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: This paper addresses test-time collective prediction. As far as I know, this is an underexplored topic, yet it is interesting. The paper proposes a decentralized consensus algorithm for multi-agent model aggregation. The paper includes both a nice theoretical analysis and empirical validation. Although I did not check the proofs, the theoretical analysis seems sound.