Gen-DFL: Decision-Focused Generative Learning for Robust Decision Making
Authors: Prince Zizhuang Wang
Shuyi Chen
Jinhao Liang
Ferdinando Fioretto
Shixiang Zhu
Venue: ICLR 2026 Poster (2026)
Keywords: decision focused learning
decision making
stochastic optimization
generative models
operational research
Primary area: optimization
TL;DR: Our paper presents a novel decision-focused learning framework that leverages generative modeling to solve robust decision-making problems.

Abstract: Decision-focused learning (DFL) integrates predictive models with downstream optimization, directly training machine learning models to minimize decision errors. While DFL has been shown to provide substantial advantages when compared to a counterpart that treats the predictive and prescriptive models separately, it has also been shown to struggle in high-dimensional and risk-sensitive settings, limiting its applicability in real-world settings. To address this limitation, this paper introduces Decision-Focused Generative Learning (Gen-DFL), a novel framework that leverages generative models to adaptively model uncertainty and improve decision quality. Instead of relying on fixed uncertainty sets, Gen-DFL learns a structured representation of the optimization parameters and samples from the tail regions of the learned distribution to enhance robustness against worst-case scenarios. This approach mitigates over-conservatism while capturing complex dependencies in the parameter space. The paper shows, theoretically, that Gen-DFL achieves improved worst-case performance bounds compared to traditional DFL. Empirically, it evaluates Gen-DFL on various scheduling and logistics problems, demonstrating its strong performance against existing DFL methods.

Published as a conference paper at ICLR 2026
GEN-DFL: DECISION-FOCUSED GENERATIVE LEARN- ING FOR ROBUST DECISION MAKING
Prince Zizhuang Wang
Shuyi Chen
Heinz College
Heinz College
Carnegie Mellon University
Camegii Mellon University
princewang@ cmu. edu
shuyic@andrew. cmu cmu.edu. edu
Jinhao Liang
Ferdinando Fioretto
Department of Computer Science University of Virginia
Department of Computer Science University of Virginia spq7wpêvi rgi iigiiii... edu
njs4malvirginit a. edu
Shixiang Zhu
Heinz College
Carnegie Mellon University
shixiangzhu@en cmu, edu
ABSTRACT
Decision-focused learning (DFL) integrates predictive models with downstream optimization, directly training machine learning models minimize decision rors. While DFL has been shown to provide substantial advantages when compared counterpart that treats the predictive and prescriptive models separately, it has also been shown to struggle in high-dimensional and risk-sensitive settings, limiting its applicability in real-world settings. To address this limitation, this paper introduces seciision-focusee generative learning (Gen-DFL) L), novel framework that leverages generative models to adaptively model uncertainty and improve decision quality. Instead of relying fixed uncertainty sets, Gen-DFL, learns a structured representation of the optimization parameters and samples from the tail regions of the learned distribution to enhance robustness against worst-case scenarios. This approach mitigates over-conservatism while capturing complex de- pendencies in the parameter space. The paper shows, theoretically, that Gen-DFL achieves improved worst-case performance bounds compared to traditional DFL. Empirically, evaluate Gen-DFL various scheduling and logisties problems, demonstrating its strong performance against existing DFL methods.
INTRODUCTION
Decision-making under uncertainty arises in many real-worie applications, including supply chain management, energy grid optimization, portfolio management, and transportation planning Sahinidis
(2004) Liu & Liu (2009); Garlappi et al. (2006); Delage & Ye (010);; Hu al. (2016); Kim et al. (2055): : Chen et al. (2025b). In these settings, Dicision-makenn act with incomplete information and use machine learning predictions estimate uncertain parameters. such as future demand, outage
risk in power grids, asset returns in portfolios, and travel times or flows in transportation systems.
