Efficient Learning of Generalized Linear and Single Index Models with Isotonic Regression (1104.2018v1)

Abstract: Generalized Linear Models (GLMs) and Single Index Models (SIMs) provide powerful generalizations of linear regression, where the target variable is assumed to be a (possibly unknown) 1-dimensional function of a linear predictor. In general, these problems entail non-convex estimation procedures, and, in practice, iterative local search heuristics are often used. Kalai and Sastry (2009) recently provided the first provably efficient method for learning SIMs and GLMs, under the assumptions that the data are in fact generated under a GLM and under certain monotonicity and Lipschitz constraints. However, to obtain provable performance, the method requires a fresh sample every iteration. In this paper, we provide algorithms for learning GLMs and SIMs, which are both computationally and statistically efficient. We also provide an empirical study, demonstrating their feasibility in practice.

• The paper proposes GLM-tron and L-Isotron, novel algorithms leveraging isotonic regression for efficient learning of GLMs and SIMs.
• These algorithms achieve provable efficiency and statistical rates without requiring new data samples at each iteration, overcoming prior limitations.
• The algorithms show potential for handling high-dimensional data and kernel spaces, opening new avenues for efficient non-convex learning and broader applications.

Efficient Learning of Generalized Linear and Single Index Models with Isotonic Regression

This paper addresses the challenge of efficiently learning Generalized Linear Models (GLMs) and Single Index Models (SIMs) through isotonic regression, offering novel insights into computational and statistical efficiency in non-convex estimation problems. The authors propose algorithms that successfully overcome limitations of existing methods by circumventing the requirement of new data samples at each iteration—a necessity in previous approaches.

Overview of GLMs and SIMs

GLMs extend linear regression by introducing a link function $g$ that models the relationship between the expected value of the target variable and the predictors in a linear form. Conversely, SIMs involve learning the linear parameter vector $w$ alongside the link function, which belongs to a vast family of monotonic functions. The authors note existing iterative approaches lack provable guarantees for computational efficiency and statistical performance, especially concerning SIMs.

Main Contributions

The paper introduces two algorithms: GLM-tron for GLMs with known link functions, and L-Isotron for SIMs with unknown monotonic link functions. These algorithms are characterized by their practical applicability and provable efficiency:

• GLM-tron algorithm provides robust performance with guarantees on statistical rates without iterative reweighted procedures while maintaining computational efficiency akin to perceptron algorithms.
• L-Isotron algorithm extends the framework to SIMs, efficiently handling non-parametric models with unknown functions that are monotonic and Lipschitz. Notably, the algorithm integrates a Lipschitz-constrained version of isotonic regression, optimizing statistical rates without fresh samples at each iteration.

Theoretical Implications

The paper asserts that leveraging isotonic regression tailored with Lipschitz constraints allows GLM-tron and L-Isotron algorithms to achieve efficient sample and computational complexities. Specifically, the L-Isotron algorithm's statistical rates are notably sharper due to the joint estimation strategy of both parameters and non-parametric link functions, under the assumption of monotonicity.

Numerical Results and Practical Implications

Empirical studies underscore the algorithms' feasibility and effectiveness, showcasing their robustness relative to established procedures. The algorithms demonstrate the potential to handle high-dimensional data and kernel feature spaces, promising scalable real-world applications.

Future Developments

This research sets a precedent for future investigations into non-convex learning models where monotonicity and Lipschitz properties facilitate efficient learning algorithms. Expanding these methods to broader non-linear models and other distributional assumptions represents a compelling direction for advancing AI capabilities in statistical learning frameworks.

Conclusion

The paper provides significant advancements in the field of statistical learning for GLMs and SIMs, mitigating the typical computational inefficiencies with a concrete empirical risk minimization approach and isotonic regression enhancements. This research opens avenues for more sophisticated algorithms, particularly in cases where traditional iterative techniques fall short, offering novel strategies for efficient non-convex optimization.