Decision-Focused Learning with Directional Gradients
Authors: Michael Huang
Vishal Gupta
Venue: NeurIPS 2024 poster (2024)
Keywords: decision-focused learning
Primary area: optimization
TL;DR: We propose a new decision-aware loss surrogate with theoretical guarantees on its accuracy with respect to the downstream decision loss.

Abstract: We propose a novel family of decision-aware surrogate losses, called Perturbation Gradient (PG) losses, for the predict-then-optimize framework. These losses directly approximate the downstream decision loss and can be optimized using off-the-shelf gradient-based methods. Importantly, unlike existing surrogate losses, the approximation error of our PG losses vanishes as the number of samples grows. This implies that optimizing our surrogate loss yields a best-in-class policy asymptotically, even in misspecified settings.  This is the first such result in misspecified settings and we provide numerical evidence confirming our PG losses substantively outperform existing proposals when the underlying model is misspecified and the noise is not centrally symmetric.  Insofar as misspecification is commonplace in practice -- especially when we might prefer a simpler, more interpretable model -- PG losses offer a novel, theoretically justified, method for computationally tractable decision-aware learning.

Decision-Focused Learning with Directional Gradients
Vishal Gupta
Michael Huang
USC Marshall School of Business Los Angeles, CA 90029
CUNY Baruch Zicklin School of Business
New York, NY 10010
guptavisQusc.eet
michael, haa.hilllhuanggbuucc.c..o cuny. edu
Abstract
We propose novel family of decision-awand surrogate losses, called Perturbation Gradient (PG) losses, for the predict-then-opimmiz framework. The key idea is to connect the expected downstream decision loss with the directional derivative of a particular plug-in in objective, and then approximate this derivative using zeroth order gradient techniques. Unlike the original decision loss which typically piecewise constant and discontinuous, our new PG losses is a Lipschitz continuous, difference of concave functions that can be optimized using off-the-shelf gradient-beat based methods. Most importantly, unlike existing surrogate losses, the approximation error of our PG losses vanishes as the number of samples grows. Hence, optimizing our surrogate loss yields best-in-class policy asymptotically, even in misspecified settings. This is the first such result in misspecified settings, and we provide numerical evidence confirming our PG losses substantively outperform existing proposals when the underlying model misspecified.
1 Introduction
We study the contextual optimization problem
π (X) E arg min (X)) (X sE2
where (X,Y) E X' x y are random variables, and C Rd is a known, potentially non-convex feasible region. We work in data-driven setting in which f" is unknown, but we observe i.i.d. draws {(x,, Y ; 1i==,..... I,...,n) n of (X, Y). Problem (T) models applications in which we observe potentially informative context X before selecting the decision =(X) such vehicle routing, portfolio allocation, and inventory management 7 32 Problem (T) has also been used as an "optimization layer" in neural network architectures model combinatorial decisions (26). Through suitable transformation. it can also represent some, but not all, nonlinear problems like personalized pricing (see Appendix A).
The predict-then-optimiiz framework focuses on plug-in policies for Problem 1 Given a function f & + Y, the corresponding plug-in policy is
* *(((X)) E arg min f(X)T
with ties broken by some pre-specified tie- breaking rule. Plug-in in policies are attractive because they separate the prediction procedure (f" from the optimization procedure (Problem ( This decoupling is especially useful when i) decisions must satisfy hard constraints (enforced by 2), or ii) one has specialized algorithm for solving instances of Problem (e-g., a custom vehicle-routing solver).
Given the form of ボ" natural approach might be learn an estimate f of f from the data, e.g.. by minimizing the mean-squared error, and then compute #(f(X). Such procedures are called decision-blind since they do not leverage Problem [] when learning f.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Figure 1: (Convergence under Misspecifi- cation). Excess regret normalized by opti- mal policy's ' performance under the misspec- ified setting of Section [.1 (a = 1. m 0). PGB is our proposed loss, ETO is a decision- blind approach that minimizes MSE. Other benchmarks include: DBB (26). FYL. [. and SPO+ 77]. Under misspecification, only the PG losses have vanishing excess regret. Error bars 95% confidence intervals the mcan over 100 trials.
Nornalized Exauss Regnat (99
60
40
DBB ETO
20
TYL
PGH
100
150
200
Training (e)
The seminal paper Elmachtoub and Grigas
7 argues decision-aware techniques be superior to decision-bling ones. Given a hypothesis class F S YAA they propose solving min Regret
where Regret(f) f) = E [Y (f(()) - E [Y (f)( This is equivalent to solving
minye 13 [e((f()), where ((t, )) = y at).
Growing empirical evidence supports the strength of decision- approaches
A challenge is that when Z is polyhedral combinatorial, + E(t. y) is piecewise constant, discontinuous map. Its gradient is either zero or undefined at all points. Hence. one cannot easily apply a first-order method like stochastic gradient descent (SGD) to optimize Problem (3).
In this paper we propose new family of surrogate losses to approximate E(1. y) by connecting ((t,)) v) to the directional derivative of a particular plug-in function and using zeroth order gradients to approximate this derivative. We call this family perturbation gradient (PG) losses. PG losses are Lipschitz continuous, general purpose, and only require black-box oracle which solves Problem (2). Most importantly, their gradients are "informative" (c.f. Lemma 2.2); after replacing with a PG loss, one can apply first order method to Problem
B or its empirica counterpart "out-of-the-box."
Previous authors have also proposed surrogates which satisfy some of these properties (see Sec- tion 1.2p. What distinguishes our work is that under mild assumptions on the distribution of (.X, Y). the error of our surrogate in approximating E(1, y) vanishes as n -> oo with a rate that depends on the complexity of F. More precisely, we prove that, for general 2, optimizing the empirical PGB loss (a particular member of the PG family) induces an excess regret over the best-in-class policy of at most Öp(VR" (( + n -1/2) where RR is the multivariate Rademacher complexity of F (Theorem 3.8 For linear hypotheses with dim(X)
this bound reduces to (dd(p//))2/4) When 2 is polyhedral, 1/2
optimizing empirical PGB loss induces an excess regret of at most Ö, (n_1/2 1/2//vll /v/log (2 where is VC linear subgraph dimension F and 2 are the extreme points of (Theorem 3.8 Both bounds vanish as xc, implying that optimizing our PGB loss yields a best-in-class policy asymptotically.
Critically - and this is the most distinctive feature of our work - our results hold even when f & F (misspecified setting). To our knowledge, these are the first result of their kind for differentiable surrogate. Existing results on the predict-then-opiime (the well-specified setting) and somewhat restrictive assumptions on the noise Section 1.2). These requirements are not simply weakness in prior analysis. As seen in Fig. []] existing methods can have very poor performance under misspecification. The key issue is that the justification for many of these losses facitly relies on the fact that optimal ʃ should be such that f(X)= R almost surely, but under misspecification, this is generally impossible. Hence, they do not well-approximate the decision loss l. See Fig. 2
framework [19 14 8) require f E F
fXXX (see
This poor performance is especially unfortunate, because misspecified settings are precisely those where decision-awant learning offers the most benefit over decision-blind approaches [8] 3 This is for at least three reasons: First, because the solution mapping - is piecewise constant, there may exist + f" such that # (f()) *(f*(x) almost surely [7] 34 (Indeed, this appears to occur Fig. [) Hence, one might still achieve (near) zero regret by learning over low-complexity F in decision-awand fashion, and, typically cow-compeextyy hypothesis classes F are preferred for tractability, interpretability, and strong generalization properties. Second, when every f must induce some error, decision-aware learning seeks an f(-) such that ((((X) disagrees with म ([["(() on regions of the covariate space X' that are not too costly in the decision-probles while decision-blind methods typically seek an f such that f(X) disagrees with f (X) on less probable regions of X
2

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1,00
1.00
DBB
DL
0.75
n=
0.75
LTR L
PYL.
0.50
PGB
SPO
30
PGB
0.25
D).
SPO+
Scaled
0.50
0.00
1.00
0.75
0.25
0.50
= 0000
0.25
0.00
0.0
2,5
5.0
7,5
10.0
0.0
2,5
5.0
7,5
10.0
Policy Parameter
Policy Parameter
Figure 2: (Comparing Surrogates under Misspecification). See Section 4.1 for setup (a = 1. m 0).
Benchmarks are decision-loss (DL) E, our PGB and PGC losses, Fenchel- Young Loss (FYL) (
SPO+ 7. and the learning-to-rane list loss ((21). Left-panel: (n 200) Only our PG losses closely track the DL. Right Panels: As n increases, the DL. and PG losses both become smoother.
3). Finally, 88 14| suggest traditional decision-bling learning strictly dominates decision- aware techniques in a speccfied setting (see Section I.2 i.c, decision-awe aware learning loses most of its advantages f Tre. F.
To be fair, the improved approximation quality of our PG losses comes the cost of computational complexity. Many existing surrogates convex if JF is linear class. We can optimize these surrogates over F in polynomial time. On the other hand, Elmachtoub and Grigas (7) shows that
solving the empirical counterpart of Problem (3) is NP-Hard (reduction from 0/1 classification). Thus, these aforementioned surrogates cannot be expected to reliably find best-in-class policies without additional assumptions on the data distribution unless P P= NP.
By contrast, our proposed PG losses are non-convex. expressible as the difference of concave functions. Optimizing such functions is well-studied [25 30 18), but is, in the worst-case, NP-Hard. This is to expected; if seek a method that finds best-in-class policy, it must contend with this hardness. Importantly, some NP-Hard problems admit algorithms that find high-quality solutions efficiently for most real-world instances. We argue our loss (combined with simple gradient descent type methods) yields such a problem. Previous authors [26) 34) have also proposed non-convex surrogates and shown that first-order methods still recover high-quality solutions.
Finally, we offer that convexity is often moot in applications. When using nonlinear hypothesis class (c.g.. neural network with more than layer), even convex surrogates induce non-convex loss functions. Optimizing these losses is theoretically no casier than optimizing our surrogate.
In summary, PG losses represent practically implementable and (the first) theoretically justified approach to decision-aware learning in misspecified setting.
1.1 Contributions
We propose new family of surrogate losses called Perturbation Gradient (PG) losses for the predict-then-optimiie approach to Problem (T). Our surrogates are Lipschitz continuous and can be expressed as the difference of concave functions.
We show that the gradient of a PG loss evaluated at a sample point is an unbiased estimate of the gradient of the expected loss (Lemma 2.2}. Thus, unlike the decision loss f, can apply first-order methods to minimize our expected surrogate or its ERM counterpart.
We bound the uniform approximation error our surrogates with respect decision loss by a term vanishing in (Theorems 3.4 3.4]am and 3.7). Thus, with more data, our loss becomes more accurate.
We prove that the empirical minimizer of our PGB loss yields best-in-class policy asymptotically, even if the underlying hypothesis class is misspecified (Theorem 3.8). To our knowledge, ours is the first surrogate for the peddictthen-optimii framework with such a performance guarantee.
We provide numerical evidence showing that minimizing our surrogate loss performs comparably other surrogates when the hypothesis class is well-specified, and substantively outperforms them when the hypothesis class is misspecified.
3

