Combining large language models during training or at inference time has shown substantial performance gain over component LLMs. This paper presents LLM-TOPLA, a diversity-optimized LLM ensemble method with three unique properties: (i) We introduce the focal diversity metric to capture the diversity-performance correlation among component LLMs of an ensemble. (ii) We develop a diversity-optimized ensemble pruning algorithm to select the top-k sub-ensembles from a pool of $N$ base LLMs. Our pruning method recommends top-performing LLM subensembles of size $S$, often much smaller than $N$. (iii) We generate new output for each prompt query by utilizing a learn-to-ensemble approach, which learns to detect and resolve the output inconsistency among all component LLMs of an ensemble. Extensive evaluation on four different benchmarks shows good performance gain over the best LLM ensemble methods: (i) In constrained solution set problems, LLM-TOPLA outperforms the best-performing ensemble (Mixtral) by 2.2\% in accuracy on MMLU and the best-performing LLM ensemble (MoreAgent) on GSM8k by 2.1\%. (ii) In generative tasks, LLM-TOPLA outperforms the top-2 performers (Llama70b/Mixtral) on SearchQA by $3.9\mathrm{x}$ in F1, and on XSum by more than $38$ in ROUGE-1. Our code and dataset, which contains outputs of 8 modern LLMs on 4 benchmarks is available at https://github.com/git-disl/llm-topla
This paper presents FusionShot, a focal diversity optimized few-shot ensemble learning approach for boosting the robustness and generalization performance of pre-trained few-shot models. The paper makes three original contributions. First, we explore the unique characteristics of few-shot learning to ensemble multiple few-shot (FS) models by creating three alternative fusion channels. Second, we introduce the concept of focal error diversity to learn the most efficient ensemble teaming strategy, rather than assuming that an ensemble of a larger number of base models will outperform those sub-ensembles of smaller size. We develop a focal-diversity ensemble pruning method to effectively prune out the candidate ensembles with low ensemble error diversity and recommend top-$K$ FS ensembles with the highest focal error diversity. Finally, we capture the complex non-linear patterns of ensemble few-shot predictions by designing the learn-to-combine algorithm, which can learn the diverse weight assignments for robust ensemble fusion over different member models. Extensive experiments on representative few-shot benchmarks show that the top-K ensembles recommended by FusionShot can outperform the representative SOTA few-shot models on novel tasks (different distributions and unknown at training), and can prevail over existing few-shot learners in both cross-domain settings and adversarial settings. For reproducibility purposes, FusionShot trained models, results, and code are made available at https://github.com/sftekin/fusionshot
We argue that there is a strong connection between ensemble learning and a delegative voting paradigm -- liquid democracy -- that can be leveraged to reduce ensemble training costs. We present an incremental training procedure that identifies and removes redundant classifiers from an ensemble via delegation mechanisms inspired by liquid democracy. Through both analysis and extensive experiments we show that this process greatly reduces the computational cost of training compared to training a full ensemble. By carefully selecting the underlying delegation mechanism, weight centralization in the classifier population is avoided, leading to higher accuracy than some boosting methods. Furthermore, this work serves as an exemplar of how frameworks from computational social choice literature can be applied to problems in nontraditional domains.
Deep neural network ensembles combine the wisdom of multiple deep neural networks to improve the generalizability and robustness over individual networks. It has gained increasing popularity to study deep ensemble techniques in the deep learning community. Some mission-critical applications utilize a large number of deep neural networks to form deep ensembles to achieve desired accuracy and resilience, which introduces high time and space costs for ensemble execution. However, it still remains a critical challenge whether a small subset of the entire deep ensemble can achieve the same or better generalizability and how to effectively identify these small deep ensembles for improving the space and time efficiency of ensemble execution. This paper presents a novel deep ensemble pruning approach, which can efficiently identify smaller deep ensembles and provide higher ensemble accuracy than the entire deep ensemble of a large number of member networks. Our hierarchical ensemble pruning approach (HQ) leverages three novel ensemble pruning techniques. First, we show that the focal diversity metrics can accurately capture the complementary capacity of the member networks of an ensemble, which can guide ensemble pruning. Second, we design a focal diversity based hierarchical pruning approach, which will iteratively find high quality deep ensembles with low cost and high accuracy. Third, we develop a focal diversity consensus method to integrate multiple focal diversity metrics to refine ensemble pruning results, where smaller deep ensembles can be effectively identified to offer high accuracy, high robustness and high efficiency. Evaluated using popular benchmark datasets, we demonstrate that the proposed hierarchical ensemble pruning approach can effectively identify high quality deep ensembles with better generalizability while being more time and space efficient in ensemble decision making.
