Original: Decision-Focused Learning without Decision-Making: Learning Locally Optimized Decision Losses
Revised: Decision-Focused Learning without Differentiable Optimization: Learning Locally Optimized Decision Losses

Original: Anonymous Author(s)

Original: AfﬁliationAddress email

Original: Abstract
Revised: Abstract

Original: Decision-Focused Learning (DFL) is a paradigm for tailoring a predictive modelto a downstream optimization task that uses its predictions in order to perform better on that speciﬁc task .
Revised: Decision-Focused Learning (DFL) is a paradigm for tailoring a predictive model to a downstream optimization task that uses its predictions in order to perform better on that speciﬁc task .

Original: The main technical challenge associated with DFL is thatit requires being able to differentiate through the optimization problem, which isdifﬁcult due to discontinuous solutions and other challenges.
Revised: The main technical challenge associated with DFL is that it requires being able to differentiate through the optimization problem, which is difﬁcult due to discontinuous solutions and other challenges.

Original: Past work has largelygotten around this this issue by handcrafting task-speciﬁc surrogates to the original optimization problem that provide informative gradients when differentiatedthrough.
Revised: Past work has largely gotten around this this issue by handcrafting task-speciﬁc surrogates to the original optimization problem that provide informative gradients when differentiated through.

Original: However, the need to handcraft surrogates for each new task limits theusability of DFL.
Revised: However, the need to handcraft surrogates for each new task limits the usability of DFL.

Original: In addition, there are often no guarantees about the convexity ofthe resulting surrogates and, as a result, training a predictive model using them canlead to inferior local optima.
Revised: In addition, there are often no guarantees about the convexity of the resulting surrogates and, as a result, training a predictive model using them can lead to inferior local optima.

Original: In this paper, we do away with surrogates altogetherand instead learn loss functions that capture task-speciﬁc information.
Revised: In this paper, we do away with surrogates altogether and instead learn loss functions that capture task-speciﬁc information.

Original: To the bestof our knowledge, ours is the ﬁrst approach that entirely replaces the optimizationcomponent of decision-focused learning with a loss that is automatically learned.
Revised: To the best of our knowledge, ours is the ﬁrst approach that entirely replaces the optimization component of decision-focused learning with a loss that is automatically learned.

Original: Our approach (a) only requires access to a black-box oracle that can solve the optimization problem and is thus generalizable , and (b) can be convex by constructionand so can be easily optimized over.
Revised: Our approach (a) only requires access to a black-box oracle that can solve the optimization problem and is thus generalizable , and (b) can be convex by construction and so can be easily optimized over.

Original: We evaluate our approach on three resourceallocation problems from the literature and ﬁnd that our approach outperformslearning without taking into account task-structure in all three domains, and evenhand-crafted surrogates from the literature.
Revised: We evaluate our approach on three resource allocation problems from the literature and ﬁnd that our approach outperforms learning without taking into account task-structure in all three domains, and even hand-crafted surrogates from the literature.

Original:  Introduction
Revised:  Introduction

Original: Predict-then-optimize [7, 8] is a framework for using machine learning to perform decision-making.
Revised: Predict-then-optimize [7, 8] is a framework for using machine learning to perform decision-making.

Original: As the name suggests, it proceeds in two stages—ﬁrst, a predictive model takes as input featuresand makes some predictions using them, then second, these predictions are used to parameterize anoptimization problem that outputs a decision .
Revised: As the name suggests, it proceeds in two stages—ﬁrst, a predictive model takes as input features and makes some predictions using them, then second, these predictions are used to parameterize an optimization problem that outputs a decision .

Original: A large number of real-world applications involve bothprediction and optimization components and can be framed as predict-then-optimize problems—fore.g., recommender systems in which missing user-item ratings need to be predicted [13], portfoliooptimization in which future performance needs to be predicted [17], or strategic decision-makingin which the adversary behavior needs to be predicted [14]. 
Revised: A large number of real-world applications involve both prediction and optimization components and can be framed as predict-then-optimize problems—for e.g., recommender systems in which missing user-item ratings need to be predicted [13], portfolio optimization in which future performance needs to be predicted [17], or strategic decision-making in which the adversary behavior needs to be predicted [14]. In addition to wide applicability, this framework also formalizes the relationship between prediction and decision-making.

Original: Historically, learning a predictive model was often done independently of the downstream optimization task. However, recent work  [8, 16, 25, 9, 3, 25, 27, 26, 6] has shown that it is possible to achievebetter task-speciﬁc performance by tailoring the predictive model to the downstream task.
Revised: This is important because such predictive models are typically learned inde- pendently of the downstream optimization task in Machine Learning-based decision-making systems, and recent work in the predict-then-optimize setting [8, 16, 25, 9, 3, 25, 27, 26, 6] has shown that it is possible to achieve better task-speciﬁc performance by tailoring the predictive model to the downstream task.

Original: A largesubset of these approaches, which we will refer to as decision-focused learning (DFL) [25], learn a predictive model by differentiating through the entire prediction and optimization pipeline end-toend.
Revised: This is often done by differentiating through the entire prediction and optimization pipeline end-to-end, leading to a family of approaches that we will refer to as decision-focused learning (DFL) [25].

Original: This procedure yields a loss function we call the decision loss , which optimizes for the qualityof decisions induced by the predictive model.
Revised: Optimizing directly for the quality of decisions induced by the predictive model in this end-to-end manner yields a loss function we call the decision loss .

Original: Training with the decision loss can be challenging because the solutions toan optimization  are oftendiscontinuous interms the predictions.
Revised: However, training with the decision loss can be challenging because the solutions to optimization problems are often discontinuous in the predictions (see Section 2.1).

Original: For example, consider the optimization arg min(ˆ y 1 , ˆ y 2 ) thatpicks the smaller of two predicted values ˆ y 1 and ˆ y 2 .

Original: The solution will rarely change for ˆ y 1 ± ✏ .

Original: This results in an uninformativedecision loss  withgradients that are zero or undeﬁned, neitherof which are useful for learning a predictive model.
Revised: This results in an uninformative loss function with zero or undeﬁned gradients, neither of which are useful for learning a predictive model.

Original: In addition, even if the gradients are nonzero, there may be local optima in the decision loss that further complicate training.

Original: To addressthese challenges, DFL approaches often leveragea handcrafted surrogate task that provides moreuseful gradients.
Revised: To address this, DFL approaches often leverage handcrafted surrogate optimization tasks that provides more useful gradients.

Original: These surrogateoptimization problems may be constructed by relaxing the originalproblem [8, 16, 25, 9], adding regularization to the objective [3, 25, 27], or even using entirely andifferent optimization problem that shares the same decision space [26].
Revised: These surrogate problems may be constructed by relaxing the original problem [8, 16, 25, 9], adding regularization to the objective [3, 25, 27], or even using entirely an different optimization problem that shares the same decision space [26].

Original: Designing good surrogates is an art, requiring manual effort and insight about each optimizationproblem.
Revised: Designing good surrogates is an art, requiring manual effort, insight into the optimization problem of interest.

Revised: In addition, there are no guarantees that the surrogates induce convex decision losses, leading to local optima that further complicate training.

Original: We propose a fundamentally different approach: learning a decision loss directly for agiven task, circumventing surrogate problem design entirely.
Revised: Instead, we propose a fundamentally different approach: to learn a decision loss directly for a given task, circumventing surrogate problem design entirely.

Original: Our framework represents the loss as afunction in a particular parametric family and selects parameters which provide an informative lossfor the optimization task.
Revised: Our framework represents the loss as a function in a particular parametric family and selects parameters which provide an informative loss for the optimization task.

Original: We call the resulting loss a locally optimized decision loss ( LODL ) .
Revised: We call the resulting loss a locally optimized decision loss (LODL) .

