Differentiable Learning Under Triage

1 Introduction . In recent years , there has been a raising interest on a new learning paradigm which seeks the development of predictive models that operate under different automation levels—models that take decisions for a given fraction of instances and leave the remaining ones to human experts . This new paradigm has been so far referred to as learning under algorithmic triage [ 1 ] , learning under human assistance [ 2 , 3 ] , learning to complement humans [ 4 , 5 ] , and learning to defer to an expert [ 6 ] . Here , one does not only has to find a predictive model but also a triage policy which determines who predicts each instance . The motivation that underpins learning under algorithmic triage is the observation that , while there are high-stake tasks where predictive models have matched , or even surpassed , the average performance of human experts [ 8 , 9 ] , they are still less accurate than human experts on some instances , where they make far more errors than average [ 1 ] . The main promise is that , by working together , human experts and predictive models are likely to achieve a considerably better performance than each of them would achieve on their own . While the above mentioned work has shown some success at fulfilling this promise , the interplay between the predictive accuracy of a predictive model and its human counterpart under algorithmic triage is not well understood . One of the main challenges in learning under algorithmic triage is that , for each potential triage policy , there is an optimal predictive model , however , the triage policy is also something one seeks to optimize , as first noted by De et al . [ 2 ] . In this context , previous work on learning under algorithmic triage can be naturally differentiated into two lines of work . The first line of work has developed rather general heuristic algorithms that do not enjoy theoretical guarantees [ 1 , 4 , 5 ] . The second has developed algorithms with theoretical guarantees [ 2 , 3 , 6 ] , however , they have focused on more 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . restrictive settings . More specifically , De et al . [ 2 , 3 ] focus on ridge regression and support vector machines and reduce the problem to the maximization of approximately submodular functions and Mozannar and Sontag [ 6 ] view the problem from the perspective of cost sensitive learning and introduce a convex and consistent surrogate loss of the objective function they consider . Our contributions . Our starting point is a theoretical investigation of the interplay between the prediction accuracy of supervised learning models and human experts under algorithmic triage . By doing so , we hope to better inform the design of general purpose techniques for training differentiable models under algorithmic triage . Our investigation yields the following insights : I . To find the optimal triage policy and predictive model , we need to take into account the amount of human expert disagreement , or expert uncertainty , on a per-instance level . II . We identify under which circumstances a predictive model that is optimal under full automation may be suboptimal under a desired level of triage . III . Given any predictive model and desired level of triage , the optimal triage policy is a deterministic threshold rule in which triage decisions are derived deterministically by thresholding the difference between the model and human errors on a per-instance level . Building on the above insights , we introduce a practical gradient-based algorithm that finds a sequence of predictive models and triage policies of increasing performance subject to a constraint on the maximum level of triage . We apply our gradient-based algorithm in a wide variety of supervised learning tasks using both synthetic and real-world data from two important applications—content moderation and scientific discovery . Our experiments illustrate our theoretical results and show that the models and triage policies provided by our algorithm outperform those provided by several competitive baselines1 . Further related work . Our work is