Active Assessment of Prediction Services as Accuracy Surface Over Attribute Combinations

1 Introduction . Increasing concentration of big data and computing resources has resulted in widespread adoption of machine learning as a service ( MLaaS ) . The best-performing NLP , speech , image and video recognition tools are now provided as network services . MLaaS comes with few accuracy specifications or service level agreements , perhaps only leaderboard numbers from benchmarks that may not be closely related to most clients ’ deployment data distributions . The client , therefore , finds it difficult to choose the best provider without extensive pilot trials [ 1 ] . Different clients may need to deploy the service on very different data distributions , with possibly widely different accuracy . In such circumstances , we propose that a service provider , or a service standardization agency , publish the accuracy of the classifier , not as one or few aggregate numbers , but as a surface defined on a space of input instance attributes that capture the variability of consumer expectations . Indoor/outdoor , day/night , urban/rural may be attributes of input images for visual object recognition tasks . Speaker age , gender , ethnicity/accent may be attributes of input audio for speech recognition tasks . We call a combination of attributes in their Cartesian space an arm ( borrowing from bandit terminology ) 2 . The labeled instances used by the service provider may not represent or cover well the space of attributes of interest to subscribers . Labeled data may be proprietary and inaccessible to prospective consumers and standardization agencies . Whoever estimates the accuracy surface , therefore , needs to ∗vihari @ cse.iitb.ac.in 2Figure 1 shows an example of diverse accuracy over arms . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . actively select instances from an unlabeled pool for labeling , presumably within a restricted budget , to adequately cover the attribute space . Several recent studies have highlighted the variability in accuracy across data sub-populations [ 2 , 3 ] , specifically in the context of fairness [ 4 , 5 , 6 ] , and also proposed active estimation techniques of sub-population accuracy [ 7 , 8 ] . We solve a more general problem where the space of arms ( subpopulation ) defined by the Cartesian space of attributes grows combinatorially . This inevitably leads to extreme sparsity of labeled instances for many arms . A central challenge is how to smooth the estimate across related arms while faithfully representing the uncertainty for active exploration . We present Attributed Accuracy Assay ( AAA ) — a practical system that estimates accuracy , together with the uncertainty of the estimate , as a function of the attribute space . AAA uses these estimates to drive the sampling policy for each attribute combination . Gaussian Process ( GP ) regression is a natural choice to obtain smooth probabilistic accuracy estimates over arm attributes . However , a straightforward GP model fails to address the challenge of heteroscedasticity that we face with uneven and sparse supervision across arms . We model arm-specific service accuracy as drawn from a Beta density that is characterized by mean and scale parameters , which are sampled from two GPs that are informed by suitable trained kernels over the attribute space . We propose two further enhancements to the training of this model . First , we recognize an over-smoothing problem with GP ’ s estimation of the Beta scale parameters , and propose a Dirichlet likelihood to supervise the relative values of scale across arms . Second , we recognize that arms with very low support interfere with learning the kernel parameters of the GPs . We mitigate this by pooling observations across related arms . With these fixes , AAA achieves the best estimation performance among competitive alternatives . Another practical challenge in our setting is that some attributes of instances are not known exactly . For example , attributes , such as camera shutter speed or speaker gender , may be explicitly provided as meta information attached with instances . But other attributes , such as indoor/outdoor , or speaker age , may have to be estimated noisily via another ( attribute ) classifier , because accurate human-based acquisition of attributes would be burdensome . AAA also tackles uncertain attribute inference . Its attribute classifiers are trained on a small amount of labeled data and their error rates are modeled in a probabilistic framework . We report on extensive experiments using several real data sets . Comparison with several estimators based on Bernoulli arm parameters , Beta densities per arm , and even simpler forms of GPs on the arm Beta distributions , shows that AAA is superior at quickly cutting down arm accuracy uncertainty . Summarizing , our contributions are : • We motivate and define the problem of accuracy surface estimation over a large space of attribute combinations . • Our proposed estimator AAA fits a Beta density for every attribute combination ( arm ) , with its parameters smoothed via two GPs to capture heteroscedastic uncertainty of each arm ’ s accuracy under limited data settings . • We propose two important components included in AAA : 1 ) a Dirichlet regularization to control over-smoothing of the Beta scale parameters , and 2 ) pooled observations to reduce over-fitting of a GP-associated kernel to sparse arms . • We show significant gains in terms of both estimation quality and the efficiency of exploration on four real classification models compared to existing methods . AAA obtains an average 80 % reduction in macro averaged square error over the existing methods . 