Ensemble techniques are frequently encountered in machine learning and engineering problems since the method combines different models and produces an optimal predictive solution. The ensemble concept can be adapted to deep learning models to provide robustness and reliability. Due to the growth of the models in deep learning, using ensemble pruning is highly important to deal with computational complexity. Hence, this study proposes a mathematical model which prunes the ensemble of Convolutional Neural Networks (CNN) consisting of different depths and layers that maximizes accuracy and diversity simultaneously with a sparse second order conic optimization model. The proposed model is tested on CIFAR-10, CIFAR-100 and MNIST data sets which gives promising results while reducing the complexity of models, significantly.
Gradient boosted decision trees are some of the most popular algorithms in applied machine learning. They are a flexible and powerful tool that can robustly fit to any tabular dataset in a scalable and computationally efficient way. One of the most critical parameters to tune when fitting these models are the various penalty terms used to distinguish signal from noise in the current model. These penalties are effective in practice, but are lacking in robust theoretical justifications. In this paper we develop and present a novel theoretically justified hypothesis test of split quality for gradient boosted tree ensembles and demonstrate that using this method instead of the common penalty terms leads to a significant reduction in out of sample loss. Additionally, this method provides a theoretically well-justified stopping condition for the tree growing algorithm. We also present several innovative extensions to the method, opening the door for a wide variety of novel tree pruning algorithms.
Ensemble learning combines multiple classifiers in the hope of obtaining better predictive performance. Empirical studies have shown that ensemble pruning, that is, choosing an appropriate subset of the available classifiers, can lead to comparable or better predictions than using all classifiers. In this paper, we consider a binary classification problem and propose an integer programming (IP) approach for selecting optimal classifier subsets. We propose a flexible objective function to adapt to desired criteria of different datasets. We also propose constraints to ensure minimum diversity levels in the ensemble. Despite the general case of IP being NP-Hard, state-of-the-art solvers are able to quickly obtain good solutions for datasets with up to 60000 data points. Our approach yields competitive results when compared to some of the best and most used pruning methods in literature.
Deep neural network ensembles have become attractive learning techniques with better generalizability over individual models. Some mission critical applications may require a large number of deep neural networks to achieve desirable accuracy and generalizability, making the ensemble execution costly with respect to runtime and space. This paper proposes a novel hierarchical ensemble pruning approach, which can effectively examine a given pool of M base models and identify smaller high quality deep ensembles of size S (<< M) with higher ensemble accuracy than the entire deep ensemble of all M models. Our hierarchical pruning approach, coined as HQ, combines three novel techniques. First, we show that the focal diversity metrics is innovative and can accurately capture the negative correlation among the member models of an ensemble, and the use of focal diversity metrics can boost ensemble accuracy. Second, we introduce a focal-diversity based hierarchical pruning algorithm to progressively identify low-cost ensembles with high ensemble diversity and accuracy. Third, we design a focal diversity consensus method to find smaller deep ensembles with low negative correlation. We demonstrate such ensembles offer high accuracy and high robustness while being more time and space efficient in ensemble decision making. Evaluated using two benchmark datasets, we show that the proposed focal diversity powered hierarchical pruning can find significantly smaller ensembles of deep neural network models while achieving the same or better classification generalizability.