Original: Our starting point is the observation that a good decision loss should satisfy three properties: itshould (i) be faithful to the original task, i.e., the decision quality is consistent with the originalproblem; (ii) provide informative gradients everywhere (i.e., deﬁned and non-zero);  (iii) be convexin prediction space to avoid local minima.
Revised: Our starting point is the observation that a good decision loss should satisfy 3 properties: it should (i) be faithful to the original task, i.e., the decision quality is consistent with the original problem; (ii) provide informative gradients everywhere (i.e., deﬁned and non-zero); and (iii) be convex in prediction space to avoid local minima.

Original: These demands are in tension—the ﬁrst requirementprevents the loss function from being modiﬁed too much to achieve the other two.
Revised: These demands are in tension—the ﬁrst requirement prevents the loss function from being modiﬁed too much to achieve the other two.

Original: It is not obviousapriori that any tractable parametric family should be able to simultaneously satisfy all three properties for the complex structure induced by many optimization tasks.
Revised: It is not obvious apriori that any tractable parametric family should be able to simultaneously satisfy all three properties for the complex structure induced by many optimization tasks.

Original: We resolve this tension byseparately modeling the loss function locally for the neighborhood around each individual trainingexample.
Revised: We resolve this tension by separately modeling the loss function locally for the neighborhood around each individual training example.

Original: Faithful representation of the decision loss is easier to accomplish locally in each individual neighborhood than globally across instances, allowing us to introduce convex parametric familiesof loss functions which capture structural intuition about properties important for optimization.
Revised: Faithful representation of the decision loss is easier to accomplish locally in each individual neighborhood than globally across instances, allowing us to introduce convex parametric families of loss functions which capture structural intuitions about properties important for optimization.

Original: Toﬁt the parameters, we sample points in the neighborhood of the true labels, evaluate the decisionloss associated with these sampled points, and then train a loss function to mimic the decision loss.
Revised: To ﬁt the parameters, we sample points in the neighborhood of the true labels, evaluate the decision loss associated with these sampled points, and then train a loss function to mimic the decision loss.

Original: We evaluate LODL s on three resource allocation domains from the literature [12, 25, 24].
Revised: We evaluate LODLs on three resource allocation domains from the literature [12, 25, 24].

Original: Perhapssurprisingly, we ﬁnd that LODL s outperforms handcrafted surrogates in two out of the three.
Revised: Perhaps surprisingly, we ﬁnd that LODLs outperform handcrafted surrogates in two out of the three.

Original: In ouranalysis, we discover a linear correlation betweendecision quality and the agreement of the learned LODL with the decision loss in a speciﬁc neighborhood of the true label.
Revised: In our analysis, we discover a linear correlation between the agreement of the learned LODL with the decision loss and the decision loss of a predictive model learned using said LODL.

Original: Our approach motivatesa new line of research on decision-focused learning.
Revised: Our approach motivates a new line of research on decision-focused learning.

Original:  Background
Revised:  Background

Original: We consider the following predict-then-optimize problem. Given features x , a predictive model M ✓produces predictions ˆ y =
Revised: In predict-then-optimize, a predictive model M ✓ ﬁrst takes as input features x and produces predictions ˆ y =

Original: M ✓ ( x ) .
Revised: M ✓ ( x ) .

Original: These predictions ˆ y are  used to parameterize an optimizationproblem that is solved to yield decision z ⇤ ( ˆ y ) :
Revised: These predictions ˆ y are then used to parameterize an optimization problem that is solved to yield decisions z ⇤ ( ˆ y ) :

Original: [Equation]
Revised: [Equation]

Original: Predictions are evaluated w.r.t. to decision loss (DL) of the decision they induce—the value of the objective function of the optimization under the ground truth parameters y : DL ( ˆ y, y )
Revised: Note that, unlike typical machine learning problems, the dimensionality of dim( ˆ y )

Original: =
Revised: =

Original: f ( z ⇤ ( ˆ y) ,  y ) .
Revised: D is likely to be large as it consists of all predictions needed to parameterize the optimization problem. Given the large dimensionality, and the similarity in the role of all the predictions, the predictive model typically predicts individual components of ˆ y , i.e. ˆ y =

Original: Thus, for a dataset [( x 1 ,  y  1 ) , . . .
Revised: [ˆ y 1 , . . . , ˆ y D ] = [ M ✓ ( x 1 ) , . . .

Original: ,  ( x N , y N )] , we aim to learn a model M ✓ that generates predictions[ ˆ y1 , . . . , ˆ y N ] that minimize DL :
Revised: , M ✓ ( x D )] . Predictions are evaluated with respect to the decision loss (DL) of the decision that they induce, i.e., the value under the objective function of the optimization under the ground truth parameters y :

Original: [Equation]
Revised: [Equation]

Original: min ✓ N N
Revised: LODL does not require backpropagating through the optimization problem.

Original: X n = DL ( M ✓ ( x n ) , y n )  .
Revised: Thus, for a dataset [( x 1 , y 1 ) , . . .

Original: LODL does not require backpropagating through the optimization problem.
Revised: , ( x N , y N )] , we aim to learn a model M ✓ that generates predictions [ ˆ y 1 , . . . , ˆ y N ] that minimize the decision loss DL : [Equation]

Original: This is in contrast with standard supervised learning approaches in which the quality of a prediction is measured by a somewhat arbitrary intermediate loss (e.g., mean squared-error) that does notcontain information about the downstream decision-making task.
Revised: This is in contrast with standard supervised learning approaches in which the quality of a prediction is measured by a somewhat arbitrary intermediate loss (e.g., mean squared error) that does not contain information about the downstream decision-making task.

Original: In this paper, we refer to modelsthat use an intermediate loss as 2-stage and those that directly optimize for DL as decision-focusedlearning (DFL) .
Revised: In this paper, we refer to models that use an intermediate loss as 2-stage and those that directly optimize for DL as decision-focused learning (DFL) .

Original: Figure 1 outlines how a predictive model is learned using these  approaches.
Revised: Figure 1 outlines how a predictive model is learned using these different approaches.

Revised: 2.1 Motivating Example

Revised: Consider an arg min optimization where the goal is to predict the dis utility y of 2 agents (A, B), e.g., y = (0 , 1) .

Original: Related Work
Revised: Now, if these parameters are predicted perfectly, the decision is z = “Pick A”, and the decision loss DL is the true disutility of agent A, i.e., DL = 0 . On the other hand, consider the set of predictions ˆ y bad = (1 ± ✏ A , 0 ± ✏ B ) for 0 < ✏ A , ✏ B < 0 . Any prediction in this set will yield the decision ‘Pick B” and a decision loss DL of 1.

Revised: Given that the decision loss is constant in this region, the gradients are all zero, i.e., r ˆ y DL

Revised: ( ˆ y , y )

Revised:  ˆ y 2 ˆ y good = 0 .

Revised: As a result, if a predictive model makes such a prediction, it cannot improve its predictions by gradient descent.

Revised: Therefore, although DL is what we want to minimize, we don’t want to ﬁt it perfectly .

Revised: [Figure]

Revised: loss functions (Blue = 1, Red = 0) as a function of the predictions of different disutilities for A and B.

Revised: If instead of minimizing DL directly, we minimized a surrogate loss SL = ˆ y A  ˆ y B , gradient descent would lead to the model to predict ˆ y = ( 1 , 1 ) regardless of the initialization (see Figure 2).

Revised: While ˆ y isn’t the true set of disutilities, it does lead to the optimal decision, i.e., “Pick A” and is thus effective from a predict-then-optimize perspective.

Revised: The challenge is then coming up with such surrogates.

Revised: Although it is easy for the problem above, it becomes more complicated as the number and types of variables and constraints grows.

Revised: Below, we propose an approach to automatically learn good surrogates.

Revised: Figure 2c shows what our method looks like for the example above.

Original: A great deal of recent work on decision-focused learning and related topics aims to incorporate information about a downstream optimization problem into the training of a machine learning models.
Revised: A great deal of recent work on decision-focused learning and related topics aims to incorporate information about a downstream optimization problem into the training of a machine learning models.