Standard methods, commonly referred to as pedict-then-optimie (2017), tackle this problem by first training predictive model to estimate the parameters of an optimization problem (e.g., expected demand or cost coefficients) and then using these estimates as inputs to optimization model. While the separation between prediction and optimization enhances efficiency, it also introduces a fundamental drawback. Predictive models are typically trained to minimize standard loss functions (e.g., mean squared error). which may not align with the true objective minimizing decision costs. As a result, small prediction errors can propagate through the optimization process, leading costly, suboptimal decisions. For instance, in power outage
(PTO) Elmachtoub Grigas

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PTO
Predictor
Predict
Predictive Loss Minimization
Optimize
Decision
maia Bi ell*)
Backprop
go (z)
DFL
Predict
Avg. Cost Minimization
Optimize
Predictor
Decision
min oo (f (c,
Backprop
Backprop
= go(2))
Gen-DFL
Generate
CVaR Minimization
Optimize
Robust
Generator
min (st
Decislon
Backprop
Backprop
[c
po(c(z)
Figure 1: Comparison of the proposed decision-focused generative learning Gen-DFL ramework with conventional predict-then-optinzze (PTO) and decision- focused learning (DFL).
management Zhu al. (2021) overestimating energy demand may lead to unnecessary resource
allocation, whereas underestimation could result in supply shortages and prolonged downtime.
To address this issue, ppcision-focuseee learning (DFL) integrates prediction and optimization into single end-to-end framework Donti et al. (2017): Mandi et al. (2024b). Instead of optimizing purely for predictive accuracy, DFL. trains machine learning models with the explicit goal of minimizing the final decision cost. This key idea is enabled by differentiating the optimization process within the learning loop, and results in an alignment of the model's predictions with their downstream impact. This approach has shown clear improvements in structured decision-making tasks where the optimization landscape is well-behaved and relatively low-dimensional. Despite these advantages, DFL suffers from several critical limitations: (1) Scalability: high- dimensional settings, the curse of dimensionality Köppen (2000) degrades the predictive model's ability to capture complex dependencies in the parameter space. Since DFL typically relies on single-point predictions, struggles to encode the full distributional uncertainty of the decision variables Mandi et al. (2024a). This leads to overconfident estimates that degrade decision quality many applications, robustness over worst-case outcomes rather than optimizing for expected performance. Traditional DFL models, however, are primarily trained to improve average-case decisions and do not explicitly model tail risks Tal et al. (2009); Beyer & Sendhoff (2007).
when uncertainty high. (ix) Risk Sensitivity:
prioritize
To overcome these challenges, this paper proposes decision-focused generative learning (Gen-DFP DFLL,, novel end-to-end framework that leverages generative models to enhance decision quality in high-dimensional and risk-sensitive settings. Unlike traditional approaches that rely on fixed un- certainty sets, Gen-DFL learns distributional representation of uncertain parameters using deep generative models. Recent advances generative modeling enable efficient learning of complex, high-dimensional distributions Dong et al. (2023); Wu al. (2024), allowing for adaptive sampling from tail regions to support risk-aware decision-making without excessive conservatism. By dy- namically balancing robustness and efficiency, Gen-DFL provides more flexible and principled approach to decision optimization. A schematic comparison of the predict-then teeettttheo-o-izzze- optimize (PTO) model, standard DFL, and Gen-DFI. L is shown Figure
Contributions. The paper makes three contributions:
It introduces Gen DFL, DFL framework that leverages generative models to capture uncertainty in high-dimensional stochastic optimization and to enable task-specific robustness control.
provides theoretical analysis that characterizes conditions under which Gen-DFL outperforms traditional DFL, with emphasis on high-dimensionn
and risk-sensitive decision problems. Through experiments on both synthetic and real-world decision- decision-making tasks, it shows that Gen-DFL improves decision quality compared to existing DFL baselines.
2
RELATED WORKS
Decision- focused learning (DFL) enhances decision decision-making under uncertainty by integrating pre- diction and optimization into single framework. [Beggio (1997) showed that optimizing predictive
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models for decision outcomes improves financial performance. Differentiable optimization layers have further expanded DFL applications Agrawal et al. (211)) For example, Amos & Kolter (2017) introduced differentiable quadratic programs, enabling backpropagation through constramed op- timization, while Agrawal et al. (2019) extended this to all convex programs. Parallel work has explored integrating integer programming into neural networks [Madii (Mandi & Guns] (2020); |Wilder et al.)