Over the last few years, the Shapley value, a solution concept from cooperative game theory, has found numerous applications in machine learning. In this paper, we first discuss fundamental concepts of cooperative game theory and axiomatic properties of the Shapley value. Then we give an overview of the most important applications of the Shapley value in machine learning: feature selection, explainability, multi-agent reinforcement learning, ensemble pruning, and data valuation. We examine the most crucial limitations of the Shapley value and point out directions for future research.
Meta-learners and ensembles aim to combine a set of relevant yet diverse base models to improve predictive performance. However, determining an appropriate set of base models is challenging, especially in online environments where the underlying distribution of data can change over time. In this paper, we present a novel approach for estimating the conceptual similarity of base models, which is calculated using the Principal Angles (PAs) between their underlying subspaces. We propose two methods that use conceptual similarity as a metric to obtain a relevant yet diverse subset of base models: (i) parameterised threshold culling and (ii) parameterless conceptual clustering. We evaluate these methods against thresholding using common ensemble pruning metrics, namely predictive performance and Mutual Information (MI), in the context of online Transfer Learning (TL), using both synthetic and real-world data. Our results show that conceptual similarity thresholding has a reduced computational overhead, and yet yields comparable predictive performance to thresholding using predictive performance and MI. Furthermore, conceptual clustering achieves similar predictive performances without requiring parameterisation, and achieves this with lower computational overhead than thresholding using predictive performance and MI when the number of base models becomes large.
Random Forests (RF) are among the state-of-the-art in many machine learning applications. With the ongoing integration of ML models into everyday life, the deployment and continuous application of models becomes more and more an important issue. Hence, small models which offer good predictive performance but use small amounts of memory are required. Ensemble pruning is a standard technique to remove unnecessary classifiers from an ensemble to reduce the overall resource consumption and sometimes even improve the performance of the original ensemble. In this paper, we revisit ensemble pruning in the context of `modernly' trained Random Forests where trees are very large. We show that the improvement effects of pruning diminishes for ensembles of large trees but that pruning has an overall better accuracy-memory trade-off than RF. However, pruning does not offer fine-grained control over this trade-off because it removes entire trees from the ensemble. To further improve the accuracy-memory trade-off we present a simple, yet surprisingly effective algorithm that refines the predictions in the leaf nodes in the forest via stochastic gradient descent. We evaluate our method against 7 state-of-the-art pruning methods and show that our method outperforms the other methods on 11 of 16 datasets with a statistically significant better accuracy-memory trade-off compared to most methods. We conclude our experimental evaluation with a case study showing that our method can be applied in a real-world setting.
Ensemble pruning is the process of selecting a subset of componentclassifiers from an ensemble which performs at least as well as theoriginal ensemble while reducing storage and computational costs.Ensemble pruning in data streams is a largely unexplored area ofresearch. It requires analysis of ensemble components as they arerunning on the stream, and differentiation of useful classifiers fromredundant ones. We present CCRP, an on-the-fly ensemble prun-ing method for multi-class data stream classification empoweredby an imbalance-aware fusion of class-wise component rankings.CCRP aims that the resulting pruned ensemble contains the bestperforming classifier for each target class and hence, reduces the ef-fects of class imbalance. The conducted experiments on real-worldand synthetic data streams demonstrate that different types of en-sembles that integrate CCRP as their pruning scheme consistentlyyield on par or superior performance with 20% to 90% less averagememory consumption. Lastly, we validate the proposed pruningscheme by comparing our approach against pruning schemes basedon ensemble weights and basic rank fusion methods.
What is the value of an individual model in an ensemble of binary classifiers? We answer this question by introducing a class of transferable utility cooperative games called \textit{ensemble games}. In machine learning ensembles, pre-trained models cooperate to make classification decisions. To quantify the importance of models in these ensemble games, we define \textit{Troupe} -- an efficient algorithm which allocates payoffs based on approximate Shapley values of the classifiers. We argue that the Shapley value of models in these games is an effective decision metric for choosing a high performing subset of models from the ensemble. Our analytical findings prove that our Shapley value estimation scheme is precise and scalable; its performance increases with size of the dataset and ensemble. Empirical results on real world graph classification tasks demonstrate that our algorithm produces high quality estimates of the Shapley value. We find that Shapley values can be utilized for ensemble pruning, and that adversarial models receive a low valuation. Complex classifiers are frequently found to be responsible for both correct and incorrect classification decisions.