Original: Some optimization problems (especially strongly convex ones) can be directly differentiated through[7, 2, 1].
Revised: Some optimization problems (especially strongly convex ones) can be directly differentiated through [7, 2, 1].

Original: For others, particularly discrete optimization problems with ill-deﬁned gradients, a varietyof approaches have been proposed.
Revised: For others, particularly discrete optimization problems with ill-deﬁned gradients, a variety of approaches have been proposed.

Original: Most of these construct surrogate loss functions via smoothingthe optimization problem [18, 25, 9, 26, 23, 15].
Revised: Most of these construct surrogate loss functions via smoothing the optimization problem [18, 25, 9, 26, 23, 15].

Original: Alternatively, Elmachtoub and Grigas [8] proposea closed-form convex surrogate loss with desirable theoretical properties; however, this loss appliesonly when the optimization problem has a linear objective function.
Revised: Alternatively, Elmachtoub and Grigas [8] propose a closed-form convex surrogate loss with desirable theoretical properties; however, this loss applies only when the optimization problem has a linear objective function.

Original: Similarly, Mulamba et al.
Revised: Similarly, Mulamba et al.

Original: [20]provide a contrastive learning-based surrogate which does not require differentiation through optimization, but which is also developed speciﬁcally for linear objectives.
Revised: [20] provide a contrastive learning-based surrogate which does not require differentiation through optimization, but which is also developed speciﬁcally for linear objectives.

Original: When the predictive modelis itself a linear function, Guler et al.
Revised: When the predictive model is itself a linear function, Guler et al.

Original: [11] propose an approach to search directly for the best model.
Revised: [11] propose an approach to search directly for the best model.

Original: Perhaps the most related work to ours is by Berthet et al.
Revised: Perhaps the most related work to ours is by Berthet et al.

Original: They differentiate through linear optimization problems during training by adding randomized perturbations to smooth the induced lossfunction.
Revised: They differentiate through linear optimization problems during training by adding randomized perturbations to smooth the induced loss function.

Original: Speciﬁcally, they randomly perturb the predictions ˆ y with random noise ✏ and solve forz ⇤ (ˆ y + ✏ ) . This can be interpreted as replacing the decision loss with its averaged value in a neighborhood around ˆ y , where the averaging smooths the function and ensures differentiability.
Revised: Speciﬁcally, they randomly perturb the predictions ˆ y with random noise ✏ and solve for z ⇤ (ˆ y + ✏ ) , which can be interpreted as replacing the decision loss with its averaged value in a neighborhood around ˆ y , where the averaging smooths the function and ensures differentiability.

Original: Thereare two key differences between our approach and theirs.
Revised: There are two key differences between our approach and theirs.

Original: First, they apply random perturbationsto optimization in the training loop in order to produce a smoother surrogate loss.
Revised: First, they apply random perturbations to optimization in the training loop in order to produce a smoother surrogate loss.

Original: By contrast, weuse random perturbations to learn a loss function prior to training ; during training optimization isremoved entirely.
Revised: By contrast, we use random perturbations to learn a loss function prior to training ; during training optimization is removed entirely.

Original: This allows us to use the same random samples to inform each training iterationinstead of drawing new samples per-iteration.
Revised: This allows us to use the same random samples to inform each training iteration instead of drawing new samples per iteration.

Original: Second, their approach applies only to linear objectivefunctions while ours applies to arbitrary optimization problems.
Revised: Second, their approach applies only to linear objective functions while ours applies to arbitrary optimization problems.

Original:  Locally Optimized Decision Losses ( LODL )
Revised:  Locally Optimized Decision Losses (LODL)

Original: In this paper, we do away with the need for custom task-speciﬁc relaxations of the optimizationproblem z ⇤ ( ˆ y ) by instead translating task-speciﬁc information from the decision loss DL into aloss function LODL  ( ˆ y , y ) that  approximates the behavior of DL  and  is convex by construction.
Revised: In this paper, we do away with the need for custom task-speciﬁc relaxations of the optimization problem z ⇤ ( ˆ y ) by instead translating task-speciﬁc information from the decision loss DL into a loss function LODL  ( ˆ y , y ) that (i) approximates the behavior of DL , and (ii) is convex by construction.

Original: The goal is to ﬁnd a set of parameters  ⇤ such that training a predictive model using LODL  ⇤yields a trained model M ✓ ⇤ LODL that performs just as well as the (hard to compute) model whichoptimizes DL directly.
Revised: Our broad strategy to do this is to learn the function LODL  ⇡ DL using supervised machine learning.

Original: We now detail the main steps in this process.
Revised: Speciﬁcally, we proceed in 3 steps:

Original: 4.1 Representing the  LODL 
Revised: We simplify the learning problem in two ways (Section 4.1) . First, we learn a separate LODL for every ( ( x , y ) ) pair in the dataset to make our learning problem easier.

Original: The key design choice in instantiating our framework is choice of the parametric family used torepresent the LODL . 
Revised: Second, we note that there’s a chicken-and-egg problem associated with learning LODLs—to train the LODL we need inputs of the form ( ˆ y , y ) , but to produce ˆ y we need a predictive model trained on said LODL. To resolve this, we make the assumption that our predictive model M ✓ will get us sufﬁciently close to the true labels y . This means: [Equation]

Original: Optimization problems can induce a complex loss landscape which is noteasily summarized in a closed-form function with concise parameterization.
Revised: and [Equation] Given these simpliﬁcations, we propose convex-by-construction parametric forms for LODL  i (Section 4.2) .

Original: Accordingly, we designa set of families which capture phenomena particularly important for common families of predictthen-optimize problems.
Revised: We subtract a constant DL ( y i , y i ) from the target to ensure that the function to be learned has a minima at ˆ y i = y i and that the result can be modeled well by a convex function: [Equation]

Original: There are two key steps: ﬁrst, creating a representation which is local toeach training instance and second, choosing the parametric family which will be optimized over toproduce each local representation.
Revised: We propose different strategies for sampling ˆ y (Section 4.3) and also describe the equation used to learn the optimal LODL parameters  ⇤ (Equation 2).

Original: Local loss functions We introduce a separate set of parameters  n for each[( x 1 , y 1 ) , . . .
Revised: We introduce a separate set of parameters  n for each [( x 1 , y 1 ) , . . .

Original: , ( x n , y n ) , . . .
Revised: , ( x n , y n ) , . . .

Original: , ( x N , y N )] in the training set.
Revised: , ( x N , y N )] in the training set.

Original: We take this step because learning a global approximation to DL ( ˆ y , y ) (for arbitrary ˆ y ) is hard; it requires learning a closed-formapproximation to the general optimization problem
Revised: We take this step because learning a global approximation to DL ( ˆ y , y ) (for arbitrary ˆ y ) is hard; it requires learning a closed-form approximation to the general optimization problem

Original: z ⇤ ( ˆ y ) which may not always exist.
Revised: z ⇤ ( ˆ y ) which may not always exist.

Original: Introducingseparate parameters per-instance gives two key advantages.1.
Revised: Introducing separate parameters per-instance gives two key advantages.

Original: Learning for a speciﬁc y n : Instead of learning a R dim( y )+dim( ˆ y ) function LODL  ( ˆ y , y ) toimitate DL ( ˆ y , y ) , we instead learn N different R dim( ˆ y ) functionsLODL  n ( ˆ y ) that imitate
Revised: Learning for a speciﬁc y n : Instead of learning a R dim( y )+dim( ˆ y ) function LODL  ( ˆ y , y ) to imitate DL ( ˆ y , y ) , we instead learn N different R dim( ˆ y ) functions

Original:  DL ( ˆ y , y n ) for each n 2 N .
Revised: LODL  n ( ˆ y ) that imitate DL ( ˆ y , y n ) for each n 2 N .

Original: Doing this signiﬁcantly reduces the dimensionality of the learningproblem—this is especially relevant when N is not very large in comparison to dim( y ) (as inour experiments).
Revised: Doing this signiﬁcantly reduces the dimensionality of the learning problem—this is especially relevant when N is not very large in comparison to dim( y ) (as in our experiments).