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1.2 Related Work
Elmachtoub and Grigas first proposed a convex, differentiable surrogate loss for Problem 3 called the SPO+ loss leveraging duality argument. Subsequent researchers have proposed other approaches to surrogate creation including replacing the plug-in policy Problem (2) with a regularized counterpart (32), creating a response-surface (28 [0]. training neural network to approximate with conjugate duality [.. A recent computational study [3] compares many of these approaches and found that SPO+ and the Senchel-Young loss of [ performed best or near-best on all benchmarks.
((t, y) panpaaaaetrrcllyy (34), linearizing ((t, y) [26]. and combining randomized-smootine
Despite the empirical strengths of decision methods, their theoretical justification is less clear. Few methods establish regret bounds. Wilder, Dilkina, and Tambe (32) and Berthet et al. (T] prove that gradients of partieular surrogates can be evaluated easily, but do not prove regret guarantee for the minimizer of those surrogates, On the other hand, El Balghiti et al. (6) and Hu, Kallus, and Mao [14) prove generalization guarantees relating E (Xe(s(X), Y)] to its empirical counterpart; hence, one finds an f E F with small empirical loss (X(((X),,) Y)) will also be small. But minimizing the empirical counterpart to Problem (3) computationally challenging. Jeong et al. (16) proposes a symbolic reduction scheme for this task. However, the method only applies to linear f and does not scale to large Most importantly, it integrated into neural architectures.
amenable to first-order methods, cannot be easily
The strongest known regret bounds are for the SPO+ loss in the well-specified setting (f E F). When the conditional distribution of Y| is centrally symmetric around its mean, Elmachtoub and Grigas (7 establish Fisher-consistency result. Liu and Grigas [1]] strengthen this result, establishing (under similar assumptions) that if the multivariate Rademacher complexity of F is O(n-1/2,, then the empirical minimizer of the SPO+ loss has regret at most Onn-1)))
That said, such results are perhaps unsatisfying because decision-bling methods typically dominate decision-ane -aais methods in well-specified settings. Hu. Kallus, and Mao (14) show that when f E F, the regret of a decision-bling -hiind approach that minimizes MSE converges to zero faster than the empirical minimizer of Problem (3). Said differently, decision-awar methods likely offer the most benefit in misspecified settings. Hence, these settings are arguably the most interesting.
Most closely related to our work are perturbation-based approaches for estimating out-of-sample performance. These works each use Danskin's theorem "debias" a naive estimate of out-of-sample
performance. Ito, Yabe, and Fujimaki (13) and Guo, Jordan, and Zhou [T]] each establish asymptotic
convergence of their estimators (without an explicit rate): Ito, Yabe, and Fujimaki (15) treats
non-contextual setting and focuses on the ERM estimator. Guo, Jordan, and Zhou causal inference setting. By contrast, Gupta, Huang, and Rusmevichientong [12 L3 establish a finite-sample regret guarantee, but in a small-data, large-scale data regime with nearly- Geasss-Ga corruptions. In this paper, we focus on the traditional large-sample regime (n + oo) with contexts. Moreover, instead of "debiasing," we perturbations to approximate a directional derivative which exactly represents our oft-of-sample loss.
treats a
1.3 Notation and Preliminaries
We write S b to mean that there exists universal constant C such that S Cb. We denote the & norm by || ||. To simplify the presentation, we also make the following boundedness assumption: Assumption 1.1 (Boundedness). There exists almost surely.
>> 0 such that maxzez |||||| s B, and ||||| < 1,
2 A New Family of Surrogate Losses
Define the plug-in policy objective:
V(I) min 1. iT =t aT())) 222
Evaluating (I) only requires a black-box oracle for Problem 2h. Since minimum of linear functions, (t) is concave.

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Our first key observation is that by Danskin's Theorem 2 Prop B.22],
(t Xy) A= y x(t) = ((t,g).
a
where the left side is a derivative if (t) is unique and a subgradient otherwise. We can form a family of PG surrogates by considering different zeroth order approximations the derivative the left
(see 20 24 for more on zeroth order gradients). We focus on two specific zeroth order gradients:
Backward Differencing (PGB): (t,t,)) v) (V(1) - V hh))
Central Differencing (PGC): (1, = 27 (V + bv) - V(t-hy))
for some user-defined h > 0. Intuitively, as h 0, both & (t, and f (t,y) should better approximate {(t, y). (We formalize the tradeoff in h below.)
For intuition the shape of PG losses, consider the special case where 2 = 1-1, and Y 1-1. Then, [(t, y) = sgn(ty), a step function. The PGB and PGC losses are both ramp losses in this case, where the width of the ramp is controlled by h.
Other zeroth order gradient schemes are possible. For example, forward differencing yields the surrogate from [26]. motivated from a different perspective. This alternate perspective sheds light on empirical performance. Indeed, our theoretical analysis suggests h should be small, tending to zero, while [26) advocates for large h. Our analysis also shows forward differencing suffers optimistic bias because it overestimates the derivative of a concave function. These features might explain the poor performance of [26] in [3]] benchmarks. We explore some of these issues in Appendix B but fully characterizing how the choice of zeroth order gradient affects surrogate quality is an open problem.
2.1 Properties of PG Losses
Using the structure of Problem
T.. we prove some key properties of our surrogates.
Lemma 2.1 (Basic Properties). Suppose Assumption I.I holds. For any I, 1' E TRd and E Y, the PG losses are
Lipschitz, i.e..
(t. v) (t', v)
I h
2B ||t Il and (e(t.)) - (t'( v) s Kllt +'|l-
b) Bounded, i.e.,
(1, v) B, and
(tc v) B.
c) Differentiable luy)).
ie.. (t, = (##(()= - -#(t-hy) and V (t, = 2 (+(1 h2y ( #(t
Finally, the backward difference upperbounds the true loss, i.e., E(t, v) S (t.
A primary advantage of our PG losses over the original loss l is that gradients are "informative." More precisely, because is discontinuous, V.E [E(t, Y)l A E ( Y)], and Vel(t, Y2) an unbiased estimate of V, [tt Y)]. Our surrogates do not have this problem.
Lemma 2.2 (Informative Gradients). Suppose Assumption
7.1 holds. For all t and Y,
V,ELC, (t, Y E e (t, (t,YY Y)]. Thus, VeCg(t. (t,Yg) Yÿ) is an unbiased estimate of VE [2 ()) The same statements also hold é
Lemma 2.2 ensures that we can apply first order methods
to optimize our PG losses.
Performance Guarantees
For brevity, we focus on the backward PG loss. Analogous results hold for the central PG loss.
Key Intuition. The key challenge is bounding the eтror between our PGB loss C) and the decision loss f. For intuition, consider the expected error fixed F. i.e., E ([ ((TYY). where
'These expressions are gradients when (t) and f + y) are unique optimizers, and elements of the Clarke subdifferential otherwise.
5