2 Problem Setup . Our goal is to evaluate a given machine learning service model S used by a diverse set of consumers . The service S : X 7→ Y could be any predictive model that , for an input instance x ∈ X , assigns an output label ŷ ∈ Y , where Y is a discrete label space . Let y ( x ) denote the true label of x and Agree ( y , ŷ ) denote the match between the two labels . For scalar classification , Agree ( y , ŷ ) is in { 0,1 } . For structured outputs , e.g. , sequences , we could use measures like BLEU scores in [ 0,1 ] . Classifiers are routinely evaluated on their expected accuracy on a data distribution P ( X , Y ) : ρ = EP ( x , y ) [ Agree ( y , S ( x ) ] ( 1 ) We propose to go beyond this single measure and define accuracy as a surface over a space of attributes of the input instances . Let A denote a list of K attributes that capture the variability of consumer expectation on which the service S will be deployed . For instance , visual object recognition is affected by the background scene , and facial recognition is affected by demographic attributes . We use A ( x ) ∈ A to denote the vector of values of attributes of input x and A to denote the Cartesian product of the domains of all attributes . An attribute could be discrete , e.g. , the ethnicity of a speaker ; Boolean , e.g. , whether a scene is outdoors/indoors ; or continuous , e.g. , the age of the speaker in speech recognition . Some of the attributes of x , for example the camera settings of an image , may be known exactly , and others may only be available as a distribution Mk ( ak|x ) for an attribute ak ∈ A , obtained from a pre-trained probabilistic classifier . Generalizing from a single global expected accuracy ( 1 ) , we define the accuracy surface ρ : A → [ 0 , 1 ] of a service S at each attribute combination a ∈ A , given a data distribution P ( X , Y ) , as ρ ( a ) = EP ( x , y|A ( x ) =a ) [ Agree ( y , S ( x ) ] ( 2 ) Our goal is to provide an estimate of ρ ( a ) given two kind of data sampled from P ( X , Y ) : a small labeled sample D , and a large unlabeled sample U . In addition , we are given a budget of B instances for which we can seek labels y from a human by selecting them from U . Applying Mk to all of U is , however , free of cost . We aim to design a probabilistic estimator for ρ ( a ) , which we denote as P ( ρ|a ) where ρ ∈ [ 0 , 1 ] and a ∈ A . This is distinct from active learning , which selects instances to train the learner toward greater accuracy , and also active accuracy estimation [ 7 ] , which does not involve a surface over as . We also show that standard tools to regress from a to ρ are worse than our proposal . We measure the quality of our estimate as the square error between the gold accuracy ρ ( a ) and the mean of the estimated accuracy distribution P ( ρ|a ) . Our estimator distribution naturally gives an idea of the posterior variance of accuracy estimate of each attribute combination , which we use for uncertainty-based exploration . 3 Proposed Estimator . We will first review recent work that leads to candidate solutions to our problem , discuss their limitations , and finally present our solution . Initially , to keep the treatment simple , we assume A ( x ) and gold y ( hence c = Agree ( S ( x ) , y ) , the service correctness bit ) is known for all instances . Later in this section , we remove these assumptions . The simplest option is to ignore any relationship between arms , and , for each arm a , fit a suitable density over ρ ( a ) . When this density is sampled , we get a number in [ 0 , 1 ] , which is like a coin head probability used to sample correctness bits c. For representing uncertainty of accuracy values ( which are ratios between two counts ) , the Beta distribution B ( · , · ) is a natural choice . We call this baseline method Beta-I . The variance of the estimated Beta density can be used for actively sampling arms . Ji et al . [ 7 ] describe a related scenario , stressing on active sampling . However , this approach can not share observations or smooth the estimated density at a sparsely-populated arm with information from similar arms . In our real-life scenario , we expect accuracy surface smoother and the number of arms to be large enough that many arms will get very few , if any , instances . The second baseline method , which we call BernGP , is to view the ( a , c ) instances inD as a standard classification data set with the binary c values as class label and a as input features . Given the limited data , we can use the well-known GP classification approach [ 9 ] for fitting smooth values ρ as a function of a . Suppose the arms a can be embedded to V ( a ) in a suitable space induced by some similarity kernel . In this embedding space , we expect the accuracy of S to vary smoothly . Given a kernel K1 ( a , a′ ) to guide the extent of sharing of information across arms , a standard form of this GP would be P ( c|a ) = Bernoulli ( c ; sigmoid ( fa ) ) ; f ∼ GP ( 0 , K1 ) . ( 3 ) The GP can give estimates of uncertainty of ρ ( a ) , which may be used for active sampling of arms . As we will demonstrate , such GP-imposed estimate of uncertainty of ρ ( a ) is inadequate , because it loses sight of the number of supporting observations at each arm , which could be very diverse . This is because the standard GP assumption of homoscedasticity , that is , identical noise around each arm is violated when observations per arm differ significantly . We therefore need a mechanism to separately account for the uncertainty at each arm , even the unexplored ones , to guide the strategy for actively collecting more labeled data .

Summary: The paper introduce a novel approach to evaluate a black-box classifier and capture it's performance in different regions of the input space. This is done by modelling an accuracy surface over the space of combinations of data attributes. The model is a beta-GP with variational inference. It can also be used for active learning. Experimental evaluation shows that the model compares favourably against competing methods and illustrates the rationale behind its gains.