Original: It also circumvents the need to enforce invariance properties on the globalloss.
Revised: It also circumvents the need to enforce invariance properties on the global loss.

Original: For example, many optimization problems are invariant to permutations of the ordering of dimensions in y , making it difﬁcult to measure the quality of ˆ y ’s across different instances.
Revised: For example, many optimization problems are invariant to permutations of the ordering of dimensions in y , making it difﬁcult to measure the quality of ˆ y ’s across different instances.

Original: In a local loss, the ordering of the dimensions is ﬁxed and so this issue is no longer relevant. 
Revised: In a local loss, the ordering of the dimensions is ﬁxed and so this issue is no longer relevant. 2.

Original: Learning for only ˆ y n ⇡ y n : In addition to the simpliﬁcation above, we don’t try to learn a {+faithful approximation 
+}
Revised: Learning for only ˆ y n ⇡ y n : In addition to the simpliﬁcation above, we don’t try to learn a {+faithful approximation 
+}

Original: faithful approximation 8 ˆ y 2 R dim( y ) —we instead limit ourselves to learning an approximation of DL ( ˆ y , y n ) only when ˆ y is in the neighborhood of y n .
Revised: ˆ y 2 R dim( y ) —we instead limit ourselves to learning an approximation of DL ( ˆ y , y n ) only when ˆ y is in the neighborhood of y n .

Original: This is because we assume that our predictive model will always get us in the neighborhood of the true labels, and the utility of 
Revised: We assume that our predictive model will always get us in the neighborhood of the true labels, and the utility of LODL is in helping distinguish between these points.

Original:  LODL  is  in helping distinguish between these points.
Revised: The combination of these two choices makes the learned LODL  n a “local” surrogate for DL , rather than a global one. 4.2 Representing the LODL The key design choice in instantiating our framework is choice of the parametric family used to represent the LODL. Optimization problems can induce a complex loss landscape which is not easily summarized in a closed-form function with concise parameterization.

Revised: Accordingly, we design a set of families which capture phenomena particularly important for common families of predict-thenoptimize problems.

Original: Parametric families for the local losses Having made the decision to allow separate parametersfor each training instance, the second design choice is how to represent each local loss, i.e., thespeciﬁc parametric family to use.
Revised: Parametric families for the local losses Having made the decision to allow separate parameters for each training instance, the second design choice is how to represent each local loss, i.e., the speciﬁc parametric family to use.

Original: Our choice must be sufﬁciently expressive to capture the localdynamics of the optimization problem while remaining sufﬁciently efﬁcient to be replicated acrossthe N training instances.
Revised: Our choice must be sufﬁciently expressive to capture the local dynamics of the optimization problem while remaining sufﬁciently efﬁcient to be replicated across the N training instances.

Original: We propose that the structure of the loss function should capture theunderlying rationale for why decision-focused learning provides an advantage over 2-stage in theﬁrst place; this is the key behavior which will underpin improved decision quality.
Revised: We propose that the structure of the loss function should capture the underlying rationale for why decision-focused learning provides an advantage over 2-stage in the ﬁrst place; this is the key behavior which will underpin improved decision quality.

Original: We identify threekey phenomena which motivate the design of the family of loss functions:
Revised: We identify three key phenomena which motivate the design of the family of loss functions:

Original: Relative importance of different dimensions: Typically, the different dimensions of a pre- diction problem are given equal weight, e.g., the MSE weights errors in each coordinate of y equally.
Revised: Relative importance of different dimensions: Typically, the different dimensions of a prediction problem are given equal weight, e.g., the MSE weights errors in each coordinate of y equally.

Original: However, there may be some dimensions along which DL is more sensitive to local per- turbations.
Revised: However, there may be some dimensions along which DL is more sensitive to local perturbations.

Original: For example, a knapsack problem may be especially sensitive to errors in the value of items which are on the cusp of being chosen.
Revised: For example, a knapsack problem may be especially sensitive to errors in the value of items which are on the cusp of being chosen.

Original: In such cases, DFL can capture the relative importance of accurately predicting different dimensions.
Revised: In such cases, DFL can capture the relative importance of accurately predicting different dimensions.

Original: Cost of correlation:
Revised: Cost of correlation:

Original: Given the possibly large dimensionality of y , in practice, the predictive model M ✓ does not typically predict y directly.
Revised: Given the possibly large dimensionality of y , in practice, the predictive model M ✓ does not typically predict y directly.

Original: Instead, the the structure of the optimization is exploited to make multiple predictions [ˆ y 1 , . . .
Revised: Instead, the the structure of the optimization is exploited to make multiple predictions [ˆ y 1 , . . .

Original: , ˆ y K ] that are then combined to create ˆ y .
Revised: , ˆ y K ] that are then combined to create ˆ y .

Original: Forexample, in a knapsack problem, we might train a model which separately predicts the value ofeach item (i.e., predicts each entry of ˆ y separately) using features speciﬁc to that item, instead ofjointly predicting the entire set of values using the features of all of the items.
Revised: For example, in a knapsack problem, we might train a model which separately predicts the value of each item (i.e., predicts each entry of ˆ y separately) using features speciﬁc to that item, instead of jointly predicting the entire set of values using the features of all of the items.

Original: However, Cameron et al.
Revised: However, Cameron et al.

Original: [5] show that ignoring the correlation between different sub- y scale predictions (as in 2stage) can result in poor optimization performance, and that DFL can improve by propagatinginformation about the interactions between entries of y to the predictive model.
Revised: [5] show that ignoring the correlation between different sub- y scale predictions (as in 2stage) can result in poor optimization performance, and that DFL can improve by propagating information about the interactions between entries of y to the predictive model.

Original: Directionality of predictions In the arg min(ˆ y 1 , ˆ y 2 ) example from the introduction, the prediction ˆ y 1 = 2 , ˆ y 2 = 1 produces the same decision as ˆ y 1 = 2000 , ˆ y 2 = 1 .
Revised: Directionality of predictions: In the arg min(ˆ y 1 , ˆ y 2 ) example from the introduction, the prediction ˆ y 1 = 2 , ˆ y 2 = 1 produces the same decision as ˆ y 1 = 2000 , ˆ y 2 = 1 .

Original: On the other hand,the prediction ˆ y 1 = 0 .
Revised: On the other hand, the prediction ˆ y 1 = 0 .

Original: 5 , ˆ y 2 = 1 leads to a different decision.
Revised: 5 , ˆ y 2 = 1 leads to a different decision.

Original: As a result, over-predicting andunder-predicting often have different associated costs for some optimization problems.
Revised: As a result, over-predicting and under-predicting often have different associated costs for some optimization problems.

Original: Predictive models trained with DFL can take into account this behavior while those trained by typicalsymmetric 2-Stage losses like MSE cannot.
Revised: Predictive models trained with DFL can take into account this behavior while those trained by typical symmetric 2-Stage losses like MSE cannot.

Original: Given these insights, we propose three corresponding families of loss functions for LODL  , eachof which is convex by design and has a global minima at the true label y n —a desirable propertybecause DL also has its minima at ˆ y = y .
Revised: Given these insights, we propose three corresponding families of loss functions for LODL  , each of which is convex by design and has a global minima at the true label y n , a desirable property because DL also has its minima at ˆ y = y .

Original: WeightedMSE: To take into account the relative importance of different dimensions, we propose a weighted version of MSE:
Revised: WeightedMSE: To take into account the relative importance of different dimensions, we propose a weighted version of MSE:

Original: [Equation]
Revised: [Equation]

Original: where ‘weights’ w l are thefree parameters in the LODL ,
Revised: where ‘weights’ w l are the parameters of the LODL, i.e.,  = w 2 R dim( y )

Original: i e.,  = w .
Revised: + (for convexity).

