Risk Bounds and Calibration for a Smart Predict-then-Optimize Method

The predict-then-optimize framework is fundamental in practical stochastic decision-making problems : first predict unknown parameters of an optimization model , then solve the problem using the predicted values . A natural loss function in this setting is defined by measuring the decision error induced by the predicted parameters , which was named the Smart Predict-then-Optimize ( SPO ) loss by Elmachtoub and Grigas [ 2021 ] . Since the SPO loss is typically nonconvex and possibly discontinuous , Elmachtoub and Grigas [ 2021 ] introduced a convex surrogate , called the SPO+ loss , that importantly accounts for the underlying structure of the optimization model . In this paper , we greatly expand upon the consistency results for the SPO+ loss provided by Elmachtoub and Grigas [ 2021 ] . We develop risk bounds and uniform calibration results for the SPO+ loss relative to the SPO loss , which provide a quantitative way to transfer the excess surrogate risk to excess true risk . By combining our risk bounds with generalization bounds , we show that the empirical minimizer of the SPO+ loss achieves low excess true risk with high probability . We first demonstrate these results in the case when the feasible region of the underlying optimization problem is a polyhedron , and then we show that the results can be strengthened substantially when the feasible region is a level set of a strongly convex function . We perform experiments to empirically demonstrate the strength of the SPO+ surrogate , as compared to standard ` 1 and squared ` 2 prediction error losses , on portfolio allocation and cost-sensitive multi-class classification problems . 1 Introduction . The predict-then-optimize framework , where one predicts the unknown parameters of an optimization model and then plugs in the predictions before solving , is prevalent in applications of machine learning . Some typical examples include predicting future asset returns in portfolio allocation problems and predicting the travel time on each edge of a network in navigation problems . In most cases , there are many contextual features available , such as time of day , weather information , financial and business news headlines , and many others , that can be leveraged to predict the unknown parameters and reduce uncertainty in the decision making problem . Ultimately , the goal is to produce a high quality prediction model that leads to a good decisions when implemented , such as a position that leads to a large return or a route that induces a small realized travel time . There has been a fair amount of recent work examining this paradigm and other closely related problems in data-driven decision making , such as the works of Bertsimas and Kallus [ 2020 ] , Donti et al . [ 2017 ] , Elmachtoub and Grigas [ 2021 ] , Kao et al . [ 2009 ] , Estes and Richard [ 2019 ] , Ho and Hanasusanto [ 2019 ] , Notz and Pibernik [ 2019 ] , Kotary et al . [ 2021 ] , the references therein , and others . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . In this work , we focus on the particular and important case where the optimization problem of interest has a linear objective with a known convex feasible region and where the contextual features are related to the coefficients of the linear objective . This case includes the aforementioned shortest path and portfolio allocation problems . In this context , Elmachtoub and Grigas [ 2021 ] developed the Smart Predict-then-Optimize ( SPO ) loss function , which directly measures the regret of a prediction against the best decision in hindsight ( rather than just prediction error , such as squared error ) . After the introduction of the SPO loss , recent work has studied its statistical properties , including generalization bounds of the SPO loss function in El Balghiti et al . [ 2019 ] and generalization and regret convergence rates in Hu et al . [ 2020 ] . Moreover , since the SPO loss is not continuous nor convex in general [ Elmachtoub and Grigas , 2021 ] , which makes the training of a prediction model computationally intractable , Elmachtoub and Grigas [ 2021 ] introduced a novel convex surrogate loss , referred to as the SPO+ loss . Elmachtoub and Grigas [ 2021 ] highlight and prove several advantages of the SPO+ surrogate loss : ( i ) it still accounts for the downstream optimization problem when evaluating the quality of a prediction model ( unlike prediction losses such as the squared ` 2 loss ) , ( ii ) it has a desirable Fisher consistency property with respect to the SPO loss under mild conditions , and ( iii ) it often performs better than commonly considered prediction losses in experimental results . Unfortunately , although a desirable property of any surrogate loss in this context , Fisher consistency is not directly applicable when one only has available a