Test-time Collective Prediction

1 Introduction . Large-scale datasets are often collected from diverse sources , by multiple parties , and stored across different machines . In many scenarios centralized pooling of data is not possible due to privacy concerns , data ownership , or storage constraints . The challenge of doing machine learning in such distributed and decentralized settings has motivated research in areas of federated learning [ Konečný et al. , 2015 , McMahan et al. , 2017 ] , distributed learning [ Dean et al. , 2012 , Gupta and Raskar , 2018 ] , as well as hardware and software design [ Zaharia et al. , 2016 , Moritz et al. , 2018 ] . While the predominant paradigm in distributed machine learning is collaborative learning of one centralized model , this level of coordination across machines at the training stage is sometimes not feasible . In this work , we instead aim for collective prediction at test time without posing any specific requirement at the training stage . Combining the predictions of pre-trained machine learning models has been considered in statistics and machine learning in the context of ensemble methods , including bagging [ Breiman , 1996a ] and stacking [ Wolpert , 1992 ] . In this work , we explore a new perspective on this aggregation problem and investigate whether insights from the social sciences on how humans reach a consensus can help us design more effective aggregation schemes that fully take advantage of ∗most of the work conducted while at UC Berkeley . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . each model ’ s individual strength . At a high level , when a panel of human experts come together to make collective decisions , they exchange information and share expertise through a discourse to then weigh their subjective beliefs accordingly . Experts have a conforming influence on each other which gives rise to a dynamic process of consensus finding that is decentralized and does not rely on any external judgement . We map this paradigm to a machine learning setting where experts correspond to pre-trained machine learning models , and show that it leads to an appealing mechanism for test-time collective prediction . 1.1 Our work . We build on a model for human consensus finding that defines an iterative process of opinion pooling based on mutual trust scores among agents [ DeGroot , 1974 ] . Informally speaking , a mutual trust score reflects an agent ’ s willingness to adapt another agent ’ s opinion . Thus , each individual ’ s impact on the final judgment depends on how much it is trusted by the other agents . Mapping the DeGroot model to our learning scenario , we develop a scheme where each agent uses its own local training data to assess the predictive capabilities of other agents ’ models and to determine how much they should be trusted . This assessment of mutual trust is designed to be adaptive to the prediction task at hand , by using only a subset of the local data in the area around the current test point . Then , building on these trust scores , a final judgment is reached through an iterative procedure , where agents repeatedly share their updated predictions . This aggregation scheme is conceptually appealing as it mimics human consensus finding , and it exhibits several practical advantages over existing methods . First , it is decentralized and does not require agents to release their labeled data or model parameters . This can be beneficial in terms of communication costs and attractive from a privacy perspective , as it allows agents to control access to their data and model at any time . Second , it does not require hold-out validation data to construct or train the aggregation function . Thus , there is no need to collect additional labeled data and coordinate shared access to a common database . Third , it does not require any synchronization across agents during the training stage , so agents are free to choose their preferred model architecture , and train their models independently . A crucial algorithmic feature of our procedure is its adaptivity to the test point at hand , which allows it to deal with inherent variations in the predictive quality of models in different regions of the input space . Figure 1 illustrates how our mechanism adaptively upweights models with lower error depending on the test point ’ s location in the input space . In fact , we prove theoretically that our aggregation procedure can recover inverse-MSE weighting in the large-sample limit , which is known to be optimal for variance reduction of unbiased estimators [ Bates and Granger , 1969 ] . To assess our procedure ’ s sensitivity to the predictions of individual agents we propose a decentralized Jackknife algorithm to compute error bars for the consensus predictions with respect to the collection of observed agents . These error bars offer an attractive target of inference since they are agnostic to how individual models have been trained , can be evaluated without additional model evaluations , and allow one to diagnose the vulnerability of the algorithm to malicious agents . On the empirical side , we demonstrate the efficacy of our mechanism through extensive numerical experiments across different learning scenarios . In particular , we illustrate the mechanism ’ s advantages over model averaging as well as model selection , and demonstrate that it consistently outperforms alternative non-uniform combination schemes that have access to additional validation data across a wide variety of models and datasets . 1.2 Related work . The most common approach for combining expert judgment is to aggregate and average individual expert opinions [ Hammitt and Zhang , 2013 ] . In machine learning this statistical approach of equally-weighted averaging is also known as the “ mean-rule ” and underlies the popular bagging technique [ Breiman , 1996a , Buja and Stuetzle , 2006 , Efron , 2014 ] . It corresponds to an optimal aggregation technique for dealing with uncertainty in settings where individual learners are homogeneous and the local datasets are drawn independently from the same underlying distribution . While principled , and powerful due to its simplicity , an unweighted approach can not account for heterogeneity in the quality , or the expertise , of the base learners . To address this issue , performance-weighted averaging schemes have been proposed . For determining an appropriate set of weights , these approaches typically rely on calibrating variables [ Aspinall , 2010 ] to detect differences in predictive quality among agents . In machine learning