Original: Quadratic: To take into account the effect of correlation of different dimensions on each other, we propose learning a quadratic function that has terms of the form (
Revised: Quadratic: To take into account the effect of correlation of different dimensions on each other, we propose learning a quadratic function that has terms of the form (

Original: ˆ y i
Revised: ˆ y i

Original:  y i )
Revised:  y i )

Original: (ˆ y j
Revised: (ˆ y j

Original:  y j ) :
Revised:  y j ) :

Original: Quadratic

Original: [Equation]
Revised: [Equation]

Original: where 
Revised: where

Original: =
Revised: =

Original: H is a learned low-rank symmetric Positive semideﬁnite (PSD) matrix.
Revised: H is a learned low-rank symmetric Positive semideﬁnite (PSD) matrix.

Original: This familyof functions is convex as long as H is PSD, which we enforce by parameterizing H =
Revised: This family of functions is convex as long as H is PSD, which we enforce by parameterizing H =

Original: L T L , where L is a low-rank triangular matrix L of dimension dim( y ) ⇥
Revised: L T L , where L is a low-rank triangular matrix L of dimension dim( y ) ⇥

Original: k and k is the desired rank.
Revised: k and k is the desired rank.

Original: This loss function family has an appealing interpretation because learning LODL is similar toestimating the partial derivative of DL with respect to its ﬁrst input ˆ y n .
Revised: This loss function family has an appealing interpretation because learning LODL is similar to estimating the partial derivative of DL with respect to its ﬁrst input ˆ y n .

Original: Speciﬁcally, considerthe ﬁrst three terms of the Taylor expansion of DL with respect to ˆ y n at ( y n , y n ) :
Revised: Speciﬁcally, consider the ﬁrst three terms of the Taylor expansion of DL with respect to ˆ y n at ( y n , y n ) :

Original: [Equation]
Revised: [Equation]

Original: Quadratic LODL  can be seen as a 2 nd -order Taylor-series approximation of DL at ( y n , y n )where the learned H approximates the Hessian of DL .
Revised: Quadratic LODL  can be seen as a 2 nd -order Taylor-series approximation of DL at ( y n , y n ) where the learned H approximates the Hessian of DL .

Original: Note that WeightedMSE is a special caseof this Quadratic loss when H = diag ( w ) .3.
Revised: Note that WeightedMSE is a special case of this Quadratic loss when H = diag ( w ) .

Original: DirectedWeightedMSE and DirectedQuadratic: To take into account the fact that overpredicting and underpredicting can have different consequences, we propose modiﬁcations to thetwo loss function families above.
Revised: DirectedWeightedMSE and DirectedQuadratic: To take into account the fact that overpredicting and underpredicting can have different consequences, we propose modiﬁcations to the two loss function families above.

Original: For WeightedMSE, we redeﬁne the weight vector w as below,and learn both w + and w  .
Revised: For WeightedMSE, we redeﬁne the weight vector w as below, and learn both w + and w  .

Original: Similarly for Quadratic, we deﬁne 4 copies of the parameter Lbased on the directionality of the predictions.[Equation]
Revised: Similarly for Quadratic, we deﬁne 4 copies of the parameter L based on the directionality of the predictions.

Original: [Equation]
Revised: [Equation]

Original: otherwise

Original: 4.2 Learning LODL 
Revised: 4.3 Learning LODL

Original: Given families proposed in Section 4.1, our goal is to learn some  ⇤ n for every n 2 N such that
Revised: Given families proposed in Section 4.1, our goal is to learn some  ⇤ n for every n 2 N such that

Original: LODL  n ( ˆ y n ) ⇡ DL ( ˆ y n , y n ) for ˆ y n “close” to y n .
Revised: LODL  n ( ˆ y n ) ⇡ DL ( ˆ y n , y n ) for ˆ y n “close” to y n .

Original: We propose a supervised approach to learning ⇤ n which proceeds in two steps (Figure 1): (1) we build a dataset mapping ˆ y n !
Revised: We propose a supervised approach to learning  ⇤ n which proceeds in two steps (Figure 1): (1) we build a dataset mapping ˆ y n !

Original: DL ( ˆ y n , y n ) in the region of y n , and then (2) we use this dataset to estimate  ⇤ n by minimizing the mean-squarederror to the true decision loss: 
Revised: DL ( ˆ y n , y n ) in the region of y n , and then (2) we use this dataset to estimate  ⇤ n by minimizing the mean squared error to the true decision loss: [Equation]

Original: [Equation]
Revised: [Equation]

Original: This framework has the key advantage of reducing the design of good surrogate tasks (a complexproblem requiring in-depth knowledge of each optimization problem) to supervised learning (forwhich many methods are available).
Revised: This framework has the key advantage of reducing the design of good surrogate tasks (a complex problem requiring in-depth knowledge of each optimization problem) to supervised learning (for which many methods are available).

Original: Indeed, future advances, e.g. in representation learning, cansimply be plugged into our framework.
Revised: Indeed, future advances, e.g. in representation learning, can simply be plugged into our framework.

Original: The major remaining step is to specify the construction of the dataset for supervised learning of  ⇤ n .
Revised: The major remaining step is to specify the construction of the dataset for supervised learning of  ⇤ n .

Original: We propose to construct this dataset by sampling a set of K points [ y 1 n , . . .
Revised: We propose to construct this dataset by sampling a set of K points [ y 1 n , . . .

Original: , y Kn ] in the vicinity of yand calculate DL ( y kn , y n ) for each.
Revised: , y K n ] in the vicinity of y and calculate DL ( y kn , y n ) for each.

Original: In this paper, we consider three sampling strategies:
Revised: In this paper, we consider three sampling strategies:

Original: All-Perturbed: Add zero-mean Gaussian noise to the true label y n :
Revised: All-Perturbed: Add zero-mean Gaussian noise to the true label y n :

Original: [Equation]
Revised: [Equation]

Original: where ↵ is a normalization factor and I is a dim ( y ) ⇥ dim ( y ) identity matrix.2.
Revised: where ↵ is a normalization factor and I is a dim ( y ) ⇥ dim ( y ) identity matrix [Equation]

Original:  1-Perturbed or 2-Perturbed: Estimating the behavior of DL ( y n + ✏ i , y n ) for small ✏ can alternatively be interpreted as estimating (  ˆ y n ) dim ( y n ) DL ( ˆ y n  y n ) (i e the dim  y n ) th partialderivative of DL w.r
Revised: 2. 1-Perturbed or 2-Perturbed: Estimating the behavior of DL ( y n + ✏ i , y n ) for small ✏ can alternatively be interpreted as estimating (  ˆ y n ) dim ( y n ) DL ( ˆ y n , y n ) (i.e., the dim ( y n ) th partial derivative of DL w.r.t.

Original: t. its ﬁrst input ˆ y n ) at ( y n  y n ) because all the dimensions of ˆ y n are beingvaried simultaneously.
Revised: its ﬁrst input ˆ y n ) at ( y n , y n ) because all the dimensions of ˆ y n are being varied simultaneously.

Original: While computing (  ˆ y n ) dim ( y n ) is computationally challenging, it can be estimated using the simpler 1 st or 2 nd partial derivatives.
Revised: While computing (  ˆ y n ) dim ( y n ) is computationally challenging, it can be estimated using the simpler 1 st or 2 nd partial derivatives.

Original: This corresponds to perturbing only 
Revised: This corresponds to perturbing only one or two dimensions at a time.

Original:  one 
Revised: The amount of time taken by each of the methods using gradient descent is: • 2-Stage = ⇥ ( T · N · T M ) , where T M is the amount of time taken to run one forward and backwards pass through the model M ✓ for one optimization instance, N is the number of optimization instances, and T is the number of time-steps M ✓ is trained for.