Ensembles, as a widely used and effective technique in the machine learning community, succeed within a key element -- "diversity." The relationship between diversity and generalization, unfortunately, is not entirely understood and remains an open research issue. To reveal the effect of diversity on the generalization of classification ensembles, we investigate three issues on diversity, i.e., the measurement of diversity, the relationship between the proposed diversity and the generalization error, and the utilization of this relationship for ensemble pruning. In the diversity measurement, we measure diversity by error decomposition inspired by regression ensembles, which decomposes the error of classification ensembles into accuracy and diversity. Then we formulate the relationship between the measured diversity and ensemble performance through the theorem of margin and generalization and observe that the generalization error is reduced effectively only when the measured diversity is increased in a few specific ranges, while in other ranges larger diversity is less beneficial to increasing the generalization of an ensemble. Besides, we propose two pruning methods based on diversity management to utilize this relationship, which could increase diversity appropriately and shrink the size of the ensemble without much-decreasing performance. Empirical results validate the reasonableness of the proposed relationship between diversity and ensemble generalization error and the effectiveness of the proposed pruning methods.
Neural architecture search (NAS) is gaining more and more attention in recent years due to its flexibility and remarkable capability to reduce the burden of neural network design. To achieve better performance, however, the searching process usually costs massive computations that might not be affordable for researchers and practitioners. While recent attempts have employed ensemble learning methods to mitigate the enormous computational cost, however, they neglect a key property of ensemble methods, namely diversity, which leads to collecting more similar sub-architectures with potential redundancy in the final design. To tackle this problem, we propose a pruning method for NAS ensembles called "Sub-Architecture Ensemble Pruning in Neural Architecture Search (SAEP)." It targets to leverage diversity and to achieve sub-ensemble architectures at a smaller size with comparable performance to ensemble architectures that are not pruned. Three possible solutions are proposed to decide which sub-architectures to prune during the searching process. Experimental results exhibit the effectiveness of the proposed method by largely reducing the number of sub-architectures without degrading the performance.
Ensemble models refer to methods that combine a typically large number of classifiers into a compound prediction. The output of an ensemble method is the result of fitting a base-learning algorithm to a given data set, and obtaining diverse answers by reweighting the observations or by resampling them using a given probabilistic selection. A key challenge of using ensembles in large-scale multidimensional data lies in the complexity and the computational burden associated with them. The models created by ensembles are often difficult, if not impossible, to interpret and their implementation requires more computational power than single classifiers. Recent research effort in the field has concentrated in reducing ensemble size, while maintaining their predictive accuracy. We propose a method to prune an ensemble solution by optimizing its margin distribution, while increasing its diversity. The proposed algorithm results in an ensemble that uses only a fraction of the original classifiers, with improved or similar generalization performance. We analyze and test our method on both synthetic and real data sets. The simulations show that the proposed method compares favorably to the original ensemble solutions and to other existing ensemble pruning methodologies.
Machine learning algorithms have been effectively applied into various real world tasks. However, it is difficult to provide high-quality machine learning solutions to accommodate an unknown distribution of input datasets; this difficulty is called the uncertainty prediction problems. In this paper, a margin-based Pareto deep ensemble pruning (MBPEP) model is proposed. It achieves the high-quality uncertainty estimation with a small value of the prediction interval width (MPIW) and a high confidence of prediction interval coverage probability (PICP) by using deep ensemble networks. In addition to these networks, unique loss functions are proposed, and these functions make the sub-learners available for standard gradient descent learning. Furthermore, the margin criterion fine-tuning-based Pareto pruning method is introduced to optimize the ensembles. Several experiments including predicting uncertainties of classification and regression are conducted to analyze the performance of MBPEP. The experimental results show that MBPEP achieves a small interval width and a low learning error with an optimal number of ensembles. For the real-world problems, MBPEP performs well on input datasets with unknown distributions datasets incomings and improves learning performance on a multi task problem when compared to that of each single model.