(2019). There is also another line of research which focuses on improving efficiency and effectiveness
of prediction-basel DFL al. (2022 2024) Kong et al. 22222).
However, existing DFL methods rely on single-point predictions. failing to capture uncertainty and leading to suboptimal decisions KKöppen (2000);[BenTTae (Ben-Tal al. (2009). Additionally, they typically optimize for average-cast performance, making them unsuitable for risk-sensitive applications Mandi al. (024b) in safety-critical or regulated domains. Approaches like Cerffrmmm--Preiict... Then- Optimize (CPO) Patel et al. (2024) attempt to address this by constructing fixed uncertainty sets but can be overly conservative, especially in high-dimensional settings.
Robust Optimization (RO) provides principled approach to decision-making under uncertainty by ensuring solutions remain feasible under the worst-case scenario BBe--Tal & Nemirovski (2002): Bertsimas & Thiele (2004); [Ben-Tal et al. (2006), with clear guarantees and scalability. Instead of relying on probabilistic assumptions about uncertain parameters, RO constructs uncertainty sets that define the range of possible parameter values Bertsimas et al. (20IT) and aims to find the decision that is robust against the worst-case in the uncertainty sets. This approach has found applications in domains such as supply chains Bertssims & Thiele (2004), currency portfolio management Fonseca et al. (011). and power system optimization Liang el al. (2024).
Despite its guarantees, the solutions suggested by RO suffer from two major limitations: (i) Un- certainty set construction usually relies on heuristic choices, making it difficult to capture the real dynamics in the real-world applications Liang et al.) (2024). (it) Such pre-specified uncertainty sets tend to be overly conservative Roos & den Hertog (2020) as it focuses solely on the worst-case out- come Wang al. (2025): Chenreddy & Delage (2024); Yeh et al. (2024), whereas many high-stakes applications require accounting for multiple adverse scenarios.
3 PRELIMINARIES
Decision-Focused Learning.
Consider a general stochastic optimization problem:
w² == arg min Ee--pte) w)],
where is random vector characterizing the problem parameters and f(c, w) is the objective function. The goal is to find the optimal decision w² that minimizes the expected decision cost under the distribution p(c), given the true data- aata-generating process.
common approach, predict-then then-optimizz (PTO), assumes linear objective, which simplifies the problem to the following deterministic surrogate:
w" (c) arg min eT w,
where is the estimate of E[c|2] conditioning on covariate x. This framework consists of two components: (i) A predictor è == ge (x), trained to minimize the standard mean squared error (MSE) El 1 (ii) An optimization model that finds the best decision w given è. As noted by /llacchoob & Grigas (2017), this approach often leads to suboptimal decisions, as minimizing prediction error does not necessarily translate to improved decision quality.
To mitigate this issue, decision-focused learning (DFL) Mandi et al. (2024b) integrates prediction with decision- decision-making by training gz) using decision regret as the loss function. The resulting objective takes the following form:
(#) E., [Regret(ge(x))c) c)], where Regret(ge c) f(c, wn (gs(z))) - f(c, w² (c)).
For notational simplicity, we use c to denote the true mean of the optimization parameters given x. By optimizing (z) directly with respect to decision performance, DFL ensures that the predicted parameters yield decisions that are robust to downstream cost objectives. We will refer to this conventional DFL approach, which relies on explicit prediction models, as Pred-DFF..
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Generative Modeling
Gen-DFL vs Pred-DFL
Low-Risk Regions
p(c|2z)
p(c/2z)
High- Risk
High-Risk Regions
Sampling {cE) ~ po(c)2)
Ground-truth p(c|2)
CVaR Optimization:
Naive Generative Model
(x;; arg min CVaR c-uccc) w)s
Gen-DFL
Pred-DFL
Learning:
Pred-DFL
arg min CVaR.-pv(nl)) (/ (x;
Risk
Figure 2:
Unlike Pred-DFL, Gen-DFL leverages a generative model to capture p(c|z) while incorpo- rating the decision-making objective which emphasizes the high-risk region.