Original: or two dimensions at a time.{+• DFL = ⇥ ( T · N · ( T M + T O + T 0 O )) , where T O is the time taken to solve the forward pass of one optimization instance and T 
 O is the time taken to compute the backward pass. • LODL = ⇥ ( K · N · T O + N · ( T · K · T LODL )+}
Revised: {+• DFL = ⇥ ( T · N · ( T M + T O + T 0 O )) , where T O is the time taken to solve the forward pass of one optimization instance and T 
 O is the time taken to compute the backward pass. • LODL = ⇥ ( K · N · T O + N · ( T · K · T LODL )+}

Original: Experiments
Revised: + T · N · T M ) , where K is the number of samples needed to train the LODL, and T LODL is the amount of time taken to run one forward and backwards pass through the LODL. The three terms correspond to (i) generating samples, (ii) training N LODLs, and (iii) training M ✓ using the trained LODLs. In practice, we ﬁnd that ( T 0 O > T O ) >> ( T M > T LODL ) . As a result, the difference in complexity of DFL and LODL is roughly ⇥ ( T · N · T O ) vs. ⇥ ( K · N · T O + N · T · K · T LODL ) .

Revised: Further, the calculation above assumes that LODLs are trained in the same way as M ✓ .

Revised: However, in practice, they can often be learned much faster, sometimes even in closed form (e.g., WeightedMSE and

Revised: DirectedWeightedMSE), leading to an effective runtime of ⇥ ( K · N · T O + N · K · T LODL )

Revised: ⇡ ⇥ ( K · N · T O ) .

Revised: Then, the difference between DFL and LODL boils down to T vs. K , i.e., the number of time-steps needed to train M ✓ vs. the number of samples needed to train the LODL.

Revised: While our approach can be more computationally expensive, it typically isn’t for two reasons:

Revised: • Amortization: We need only sample candidate predictions once, to then train any number of LODLs (e

Revised: g., WeightedMSE, DirectedQuadratic) without ever having to call an optimization oracle.

Revised: Once the LODLs have been learned, you can train any number of predictive models M ✓ based on said LODLs—in contrast to DFL, which requires calling the oracle to train each model.

Revised: DFL

Revised: is

Revised: thus

Revised: more

Revised: expensive

Revised: when training a large number of models (

Revised: e

Revised: g., for

Revised: hyperparame ter

Revised: /architecture

Revised: search trading off

Revised: performance vs. inference

Revised: time

Revised: vs. interpretability,

Revised: etc

Revised: In the

Revised: future we imagine

Revised: that datasets could be shipped with not only features and labels, but also LODLs associated with downstream tasks!

Revised: Parallelizability: The sample generation process for LODL is completely parallelizable, resulting in an ⌦ ( T O ) lower-bound wall-clock complexity for our approach.

Revised: In contrast, the calls to the optimization oracle in DFL are interleaved with the training of M ✓ and, as a result, cannot be parallelized with respect to T , resulting in an ⌦ ( T · T O ) wall-clock complexity.

Revised: We demonstrate this empirically in Section 5.3.

Original: To validate the efﬁcacy of our approach, we run experiments on three resource allocation tasksfrom the literature.
Revised: To validate the efﬁcacy of our approach, we run experiments on three resource allocation tasks from the literature.

Original: Because these problems are maximizations, we use the term decision quality(DQ) to represent the negative decision loss (higher is better).
Revised: We use the term decision quality DQ (higher is better) instead of DL because these are all maximization problems.

Original: A more complete description of theexperimental setup is provided in the appendix.

Original: Linear Model
Revised: Linear Model

Original: This domain involves learning a linear model when the underlying mapping between features and predictions is cubic.
Revised: This domain involves learning a linear model when the underlying mapping between features and predictions is cubic.

Original: With limited model capacity, one cannot model all thedata accurately and so the loss function must prioritize where to make errors.

Original: Such problems arecommon in the explainable AI literature [21, 10, 12] where predictive models must be interpretable.
Revised: Such problems are common in the explainable AI literature [21, 10, 12] where predictive models must be interpretable.

Original: Concretely, the aim is to choose the top B = 1 out of N = 50 resources using a linear model.

Original:  Predict:
Revised: • Predict:

Original: Given a feature x n ⇠ U [0 , 1] , use a linear model to predict the utility ˆ y of resource n ,where the true value is y n = 10 x 3 n  6 . 5 x n .
Revised: Given a feature x n ⇠ U [0 , 1] , use a linear model to predict the utility ˆ y of resource n , where the true utility is y n = 10 x 3 n  6 . 5 x n .

Original: Combining predictions yields ˆ y =
Revised: Combining predictions yields ˆ y =

Original: [ˆ y 1 , . .
Revised: [ˆ y 1 , . .

Original: , ˆ y N ] .
Revised: , ˆ y N ] .

Original:  Optimize: Choose the B = 1 (budget) resources with highest utility: z ⇤ ( ˆ y ) =
Revised: • Optimize: Choose the B = 1 out of N = 50 resources with highest utility: z ⇤ ( ˆ y ) =

Original: arg topk( ˆ y )
Revised: arg topk( ˆ y )

Original:  Surrogate: Because the arg max operation is piecewise constant, DFL requires a surrogate—we use the soft Top-K proposed by Xie et al.
Revised: [Table] • Surrogate: Because the arg max operation is piecewise constant, DFL requires a surrogate—we use the soft Top-K proposed by Xie et al.

Original: [27] that reframes the Top-K problem with entropyregularization as an optimal transport problem.

Original: This surrogate is  not convex in the predictions.
Revised: Although this surrogate is convex in the decision variables, it is not convex in the predictions.

Original: Web Advertising
Revised: Web Advertising

Original: This is a submodular optimization task taken from [25].
Revised: This is a submodular optimization task taken from Wilder et al.

Original: The aim is to determineon which websites to advertise given features about differentwebsites.
Revised: The aim is to determine on which websites to advertise given features about different

Revised: websites

Original:  Predict:
Revised: • Predict:

Original: Given features x m associated with some website m , predict the click-through rates(CTRs) for a ﬁxed set of N = 10 users on M = 10 websites ˆ y m =
Revised: Given features x m associated with some website m , predict the click-through rates (CTRs) for a ﬁxed set of N = 10 users on M = 5 websites ˆ y m =

Original: [ˆ y m, 1 , . . .
Revised: [ˆ y m, 1 , . . .

Original: , ˆ y m,N ] .
Revised: , ˆ y m,N ] .

Original: The task isbased on the Yahoo! Webscope Dataset [28] which contains multiple CTR matrices. To obtain thefeatures x m for each website m ,the true CTRs y m for the website are scrambled by multiplyingwith a random N ⇥ N matrix A , i.e., x m =
Revised: To obtain the features x m for each website m , true CTRs y m from the Yahoo! Webscope Dataset [28] are scrambled by multiplying with a random N ⇥ N matrix A , i.e., x m =

Original: A y m .
Revised: A y m .

Original:  Optimize: Given the matrix of CTRs, determine on which B = 2 (budget) websites to advertisesuch that the expected number of users that click on the ad at least once is maximized, i.e., z ⇤ ( ˆ y )
Revised: • Optimize: Given the matrix of CTRs, determine on which B = 2 (budget) websites to advertise such that the expected number of users that click on the ad at least once is maximized i e. z ⇤ ( ˆ y )

Original: =
Revised: =

Original: arg max z
Revised: arg max z

Original: [Equation]
Revised: [Equation]

Original: 1  [Equation] ˆ y ij )) , where all the z
Revised: , where all the z

Original: i2 { 0 , 1 } .
Revised: i

Original:  Surrogate: Instead of requiring that z i 2 { 0 , 1 } , the multi-linear relaxation from Wilder et al.[25] allows fractional values.
Revised: • Surrogate: Instead of requiring that z i 2 { 0 , 1 } , the multi-linear relaxation from Wilder et al. [25] allows fractional values.

Original: The DL induced by the relaxation is non-convex .
Revised: The DL induced by the relaxation is non-convex .