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= )X). Define the auxiliary function (() - [V + XY).. When #(T + XY) is unique,
Lemma E.1 justifies switching the derivative and expectation yielding
d
(A) =
V(T+
B +T +
da
where the last equality is Danskin's 's theorem [2
(T. Y)] (H - H - (0), i.e., the expected approximation enror equals the error in estimating the derivative of H.
Prop B.22]. Thus, E (T,
If H is not sufficiently well-behaved, this error may not be small. Lemma E.2 proves that H is 3-smooth, i.e., (A) is 3-Lipschitz, then this error is at most Bh. Since H entails expectation, we intuit that it should be smooth if (T, Y) has a "nice" density, similar to the intuition behind randomized smoothing.
To quantify what "nice" might mean, write
(A) H()
([ E E A(T +TT +X)]|-
Since (t, H A(T + is B-bounded by Lemma 2.1 the last difference is at most where (V.,,) is the total variation distance between the two random vectors. Hence, a "nice" density is any density such that distributions of (Y, XY) and (Y, XY) close whenever À and À are close. We expect this generally occurs whenever (T admit Lipschitz continuous densities, but can be shown to fail if, e.g., concentrated at single point.
AY),
We make the above intuition formal in the next section.
3.1 Expected Approximation Error
We make the following assumption:
Assumption 3.1 (Lipschitz Log Conditional Density). Let g(- Y) be the conditional density of f(X) Y. We assume that there exists constant L >> 0 such that log g(: f ) is L-Lipschitz for all F and all almost surely.
As discussed above, Assumption 3.1 sufficient to ensure the requisite TV distance is small, but not necessary. We prefer Assumption 3.1 it facilitates short proof. Under this assumption, have: Lemma 3.2 (Expected Approximation Error). Suppose Assumptions 1. and 3.1 hold and 1 Then, for any f E F, 0 <
YI S (e 11)B L h.
3.2 Uniform Error Bounds
Combining Lemma 3.2 and Hoeffding's inequality, yields pointwise bound: Corollary 3.3 (Pointwise Approximation Error). Fix some f E F. Suppose Assumptions 1.1 and hold and h < I. Then, for any 0 < 5 < 2. with probability at least 1 - 8,
E (L(F( - & BLh Bv/lo((1/0//n.
As seen in Lemma 2.1 the Lipschitz constant of 2 scales like 1/h. Henee, unlike other learning methods, /1 does not control a bias-variance tradeoff;; rather h controls bias-computational com- plexity tradeoff. Practically, we suggest taking as large as the next largest term in the bound, i.e. O(n 1/2) above, to maximize the smoothness without compromising the rate.
Corollary 3.3 captures the key ideas of our approach, but insufficient to establish a regret guarantee; need uniform error bound. To that end, we prove two results:
Our first generalization bound applies to any choice of Z. We leverage the Lipschitzness of 2 (Lemma)) 4.) to apply a vector contraction inequality from Maurer (22 and bound the Rademacher complexity of our sample surrogate loss. A similar strategy is used in [19
More specifically, define the multivariate Rademacher complexity
RR (F)
E 5r (F)
E sup 1 n of
(5)
where o; (o.1..... dta) and oie are i..dd Rademacher random variables. Then, have

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Theorem 3.4 (Uniform Error Bound for General 2). Suppose Assumptions 1.7 and 3.1 hold. For any 8 and 0 < h .. with probability at least
SuP E th x)
S BLh = B b (F) + B (0gT))+B/Bg()))))))
(og(1//)/m
If dim(X) and Fis a linear class, (F) O(Vdp/n) [6]
Choosing ((dp/n))//)
yields an error of size (p((dp/)))//) This is same rate as Liu and Grigas but also holds the misspecified setting where f F.
Theorem 3.4 applies general 2, but may loose. We next present stronger result when Z is polyhedral by leveraging results from Hu, Kallus, and Mao [14] based on VC dimension:
Definition 3.5 VVC-Linear-Subgraph Dimension). The VC-linear-subgraph dimension of a class of functions J S ) is the VC dimension of the sets F = (1, ()
[(x) S 1 f E F)
in X' Rd+1, that is, the largest integer v for which there exist 21, 21..... Tv E X, ...... By E Rd.
E R. ..... tu E IR such that { (I (3) f (x²) <tz} :j -....., v) E F) =
We make the following assumption:
Assumption 3.6 (Bounded VC Dimension). The VC-linear-suubraph dimension of the class F
{f [z, y) f() + hy, for E F, R is at most v.
We obtain the following bound for polyhedral 2, where 2 is the set of extreme points of 2.
Theorem 3.7 (Uniform Error Bound for Polyhedral 2). Suppose Assumptions 1.7 3.7 and 3.6 hold.
For any 0 < 8 < I and 0 < h < I. with probability at least 1 5,
SuP
1 E 2R - YOI| & BLh + BV logg( [ 22 2z| +1) log(1/8)
[(((s()))
Choosing h = O(n 1/2) yields a bound of size (112) which matches the generalization error of from [14] 6). Thus, for polyhedral 2, our surrogate converges no slower than the empirical loss, but is more computationally tractable.
3.3 Excess Regret Bounds
We next transform the uniform bounds of the previous section to bounds on excess regret. Define
EReegre(f)) = E ((((()))) - E TTTT(FORR) (x)). where jOR E argmin Regret(f).
Excess regret measures regret relative to the best-in-class policy foR, not the full-information optimum S" For fixed h < +. define the empirical minimizer of PGB loss fn. argmin JEF jEF 1 Σ ii / (f(X), Y2). Then, we have the following:
Theorem 3.8 (Excess Regret Bounds).
i) Suppose the assumptions of Theorem 3.4 hold. Then, ERegret(ff) S (BAL + B
Vn
ii) Suppose the assumptions from Theorem 3.7 hold. Then, FRegret((h) S By (1og((22+t)
For many hypothesis classes, the multivariate Rademacher complexity is vanishing in n. Hence, both bounds are vanishing in n and fn achieves best-in-class performance asymptotically.
Numerical Experiments
We compare learning a linear hypothesis class with our PG losses (PGB and PGC) to surrogates currently implemented in the PyEPO Python package (31). Specifically, we benchmark against: SPO+ 77. DBB [26). FYL. (T]. and the family of LTR losses [2T]. Additionally, we also benchmark against b'decision-blin" two stage policy that first minimizes the least-squares loss and then implements

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the corresponding plug-in policy (ETO). We optimize each surrogate using ADAM via the PyEPO framework. All methods are trained for a total of 100 epochs. and we select the best model found in those 100 epochs based on validation set of size 200. For PG losses, we initialized at the SPO+ solution and choose from a small grid of values based on validation set performance. Future computational experiments might explore the effect of alternate initializations. We do not add additional regularization or smoothing to any of the surrogates. See Appendix C implementation details.
for other
Our metric of interest is the normalized excess regret (E Y (x) / xY)x(())))
where we have normalized by the optimum policy (c.f. Problem (T) for interpretabilityy.
4.1 Simple Misspecification Experiment
In our first experiment, we let 2 = =00, 1}. We let X ~ Unif(0, 2) and
(x) [-4r + 2 (m(r - 0.55) - 0.2, for x E [0.55. 2]
for x E [0, 0.55)
The function piecewise linear with one piece that has a slope of -4 and another piece with a slope of m [0. -4] (an elbow). The change point is at = 0.55 where the two functions meet at -0.2 (see Fig. | in Appendix [D). Intuitively, m controls the degree of misspecification; at m -4,
f E F and the problem is well-specified. m O, the problem is maximally misspecified.
We generate synthetic data Y (X) fa. We define e vā 0.5) + vI a'y where E [0, 1]. Ç is an exponential random variable with mean 0.5, and 7 ~ .N 0.25). By construction e mean-zere noise with variance 0.25. The value of + 0, controls how asymmetric the noise is.
Note, when a * 0, the theoretical performance guarantees on SPO+ from [9 do not apply.
that is, the most misspecified setting with the most asymmetric noise c. Beyond highlighting the superior performance of the PG losses in misspecified settings, Fig. also shows the choice of finite difference approximation (backward or central) also impacts performance. Intuitively, central differencing likely outperforms backward differencing because in standard, deterministic settings, central finite differencing has error Ohh)) relative to the true derivative, while backward differencing has error ((h) (17). This intuition can be made formal in our setting by adapting Lemma 3.2 but omit the details for brevity.
Results. Figure 1 plots the relative regret for 0 and
30%
30%
Nemmiited Excess Regret
PYB
Noemmstzed Exsos Regret
PGC
20%
20%
10%
10%
PG2
Level Misspccifiationn (m))
Asymmetry (o)
Figure 3: (SPO+ Comparisons) The left figure plots the excess regret normalized by the optimal policy's performance as we vary m for we vary for = 200. When = 0 the noise is centrally symmetric and when a = 1 the noise is the most asymmetric. Error bars are 95% confidence intervals on the mean over 100 trials.
80 and 1. The right figure plots the same value as
The left panel of Fig. 3 in Appendix studiee the effect of increasing degrees of misspecification. We limit the benchmarks to ETO and SPO+ as other methods are qualitatively similar. We find that (as argued in the introduction), in well-specified settings (m -4), the benefits of decision-aware learning may be small. All methods (including decision- ETO) achieve low regret, even for small In our experiments, even for 20 the relative regret less than 0.6% across all methods.
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On the other hand, as the degree of misspecificatiion grows, decision recision-aware aware methods have an advantage. However, we see that SPO+ nearly as susceptible as misspecification decision- -lind approaches since the relative regret also increases rapidly. By contrast, the relative regret for our PG losses increases more slowly. We stress, this experiment fixes n. As + oc, our theory shows the regret of the PG losses tends to best-in-class as in Fig.
The right panel of Fig. studies how the noise distribution impacts the relative regret since all prior known performance guarantees for SPO+ require strong assumptions on the noise 7 19 The plot suggests that requiring a symmetric noise not simply weakness in the analysis of SPO+, but fundamental to the method. As the noise becomes less symmetrie, the performance of SPO+ degrades. Even when the assumption is satisfied (a 0). we SPO+ is still significantly impacted by misspecification. By contrast, the PG losses perform similarly as the shape of the noise varies.
4.2 Shortest Path Experiments
Random Are Costs. We first replicate the shortest path experiment from graph. We let X [O, Is) and for each edge j. and take
31 on 5 5 grid
1
[J/ x), +3) +1
fz(z)
xB
3.56
where 8* E 10, 10x5 has i.d. Bermouli(005)) entries (drawn once and fixed throughout). We consider two different data generation mechanisms: i) Multiplicative noise, i.c. Y, = fj (1) where Ej are i.d Unif[ 33 .3.. This choice closely mirrors the original experiment 7 ii) Additive Gaussian noise, i.e., Y, = [ (X) + E where E, ~ (0, 0.3²).
Figure 8 in the Appendix D compares the PG losses to the best two surrogates in our experiments, FYL [I] and SPO+ [7]. Here PGF represents zeroth order gradient using forward differeneing and is equivalent to the method of (26) but with a small / as opposed to large h. Despite the non-convexity, minimizing our PG losses with first order methods yields comparable performance to FYL and SPO+ (convex methods). In other words, they do not seem to get stuck in local minimima. For small we do seem some distinction, which likely because our losses are less smooth (see the right figure of Fig. 2
Harder Example with Planted Ares. Because are costs completely random in the previous example, there are likely many paths with near-optimal length. We next consider a harder instance where we hide unique good path.
Gaussian, Noise-Halfwide 0.3
Uniform. Koise-Halfwidde = 0.3
y
Normalized Excess Rogret (%)
2.Stogs I.R
PCC
200
400
800
1600
200
400
800
1600
Training Size (8)
(a) Safe and Risky Path
(b) Performance
Figure 4: Harder Shortest Path. a) One of the two planted paths will be optimal depending on value of Xg. All other ares strictly worse. b) Normalized Excess Regret we vary the training samples. Error bars are 95% confidence intervals on the mean over 100 trials.
Specifically, we now take X R where X ~ (O0 Is) and X, ~ Unif(0, 2). Fig. 4a we have safe (red) path and risky (blue) path. For red ares, (x) = 2 for all æ. For the blue ares (risky