these calibrating variables correspond to labeled cross-validation samples , while in areas such as expert forecasting and risk analysis these variables take the form of a set of agreed-upon seed questions [ Cooke , 1991 ] . These weights can be updated over time using statistical information about the relative predictive performance [ Rogova , 2008 ] or the posterior model probability [ Wasserman et al. , 2000 ] . More recently there have been several applications to classification that focus on dynamic selection techniques [ Cruz et al. , 2018b ] , where the goal is to select the locally best classifier from a pool of classifiers on the fly for every test point . At their core , these approaches all rely on constructing a good proxy for local accuracy in a region of the feature space around the test point and to use this proxy to weight or rank the candidate models . Various versions of distance and accuracy metrics have been proposed in this context [ eg . Woods et al. , 1997 , Didaci et al. , 2005 , Cruz et al. , 2018a ] . However , all these methods rely crucially on shared access to labeled validation data to determine the weights . Similarly , aggregation rules such as stacking [ Breiman , 1996b , Coscrato et al. , 2020 ] or meta-models [ Albardan et al. , 2020 ] , both of which have proved their value empirically , require a central trusted agency with additional data to train an aggregation function on top of the predictor outputs . A related model-based aggregation method , the mixture-of-experts model [ Jacobs et al. , 1991 ] , requires joint training of all models as well as data pooling . To the best of our knowledge there is no approach in the literature that can perform adaptive performance weighting without relying on a central agent to validate the individual models . Notably , our approach differs from general frameworks such as model parameter mixing [ Zhang et al. , 2013 ] , model fusion [ Leontev et al. , 2018 , Singh and Jaggi , 2020 ] in that agents are not required to release their models . Outside machine learning , the question of how to combine opinions from different experts to arrive at a better overall outcome has been studied for decades in risk analysis [ Hanea et al. , 2018 , Clemen and Winkler , 1999 ] and management science [ Morris , 1977 ] . Simplifying the complex mechanism of opinion dynamics , the celebrated DeGroot model [ DeGroot , 1974 ] offers an appealing abstraction to reason mathematically about human consensus finding . It has been popular in the analysis of convergent beliefs in social networks and it can be regarded as a formalization of the Delphi technique , which has been used to forecast group judgments related to the economy , education , healthcare , and public policy [ Dalkey , 1972 ] . Extensions of the DeGroot model have been proposed to incorporate more sophisticated socio-psychological features that may be involved when individuals interact , such as homophily [ Hegselmann and Krause , 2002 ] , competitive interactions [ Altafini , 2013 ] and stubbornness [ Friedkin and Bullo , 2017 ] . 2 Preliminaries . We are given K independently trained models and aim to aggregate their predictions at test time . Specifically , we assume there are K agents , each of which holds a local training dataset , Dk , of size nk , and a model fk : X → Y . We do not make any prior assumption about the quality or structure of the individual models , other than that they aim to solve the same prediction task . At test time , let x′ ∈ X denote the data point that we would like to predict . Each agent k ∈ [ K ] has her own prediction for the test point corresponding to fk ( x′ ) . Our goal is to design a mechanismM that combines these predictions into an accurate single prediction , p∗ ( x′ ) =M ( f1 ( x′ ) , ... , fK ( x′ ) ) . In the following we focus on regression and use the squared loss as a measure of predictive quality . The requirements we pose on the aggregation mechanismM are : ( i ) agents should not be required to share their labeled data or model parameters with other agents , or any external party ; ( ii ) the procedure should not assume access to additional training data for an independent performance assessment or training of a meta-model . 2.1 The DeGroot consensus model . The consensus model proposed by DeGroot [ 1974 ] specifies how agents in a group influence each other ’ s opinion on the basis of mutual trust . Formally , we denote the trust of agent i towards the belief of agent j as τij ∈ ( 0 , 1 ] , with ∑ j∈ [ K ] τij = 1 for every i . Then , DeGroot defines an iterative procedure whereby each agent i repeatedly updates her belief pi as p ( t+1 ) i = K∑ j=1 τij p ( t ) j . ( 1 ) The procedure is initialized with initial beliefs p ( 0 ) i and run until p ( t ) i = p ( t ) j for every pair i , j ∈ [ K ] , at which point a consensus is deemed to have been reached . We denote the consensus belief by p∗ . The DeGroot model has a family resemblance to the influential PageRank algorithm [ Page et al. , 1999 ] that is designed to aggregate information across webpages in a decentralized manner . Similar to the consensus in DeGroot , the page rank corresponds to a steady state of equation ( 2 ) , where the weights τij are computed based on the fraction of outgoing links pointing from page i to page j .

Summary: The authors present a decentralized algorithm for test-time model combination (collective prediction) wherein each party has an private model and dataset, and only has query access to other party's models. The algorithm is a novel application of DeGroot's consensus model which models how several agents iteratively influence each other's predictions based on mutual trust. The authors use the mean squared error on a local set of points as a proxy for mutual trust: each party queries each other party's models using points neighboring the test instance from its private dataset and measures the MSE on this set. The authors also show that the algorithm easily lends itself to a decentralized jackknife algorithm.  The authors prove several intuitive properties of their algorithm and prove that, asymptotically, it recovers an optimal inverse-MSE weighting. Empirically, they show that the method outperforms a naive equally-weighted average of the classifier predictions on synthetic and real data. Through synthetic experiments, the authors show that the proposed algorithm is particularly beneficial when there is significant heterogeneity in the private data or learners because of the adaptively computed weighting (based on MSE in a neighborhood around the test instance).