Classifier ensemble generally should combine diverse component classifiers. However, it is difficult to give a definitive connection between diversity measure and ensemble accuracy. Given a list of available component classifiers, how to adaptively and diversely ensemble classifiers becomes a big challenge in the literature. In this paper, we argue that diversity, not direct diversity on samples but adaptive diversity with data, is highly correlated to ensemble accuracy, and we propose a novel technology for classifier ensemble, learning to diversify, which learns to adaptively combine classifiers by considering both accuracy and diversity. Specifically, our approach, Learning TO Diversify via Weighted Kernels (L2DWK), performs classifier combination by optimizing a direct but simple criterion: maximizing ensemble accuracy and adaptive diversity simultaneously by minimizing a convex loss function. Given a measure formulation, the diversity is calculated with weighted kernels (i.e., the diversity is measured on the component classifiers' outputs which are kernelled and weighted), and the kernel weights are automatically learned. We minimize this loss function by estimating the kernel weights in conjunction with the classifier weights, and propose a self-training algorithm for conducting this convex optimization procedure iteratively. Extensive experiments on a variety of 32 UCI classification benchmark datasets show that the proposed approach consistently outperforms state-of-the-art ensembles such as Bagging, AdaBoost, Random Forests, Gasen, Regularized Selective Ensemble, and Ensemble Pruning via Semi-Definite Programming.
Random Forest (RF) is an ensemble classification technique that was developed by Breiman over a decade ago. Compared with other ensemble techniques, it has proved its accuracy and superiority. Many researchers, however, believe that there is still room for enhancing and improving its performance in terms of predictive accuracy. This explains why, over the past decade, there have been many extensions of RF where each extension employed a variety of techniques and strategies to improve certain aspect(s) of RF. Since it has been proven empirically that ensembles tend to yield better results when there is a significant diversity among the constituent models, the objective of this paper is twofolds. First, it investigates how an unsupervised learning technique, namely, Local Outlier Factor (LOF) can be used to identify diverse trees in the RF. Second, trees with the highest LOF scores are then used to produce an extension of RF termed LOFB-DRF that is much smaller in size than RF, and yet performs at least as good as RF, but mostly exhibits higher performance in terms of accuracy. The latter refers to a known technique called ensemble pruning. Experimental results on 10 real datasets prove the superiority of our proposed extension over the traditional RF. Unprecedented pruning levels reaching 99% have been achieved at the time of boosting the predictive accuracy of the ensemble. The notably high pruning level makes the technique a good candidate for real-time applications.
Random Forest (RF) is an ensemble supervised machine learning technique that was developed by Breiman over a decade ago. Compared with other ensemble techniques, it has proved its accuracy and superiority. Many researchers, however, believe that there is still room for enhancing and improving its performance accuracy. This explains why, over the past decade, there have been many extensions of RF where each extension employed a variety of techniques and strategies to improve certain aspect(s) of RF. Since it has been proven empiricallthat ensembles tend to yield better results when there is a significant diversity among the constituent models, the objective of this paper is twofold. First, it investigates how data clustering (a well known diversity technique) can be applied to identify groups of similar decision trees in an RF in order to eliminate redundant trees by selecting a representative from each group (cluster). Second, these likely diverse representatives are then used to produce an extension of RF termed CLUB-DRF that is much smaller in size than RF, and yet performs at least as good as RF, and mostly exhibits higher performance in terms of accuracy. The latter refers to a known technique called ensemble pruning. Experimental results on 15 real datasets from the UCI repository prove the superiority of our proposed extension over the traditional RF. Most of our experiments achieved at least 95% or above pruning level while retaining or outperforming the RF accuracy.