---

Normalized Expected Regret
0.7
Figure 5: (Portfolio Optimization) We plot the excess regret normalized by optimal pol- iey's performance as vary the number of training samples. Error bars are 95% confi- dence intervals on the mean over 100 trials.
0.6
2-Stage I.R
FYL
SPO+
PGB
PGC
0.5
200
400 Training Size (n)
800
1600
path), fz(x)) 4xg
if S z6 $ 0.55 and fj (26) 2.2 otherwise. For all other ares, we take
(z(x) 3.56
(55 1 x), (B*
+ + 2.2,
which is similar to previous experiment but shifted up by 2.2. Consequently, either the red path or the blue path is optimal, depending the value of Xg. The observed Y values generated as before by adding either multiplicative uniform or additive Gaussian noise. A good method thus must first identify these two paths as the best options (despite the noise) and choose between them (by learning the relationship to X6). In this harder setting, PG losses offer significant benefit. Figure in Appendix D shows this performance is relatively robust the choice of h.
4.3 Portfolio Experiment
We study the same portfolio optimization problem as 28 12 Fama French Industry Sector Portfolios from the Fama French Library (9]. These portfolios are indices representing monthly returns of different asset classes and realistically mirror the asset allocation problem faced by wealth managers. We sample month t at random from the last 10 years, and let Y = be the return the = 12 indices, and let X = Ft-1 + WO 0.5Σ) (p = 12) where E is the covariance of r. over those 10 years. The additional noise lowers the signal-to-noise ratio while maintaining the correlation matrix X and makes the problem harder.
but use real data, specifically the
As one can see in Fig- because of the low signal-to -ignse-t--ooie ratio, all methods induce significant regret to the optimum, but both PGB and PGC are notably stronger.
Conclusion
In this paper we proposed novel family of surrogate losses for the predict-then-optimize framework that can be optimized using off-the-shelf gradient methods. Most importantly, the approximation error of these surrogates vanishes - Hence, optimizing our surrogate yields best-in-class policy asymptotically, even misspecified settings. Our PG losses are the first proposed surrogates with this property and substantively outperform other methods in misspecified settings.
The family of PG losses arises from different approaches to approximating a derivative. As mentioned, an interesting open question is identifying the best-possible choice of approximation. We also believe that better understanding the role of h in trading off between bias and computational complexity might shed light improve algorithms and tuning procedures.
10

---

Acknowledgments and Disclosure of Funding
The authors have no competing interests disclose. The authors would like to thank Hamsa Bastani. Osbert Bastani, Adam Elmachtoub, Paul Grigas, Ziyu He, and Angela Zhou for feedback on an initial draft of this manuscript. VG was partially funded by the Institute for Outlier Research in Business.
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12

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Figure (Comparing Zeroth Order Gradi- ents), PGC, PGB, and PGF all approximate the decision loss (DL), but PGB is a pes- simistic bound, while PGF is optimistic bound. Here optimism causes PGF to choose the wrong policy.
PGF
PGC
-02
PGB
-1.0
-0.
0.0
0.5
1.0
8b
Appendix Supplemental Material
A
Reformulating Nonlinear Problems
Through an appropriate transfomation of variables, some nonlinear optimization problems can be rewritten in the form Problem (T), and, thus, are amenable our approach.
Consider the problem
x" (X) E argmin ("))
=EZ
where f"(X) E E X] and g(-) fixed, known, vector-valued function. This problem is equivalent to the problem
min f(x)²
s.t. Z {g(2) :2 2}.
which is of the requisite form for analysis. Moreover, our algorithms only require access an oracle which can compute # (J(()) E argminge s(X)== for any f. Often, this is accomplished by solving x (ff()) argmin.ez f(X) and then returning (f()))))
Gupta, Huang, and Rusmevichientong (see Example 2.3 of their paper).
12 this reduction to model personalized pricing problem
B Comparing Zeroth Order Gradient Schemes
In this section we provide a brief comparison of the forward differencing scheme to backwards and central differencing. The key distinction is that since V--) is concave, forward differencing creates a surrogate that optimistically underestimates the true loss (forward differences underestimate the derivative of concave functions) whereas backward differeneing pessimistically overestimates the true loss. Some authors 4 33 have shown that pessimism can improve learning, and we observe a similar phenomenon.
Figure an illustration. We consider the same misspecified data setup as Section 4.1 (a 1, m 0) and take 200. We plot the decision loss (DL) f, and our PGB, PGF, and PGC losses, for the plug-in class = (-0.1x + Bo Bo E [-1,1]] 1]}. Because PGF optimisticaly underestimates loss, it suggests the policy Bn .1, which actually induces significant regret. By contrast, backwards differencing pessimistic and suggests the policy 3n = .98 which is essentially optimal. Central differencing is neither optimistic pessimistie, but still suggests good policy Bo .99.
C Implementation Details
For our numerical experiments we leverage the PyEPO framework which was developed using PyTorch. For our experiments, we utilize Adam with learning rate 0.01 to optimize the training losses. We run Adam over 100 epochs with batch size of 32 for each surrogate loss, For non-PG loss
13

---

surrogates we use the recommended parameters provided by PyEPO. For our PG losses, we tune by validating with hold out set of training 200 samples. We note that similar results were obtained by validating against the training decision loss. Additionally, we initialize the PG losses at the SPO+ solution.
To compute the expected regret, we generated test set of 10000 samples and use it to estimate the relative regret described in Section
Some our experiments were run on a high performance computing cluster administred by the University of Southern California's ' Center for Advanced Research Computing (CARC). The cluster facilitated multiple simulation runs of the experiments. However, a significant portion of the experi- ments in the paper (that did not require multiple Monte Carlo runs) were Macbook Pro with an Apple M3 Max Chip with 96 GB Memory.
D Additional Figures
Well-Spocifled (m
Misspocified
2.5
Figure 7: (Synthetic Data Generation from
(X,, Y.) m 4 (left) and m = 0 (right). Red line is (X) for cach setting.
Section 4.1 Observations
-5.0
-7.5
0.0
0.5
1.0
1.5
0.0
0.5
2.0
X
Gaussian, Noise Halfwidth Noise-Hallwidhh
0.3
Uniforn, Noise- Halfwidth Noise-Halfwidte 0.3
Normalized Excess Regret (%)
I
E
SPO+ PGC
Training Size (x)
Figure 8: (Shortest Path, Random Are Costs) Excess regret normalized by optimal policy's 's perfor- mance vary the number of training samples. Error bars 95% confidence intervals on the mean over 100 trials.
Omitted Proofs
E.1
Proof for Lemma 2.1
Proof. We first prove (a), the Lipschitz property. We first claim V(.) B Lipschitz, since
v()) V(s)
1 (1) -s =T()))
( (#(1) #s)) (l 8) #s))
<0, by optimality #())
s || - |||
B It a||.
14

---

n
ep_type
h
avg
std
ep_type
avg
std
800
normal
0.001
0.050
0.008
800
normal
0.001
0.006
0.006
800
normal
0.035
0.049
0.008
800
normal
0.035
0.005
0.006
800
normal
0.188
0.048
0.008
800
normal
0.188
0.004
0,006
800
normal
0.434
0.049
0.013
800
normal
0,434
0.005
0.011
800
unif
0.001
0.074
0.009
800
unif
0,001
0.007
0.007
800
unif
0.035
0.073
0.009
800
unif
0.035
0.006
0.007
800
unif
0.188
0.072
0.010
800
unif
0.188
0.006
0.008
800
unif
0.434
0.076
0.017
800
unif
0,434
0.009
0.015
1600
normal 0.001
0.048
0.005
1600
normal
0.001
0.003
0.002
1600
normal
0.025
0.047
0.004
1600
normal
0.025
0.003
0.002
1600
normal
0.158
0.046
0.004
1600
normal
0.158
0.002
0.002
1600
normal
0.398
0.048
0.012
1600
normal
0.398
0.003
0.010
1600
unif
0.001
0.070
0.006
1600
unif
0.001
0.004
0.003
1600
unif
0.025
0.070
0.006
1600
unif
0.025
0.004
0.003
1600
unif 0.158
0.068
0.006
1600
unif 0.158
0.002
0.003
1600
unif 0.398
0.071
0.015
1600
unif 0.398
0.005
0.011
(a) Validation Performance
(b) Normalized Excess Regret
Figure 9: (Dependence on h, Planted Shortest Path Experiment.) We compare the performance of the policy learned by the PGB loss for different values of h across the 100 runs. a) Shows performance on Valiation set. For ease of comparison, we have scaled the validation performance and presented (Eiee Y, (#))) # ((f(((,))) (x;)))) / (Etal Y; (X.))) b) Shows performance out of sample. This out-of sample performance is relatively flat h, suggesting the precise choice of h does not matter much in this example.
where the last inequality follows from Assumption I.I proving V is B Lipschitz.
A symmetric argument holds for V(s) - (t)
Returning to (t, w). write
V(() V(L hy) V(t) - VV('' hy)
(t, v) -2h 22)(',,))
h
(t) - -V(t)I
[V hy) - V hy)
h
h
2B It t'l
An entirely analogous argument holds for ég (t., y)- We next prove (b), the boundededness property. Write
&x (t. v) VV() - V(L hy))
B Bllhyl
B
[|vl|
Again, an analogous argument holds for (t, v). This completes the proof for (b)
The proof of (c) follows directly from applying Danskin's Theorem To prove (d), we see
Prop B.22].
V(t hy) - V(I) y #(t)
C (t.v) e (t.
(t + hy) T *(t hy) (t hy) +(t)
0
where the last inequality holds by optimality of #(t + hyy). Rearranging proves the result for (d).
15

