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

Sham Kakade; Adam Tauman Kalai; Varun Kanade; Ohad Shamir

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

Sham Kakade, Adam Tauman Kalai, Varun Kanade, Ohad Shamir

arXiv:1104.2018v1Apr 11, 2011cs.AIcs.LGstat.ML

TL;DR

The paper addresses the lack of computationally and statistically efficient estimation procedures for GLMs and SIMs, especially when the link and predictor must be jointly estimated. It introduces GLM-tron and L-Isotron, using Lipschitz-constrained link fitting to avoid the fresh-sample requirement of Isotron, and reports theoretical guarantees plus empirical feasibility.

Problem
Existing GLM and SIM estimation procedures are often non-convex heuristics, while Isotron’s provable guarantees require discarding previous data and requesting fresh examples each iteration.
Method
The paper introduces GLM-tron for known monotonic Lipschitz links and L-Isotron for unknown links, using Lipschitz-constrained isotonic fitting.
Results
The algorithms are provably efficient statistically and computationally, and experiments find them feasible and comparable to, or in many cases slightly better than, other regression techniques.
Takeaways & Limitations
Lipschitz-constrained link fitting provides a practical route to learning GLMs and SIMs while retaining non-trivial theoretical guarantees.
Takeaways & Limitations
The L-Isotron rates are worse than the GLM-tron rates because simultaneously estimating the link and predictor is more difficult.

Abstract

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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.

1 Introduction

GLMs and SIMs extend linear regression through link functions, but estimating their parameters can be computationally difficult. This work develops efficient algorithms addressing statistical waste and non-convex joint estimation.

Models and challenges: GLMs link the conditional expectation of Y to a linear predictor, while SIMs require jointly estimating an unknown link function and vector.In SIMs, the link may belong to a large non-parametric family such as all monotonic functions.
Models and challenges: Even with a known link function, classical iterative estimation lacks both computational efficiency and provable good statistical rates.Iteratively reweighted least squares is the standard procedure described for this setting.
Models and challenges: Joint estimation of the link and vector is highly non-convex, so existing practical methods are usually heuristics without guarantees of global convergence.A kernel-based agnostic method provides formal guarantees but has super-polynomial complexity dependence on the norm of w.
Prior approach: The Isotron algorithm was the first provably efficient method for these models under monotonicity, Lipschitz, and realizability assumptions, but it discards previous data at every iteration.Its isotonic-regression step uses PAV, which does not impose a Lipschitz constraint.
This work: GLM-tron and L-Isotron provide practical, parameter-free algorithms with provable statistical and computational efficiency, kernelization, and empirical feasibility.L-Isotron learns an unknown monotonic Lipschitz link, while GLM-tron handles a known one; the original Isotron also performs effectively in several cases.

2 Setting

The setting assumes bounded inputs and responses generated through a bounded-norm linear predictor and a non-decreasing 1-Lipschitz function. The algorithms construct link-composed linear predictors and evaluate them using squared loss and excess error.

Assumptions: Data are sampled i.i.d. from the unit Euclidean ball paired with responses in [0,1].The framework also applies to high- or infinite-dimensional kernel feature spaces.
Assumptions: The regression function has the form E[y|x] = u(w · x), with ||w||≤W and u non-decreasing, 1-Lipschitz, and valued in [0,1].The function u plays the role of g^-1 in generalized linear models.
Objectives: Performance is measured by squared loss, including excess error relative to the Bayes-optimal predictor x 7→u(w · x).The goal is to make excess error, equivalently the stated regression error, as small as possible.
Objectives: The empirical error quantities are computed on a sample and include a fixed-design error that conditions on the observed inputs.The empirical excess error is the counterpart of the population excess error.
Algorithmic form: Each hypothesis has the form ht(x)=ut(wt · x), where ut is non-decreasing and 1-Lipschitz and wt is a linear predictor.The analysis connects small empirical excess error to small population excess error through statistical arguments.

3 The GLM-tron Algorithm

GLM-tron addresses known-link GLMs with a parameter-free perceptron-like procedure. Under the stated realizable assumptions, one iterate is nearly optimal and achieves excess error O(1/√m).

Algorithm: GLM-tron estimates the linear predictor when u is known, while supporting arbitrary non-decreasing Lipschitz transfer functions.The algorithm is presented as a simple, parameter-free variant of the perceptron.
Guarantee: The analysis guarantees that, after sufficiently many iterations, at least one hypothesis is nearly optimal relative to the Bayes-optimal predictor.The proof uses a decrease in squared distance to the true vector when empirical excess error is not small.
Guarantee: With probability at least 1−δ, some iteration t<O(W^2(m/log(1/δ))) satisfies the theorem’s accuracy condition.The guarantee assumes i.i.d. data generated by a known non-decreasing 1-Lipschitz u and a vector with norm at most W.
Guarantee: O(1/√m) is the resulting excess-error rate for at least one GLM-tron hypothesis.A hold-out set can select an appropriate iterate by estimating its error.

4 The L-Isotron Algorithm

L-Isotron learns an unknown non-decreasing 1-Lipschitz transfer function together with the index direction in single index models. Its LPAV-based update avoids the fresh-sample requirement of Isotron, while retaining practical and theoretical guarantees.