finite dataset , which is always the case in practice , because it relies on full knowledge of the underlying distribution . Motivated thusly , it is desirable to develop risk bounds that allow one to translate an approximate guarantee on the risk of a surrogate loss function to a corresponding guarantee on the SPO risk . That is , risk bounds ( and the related notion of calibration functions ) answer the question : to what tolerance δ should the surrogate excess risk be reduced to in order to ensure that the excess SPO risk is at most ? Note that , with enough data , it is possible in practice to ensure a ( high probability ) bound on the excess surrogate risk through generalization and optimization guarantees . The main goal of this work is to provide risk bounds for the SPO+ surrogate loss function . Our results , to the best of our knowledge , are the first risk bounds of the SPO+ loss , besides the analysis of the 1-dimensional scenario in Ho-Nguyen and Kılınç-Karzan [ 2020 ] . Our results consider two cases for the structure of the feasible region of the optimization problem : ( i ) the case of a bounded polyhedron , and ( ii ) the case of a level set of a smooth and strongly convex function . In the polyhedral case , we prove that the risk bound of the SPO+ surrogate is O ( 2 ) , where is the desired accuracy for the excess SPO risk . Our results hold under mild distributional assumptions that extend those considered in Elmachtoub and Grigas [ 2021 ] . In the strongly convex level set case , we improve the risk bound of the SPO+ surrogate to O ( ) by utilizing novel properties of such sets that we develop , namely stronger optimality guarantees and continuity properties . As a consequence of our analysis , we can leverage generalization guarantees for the SPO+ loss to obtain the first sample complexity bounds , with respect to the SPO risk , for the SPO+ surrogate under the two cases we consider . In Section 5 , we present computational results that validate our theoretical findings . In particular we present results on entropy constrained portfolio allocation problems which , to the best of our knowledge , is the first computational study of predict-then-optimize problems for a strongly convex feasible region . Our results on portfolio allocation problems demonstrate the effectiveness of the SPO+ surrogate . We also present results for cost-sensitive multi-class classification that illustrate the benefits of faster convergence of the SPO risk in the case of strongly convex sets as compared to polyhedral ones . Starting with binary classification , risk bounds and calibration have been previously studied in other machine learning settings . Pioneer works studying the properties of convex surrogate loss functions for the 0-1 loss include Zhang et al . [ 2004 ] , Bartlett et al . [ 2006 ] , Massart et al . [ 2006 ] and Steinwart [ 2007 ] . Works including Zhang [ 2004 ] , Tewari and Bartlett [ 2007 ] and Osokin et al . [ 2017 ] have studied the consistency and calibration properties of multi-class classification problems , which can be considered as a special case of the predict-then-optimize framework [ El Balghiti et al. , 2019 ] . Most related to the results presented herein is the work of Ho-Nguyen and Kılınç-Karzan [ 2020 ] , who study the uniform calibration properties of the squared ` 2 loss , and the related work of Hu et al . [ 2020 ] , who also develop fast sample complexity results for the SPO loss when using a squared ` 2 surrogate . Notation . Let represent element-wise multiplication between two vectors . For any positive integer m , let [ m ] denote the set { 1 , . . . , m } . Let Ip denote the p by p identity matrix for any positive integer p. For c̄ ∈ Rd and a positive semi-definite matrix Σ ∈ Rd×d , let N ( c̄ , Σ ) denote the normal distribution P ( c ) = e − 1 2 ( c−c̄ ) TΣ−1 ( c−c̄ ) √ ( 2π ) d det ( Σ ) . We will make use of a generic given norm ‖ · ‖ on w ∈ Rd , as well as its dual norm ‖ · ‖∗ which is defined by ‖c‖∗ = maxw : ‖w‖≤1 cTw . For a positive definite matrix A , we define the A-norm by ‖w‖A : = √ wTAw . Also , we denote the diameter of the set S ⊆ Rd by DS : = supw , w′∈S ‖w − w′‖2 . 