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Explaining the Success of AdaBoost and Random Forests as Interpolating Classifiers
Abraham J. Wyner, Matthew Olson, Justin Bleich, David Mease
TL;DR
Existing explanations have been questioned for not fully accounting for AdaBoost’s success, including what it does after perfectly fitting training data. The paper decomposes AdaBoost and examines averaging in interpolating classifiers, finding that generalization error can continue declining after interpolation and that random forests is at least as good as AdaBoost.
Problem
Existing explanations have been questioned as unable to fully account for AdaBoost’s success, including its behavior after perfect training fit.
Method
The paper presents a decomposition of AdaBoost as a weighted sum and examines averaging’s localizing effect in interpolating classifiers.
Results
Generalization error continues to drop after interpolation is achieved and maintained, while random forests is at least as good as AdaBoost.
Takeaways & Limitations
The findings support examining what interpolating classifiers do after fitting the training data rather than stopping analysis at interpolation.
Takeaways & Limitations
Some research has cast doubt on whether existing explanations can fully account for the algorithms’ behavior.
Abstract
from arXiv · showhide
There is a large literature explaining why AdaBoost is a successful classifier. The literature on AdaBoost focuses on classifier margins and boosting's interpretation as the optimization of an exponential likelihood function. These existing explanations, however, have been pointed out to be incomplete. A random forest is another popular ensemble method for which there is substantially less explanation in the literature. We introduce a novel perspective on AdaBoost and random forests that proposes that the two algorithms work for similar reasons. While both classifiers achieve similar predictive accuracy, random forests cannot be conceived as a direct optimization procedure. Rather, random forests is a self-averaging, interpolating algorithm which creates what we denote as a "spikey-smooth" classifier, and we view AdaBoost in the same light. We conjecture that both AdaBoost and random forests succeed because of this mechanism. We provide a number of examples and some theoretical justification to support this explanation. In the process, we question the conventional wisdom that suggests that boosting algorithms for classification require regularization or early stopping and should be limited to low complexity classes of learners, such as decision stumps. We conclude that boosting should be used like random forests: with large decision trees and without direct regularization or early stopping.
1. Introduction
The paper challenges conventional explanations of AdaBoost by arguing that interpolation, combined with self-averaging, helps explain why AdaBoost and random forests generalize well. It develops this perspective theoretically and empirically, motivating deep trees and many boosting iterations rather than direct regularization or early stopping.
- Motivation and perspective: The paper critiques statistical optimization accounts of AdaBoost as incomplete and notes that their stage-wise framework does not apply directly to random forests.Random forests construct trees at random and average them without stage-wise optimization.
- Motivation and perspective: AdaBoost’s interpolation of training data does not prevent continued generalization improvement after perfect training accuracy is reached.Interpolation is achieved after relatively few iterations, while generalization error continues to decline and remains reduced afterward.
- Motivation and perspective: The paper proposes that AdaBoost and random forests succeed for similar reasons: both interpolate training data and self-average component classifiers.Random forests average randomly generated trees, while AdaBoost is decomposed as a weighted sum of interpolating classifiers.
- Mechanism: Interpolation combined with averaging can make noise effects highly local, allowing the fit to stabilize in signal regions while reducing noise-point influence.The paper argues that later AdaBoost iterations localize errors around noise points, producing a smoothing effect near those points.
- Implications: The authors conclude that boosting should use deep trees and many iterations so component classifiers interpolate the data and averaging improves performance.This challenges the conventional preference for low-complexity learners, direct regularization, or early stopping.
2. Competing Explanations for the Effectiveness of Boosting
The literature explains boosting through margins and exponential-loss optimization, but several empirical and conceptual findings challenge whether either view fully accounts for AdaBoost’s effectiveness. The paper instead motivates viewing AdaBoost as a smoothed, interpolating classifier and argues for deep trees and many iterations.
- Established explanations: AdaBoost is commonly explained through classifier margins and the statistical optimization of an exponential loss.These perspectives have generated substantial theoretical analysis and variants of boosting algorithms.
- Established explanations: Many boosting rounds often reduce test error even though continued fitting would conventionally suggest overfitting.Empirically, AdaBoost’s test error decreases while its empirical margins increase over many iterations.
- Challenges to the margin view: Margin maximization alone does not explain AdaBoost’s superiority: arc-gv and LP-Boost performed worse than AdaBoost despite targeting margins.The literature notes that the full margin distribution may matter more than only the minimum margin.
- Paper’s perspective: The paper challenges limiting boosting to weak learners, reporting that strong learners such as C4.5 and CART can work well, particularly in performance.It also identifies the inability of optimization theory to explain zero training error as an unresolved issue.
- Challenges to the optimization view: Exponential-loss behavior likewise does not consistently track generalization, and Beta-Boost achieved similar simulated performance despite holding that loss constant.Other experiments found AdaBoost significantly better than variants minimizing exponential loss at different rates.
- Paper’s perspective: The paper departs from direct optimization by treating AdaBoost as a self-smoothed, interpolating classifier analogous to random forests.This perspective motivates running AdaBoost for many iterations with deep trees rather than relying on direct regularization or early stopping.