Original: Portfolio Optimization
Revised: Portfolio Optimization

Original: This is a Quadratic Programming domain popular inthe literature [7, 24].
Revised: This is a Quadratic Programming domain [ in

Original: The aim is to choose a distribution over N stocks in Markowitz portfolio optimization [17, 19] thatmaximizes the expected proﬁt minus a quadratic risk penalty.
Revised: which the aim is to choose a distribution over N stocks that maximizes the expected proﬁt minus a quadratic risk penalty.

Revised: We

Original: We choose this domain as a stresstest—it is highly favorable for DFL because the optimization problem naturally provides informativegradients and thus requires no surrogate.
Revised: choose this domain as a stress test—it is highly favorable for DFL because the optimization problem naturally provides informative gradients and thus requires no surrogate.

Original:  Predict:
Revised: • Predict:

Original: Given historical data x n about stock n , predict the future stock price y n .
Revised: Given historical data x n about stock n , predict the future stock price y n .

Original: Combining thepredictions ˆ y n across a consistent set of N = 50 stocks together yields ˆ y =

Original: [ˆ y 1 , . . , ˆ y N ] . We usehistorical price and volume data from 2004 to 2017 downloaded from  QuandlWIKI dataset [22].
Revised: We use historical data from 2004 to 2017 for a set of N = 50 stocks from the QuandlWIKI dataset [22].

Original:  Optimize: Given a historical correlation matrix Q between pairs of stocks, choose a distributionz over stocks that maximizes z T y   · ˆ y T Q ˆ y , where  = 0 .
Revised: • Optimize: Given a historical correlation matrix Q between pairs of stocks, choose a distribution z over stocks that maximizes z T y   · ˆ y T Q ˆ y , where  = 0 .

Original: 1 is the risk aversion constant.
Revised: 1 is the risk aversion constant.

Revised: More experimental setup details are provided in Appendix A.

Original: 5.1 Results
Revised: 5.1 Results

Original: We train either a linear model (for the Linear Model domain) or a 2-layer fully-connected neuralnetwork (for the other domains) using the learned loss functions LODL  ⇤ approach and compareit to the following baselines:
Revised: We train either a linear model (for the Linear Model domain) or a 2-layer fully-connected neural network with 500 hidden units (for the other domains) using LODLs and compare it to:

Original: Random :
Revised: Random :

Original: The predictions are sampled uniformly from [0 , 1] dim ( y ) .2.
Revised: The predictions are sampled uniformly from [0 , 1] dim ( y ) .

Original: Optimal : The predictions are equal to the true labels y .3.
Revised: Optimal : The predictions are equal to the true labels y .

Original: 2-Stage : The model is trained on the standard MSE loss ( 1 N[-P Nn =
-]
Revised: 2-Stage : The model is trained on the standard MSE loss ( 1 N[-P Nn =
-]

Original: || ˆ y n  y n || 2 ).4.
Revised: [Equation]

Original: DFL : Decision-focused learning using the speciﬁed surrogate. 
Revised: DFL : Decision-focused learning using the speciﬁed surrogate. 5.

Original: NN: To determine how important convexity is for LODL we deﬁne LODL 
Revised: NN: To determine how important convexity is for LODL we deﬁne LODL

Original: = NN  ( ˆ y ) inwhich NN  is a 4-layer fully-connected Neural Network (NN).
Revised: = NN  ( ˆ y ) in which NN  is a 4- layer fully-connected Neural Network (NN) with 100 hidden units.

Original: There are no guarantees aboutthe convexity or minima of the

Original: loss learned using this family.

Original: [Table]

Original: Table 1 shows the main results.
Revised: Table 1 shows the main results.

Original: We ﬁnd that, in all domains, training predictive models with LODLoutperforms training them with a task-independent 2-stage loss.
Revised: We ﬁnd that, in all domains, training predictive models with LODL outperforms training them with a task-independent 2-stage loss.

Original: Surprisingly, it also outperforms
Revised: Surprisingly, it also outperforms

Original: DFL which has the beneﬁt ofhaving handcrafted surrogates.
Revised: DFL, which has the beneﬁt of handcrafted surrogates, in two of the three.

Original: This is strong evidence in favor of ourhypothesis that we can automate away the need for handcrafting surrogates.
Revised: This is strong evidence in favor of our hypothesis that we can automate away the need for handcrafting surrogates.

Original: We now analyze theresults in terms of the domains:
Revised: We ﬁrst analyze the results in terms of the domains:

Original:  Linear Model: In this domain, the directionality of predictions is very important.
Revised: [Figure] Linear Model: In this domain, the directionality of predictions is very important.

Original: As we describe in Section 4.1, predicting values higher than the true utilities for the B resources withhighest utility, does not change the decision.
Revised: As we describe in Section 2.1, predicting values higher than the true utilities for the B resources with highest utility, does not change the decision.

Original: However, if their predicted utility is lower thanthat of the B  1 th best resource, the decision quality is affected.
Revised: However, if their predicted utility is lower than that of the B  1 th best resource, the decision quality is affected.

Original: As a result, we see that the“Directed” methods perform signiﬁcantly better than their competition.
Revised: As a result, we see that the “Directed” methods perform signiﬁcantly better than their competition.

Original: Web Advertising : In this domain, Wilder et al.
Revised: Web Advertising : In this domain, Wilder et al.

Original: [25] suggest that the decision quality is linked tobeing able to predict the quantity 
Revised: [25] suggest that the decision quality is linked to being able to predict the quantity P

Original: P j ˆ y ij (the sum of CTRs across all users j for a given website i ).
Revised: j ˆ y ij (the sum of CTRs across all users j for a given website i ).

Original: However, because the input features for every ˆ y ij are the same x
Revised: However, because the input features for every ˆ y ij are the same x

Original: i , the errors can be correlatedand so the sum can be biased.
Revised: i , the errors can be correlated and so the sum can be biased.

Original: As a result, the ability to penalize the correlations between twopredictions ˆ y ij and ˆ y jk is important to being able to perform well on this task—which results inthe Quadratic methods outperforming the others.
Revised: As a result, the ability to penalize the correlations between two predictions ˆ y ij and ˆ y jk is important to being able to perform well on this task—which results in the Quadratic methods outperforming the others.

Original: Portfolio Optimization : Given that this stress-test domain was built to be favorable to DFL , itoutperforms all other approaches with statistical signiﬁcance.
Revised: Portfolio Optimization : Given that this stress-test domain was built to be favorable to DFL , it outperforms all other approaches with statistical signiﬁcance.

Original: While the directed LODL methodsdo not outperform DFL, they nonetheless signiﬁcantly outperform 2-stage at p < 0 .
Revised: While the directed LODL methods do not outperform DFL, they nonetheless signiﬁcantly outperform 2-stage at p < 0 .

Original: We now analyze the results in terms of the methods:
Revised: We now analyze the results in terms of the methods:

Original: Our DirectedQuadratic LODL consistently does well:
Revised: Our DirectedQuadratic LODL consistently does well:

Original: In addition to consistently high expected values (always better than 2-stage), the associated variance is lower as well.2.
Revised: In addition to consistently high expected values (always better than 2-stage), the associated variance is lower as well.

Original: Lack of convexity can cause inconsistent results : While NN does well in the ﬁrst two domains,it fails catastrophically in the Portfolio Optimization domain.3.
Revised: Lack of convexity can cause inconsistent results : While NN does well in the ﬁrst two domains, it fails catastrophically in the Portfolio Optimization domain.

Original: DFL has a large variance in performance :
Revised: DFL has a large variance in performance :

Original: This is best illustrated by Figure 2b. Linear modelswith a positive slope pick the largest B values whereas those with a negative slope choose thesmallest.
Revised: In both the ”Web Advertising” and ”Linear Model” domain, DFL has

Original:  We can see that, while DFL generally does well (induces linear models with positiveslope), it can get stuck in local optima (induces stray lines with negative slope).
Revised: higher variation than the best performing LODLs. We posit that this is also because of the lack of convexity of the surrogates that DFL uses in these two domains.