also related to the areas of learning to defer and active learning . In learning to defer , the goal is to design machine learning models that are able to defer predictions [ 10– 18 ] . Most previous work focuses on supervised learning and design classifiers that learn to defer either by considering the defer action as an additional label value or by training an independent classifier to decide about deferred decisions . However , in this line of work , there are no human experts who make predictions whenever the classifiers defer them , in contrast with the literature on learning under algorithmic triage [ 1–6 ] —they just pay a constant cost every time they defer predictions . Moreover , the classifiers are trained to predict the labels of all samples in the training set , as in full automation . In active learning , the goal is to find which subset of samples one should label so that a model trained on these samples predicts accurately any sample at test time [ 19–26 ] . In contrast , in our work , the trained model only needs to predict accurately a fraction of samples picked by the triage policy at test time and rely on human experts to predict the remaining samples . 2 Supervised Learning under Triage . Let X ⊆ Rm be the feature domain , Y be the label domain , and assume features and labels are sampled from a ground truth distribution P ( x , y ) = P ( x ) P ( y |x ) . Moreover , let ŷ = h ( x ) be the label predictions provided by a human expert and assume they are sampled from a distribution P ( h |x ) , which models the disagreements amongst experts [ 27 ] . Then , in supervised learning under triage , one needs to find : ( i ) a triage policy π ( x ) : X → { 0 , 1 } , which determines who predicts each feature vector—a supervised learning model ( π ( x ) = 0 ) or a human expert ( π ( x ) = 1 ) ; ( ii ) a predictive model m ( x ) : X → Y , which needs to provide label predictions ŷ = m ( x ) for those feature vectors x for which π ( x ) = 0 . Here , similarly as in standard supervised learning , we look for the triage policy and the predictive model that result into the most accurate label predictions by minimizing a loss function ` ( ŷ , y ) . More formally , let Π be the set of all triage policies , then , given a hypothesis class of predictive modelsM , 1Our code and data are available at https : //github.com/Networks-Learning/differentiable-learning-under-triage our goal is to solve the following minimization problem2 : minimize π∈Π , m∈M L ( π , m ) subject to Ex [ π ( x ) ] ≤ b ( 1 ) where b is a given parameter that limits the level of triage , i.e. , the percentage of samples human experts need to provide predictions for , and L ( π , m ) = Ex , y , h [ ( 1− π ( x ) ) ` ( m ( x ) , y ) + π ( x ) ` ( h , y ) ] . ( 2 ) Here , one might think of replacing h with its point estimate µh ( x ) = argminµh Eh|x [ ` ′ ( h , µh ) ] , where ` ′ ( · ) is a general loss function . However , the resulting objective would have a bias term , as formalized by the following proposition3 for the quadratic loss ` ′ ( h , µh ) = ( h− µh ) 2 : Proposition 1 Let ` ′ ( ŷ , y ) = ( ŷ − y ) 2 and assume there exist x ∈ X for which the distribution of human predictions P ( h |x ) is not a point mass . Then , the function L ( π , m ) = Ex , y [ ( 1− π ( x ) ) ` ′ ( m ( x ) , y ) + π ( x ) ` ′ ( µh ( x ) , y ) ] is a biased estimate of the true average loss defined in Eq . 2 . The above result implies that , to find the optimal triage policy and predictive model , we need to take into account the amount of expert disagreement , or expert uncertainty , on each feature vector x rather than just an average expert prediction . 