3. Interpolating Classifiers
AdaBoost and random forests can generalize well despite perfectly fitting training data because local interpolation and self-averaging limit the influence of noise. The paper presents them as spiked-smooth classifiers whose fit remains smooth overall while adapting sharply near training points.
- Interpolating Classifiers: Interpolating classifiers can achieve good out-of-sample performance despite fitting every training example without error.The paper uses AdaBoost and random forests as leading examples that challenge the conventional belief that interpolation implies poor generalization.
- Random Forests: Random forests combine deep-tree interpolation with self-averaging rather than directly optimizing a stated objective.Bootstrap sampling, randomized variable selection, and majority voting produce the ensemble prediction.
- Local Robustness of Interpolating Classifiers: Local interpolation confines noise effects to very small regions, allowing a classifier to fit noisy points without corrupting nearby predictions.The paper describes this overall behavior as spiked-smooth: a mostly smooth decision surface with localized spikes.
- Self-Averaging: Averaging shrinks erroneous regions from individual trees, producing a decision surface minimally affected by training noise.Majority voting turns relatively poor individual trees into an ensemble that agrees more closely with the Bayes rule.
- Twenty-Dimensional Example: On the circular response surface, AdaBoost and random forests had error rates around 0.13, versus 0.20 for one-NN and 0.18 for CART.CART lacked enough complexity to capture the circular structure, whereas deep trees in the ensembles could do so.
4. Self-Averaging Property of Boosting
The paper interprets AdaBoost as a self-averaging interpolating classifier whose additional iterations smooth and localize noise, paralleling random forests. Decomposition experiments show that averaging individually interpolating classifiers can improve performance after perfect training fit.
- Self-averaging decomposition: AdaBoost can be decomposed into multiple classifiers, each of which eventually perfectly separates the training data and therefore interpolates.The decomposition treats each classifier as a linear combination of classification trees.
- Self-smoothing mechanism: Additional boosting iterations average and smooth localized errors, reducing overfitting while preserving interpolation of the training data.The paper describes this as self-smoothing, self-averaging, or self-bagging.
- Simulation evidence: AdaBoost’s sub-ensemble error correlations ranged from 0.4 to 0.56, averaging 0.488, below random forests’ 0.81 and bagged trees’ 0.92.The authors use these lower correlations as evidence of stronger decorrelation in AdaBoost’s component classifiers.
- Comparison with random forests: With large trees, boosting remains interpolating while localizing noise, and its behavior is empirically similar to random forests.The paper therefore attributes their success to the same self-averaging mechanism.
5. Real Data Example
The paper tests spiked smoothing on a real speech-recognition dataset and on deliberately corrupted labels. Deep boosted trees and random forests achieve comparable test errors while continuing to improve after interpolation.
- Speech-recognition data: The speech dataset contains 5404 spoken-vowel examples with five harmonic-frequency covariates for nasal-versus-oral classification.The analysis uses 70% of the examples for training and 30% for testing.
- Speech-recognition results: Testing error steadily decreased as AdaBoost tree depth increased, while boosted trees of all depths were slow to overfit.Depth-seven and depth-eight trees quickly interpolated the training data.
- Label-noise experiment: After flipping about 2% of training labels, depth-eight boosting and random forests produced comparable test errors of 10.2% and 10.0%, respectively.Test error continued decreasing for boosted trees even after thousands of rounds.
- Comparison with one-nearest-neighbor: One-nearest-neighbor error rose from 10.5% on the original data to 12.6% after label noise, a larger increase than for AdaBoost or random forests.The increase was statistically significant at the 0.01 level in both cases, with p-values below 1e−16.
- Label-noise experiment: In neighborhoods around flipped points, agreement with the original labels increased across iterations despite perfect fitting of the flipped training labels.The authors interpret this increasing homogeneity as localization of noise influence through averaging.
6. Concluding Remarks
The paper argues that AdaBoost and random forests succeed as self-averaging interpolating classifiers, challenging conventional calls for regularization, early stopping, and shallow trees.
- Implications for boosting: The paper questions whether boosting requires regularization, iteration control, or low-complexity learners such as decision stumps.Its discussion contrasts the statistical view of exponential-loss optimization with random forests as a non-optimization procedure.
- Shared mechanism: AdaBoost and random forests are presented as successful for similar reasons: both combine interpolation with averaging.Random forests is a weighted ensemble of interpolating classifiers, while AdaBoost is characterized as a weighted ensemble and “random” forest of forests.
- Shared mechanism: AdaBoost’s later iterations are argued to create an averaging effect rather than merely overfit already-interpolated training data.This effect makes AdaBoost behave similarly to random forests while continuing to interpolate.
- Interpolation and averaging: Local interpolation with deep trees is argued to prevent overfitting by limiting noisy points’ influence outside their local neighborhoods.Both random forests and AdaBoost achieve this local interpolation by fitting deep trees.
- Implications for boosting: The authors broaden explanations of AdaBoost beyond margins and exponential-loss minimization by emphasizing its self-averaging and interpolating behavior.They support this perspective with examples and theoretical justification and connect it to Breiman’s conjecture about AdaBoost emulating a random forest.