---

E.2
Proof of Lemma 2.2
Proof. We apply the dominated convergence theorem. Let 6 IR be the tth coordinate vector. Then,
1 (( - (1((,))))
dr E (t,
lim E 8
(6)
Let W/ 3[C, + 6, (t, Then, by the Lipschitz property of Lemma 2.1 [Wl I 2B
h
and limg-so Wa = 0e (t, Y) almost surely. The result then holds for the hth partial derivative of from the dominated convergence theorem. Since i was arbitrary, it holds for all 1,...,d d, and thus holds for the gradient. An analogous proof holds for Eg.
E.3 Auxiliary Lemmas from Section
Lemma E.1 (Interchange Derivative for H). Suppose Assumption 1.1 holds and that the optimizer (T + XY) unique almost surely. Then (A) = (
Proof. We use the bounded convergence theorem. Write
H(A - 8) H(A)
(A(A)
lim 6-+0
5
V(T (A + 8)V) V(T AY)
lim 6
6
Because V is B-Lipschitz, theorem we can interchange the limit and expectation yielding,
((T+(A+0))-V((TTA < ||l| < 1. By the bounded convergence
+ (1+))))) -(T
(A)
lim 6-50
6
Since (T + XY) is unique, Danskin's 's theorem Prop B.22] confirms V(T + XY) is differentiable, and the above inner limit converges to the derivative ax V + XY).
Lemma E.2 (Error of Backward Finite Difference). Suppose H is differentiable on [A - h, Al, and 3-smooth. Then,
H'(A) (H(()) H(A - 11)) < Bh.
Proof. By the mean value theorem, (H((() H(A - h)) H' h) for some 0 E h < h. Thus,
H' 1 1 (HT(2) H(^ 1n)) = (T - H' - Ti) S 3h,
by 8-smoothness. Upper bounding h by h completes the proof.
E.4 Proof for Lemma 3.2
Our first observation is that the error in our surrogate is bounded by the solution stability of the policy, A similar bound is used in Gupta, Huang, and Rusmevichientong 13 in a different context:
Lemma E.3 (Solution Stability Bounds Error). For any f, y. h,
0 (t, E(t, v) < y (f(tt hy) +()))
Solution Stability
In words, solution stability measures how much the policy changes given small perturbation by. No-
tions of stability appear throughout the machine learning literature and are fundamental to learnability [29]. Lemma E.3 relates the error of our surrogate this fundamental quantity. We stress the relation holds for any t, h, y.
16

---

Proof. The first inequality was proven in Lemma 2.1 For the second, note that Vtt) = t (t)..
Hence by rearranging,
(1.1)) - -((,,)) H (V(()) V(L hyy) y (()
(1T((()) - #(t -by)) + y (#(t - hy) - -()))
Sy (f(t - -hy) =x(t),,
by the optimality of ftt).
To bound the expected approximation error in Lemma 3.2 we require the following elementary result:
Lemma E,4 (Density Ratio Bound). Suppose Assumption 3.1 holds. Then, for any t, t' such that ||e 1'll S I/L, we have
g(V:J,Y) g(t;f.V) -1 s (e 1)1 1'1
Proof. Let g(t) g(t;) By the convexity of the exponential,
exp(x) s 1+ (e 1)x vo 5 and
exp(x) 2 + x Va.
(7)
Let st)) log.ggg)))... (t). Then,
g(t')
log
s(t') s(t) s Lllt' t| g(t)
Taking the exponential of both sides and subtracting 1, we have
g((")
S exp (T.llt' +|]) S (e - 1- +|]-
g(t)
where the last inequality follows from Eq. have,
7 and our assumption that ||t t'|l S 1/L. Similarly, we
g(t')
log
N LIlt' t|l
9(1)
9('')
2 exp (- -lll' (|])
(())
2 -1ltt - t|l
2 -(E - 1)L(((' 41l
Hence,
s(())
S (e - 1)Lllt' - t||-
g(t)
This completes the proof.
Proof of Lemma 3.2 Let T'
f(X). Condition on Y and let g(t) = g(t:f.Y)
Then, by
Lemma
we have
0 < E
< SEE - -
17

---

We bound this last quantity as follows:
E Y (+(T hY) - x(T))
(8)
= / ((U) - - - g(t)) #(()ddt
/ Y ((t) + hY) g(t)) dt
Σ / | x(()
ga(t + +hY) g(t) dt
s / g(t)
g(t +hY hY)
dt
g(t)
$ (e 1)BLI /
g(t: f, d)
$ (e - B)h
Taking the expectation over Y completes the proof.
E.5 Proof for Theorem 3.4
Proof. We bound the uniform errоr as follows:
sup fEF
(I(X)), ((e(((X)),
sup fEF
(f(x)) YJ E (f(G.)))
CA (f(x), Xi) e ((f(x)).
+ sup fEF
71
(2k)
We first bound (i). Let
A RSL (F) =E 18
E
E
Σ ( fEr
sup
By Lemma 2.1b b. 0 < BY) S Hence, we can apply a standard Rademacher complexity
2B
result (23 Theorem 3.3J to show for any 0 < < $. with probability at least 1 - ô, the following holds for all f e F simultaneously:
12
E 2 (x)
Y.)
n (f(,))) Y. +B
1 3
(F) + log
2B
2B
We can apply an identical argument to -(X))) to obtain a similar lower bound. Combining the two inequalities and taking the union bound, we have that with probability at least 25, the following holds for all f E F simultaneously:
R (S(X)) Y.) E (I(.)), Yd]
< 4B (F) 2B,
log
8
We next bound Ri (F) by applying Corollary 4 of Maurer 22 show
9y (F) = E sup fEr
Σ
(((x))))) <
12
B E
sup fer n
o. [(x))
v2 f rh
B
(F (F).
h
Here have used the Lipschitz constant from Lemma 2.1
18

---

Substituting this bound above and collecting constants shows that with probability at least 1 5,
B
log (1/6)
-
(F)++
BV
h
Finally, we use Lemma 3.2 bound (ti). Combining proves the result.
E.6 Proof for Theorem 3.7
Proof.
We develop an alternative decomposition of the uniform error. Write
(f) -EI((X)))
& -
-
(9)
π
+
E -L(U())))))
Consider (1). We write
E. -
(+( - - -(F((x)))
(Tx(((X)) - AY) E -(f(()))))))) - -xr]]
(Tx(((x)) [((())))
E [.. (#(S(x)) NY. - +(((X)))
2 sup
-Y)) - =((()))))))----)) AYAI
h
E Yr. (=()(X,) -
- - +(((x.)))
where the first inequality applies the triangle inequality, the second inequality applies Lemma E.3 and the last inequality combines similar terms by taking the supremum over h.
Applying this bound in Eq. (9) shows
X) - (((())))) XiIl
sup SEF
< 3 sup fer
Σ
YTt(ff(x, Y2) E ax YDDI
1=1
(a)
Σ E I. (*(s(x)) NYA - #(S(x.)))
+ sup fEF
1=1
(8)
where we recall that
F= (f f(x,)) = f(x) by for f F, R).
19

---

Component (a) is bounded using Theorem and Theorem 2 of Hu, Kallus, and Mao 114 that with probability least
showing
8,
log (|3 + log((5)))
(a) S B
n
Component (6) bounded by Eq. proves the result.
8 in the proof of Lemma
3.2 Combining (a) and (b) components
E.7 Proof of Theorem 3.8
Proof. Both proofs follow the same general strategy. We start with the first statement. Let
(n) Σ & (I(X)) and = (f(x), Y)
Since the fo minimizes (f) over F and fOR minimizes L(f) over F, we see,
I(f)) L(JOR) = L(fo) Ln + (n, Ln + Ln L(FOR)
< (n L (5O0 +2 sup (n) L())
JEF
50, by optimality of
< 2 sup [n((f) L(f)l feJ
where the first inequality holds by taking the supremum of the first two and last two pairs, and the second inequality holds by optimality of f. Taking the expectation of both sides, we see
ERegret (b) S 21E
sup [n) - -LOD fEF
To compute the expectation, we see by Theorem 3.4 and choosing h / Ho (F) that
1
sup [n((f) - L(f) v (F) + BV FEF
log
(10)
n
8
with probability at least 8. Rearranging, we have
nt²²
P sup fer
[L.()) L()) VBLL 21 S exp
B²
By tail integration over and adding back /BBL (F), it follows that
B sup Ln. - S (F Vn
(Regret(fo) S 21E
JEF
completing the proof of the first statement.
We now proceed to the second statement, We follow the same line of argument until Eq. 10 Then, we instead use Theorem 3.7 with h = LVn S I to obtain
log (|3z log01/0)
sup (n) L(f) < C FEF
n
for universal constant Co with probability at least - 6. Rearranging we have
( log (12, (12
nt?
sup
- S exp (-C
/EF
Applying the tail integral gives
v log (|32) +)
(Regret(fo) < 2
sup fEF
S BV Ln(f) L(BI
n
completing the proof.
20