L-Isotron targets single index models where the transfer function is unknown but non-decreasing and 1-Lipschitz.
LPAV replaces PAV by fitting a non-decreasing 1-Lipschitz function to the current linear predictions and responses.The fit minimizes squared error over the one-dimensional inputs.
Unlike Isotron, L-Isotron can obtain similar guarantees without requesting a fresh training sample at every iteration.
The algorithm is simple and parameter-free, but simultaneous estimation of the transfer function and direction yields somewhat worse rates.
Theorem 2 provides both dimension-dependent and dimension-independent bounds for L-Isotron under the stated single-index assumptions.A hold-out set can select an iteration whose hypothesis satisfies the theorem’s conditions.

5 Proofs

The proofs formulate LPAV as constrained one-dimensional fitting and establish how its computed outputs support progress of the L-Isotron direction updates. Uniform convergence then converts empirical error and iteration progress into the theorem’s population guarantees.

LPAV fits values constrained to arise from a non-decreasing 1-Lipschitz function, then linearly interpolates them into the full transfer function.
At each iteration, LPAV is run on the training sample using the current direction w_t to obtain the transfer function u_t.
The proof compares computable LPAV predictions with hypothetical predictions based on conditional means through Proposition 1.The proposition’s proof is deferred to Appendix B.
The analysis shows that squared distance to the true direction decreases when the hypothesis error is not already small.
Bounding the number of such decreases yields an iteration after which empirical error is controlled, and uniform convergence transfers this control to population error.

6 Experiments

The experiments compare L-Isotron and GLM-tron with Isotron and other regression methods on synthetic and UCI data. They examine overfitting from unconstrained transfer functions and report competitive real-world performance.

Experimental design: The study tests L-Isotron and GLM-tron against Isotron and several regression techniques using synthetic data and five UCI datasets.The UCI datasets are communities, concrete, housing, parkinsons, and wine-quality; results are averaged over 10-fold cross-validation.
L-Isotron vs Isotron: L-Isotron uses LPAV to fit a Lipschitz monotonic transfer function, whereas Isotron uses PAV to fit a generally non-Lipschitz monotonic function.The comparison is intended to assess whether the Lipschitz constraint reduces overfitting.
L-Isotron vs Isotron: 0.338±0.058 versus 0.526±0.175 normalized error: L-Isotron outperformed Isotron on the synthetic experiment.Across folds, the normalized-error difference between Isotron and L-Isotron was 0.189 ± 0.139.
L-Isotron vs Isotron: In the synthetic construction, Isotron overfits irrelevant attributes, while L-Isotron predicts the true mean near 0.5 because of its Lipschitz constraint.The first attribute is the only relevant attribute, while the remaining coordinates introduce irrelevant noisy features.
Real-world datasets: On the UCI datasets, L-Isotron, GLM-tron, and Isotron had performance comparable to other regression techniques and were sometimes slightly better.Table 1 reports mean squared error normalized by the variance across 10 folds; Table 2 reports differences relative to L-Isotron.
Real-world datasets: For concrete and communities, Isotron repeatedly produced piecewise-constant transfer functions, whereas L-Isotron produced smoother Lipschitz functions.The transfer-function plots illustrate the qualitative difference between the two fitting procedures.

A.1 Proof of Thm. 1

The proof analyzes GLM-tron by showing that large prediction error decreases the distance between the iterate and the target direction. This yields an iteration bound and a high-probability error guarantee.

Progress lemma: When the current hypothesis has large error, the distance between w_t and the ideal direction w decreases.This is the central progress property used in the proof.
Progress lemma: At each iteration, either the direction error decreases by Wη or the empirical error satisfies ˆε(h_t) ≤ 6Wη.The alternative follows from the main lemma under the stated norm condition.
Iteration bound: There can be at most W/η iterations before some iterate satisfies ˆε(h_t) ≤ 6Wη.This follows because the squared distance is bounded by W^2 and decreases by Wη whenever the error alternative does not hold.
Generalization: A concentration bound and a union bound convert the empirical-error guarantee into a high-probability bound on ε(h_t).The concentration lemma applies because every iterate satisfies ∥w_t∥≤W.

B Proof of Proposition 1

The proof establishes uniform guarantees by combining covering-number bounds, concentration, empirical-risk minimization, and convexity arguments. It derives both dimension-dependent and dimension-independent bounds, with the latter having worse dependence on sample size.

Covering-number bounds: Covering numbers quantify the complexity of the function classes for link functions, predictors, and their compositions.The proof defines uniform and empirical ∞-norm coverings, constructs covers for the link class by discretization, and combines covers using Lipschitzness.
Uniform convergence: Uniform convergence converts empirical-risk minimization into guarantees for squared loss over the link and predictor classes.The argument uses Rademacher-complexity bounds and entropy integrals, together with concentration lemmas that hold simultaneously over the relevant classes.
Dimension-dependent guarantee: A covering set for the predictor class extends fixed-predictor bounds to all predictors, adding an O(ϵ) approximation term before choosing ϵ = 1/m.The union bound is applied over a finite cover, and Lipschitzness transfers the result from covered predictors to arbitrary predictors.
Risk decomposition: The squared-loss decomposition identifies conditional-mean error with excess prediction risk and supports the convexity argument for the optimal function.The proof shows that minimizing squared loss is equivalent to minimizing distance from E[y|x], then uses convexity to establish the required inner-product inequality.
Dimension-independent guarantee: The dimension-independent analysis is weaker because the proof lacks an analogue of the faster fixed-predictor bound, yielding worse dependence on m.The paper notes that simultaneously estimating both u and w introduces additional difficulty and produces somewhat worse rates.