[Paper Review] Joint Training of Deep Ensembles Fails Due to Learner Collusion

Alan Jeffares, Tennison Liu+2 more|arXiv (Cornell University)|Jan 26, 2023

Adversarial Robustness in Machine Learning4 citations

TL;DR

This paper demonstrates that joint training of deep ensembles fails due to 'learner collusion,' a phenomenon where base models artificially inflate diversity in prediction space to minimize ensemble loss, leading to poor generalization. Despite optimizing the true ensemble objective, this collusion causes a larger generalization gap than independent training, which remains more effective in practice.

ABSTRACT

Ensembles of machine learning models have been well established as a powerful method of improving performance over a single model. Traditionally, ensembling algorithms train their base learners independently or sequentially with the goal of optimizing their joint performance. In the case of deep ensembles of neural networks, we are provided with the opportunity to directly optimize the true objective: the joint performance of the ensemble as a whole. Surprisingly, however, directly minimizing the loss of the ensemble appears to rarely be applied in practice. Instead, most previous research trains individual models independently with ensembling performed post hoc. In this work, we show that this is for good reason - joint optimization of ensemble loss results in degenerate behavior. We approach this problem by decomposing the ensemble objective into the strength of the base learners and the diversity between them. We discover that joint optimization results in a phenomenon in which base learners collude to artificially inflate their apparent diversity. This pseudo-diversity fails to generalize beyond the training data, causing a larger generalization gap. We proceed to comprehensively demonstrate the practical implications of this effect on a range of standard machine learning tasks and architectures by smoothly interpolating between independent training and joint optimization.

Motivation & Objective

• To investigate why joint optimization of deep ensemble models performs worse than independent training, despite the former being the true objective.
• To identify the root cause of poor generalization in jointly trained ensembles, particularly the role of artificial diversity.
• To formally decompose the ensemble objective into individual model performance and a diversity term, revealing the mechanism behind degenerate behavior.
• To empirically validate the existence and impact of learner collusion on standard benchmarks and architectures.
• To explore potential remedies, such as learner dropout, and demonstrate their limitations in mitigating the issue.

Proposed method

• The authors decompose the joint ensemble loss into individual model losses and a diversity term, using a second-order Taylor expansion to formalize the relationship between joint and independent training objectives.
• They introduce a scalar weighting hyperparameter β to smoothly interpolate between independent training (β=0) and full joint optimization (β=1), enabling controlled analysis of the training regime.
• The diversity term is defined as the difference between joint and independent training objectives, quantifying the contribution of inter-learner disagreement to the overall loss.
• Learner collusion is diagnosed as a strategy where base models shift predictions in opposing directions to increase apparent diversity while preserving ensemble output, thus exploiting the diversity term.
• Experiments are conducted across multiple architectures (ResNet-18, VGG) and datasets (CIFAR-10, SVHN), measuring generalization gap, diversity scores, and test error under varying β.
• A dropout-based mitigation strategy is evaluated, where a random subset of models is dropped during each batch to disrupt collusion, though this is shown to degrade performance due to bootstrapping effects.

Figure 2 : Diversity explosion. Test set diversity explodes relative to MSE across four datasets. We note two further empirical phenomena: (1) the non-linear relationship between $\beta$ and diversity which we discuss further in Appendix D and (2) the minimal effect diversity has on the test MSE on

Experimental results

Research questions

• RQ1 Why does joint training of deep ensembles fail to generalize, despite optimizing the true ensemble objective?
• RQ2 What specific mechanism causes the performance degradation in jointly trained ensembles compared to independently trained ones?
• RQ3 To what extent can the diversity term in the ensemble objective be manipulated through artificial collusion between base learners?
• RQ4 How does the generalization gap vary across different interpolation levels between independent and joint training?
• RQ5 Can regularization techniques like learner dropout effectively mitigate the negative effects of learner collusion?

Key findings

• Joint training of deep ensembles results in a larger generalization gap than independent training, despite optimizing the true ensemble objective.
• Learner collusion emerges as a key failure mode, where base models artificially increase diversity in prediction space without improving actual generalization.
• The diversity term in the ensemble loss is shown to be the primary driver of performance disparity between joint and independent training.
• Experiments on CIFAR-10 and SVHN show that performance degrades monotonically as β increases toward 1, indicating worse generalization under full joint optimization.
• Learner dropout reduces collusion-induced diversity but simultaneously weakens individual models due to bootstrapping, negating any performance benefit.
• The phenomenon is consistent across architectures (ResNet-18, VGG) and datasets, indicating it is a fundamental issue in joint ensemble training.

Figure 3 : Bias removal. The percentage of diversity explained by the most simple form of learner collusion across each dataset. In the region where learner collusion appears to occur ( $\beta\to 1$ ), a large proportion of the diversity can be explained by simple additive bias in the predictions.