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Reviews:
[{"review_id": "PWvzJRIpoo", "reviewer": "Reviewer_TiWp", "summary": "This paper proposes a family of \"perturbation gradient\" losses for Predict-than-Optimize (PtO) that, if optimized for, can lead to best-in-class performance, even under model misspecification. On the theoretical side, this paper provides risk bounds that build on past theoretical work in PtO + the literature on using perturbation-based approaches for estimating out-of-sample performance. Importantly, it shows that the excess risk goes to zero when the number of data points $n \\to \\infty$. On the empirical side, they show that their loss functions outperform others from the literature in one synthetic domain and perform comparably to other loss functions in one domain from a popular benchmark.", "questions": "Could you address my comments in the weaknesses section?", "limitations": "The paper does an okay job of addressing the limitations of the proposed approach.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "* The paper identifies and addresses an important problem in the PtO literature--performance guarantees under model misspecification.\n* I'm not a theoretician, but the theoretical results seem non-trivial and relevant to practice.\n* On the empirical side, they compare to a reasonable set of baselines from the literature.", "weaknesses": "I have reviewed this paper in the past, and my two major issues were (a) that it overclaimed and (b) it had weak empirical analysis. While the paper has improved significantly on both fronts, I still have some gripes:\n* _Regarding Claims:_ While the paper has added a paragraph about the approach's limitations in the introduction, I'm not sure that I understand it. My issue with the approach is that if the _true_ loss is non-convex in the policy parameters, it will be hard to optimize for, even in the limit of infinite data when $n \\to \\infty$. This isn't the same as the issue you describe for small $n$ in Figure 2(b), or even the statistical complexity issue of cleverly choosing $h$ as discussed in Section 3.2. It's that if the true loss $\\ell(t(\\theta), y)$ is piecewise constant, then for small $h$ the PG-loss $\\hat{\\ell}^b_h$ will indeed be close to $\\ell$ but that means it will be close to piecewise constant and, as a result, hard to optimize for. The current theory assumes that you can optimize for $\\hat{\\ell}^b_h$ but not $\\ell$, but that's a big assumption that should be discussed. Using the analogy of the ramp loss from the introduction, if your initial prediction is sufficiently far from $t = 0$, the gradient of the ramp loss will still be 0 (same as the $sgn$), and you won't be able to use first-order methods to learn a good model $\\hat{f}$. Perhaps this is something that comes under your buckets of (a) the difficulty of optimizing for a \"difference of convex functions\" in the introduction or (b) the bias-complexity tradeoff in choosing $h$ that you allude to in the conclusions, but I couldn't immediately see the connection. Could you talk more about this?\n* _Empirical Evaluation:_ I appreciate that you have included $SPO+$ and $PFYL$ as baselines, and also added a PyEPO domain. However, the results aren't conclusive even under significant model misspecification (e.g., you do no better than PFYL in either case, even when PFYL has no guarantees under misspecification, and SPO+ does roughly the same for uniform noise). Given that you've implemented one domain, running tests on the other domains in the benchmark should be fairly easy. Have you run those experiments? What do they look like?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### **Overview**\nWe’d like to recall it is NP-Hard to optimize the (true) decision-loss over linear functions [6], essentially because it generalizes binary classification. Hence, any method (including ours) that aims to learn a best-in-class policy for all data generation mechanisms MUST also be NP-Hard.  That said, theory aside, not all NP-Hard problems are created equal. Some, e.g., knapsack or bin-packing, admit “practically efficient” algorithms that solve most real-world instances in reasonable amounts of time. In contrast, others (e.g., non-metric TSP) are so hard we do not even have reliably good heuristics for large-scale instances. What we are trying to argue in our paper is that although optimizing our surrogate is NP-Hard (as it MUST be), even simple gradient descent algorithms recover very high-quality local minima that are suitable for applications.\n\n### **Specific Questions Re Claims**\nYou are correct that part of the difficulty is that far away from the “heart” of the function, the true loss is flat, and, hence, our loss is flat. To be more concrete, let’s focus on the ramp loss. Basically, for a single data point $(X_i, Y_i)$, if $T(X_i)$ is more than $O(h)$ from zero, both losses will be flat. This challenge is shared by other losses, e.g., the losses proposed in [30, 24].  \n\nAs you’ve also observed, this issue connects both to a) the difference-of-convex function representation and b) the bias-complexity tradeoff of choosing h. \n\n#### _The Bias-Complexity Tradeoff_ \nAs above, in the case of the ramp loss, if we have a single data point $(X_i,  Y_i)$ and if $T(X_i)$ is more than $O(h)$ from zero, the loss is flat. However, when we have $n$ data points, the empirical loss is only flat if we are more than $O(h)$ away from ALL data points. Hence, for large $n$, our loss is unlikely to be flat in regions of high data density. Moreover, the larger we make $h$, the less likely we’ll end up in a bad region during gradient descent (provided we initialize at a “smart” point, see below). One of the takeaways of our theoretical analysis is characterizing precise conditions and how large we can make $h$ (to minimize the chances of reaching a flat region) while still guaranteeing a good enough approximation to learn the best-in-class policy, and how this should scale with $n$.\n\n#### _Difference of Convex Function Representation_ \nWe represent our loss explicitly as a difference of convex functions. This means we know A LOT about the structure of the loss landscape.  DC optimization is a growing field [F1], [F2],  and there are recent works on how to smooth DC functions [F3], [F4], [F5] to improve computational performance, and even new algorithms for identifying smart starting points for multistart gradient ascent [F6].  (These could serve as the aforementioned “smart” starting points to ensure your algorithm doesn’t get stuck in the flat parts far away from the “heart” of the function.)  By contrast, the loss landscape of the original decision loss is much less understood, and so it’s less obvious how to optimize it directly (even heuristically).\n\n#### **Summary**\nWe are not trying to mislead; we ARE replacing one NP-Hard optimization problem with a different NP-Hard optimization problem. However, in practice, some NP-Hard problems admit algorithms that find high-quality solutions very efficiently for practical instances, and we argue our loss leads to one such problem. Trying to quantify this “improved tractability” is necessarily subtle.  \n\n#### _Addendum on Non-Convexity_ \nFinally, we’d politely point out that the issue of convexity vs. non-convexity is often moot in applications. When using a nonlinear hypothesis class (e.g., a neural network with more than 1 layer), even surrogates like SPO+ and PYFL induce non-convex loss functions. These more powerful hypothesis classes are often preferred in practice, and, for these settings, optimizing these losses is theoretically no easier than optimizing our surrogate.  \n\n### **Specific Questions Regarding Empirical Evaluation**\nThank you for pushing us.  Based on feedback from you and the review team, we have added two experiments: 1) A Harder instance of Shortest Path where we've hidden good paths in the network and 2) A Portfolio Optimization Example with low signal to noise ratio.  In both of these \"harder\" settings, simple gradient descent procedures (ADAM) on our loss recovers local minima that substantively outperform benchmarks.  ***See Global Rebuttal Document.***  Obviously, leveraging tools from the DC literature, one might be able to further improve upon these solutions.\n\nFinally, in addition to the observed empirical benefits of our methods, we stress that existing \nbenchmarks have NO theoretical guarantees in misspecified settings. Offering a theoretically justified surrogate for misspecified settings is interesting in its own right.\n\n\n - F1: https://link.springer.com/article/10.1007/s11081-015-9294-x\n - F2: https://link.springer.com/article/10.1007/s10107-018-1235-y\n - F3: https://arxiv.org/abs/2104.01470\n - F4: https://ieeexplore.ieee.org/document/9304514\n - F5: https://www.sciencedirect.com/science/article/pii/0022247X91901875\n - F6: https://pubsonline.informs.org/doi/abs/10.1287/ijoc.2022.1238", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "elTxyubv3A", "reviewer": "Reviewer_CTUs", "summary": "This paper addresses the predict-then-optimize problem by proposing a new family of surrogate loss functions. The key motivation is derived from Danskin's Theorem, which connects the expected downstream decision loss with the directional derivative of a particular plug-in objective. This objective is then approximated using zero-order gradient methods. The paper includes numerical experiments conducted on both a synthetic environment and a shortest path problem.", "questions": "1. How should the parameter h be selected in practice? Have you empirically studied how it affects performance?\n\n2. In the shortest path experiment, why does FYL perform so well? Theoretically, the PG losses should be superior to FYL. Could you provide insights into this discrepancy?", "limitations": "I would suggest the paper add a limitation section.", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The paper is well-motivated.\n\n2. The properties of the proposed surrogate loss are thoroughly derived.\n\n3. Theoretical analysis shows that the approximation error of the proposed loss diminishes as the number of samples increases. Consequently, it can outperform existing surrogate losses even in misspecified settings.", "weaknesses": "1. The experimental section of the paper is relatively weak.\n\n2. While the proposed method performs well in a simple synthetic environment, it does not demonstrate a clear advantage over FYL and SPO+ in the shortest path problem.\n\n3. More experiments are needed to illustrate the advantages of the proposed method in real-world scenarios.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### **Re Weaknesses: Experimental Evaluation** \nThank you for pushing us in this direction. In the Global Response Document, we’ve added two additional experiments: i) a harder instance of a shortest-path problem and ii) a portfolio optimization problem with **real data** and a low signal-to-noise ratio. In both cases, our method has an advantage over all baseline methods. \n\n#### ***New, Harder Shortest Path Instance***\n\nThe difference between this new shortest path and the original shortest path instance is the data generation. In the original shortest path instance (following [6] and others), the costs of the arcs are exchangeable with respect to the network. There’s no special relationship between these costs and their location in the network. (For example, the edges of the square aren't systematically more expensive than internal roads.)  Consequently, many candidate paths have similar costs, and the problem is arguably not too difficult. That is why many baselines perform similarly.  \n\nIn our new instance, we generate the arc costs in a way that depends on the network. Specifically, we first embed two “good” paths along the diagonal (a safe one and a risky one) (see Global Response Doc), then ensure that any other path has a high cost, and then add noise to try and hide the “good” paths and confuse the safe and risky one. This is a more challenging setting because each method must first identify the good paths, and then choose between them to do well. As you can see, our approach has an edge in performance for large enough n.\n\n#### ***Portfolio Optimization***\nWe study the same problem as [6, 26, 32] but use **real data**, specifically the 12 Fama French Industry Sector indices from the [Fama French Library](https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html). These indices represent returns of different asset classes and realistically mirror the asset allocation problem faced by wealth managers. We sample a month $t$ at random from the last 10 years, let $Y = r_t$ be the return of the $d=12$  indices, and let $X=r_{t-1} + \\mathcal N(0, 0.5 \\Sigma)$ be the previous month return plus Gaussian noise ($p=12$). Here $\\Sigma$ is the covariance of $X$.  The additional noise lowers the signal-to-noise ratio while maintaining the correlation matrix of $X$.   See Global Response document.  \n\nBecause of limited computational resources, we only present the strongest benchmarks (SPO+, FYL, 2Stage PtO, and our method).  We again see that we have a distinct advantage.\n\nWe're happy to include these additional experiments in a galley version to strengthen the empirical evaluation of the methods.\n\n#### ***Aside: On Value of Empirical Evaluation***\n\nFinally, even if the reviewer feels our method performs comparably to existing benchmarks, it should be stressed that those benchmarks have NO theoretical guarantees in misspecified settings. We believe offering a theoretically justified surrogate for this setting (even with comparable performance) is interesting in its own right.\n\n### **Question: How to select $h$?**\nWe selected $h$ using hold-out validation set of size $200$ in our experiments.  In general, we found the method insensitive to choice of $h$ as long as it was reasonably small. Please see plot on Global Response Doc. \n\n### **Question: Why does FYL perform well in (original) Shortest Path?**\nThis is a great question.  Our current conjecture is that it is a combination of two features:\n1. Since arc costs are exchangeable across the network, there are many good candidate paths, and FYL is finding one of them.  \n2. By looking at its gradient, we argue that FYL essentially searches for a policy $T(\\cdot)$ such that $\\hat \\pi(T(X)) \\approx \\hat \\pi ( Y) $.  Notice, this isn't the same as the oracle optimality condition, which would seek a policy such that $\\hat \\pi(T(X)) \\approx \\hat \\pi (f^*(X))$. \n\nThese two conjectures informed our new (harder) shortest-path example where we i) embed only two good paths (risky and safe,) so that to perform well, a method must identify these two paths among all other paths and choose between them and ii) Lower the signal-to-noise ratio so that $Y$ is further from $f^*(X)$, and, hopefully, $\\hat\\pi(Y)$ is more distinct from $\\hat\\pi(f^*(X))$.  As seen in Global Response doc, this does seem to affect FYL's performance.