[Paper Review] Learning to Benchmark: Determining Best Achievable Misclassification Error from Training Data

Morteza Noshad, Li Xu+1 more|arXiv (Cornell University)|Sep 16, 2019

Imbalanced Data Classification Techniques20 references4 citations

TL;DR

This paper proposes a learning-to-benchmark framework that estimates the exact Bayes misclassification error rate directly from training data using an ensemble of ε-ball estimators and Chebyshev approximation. Under smoothness assumptions, the method achieves a parametric mean squared error rate of O(N⁻¹), outperforming prior bounds-based approaches in accuracy and convergence speed on simulated and real datasets.

ABSTRACT

We address the problem of learning to benchmark the best achievable classifier performance. In this problem the objective is to establish statistically consistent estimates of the Bayes misclassification error rate without having to learn a Bayes-optimal classifier. Our learning to benchmark framework improves on previous work on learning bounds on Bayes misclassification rate since it learns the {\it exact} Bayes error rate instead of a bound on error rate. We propose a benchmark learner based on an ensemble of $ε$-ball estimators and Chebyshev approximation. Under a smoothness assumption on the class densities we show that our estimator achieves an optimal (parametric) mean squared error (MSE) rate of $O(N^{-1})$, where $N$ is the number of samples. Experiments on both simulated and real datasets establish that our proposed benchmark learning algorithm produces estimates of the Bayes error that are more accurate than previous approaches for learning bounds on Bayes error probability.

Motivation & Objective

• To develop a statistically consistent estimator for the exact Bayes misclassification error rate without requiring a Bayes-optimal classifier.
• To improve upon existing methods that only provide bounds on the Bayes error by learning the exact error rate instead.
• To achieve optimal parametric convergence rates for Bayes error estimation under smoothness assumptions on class densities.
• To design an ensemble estimation framework with closed-form optimal weights using Chebyshev nodes for improved finite-sample performance.
• To extend the framework to multiclass classification with provable convergence rates.

Proposed method

• Proposes a base estimator based on ε-ball density ratio estimation for binary classification, leveraging local neighborhood statistics.
• Employs Chebyshev nodes to construct a weighted ensemble of base estimators, minimizing approximation error in the estimation process.
• Uses Chebyshev polynomial approximation to derive closed-form optimal weights for the ensemble, ensuring minimal mean squared error.
• Derives theoretical convergence rates for the base and ensemble estimators, proving an optimal O(N⁻¹) mean squared error rate under smoothness conditions.
• Extends the framework to multiclass classification by generalizing the ε-ball and ensemble estimation approach to λ ≥ 3 classes.
• Employs a consistent empirical estimator for the density ratio within ε-balls, enabling nonparametric estimation from finite training samples.

Experimental results

Research questions

• RQ1 Can we estimate the exact Bayes error rate directly from training data without learning a Bayes-optimal classifier?
• RQ2 Does an ensemble of ε-ball estimators with Chebyshev node-based weighting achieve faster convergence than existing bound-based methods?
• RQ3 What is the optimal mean squared error rate achievable for Bayes error estimation under smoothness assumptions on class densities?
• RQ4 How does the performance of the proposed benchmark learner compare to traditional classifiers like XGBoost and Random Forest in estimating achievable error rates?
• RQ5 Does the choice of scaling coefficient in the Chebyshev-based ensemble affect estimation accuracy across different data distributions and dimensions?

Key findings

• The proposed benchmark learner achieves a parametric mean squared error rate of O(N⁻¹), which is optimal under smoothness assumptions on the class densities.
• In experiments with 4-dimensional and 100-dimensional isotropic normal distributions, the Chebyshev-based ensemble method reduced MSE by up to 92% compared to the arithmetic weighting method at N=1600 samples.
• For 100-dimensional isotropic normal data, the Chebyshev method achieved significantly lower MSE than both arithmetic and uniform weighting schemes, demonstrating superior finite-sample convergence.
• The estimator's performance was stable across different scaling coefficients α ∈ [0.3, 0.5], with minimal variation in error, indicating robustness to hyperparameter choice.
• On a 3-class problem with 50-dimensional beta distributions, the benchmark learner converged to the true Bayes error, while HP-divergence bounds exhibited large bias even at large sample sizes.
• In a 4-class Gaussian problem with 100-dimensional features, the benchmark learner predicted the Bayes error more accurately than XGBoost and Random Forest classifiers, which overfit and failed to capture the true error bound.