3 On the Interplay Between Prediction Accuracy and Triage . Let m∗0 be the optimal predictive model under full automation , i.e. , m ∗ 0 = argminm∈M L ( π0 , m ) , where π0 ( x ) = 0 for all x ∈ X . Then , the following proposition tells us that , if the predictions made by m∗0 are less accurate than those by human experts on some instances , the model will always benefit from algorithmic triage : Proposition 2 If there is a subset V ⊂ X of positive measure under P such that∫ x∈V Ey|x [ ` ( m∗0 ( x ) , y ) ] dP > ∫ x∈V Ey , h|x [ ` ( h , y ) ] dP , then there exists a nontrivial triage policy π 6= π0 such that L ( π , m∗0 ) < L ( π0 , m∗0 ) . Moreover , if we rewrite the average loss as L ( π , m ) = Ex [ ( 1− π ( x ) ) Ey|x [ ` ( m ( x ) , y ) ] + π ( x ) Ey , h|x [ ` ( h , y ) ] ] , it becomes apparent that , for any modelm ∈M , the optimal triage policy π∗m = argminπ∈Π L ( π , m ) is a deterministic threshold rule in which triage decisions are derived by thresholding the difference between the model and human loss on a per-instance level . Formally , we have the following Theorem : Theorem 3 Let m ∈M be any fixed predictive model . Then , the optimal triage policy that minimize the loss L ( π , m ) subject to a constraint Ex [ π ( x ) ] ≤ b on the maximum level of triage is given by : π∗m , b ( x ) = { 1 if Ey|x [ ` ( m ( x ) , y ) − Eh|x [ ` ( h , y ) ] ] > tP , b , m 0 otherwise , ( 3 ) where tP , b , m = argmin τ≥0 Ex [ τ b+ max ( Ey|x [ ` ( m ( x ) , y ) − Eh|x [ ` ( h , y ) ] ] − τ , 0 ) ] . Then , if we plug in Eq . 3 into Eq . 1 , we can rewrite our minimization problem as : minimize m∈M L ( π∗m , b , m ) ( 4 ) where L ( π∗m , b , m ) = Ex [ Ey|x [ ` ( m ( x ) , y ) ] − THREStP , b , m ( Ey|x [ ` ( m ( x ) , y ) − Eh|x [ ` ( h , y ) ] ] , 0 ) ] ( 5 ) 2One might think that minimizing over the set of all possible triage policies is not a well-posed problem , i.e. , the optimal triage policy is an extremely complex function . However , our theoretical analysis will reveal that the optimal triage policy does have a simple form and its complexity depends on the considered hypothesis class of predictive models ( refer to Theorem 3 ) . 3All proofs can be found in Appendix A. with THRESt ( x , val ) = { x if x > t val otherwise . Here , note that , in the unconstrained case , tP , b , m = 0 and THRES0 ( x , 0 ) = max ( x , 0 ) . Next , building on the above expression , we prove that the optimal predictive model under full automation mθ∗0 is suboptimal under algorithmic triage as long as the average gradient across the subset of samples which are assigned to the human under the corresponding optimal triage policy π∗mθ∗0 , b is not zero . Formally , our main result is the following Proposition : Proposition 4 Let mθ∗0 be the optimal predictive model under full automation within a hypothesis class of parameterized modelsM ( Θ ) , π∗mθ∗0 , b the optimal triage policy for mθ∗0 defined in Eq . 3 for a given maximum level of triage b , and V = { x |π∗mθ∗0 , b ( x ) = 1 } . If∫ x∈V Ey|x [ ∇θ ` ( mθ ( x ) , y ) |θ=θ∗0 ] dP 6= 0 . ( 6 ) then it holds that L ( π∗mθ∗0 , b , mθ∗0 ) > minθ∈Θ L ( π ∗ mθ , b , mθ ) . Finally , we can also identify the circumstances under which any predictive model mθ′ within a hypothesis class of parameterized predictive modelsM ( Θ ) is suboptimal under algorithmic triage : Proposition 5 Let mθ′ be a predictive model within a hypothesis class of parameterized models M ( Θ ) , π∗mθ′ , b the optimal triage policy for mθ′ defined in Eq . 3 for a given maximum level of triage b , and V = { x |π∗mθ′ , b ( x ) = 0 } . If∫ x∈V Ey|x [ ∇θ ` ( mθ ( x ) , y ) |θ=θ′ ] dP 6= 0 . ( 7 ) then it holds that L ( π∗mθ′ , b , mθ′ ) > minθ∈Θ L ( π ∗ mθ , b , mθ ) . The above results will lay the foundations for our practical gradient-based algorithm for differentiable learning under triage in the next section .

Summary: The paper proposes a methodology for learning models when some of the prediction tasks are deferred to human experts (i.e., triage).  In the paper, the authors: (i) first derive the optimal level of triage for a given a model; (ii) then, they provide conditions under which models learnt under full automation will achieve lower predictive performance in presence of triage; (iii) then, they propose an iterative procedure to learn model and optimal level of triage jointly; (iv) lastly, they empirically verify the proposed algorith via a simulation and two case studies of real-world datasets.  The key contributions of this paper are: (1) theorem 3, i.e., the analytical derivation of the optimal triage level; (2) the proposed methodology to learn the model under triage.