\n\nWe also note that FYL performs surprisingly poorly in our portfolio allocation experiment. (See Global Response doc)\n\n\n### **Limitations**\nWe discuss limitations on pg. 3 Line 76. We’re happy to label this discussion more clearly as with a section heading saying “Limitations” in a gallery proof if it would help it to stand out.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "T3LleHgv33", "reviewer": "Reviewer_LgZs", "summary": "This paper considers a predict-then-optimize framework for solving contextual optimization problems, in particular for the case where the set of decisions is combinatorial or polyhedral, or when the loss is non-differentiable. They define a family of surrogate losses that connect the loss to the directional derivative of a plug-in function, and use zero-th order gradients to approximate the derivative. Simple numerical examples are provided.", "questions": "Can you please address the issues above?", "limitations": "Limitations are not addressed. Potential negative societal impact is negligible.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper is well written and considers a principled approach.", "weaknesses": "The role of $h$ in the approximations of $\\ell$ appearing in Theorem 2.1 should be made clear -- $\\hat\\ell^b$ and $\\hat\\ell^c$ have not been defined. Please give details of the proof of part c).\n\nThe main weakness is that the method appears to work well on a very simple synthetic problem. The performance appears to deteriorate on a slightly more complex problem, and then no interesting example is provided. I feel that for this outlet, this is a major weakness and would like to see how the method performs on a real world and/or large-scale problem. There are a number of such examples in the cited supporting literature.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### **Clarifying Minor Weaknesses**\nWe believe the reviewer meant Lemma 2.1, as there is no Theorem 2.1.  \n\n#### **Role of $h$**\nThe role of h is intuitively described on top of pg. 5 Line 154 (i.e. before Lemma 2.1).  This is further elaborated (quantitatively) after Corollary 3.3 (pg. 6 Line 214).\n\n#### **Definitions of Key Surrogates**\nThe quantities $\\hat \\ell^b$  and $\\hat \\ell^c$ are defined on the bottom of pg. 4 (line 152), which occurs just before Lemma 2.1. \n\n#### **Proof of part c**\nWe are happy to add the details of part c) in a gallery version. Here are the details for you to verify:  (As an aside, there are many versions of Danskin’s Theorem under different regularity conditions.  [This version](https://statisticaloddsandends.wordpress.com/2022/11/10/what-is-danskins-theorem/) is sufficient for the proof below.\n\nFrom the definition of $\\hat \\ell^b(t, y)$,  we have\n$\\nabla \\hat{\\ell}^b(t, y) = \\frac{1}{h} ( \\nabla_t V(t) - \\nabla V(t - hy) )$. \n\nWe next evaluate each of the \"gradients\" on the right using Danskin’s Theorem. We say \"gradients\" because, as we will show, these are gradients when $\\hat \\pi(t)$ and $\\hat \\pi(t -hy)$ are unique, and are subgradients otherwise.  First, we validate the conditions of Danskin's Theorem for the first gradient.  Specifically, \n$V(t) = \\max_{z \\in \\mathcal Z} \\langle t, z \\rangle$.  \nThe function $\\phi(t, z) = \\langle t, z \\rangle$ is continuous and differentiable by inspection, and $\\mathcal Z$ is compact by assumption.\nThus, the conditions of Danskin’s theorem are met, and \n$\\nabla_t V(t) = \\hat\\pi(t)$, where the left side is a gradient if $\\hat \\pi(t)$ is unique, and a subgradient otherwise (see part iv of above mentioned reference).   \n\nWe can treat $\\nabla_t V(t - hy)$ in a similar fashion. Combining proves part c) for $\\hat \\ell^b$.  The proof for $\\hat \\ell^c$ is similar.\n\n### **Re Weaknesses: Experimental Evaluation** \nThank you for pushing us in this direction. In the Global Response Document, we’ve added two additional experiments: i) a harder instance of a shortest-path problem and ii) a portfolio optimization problem with **real data** and a low signal-to-noise ratio. In both cases, our method has an advantage over all baseline methods. \n\n#### ***New, Harder Shortest Path Instance***\n\nThe difference between this new shortest path and the original shortest path instance is the data generation. In the original shortest path instance (following [6] and others), the costs of the arcs are exchangeable with respect to the network. There’s no special relationship between these costs and their location in the network. (For example, the edges of the square aren't systematically more expensive than internal roads.)  Consequently, many candidate paths have similar costs, and the problem is arguably not too difficult. That is why many baselines perform similarly.  \n\nIn our new instance, we generate the arc costs in a way that depends on the network. Specifically, we first embed two “good” paths along the diagonal (a safe one and a risky one) (see Global Response Doc), then ensure that any other path has a high cost, and then add noise to try and hide the “good” paths and confuse the safe and risky one. This is a more challenging setting because each method must first identify the good paths, and then choose between them to do well. As you can see, our approach has an edge in performance for large enough n.\n\n#### ***Portfolio Optimization***\nWe study the same problem as [6, 26, 32] but use **real data**, specifically the 12 Fama French Industry Sector indices from the [Fama French Library](https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html). These indices represent returns of different asset classes and realistically mirror the asset allocation problem faced by wealth managers. We sample a month $t$ at random from the last 10 years, and let $Y = r_t$ be the return of the $d=12$  indices, and let $X = r_{t-1} + \\mathcal N(0, 0.5 \\Sigma)$  ($p=12$) where $\\Sigma$ is the covariance of $r_t$.  The additional noise lowers the signal-to-noise ratio while maintaining the correlation matrix of $X$ and makes the problem harder.  See Global Response document.  \n\nBecause of limited computational resources, we only present the strongest benchmarks (SPO+, FYL, 2Stage PtO, and our method).  We again see that we have a distinct advantage.\n\nWe're happy to include these additional experiments in a galley version to strengthen the empirical evaluation of the methods.\n\n#### ***Aside: On Value of Empirical Evaluation***\n\nFinally, we stress that the benchmarks above have NO theoretical guarantees in misspecified settings. We believe offering a theoretically justified surrogate for misspecified settings is interesting in its own right, beyond its empirical evaluation.\n\n\n### Limitations\nLimitations are discussed on pg. 3 Line 76. We’re happy to label this discussion more clearly with a section header “Limitations” in a gallery proof it would help it to stand out.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "IPGMNUhtCB", "reviewer": "Reviewer_oPp7", "summary": "The paper introduces a new family of surrogate losses to the DFL with linear costs, called perturbation gradient losses (PG loss). It provides theoretical analysis to bound the approximation errors and regret bounds and uses extensive experiments to demonstrate the advantages of the proposed method.", "questions": "- Lemma 2.2 Can you develop the result when $\\ell_h^b$ is nonsmooth?\n- Line 176, what is $Y_j$?", "limitations": "Not applicable", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "- The proposed loss is more efficient than the standard DFO loss. It only requires the computation of the optimal decision variable without the need for implicit differentiation through $\\hat{\\pi}(t)$.\n- The approximation error of the surrogate functions decreases as the number of samples increases, which is advantageous for large-sample applications.\n- In the numerical experiment section, the PG losses perform very well even in the misspecification case.", "weaknesses": "- The (sub)differentiability of the PG loss is not well-established. In Line 172, the differentiability is doubtful. I don’t think the subgradient is well defined when $\\hat{\\pi}$ is not unique. The paper should make more efforts to develop the theory rigorously and in writing.\n- In Line 148, the directional derivative is not well-defined. If $V(t)$ is non-smooth, you should take infimum over all the $\\hat{\\pi}(t)$.\n- The paper presents a continuous relaxation of the DFL objective, with the error bounded by the finite difference error h. However, the resulting problem is still nonconvex and nonsmooth, which may not be efficiently solved in theory. It seems more effective to use direct smoothing by random perturbation or by adding a regularization term (e.g., [30]). I encourage the authors to compare their approach with other smoothing methods in theory.\n- Both experiments are for discrete decision variables and simulated data.  The paper does not sufficiently address the performance of the proposed method when decisions are continuous.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "### **Danskin's Theorem**\nThere are many versions of Danskin’s Theorem that apply under different regularity conditions. The version we use is well-summarized [here](https://statisticaloddsandends.wordpress.com/2022/11/10/what-is-danskins-theorem/).  The key is part 4:  When there are multiple solutions, the theorem holds but with “gradient” replaced by “subgradient.” An intuitive example tis $f(t) = \\max_{-1 \\leq z \\leq 1} t z = | t |$. Then, at $t = 0$, the set of optimal solutions is $[-1, 1]$ (non-unique), and thus no derivative. Still, there are subgradients, and any optimal solution gives a subgradient. \n\nGiven your excellent questions we intend to add a statement of Danskin’s Theorem at the end of Section 1.3.\n\n### **Addressing Weaknesses (1-2)**\n1. As in part 4 of reference above, Danskin’s theorem holds when $\\hat \\pi(\\cdot)$ is not unique. Since $\\hat \\pi(\\cdot)$ is AN optimal solution, it is a subgradient of V(t).  See also footnote 1 of pg. 5 of our paper.  \n\n2. Again, we are invoking part 4.  The “grad”  operator should be interpreted as a directional gradient if \n$\\hat \\pi(t)$ is unique and a Clarke Differential otherwise.  \n\n### **Addressing Weakness 3: Comparison to Smoothing and Regularization**\n#### ***NP Hardness*** \nOptimizing decision-loss (DL) generalizes binary classification and is NP-Hard [6]. Hence, ANY method (including smoothing and regularization) that aims to learn a best-in-class policy must either suffer similar computational challenges as our method or else fail to recover a best-in-class policy.\n\n#### ***Regularization [30]***  In [30], the authors \ni) relax combinatorial constraints,\nii) smooth by introducing a regularized policy class $\\hat \\pi^\\rho(t) \\in \\arg\\min_{z \\in \\mathcal Z} \\langle t, z \\rangle + \\rho || z ||^2$ \niii) approximately optimize the decision-loss over these regularized policies \niv) implement the non-regularized $\\hat \\pi(t)$ using the parameters learned from the regularized $\\hat \\pi^\\rho(t)$. \n\nThey provide no theoretical analysis of the approximation error. (For combinatorial problems, intuition suggests the error from the first step might be very large.) Even in very simple settings -- take binary classification $\\mathcal Z = [-1, 1]$, $Y \\in \\{-1, 1\\}$ --  their loss can have large flat regions and is non-convex. Hence step iii) does not appear (to us) to be any less challenging than our optimization problem. \n\nThe method of [30] is also computationally expensive. Evaluating gradients in step iii) entails solving a QP. By contrast, our method only involves solving the nominal LP. In our Shortest-Path example, there are specialized algorithms that solve that LP problem EXTREMELY fast (e.g. a vectorized Dijkstra’s algorithm), but we do not know of any specialized algorithms for solving the QP to high accuracy. Generic solvers are orders of magnitude slower.\n\n#### ***Randomized Smoothing*** \nWe have already compared to a convexification of a randomized smoother (the Fenchel-Young Loss from [1]).  As seen in Fig. 2 and the new experiments (See Global Response Doc) our method can outperform this method empirically because it does not guarantee learning a best-in-class policy.  \n\nOn the other hand, the reviewer may have instead intended the “DPO” procedure from [1] (see also [29] in Section 3.4.3.), which replaces $\\hat \\pi(t)$ with a smoothed version \n$\\mathbb E_\\xi[\\hat \\pi(t + \\sigma \\xi)]$ and then attempts to solve the decision-loss with this smoothed version.  [1] provides no theoretical guarantees on performance. Moreover, one can check the resulting loss is still non-convex (take $\\mathcal Z = [-1, 1]$, $Y = \\{-1, 1\\}$) . In THEORY, its gradients are Lipschitz (an advantage).  We say in theory, because in practice, the method uses monte carlo (see “Practical Implementation” on pg. 5 of [1]) with a small number of samples, and when replacing the expectation with a finite sum, the smoothness is lost.\n\nThus, it is not clear that this optimization is better behaved than our proposal. Numerical experiments from [29] suggest that the performance is poor because of this monte carlo sampling. They write in Section 6 “Notably DPO was not shown because its overall subpar performance,” i.e., it was omitted from all plots.\n\nFinally, if the reviewer's concern is our lack of Lipschitz gradients, note that since we represent our loss as a difference-of-convex functions, we can “out-of-the-box” leverage [existing results on smoothing for DC functions](https://arxiv.org/abs/2104.01470). These results relate stationary points, and the global minimum of the smoothed function to the original function.\n\n**To summarize**\n - NP-Hardness shows that ANY approach to this problem must either solve a hard optimization problem or cannot consistently learn best-in-class policies. We argue that although the problem is NP-Hard, our method finds high-quality, sub-optimal solutions.  \n - Regularization approaches [30] introduce significant computational challenges, are still non-convex, and do not induce Lipschitz gradients.\n - We have already compared to the Fenchel Young Loss in the paper (a convexified randomized smoothing approach) and found this loss might not find a best-in-class policy (Fig 2 and Global Response Doc).\n - Direct randomized-smoothing of the policy (DPO) induces a non-convex loss and does induce Lipschitz gradients (in theory).  However, prior work suggests that the challenges of Monte Carlo outweigh this benefit, and the method has poor performance. \n - We can combine our method with existing approaches for smoothing DC function to recover Lipschitz gradients out of the box.\n\n### ***Weakness 4: Additional experiments***\n - Fig. 1 has a continuous region, synthetic data\n - Shortest path experiments are combinatorial, synthetic data\n - *NEW* portfolio experiment (continuous region), real data\n\n### Questions:\n1. Yes, the lemma already holds in this case (see our previous response on Danskin’s theorem).\n2. Typo: Should say $Y$, not $Y_j$", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
Attached is our global response document, which includes the following:

i) An updated shortest path experiment that embeds two "good" paths that methods must identify and choose between based on the context. This experiment increases the difficulty and reward of finding the oracle policy compared to the initial shortest path experiment. This allows us to show our PG Losses can learn better policies compared to existing surrogate benchmarks.

ii) We highlight how the choice of h affects the learned policy for the shortest path problem and see that for our choices it had minimum effect as long as h was sufficiently small. 

iii) We introduce a new portfolio optimization experiment that was generated with **real** data. Our formulation follows existing benchmarks [6, 26, 32].  We again consider a linear objective of maximizing returns, but our feasible set is constructed with both a quadratic constraint and linear constraint. We plot the relative regret (lower is better) and show our approaches significantly out-perform existing benchmarks.

Meta-review: This paper examines the prediction-then-optimize problem and introduces a novel Perturbation Gradient (PG) loss for decision-aware learning. The approach is innovative and shows promising performance compared to existing methods. The authors' rebuttals have successfully addressed most of the reviewers' questions.

For the camera-ready version, the authors should incorporate the following improvements: 
- Incorporate the additional experimental results.
- Expand on the discussions regarding the complexity of optimizing the PG loss, as discussed with reviewers TiWp and oPp7.
- Discuss the choice of the $h$ parameter.
- Clarify the sub-differentiability issue raised by reviewer oPp7.
- Address the initialization for optimizing the PG loss, given its non-convex nature. The appendix notes that the SPO+ solution is used for initialization. This should be clarified in the experiment section, and the impact of different initializations (e.g., random initialization or initialization at the ETO or FY solutions) should be empirically demonstrated.
Decision: Accept (poster)