Original: [Figure]
Revised: 5.2 Ablations

Original: We study the impact of the sampling strategy and number of samples on the performance of the
Revised: We study the impact of the sampling strategy and number of samples on the performance of the

Original: LODL methods in the Web Advertising domain.
Revised: LODL methods in the Web Advertising domain in Table 3.

Original: Varying the sampling strategy: In
Revised: We ﬁnd:

Original: Table 2, we ﬁnd that the the best sampling strategy is loss family-speciﬁc .
Revised: Sampling strategy (Table 3a): The best sampling strategy is loss family-speciﬁc .

Original: Speciﬁcally, NN and DirectedWeightedMSE perform best with the “2Perturbed” strategy, while the remaining LODL s perform best with the “All-Perturbed” strategy.
Revised: Speciﬁcally, NN and DirectedWeightedMSE perform best with the “2-Perturbed” strategy, while the remaining LODLs perform best with the “All-Perturbed” strategy.

Original: [Table] Varying the number of samples used: Table 4 (in the appendix) shows that all models performbetter with more samples.
Revised: Number of samples (Table 3b): All models perform better with more samples.

Original: The variance reduces as the number of samples increases (especiallyfor Quadratic), suggesting that better approximations of DL lead to more consistent outcomes.
Revised: In addition, the variance reduces as the number of samples increases (especially for Quadratic), suggesting that better approximations of DL lead to more consistent outcomes.

Revised: 5.3 Computational Cost of Learning with LODLs

Revised: We measure the time taken to learn predictive models M ✓ with LODLs in the Web Advertisement domain.

Revised: We train each LODL for 100 gradient descent steps using 5000 samples and train the predictive model for 500 steps (the same as the setup as Table 1).

Revised: We ﬁnd:

Original: 5.3 Correlation between  ‘quality’ of LODL and decision quality
Revised: Learning LODLs is parallelizable: Figure 3a shows that the cost of training a predictive model using LODLs decreases near-linearly in the number of cores used. With more than 20 cores, training with LODLs can be cheaper than training with DFL for this domain. If LODLs can be reused, their (already low) overhead quickly diminishes : From Figure 3b we see that even for a modest amount of amortization (over 5-10 predictive models), the training time using LODLs converges to that of two-stage (shown using parallelization with 40 cores). 5.4 Correlation between the ‘quality’ of LODL and decision quality

Original: Recall that LODL losses are ﬁt using a Gaussian sampling strategy centered around the true labels.
Revised: Recall that LODL losses are ﬁt using a Gaussian sampling strategy centered around the true labels.

Original: It is natural to ask how well this proxy loss correlates with the decision quality on test data.
Revised: It is natural to ask how well this proxy loss correlates with the decision quality on test data.

Original: We dothis by measuring the mean absolute error (MAE) of LODL relative to the ground truth decision lossfor points in the Gaussian neighborhood around the true labels.
Revised: We do this by measuring the mean absolute error (MAE) of LODL relative to the ground truth decision loss for points in the Gaussian neighborhood around the true labels.

Original: This Gaussian neighborhood is only an approximation of the true distribution of interest—the distribution of predictions generated by the predictive model that is trained using the LODL loss.
Revised: This Gaussian neighborhood is only an approximation of the true distribution of interest—the distribution of predictions generated by the predictive model that is trained using the LODL loss.

Original: Wecan measure the MAE on this distribution, which we call the empirical neighborhood .
Revised: We can measure the MAE on this distribution, which we call the empirical neighborhood .

Original: [Table]
Revised: [Table]

Original: Table 3 shows the results.
Revised: Table 2 shows the results for the Budget Allocation domain, while the remaining graphs are in Appendix B.3.

Original: All methods are able to approximate the DL comparably well in the Gaussian neighborhood, but this does not correlate well with decision quality.
Revised: All methods are able to approximate the DL comparably well in the Gaussian neighborhood, but this does not correlate well with decision quality.

Original: In contrast, the erroron the empirical neighborhood is tightly linearly correlated with decision quality.
Revised: In contrast, the error on the empirical neighborhood is tightly linearly correlated with decision quality.

Original: Furthermore, ifwe extrapolate the line of best ﬁt to where the MAE is 0, i.e., when there is no discrepancy LODLand DL , we ﬁnd that the trend predicts the normalized DQ would be 1 .
Revised: Furthermore, if we extrapolate the line of best ﬁt to where the MAE is 0, i.e., when there is no discrepancy LODL and DL , we ﬁnd that the trend predicts the normalized DQ would be 1 .

Original: This suggests that MAE onthe empirical neighborhood is a useful metric to target—see Section 6 for more discussion.

Original:  Discussion and Conclusion
Revised:  Discussion and Conclusion

Original: Our work proposes a conceptual shift from hand-crafting surrogate losses for decision problems toautomatically learning them, and demonstrates experimentally that the LODL paradigm enables usto learn high-quality models without such manual effort.
Revised: Our work proposes a conceptual shift from hand-crafting surrogate losses for decision problems to automatically learning them, and demonstrates experimentally that the LODL paradigm enables us to learn high-quality models without such manual effort.

Original: Nevertheless, our current instantiation ofthis framework has limitations which are areas for future work.
Revised: Nevertheless, our current instantiation of this framework has limitations which are areas for future work.

Original: First, to simplify the analysis, we considered LODL that additively decompose across the dimensions of y . Doing so allows us to isolate the effects of ﬁt to the decision loss from the generalizationperformance across the dimensions of y and leads to a conceptually simple pipeline.
Revised: We considered LODLs that additively decompose across the dimensions of y , allowing us to isolate the effects of ﬁt to DL from the generalization performance across the dimensions of y .

Original: Nevertheless, such LODLs require linearly many parameters in dim( y ) (and quadratically many for DirectedQuadratic). This scaling maybe undesirable in practice. However, we note that our main conceptual contribution—directly learning a loss for decision making—opens up the potential for futurework to learn models that generalize across dimensions, which could offer beneﬁts in scalability inaddition to LODL ’s current advantages in ease of use and model quality.
Revised: Future work may learn models that generalize across dimensions, allowing for even greater scalability.

Original: Second, in Section 5.3, we demonstrate that the ﬁt of a LODL to the empirical neighborhood aroundthe ground truth label is highly correlated with the eventual decision quality.
Revised: We demonstrate that the ﬁt of a LODL to the empirical neighborhood around the ground truth label is highly correlated with the eventual decision quality.

Original: However, this metric isnot accessible as a target during training—thus, we use the more tractable Gaussian neighborhoodinstead.

Original: While the Gaussian neighborhood method does yield models that perform well, it does notcorrelate well with the decision quality across LODL parameterizations.
Revised: While the Gaussian neighborhood method does yield models that perform well, it does not correlate well with the decision quality across LODL parameterizations.

Original: It would be valuable tostudy the empirical neighborhoodMAE to better understand the reasons for this discrepancy andpotentially develop LODL s with even stronger performance.
Revised: It would be valuable to study the empirical neighborhood to better understand the reasons for this discrepancy and potentially develop LODLs with even stronger performance.

Original: In summary, LODL provides an alternate framework for machine learning which informs decisionmaking, opening up new avenues towards models which are both high-performing and easily trained. 
Revised: In summary, LODL provides an alternate framework for machine learning which informs decision making, opening up new avenues towards models which are both high-performing and easily trained. Acknowledgments and Disclosure of Funding

Revised: Research was sponsored by the ARO and was accomplished under Grant Number:

Revised: W911NF-18-10208.

Revised: Wilder was supported by the Schmidt Science Fellows program.

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Original: [Yes] As supplemental material.(b) Did you specify all the training details (e.g., data splits, hyperparameters, how theywere chosen)?
Revised: [Yes] As supplemental material. (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?

Original: [Yes] In the appendix(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?
Revised: [Yes] In the appendix (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?

Original: [Yes]
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Revised: [Yes] (b) Did you mention the license of the assets?

Original: [No] It isn’t clear exactly what the license is, but all the datasets are publicly available.
Revised: [No] It isn’t clear exactly what the license is, but all the datasets are publicly available.

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