2 Predict-then-optimize framework and preliminaries . We now formally describe the predict-then-optimize framework , which is widely prevalent in stochastic decision making problems . We assume that the problem of interest has a linear objective , but the cost vector of the objective , c ∈ C ⊆ Rd , is not observed when the decision is made . Instead , we observe a feature vector x ∈ X ⊆ Rp , which provides contextual information associated with c. Let P denote the underlying joint distribution of the pair ( x , c ) . Let w denote the decision variable and assume that we have full knowledge of the feasible region S ⊆ Rd , which is assumed to be non-empty , compact , and convex . When a feature vector x is provided , the goal of the decision maker is to solve the contextual stochastic optimization problem : min w∈S Ec∼P ( ·|x ) [ cTw ] = min w∈S Ec∼P ( ·|x ) [ c ] Tw . ( 1 ) As demonstrated by ( 1 ) , for linear optimization problems the predict-then-optimize framework relies on a prediction of the conditional expectation of the cost vector , namely E [ c|x ] = Ec∼P ( ·|x ) [ c ] . Let ĉ denote a prediction of the conditional expectation , then the next step in the predict-then-optimize setting is to solve the deterministic optimization problem with the cost vector ĉ , namely P ( ĉ ) : min w∈S ĉTw . ( 2 ) Depending on the structure of the feasible region S , the optimization problem P ( · ) can represent linear programming , conic programming , and even ( mixed ) integer programming , for example . In any case , we assume that we can solve P ( · ) to any desired accuracy via either a closed-form solution or a solver . Let w∗ ( · ) : Rd → S denote a particular optimization oracle for problem ( 2 ) , whereby w∗ ( ĉ ) is an optimal solution of P ( ĉ ) . ( We assume that the oracle is deterministic and ties are broken in an arbitrary pre-specified manner . ) In order to obtain a model for predicting cost vectors , namely a cost vector predictor function g : X → Rd , we may leverage machine learning methods to learn the underlying distribution P from observed data { ( x1 , c1 ) , . . . , ( xn , cn ) } , which are assumed to be independent samples from P. Most importantly , following ( 1 ) , we would like to learn the conditional expectation and thus g ( x ) can be thought of as an estimate of E [ c|x ] . We follow a standard recipe for learning a predictor function g where we specify a loss function to measure the quality of predictions relative to the observed realized cost vectors . In particular , for a loss function ` , the value ` ( ĉ , c ) represents the loss or error incurred when the cost vector prediction is ĉ ( i.e. , ĉ = g ( x ) ) and the realized cost vector is c. Let R ` ( g ; P ) : = E ( x , c ) ∼P [ ` ( g ( x ) , c ) ] denote the the risk of given loss function ` and let R∗ ` ( P ) = infg′ R ` ( g′ ; P ) denote the optimal ` -risk over all measurable functions g′ . Also , let R̂n ` ( g ) : = 1 n ∑n i=1 ` ( g ( xi ) , ci ) denote the empirical ` -risk . Most commonly used loss functions are based on directly measuring the prediction error , including the ( squared ) ` 2 and the ` 1 losses . However , these losses do not take the downstream optimization task nor the structure of the feasible region S into consideration . Motivated thusly , one may consider a loss function that directly measures the decision error with respect to the optimization problem ( 2 ) . Elmachtoub and Grigas [ 2021 ] formalize this notion in our context of linear optimization problems with their introduction of the SPO ( Smart Predict-then-Optimize ) loss function , which is defined by ` SPO ( ĉ , c ) : = c Tw∗ ( ĉ ) − cTw∗ ( c ) , where ĉ ∈ Rd is the predicted cost vector and c ∈ C is the realized cost vector . Due to the possible non-convexity and possible discontinuities of the SPO loss , Elmachtoub and Grigas [ 2021 ] also propose a convex surrogate loss function , the SPO+ loss , which is defined as ` SPO+ ( ĉ , c ) : = max w∈S { ( c− 2ĉ ) Tw } + 2ĉTw∗ ( c ) − cTw∗ ( c ) . Importantly , the SPO+ loss still accounts for the downstream optimization problem ( 2 ) and the structure of the feasible region S , in contrast to losses that focus only on prediction error . As discussed by Elmachtoub and Grigas [ 2021 ] , the SPO+ loss can be efficiently optimized via linear/conic optimization reformulations and with ( stochastic ) gradient methods for large datasets . Elmachtoub and Grigas [ 2021 ] provide theoretical and empirical justification for the use of the SPO+ loss function , including a derivation through duality theory , promising experimental results on shortest path and portfolio optimization instances , and the following theorem which provides the Fisher consistency of the SPO+ loss . Theorem 2.1 ( Elmachtoub and Grigas [ 2021 ] , Theorem 1 ) . Suppose that the feasible region S has a non-empty interior . For fixed x ∈ X , suppose that the conditional distribution P ( ·|x ) is continuous on all of Rd , is centrally symmetric around its mean c̄ : = Ec∼P ( ·|x ) [ c ] , and that there is a unique optimal solution of P ( c̄ ) . Then , for all ∆ ∈ Rp it holds that Ec∼P ( ·|x ) [ ` SPO+ ( c̄+ ∆ , c ) − ` SPO+ ( c̄ , c ) ] = Ec∼P ( ·|x ) [ ( c+ 2∆ ) T ( w∗ ( c ) − w∗ ( c+ 2∆ ) ) ] ≥ 0 . Moreover , if ∆ 6= 0 , then Ec∼P ( ·|x ) [ ` SPO+ ( c̄+ ∆ ) − ` SPO+ ( c̄ ) ] > 0 . Notice that Theorem 2.1 holds for arbitrary x ∈ X , i.e. , it employs a nonparametric analysis as is standard in consistency and calibration results , whereby there are no constraints on the predicted cost vector associated with x . Under the conditions of Theorem 2.1 , given any x ∈ X , we know that the conditional expectation c̄ = Ec∼P ( ·|x ) [ c ] is the unique minimizer of the SPO+ risk . Furthermore , since c̄ is also a minimizer of the SPO risk , it holds that the SPO+ loss function is Fisher consistent with respect to the SPO loss function , i.e. , minimizing the SPO+ risk also minimizes the SPO risk . However , in practice , due to the fact that we have available only a finite dataset and not complete knowledge of the distribution P , we can not directly minimize the true SPO+ risk . Instead , by employing the use of optimization and generalization guarantees , we are able to approximately minimize the SPO+ risk . A natural question is then : does a low excess SPO+ risk guarantee a low excess SPO risk ? More formally , we are primarily interested in the following questions : ( i ) for any > 0 , does there exist δ ( ) > 0 such that RSPO+ ( g ; P ) − R∗SPO+ ( P ) < δ ( ) implies that RSPO ( g ; P ) −R∗SPO ( P ) < ? , and ( ii ) what is the largest such value of δ ( ) that guarantees the above ? Excess risk bounds via calibration . The notions of calibration and calibration functions provide a useful set of tools to answer the previous questions . We now review basic concepts concerning calibration when using a generic surrogate loss function ` , although we are primarily interested in the aforementioned SPO+ surrogate . An excess risk bound allows one to transfer the conditional excess ` -risk , E [ ` ( ĉ , c ) |x ] − infc′ E [ ` ( c′ , c ) |x ] , to the conditional excess ` SPO-risk , E [ ` SPO ( ĉ , c ) |x ] − infc′ E [ ` SPO ( c′ , c ) |x ] . Calibration , which we now briefly review , is a central tool in developing excess risk bounds . We adopt the definition of calibration presented by Steinwart [ 2007 ] and Ho-Nguyen and Kılınç-Karzan [ 2020 ] , which is reviewed in Definition 2.1 below . Definition 2.1 . For a given surrogate loss function ` , we say ` is ` SPO-calibrated with respect to P if there exists a function δ ` ( · ) : R+ → R+ such that for all x ∈ X , ĉ ∈ C , and > 0 , it holds that E [ ` ( ĉ , c ) |x ] − inf c′ E [ ` ( c′ , c ) |x ] < δ ` ( ) ⇒ E [ ` SPO ( ĉ , c ) |x ] − inf c′ E [ ` SPO ( c′ , c ) |x ] < . ( 3 ) Additionally , if ( 3 ) holds for all P ∈ P , where P is a class of distributions on X × C , then we say that ` is uniformly calibrated with respect to the class of distributions P . A direct approach to finding a feasible δ ` ( · ) function and checking for uniform calibration is by computing the infimum of the excess surrogate loss subject to a constraint that the excess SPO loss is at least . This idea leads to the definition of the calibration function , which we review in Definition 2.2 below . Definition 2.2 . For a given surrogate loss function ` and true cost vector distribution Pc , the conditional calibration function δ̂ ` ( · ; Pc ) is defined , for > 0 , by δ̂ ` ( ; Pc ) : = inf ĉ∈Rd { E [ ` ( ĉ , c ) ] − inf c′ E [ ` ( c′ , c ) ] : E [ ` SPO ( ĉ , c ) ] − inf c′ E [ ` SPO ( c′ , c ) ] ≥ } . Moreover , given a class of joint distributions P , with a slight abuse of notation , the calibration function δ̂ ` ( · ; P ) is defined , for > 0 , by δ̂ ` ( ; P ) : = inf x∈X , P∈P δ̂ ` ( ; P ( ·|x ) ) . If the calibration function δ̂ ` ( · ; P ) satisfies δ̂ ` ( ; P ) > 0 for all > 0 , then the loss function ` is uniformly ` SPO-calibrated with respect to the class of distributions P . To obtain an excess risk bound , we let δ∗∗ ` denote the biconjugate , the largest convex lower semi-continuous envelope , of δ ` . Jensen ’ s inequality then readily yields δ∗∗ ` ( RSPO ( g , P ) −R∗SPO ( P ) ) ≤ R ` ( g , P ) −R ` ( P ) , which implies that the excess surrogate risk R ` ( g , P ) −R ` ( P ) of a predictor g can be translated into an upper bound of the excess SPO risk RSPO ( g , P ) −R∗SPO ( P ) . For example , the uniform calibration of the least squares ( squared ` 2 ) loss , namely ` LS ( ĉ , c ) = ‖ĉ − c‖22 , was examined by Ho-Nguyen and Kılınç-Karzan [ 2020 ] . They proved that the calibration function is δ ` LS ( ) = 2/D2S , which implies an upper bound of the excess SPO risk by RSPO ( g , P ) −R∗SPO ( P ) ≤ DS ( RLS ( g , P ) −R∗LS ( P ) ) 1/2 . In this paper , we derive the calibration function of the SPO+ loss and thus reveal the quantitative relationship between the excess SPO risk and the excess surrogate SPO+ risk under different circumstances .

Summary: The paper studies the excess SPO risk of the empirical SPO+ risk minimizers. When the feasible region of the downstream planning task is a bounded polyhedron, the excess risk is on the order of O(n^{-1/4}). When the feasible region is the level set of a strongly convex and smooth function, the excess risk is on the order of O(n^{-1/2}). Empirically, the authors show that the empirical minimizers of SPO+ perform better than the ones for \ell_1, \ell_2 for more complex hypothesis class.