance comparable to existing boosting methods, validating its practical effectiveness.
• The study reveals that primal-based optimization in boosting can yield strong results without sacrificing theoretical rigor or convergence guarantees.
• The integration of total correction with primal optimization leads to a more coherent and extensible algorithmic design paradigm for regularized risk minimization.