Robust Optimization. In some real-world applications, the expectation-based optimization in equation 2 may fail to provide reliable decisions under adverse conditions, potentially leading to severe consequences Ben-Tal optimization (RO) Kouvelis & Yu [997); Ben-Tal Tal et al. (2009); ( Shalee-Shwarrd & Wexler (2016)
al. (2009); Beyer & Sendhoff (2007). To mitigate this risk, robust
seeks decisions that pertorm well the worst-case scenario within an uncertainty set U(x). by solving the min-max formulation below:
w (x) = arg min max f(c, w).
cEl(w))
This formulation ensures robustness against the most adverse realization of c, providing worst-case protection, However, it can be overly conservative, potentially leading to suboptimal decisions in typical scenarios. In many risk-sensitive applications, more nuanced approach is required one that balances robustness and flexibility by considering a broader range of adverse outcomes beyond just the extreme worst case Sarykalin et al. (2008). This has led to the development of alternative robust and risk-aware optimization frameworks, such as distributionally robust optimization (DRO) Gao Chen et al. (2025a) and conditional value-at-rist (CVaR) optimization Duffie & Pan (1997); Rockafellar et al. (2000) Rockafellar & Uryasev (2002). which offer a more refined trade-off between robustness and performance.
et al. (2018); Zhu al. [henn
PROPOSED FRAMEWORK: GEN-DFT
This section presents the proposed decision- focused generative learning (Gen-DFL) framework. Specifically, we develop novel decision-making paradigm, cemeratethenn-opimize (GTO), designed for risk-sensitive decision problems. Our approach frames the problem as a conditional value-at-risk (CVaR) optimization, leveraging generative model to produce plausible samples that capture the dynamics of high-risk regions. To effectively learn the generative model, we propose a new loss function that integrates both decision-focused learning and generative modeling objectives, ensuring that the generated samples not only reflect the underlying data distribution but also lead to robust, high-quality decisions. Figure provides an overview of the proposed framework.
Problem Setup. We seek robust decisions that effectively manage risk by minimizing the percentiles of loss distributions. This approach has been widely adopted risk-sensitive domains such as financial portfolio optimization, where regulatory frameworks often define risk management requirements
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in terms of loss percentiles Sarykalin et al. (2008) A widely used measure for quantifying high- loss scenarios is conditional value-at-risk (CVaR) Duffie & Pan (1997): Rockafellar et al.] (2000); Rockafellar & Uryasev (2002), which provides characterization of tail risk by capturing the expected loss beyond a given percentile threshold. Formally, given a confidence level a, CVaR is defined as: CVaRif(c, w); ai = E [f w) f(c, w) > VaR.1,
(4)
where VaRa represents the value-at-risk threshold; the probability of exceeding the threshold is at most a. Our objective is to find the decision w" that minimizes the costs in the worst-a% of outcomes. This leads to the following risk-sensitive optimization formulation Krokhmal et al. (2002):
w* (z; a) arg min CVaRe~p(cls> a)).
(5)
We note that the is defined over the high-risk region of the distribution p(c|ar,, allowing for more flexible and probabilistic characterization of uncertainty compared to the "hard" uncertainty set used in equation 3 This formulation bridges robust and expectation-base
optimization: (i) As - 0. the problem reduces to robust optimization, focusing exclusively on the worst-case scenario in equation (ii) As a + 1, it converges to standard expectation-basee optimization in equation [ minimizing the expected cost across all possible outcomes. Thus, our approach generalizes robust optimization by ensuring resilience against adverse outcomes beyond a single worst-case scenario, balancing conservatism and probabilistic risk awareness in decision-making.
4.1 GENERATE- CENERATE-THEN-OO OHENIIOEI
To solve equation | we introduce novel generate-then-optinne (GTO) paradigm, which leverages generative modeling to approximate the risk-sensitive optimization problem. Conventional decision- focused learning (Pred-DFL) relies on a point estimate of the optimization parameters. While effective in some cases, this approach fails to capture the full distribution p(c|ær,, particularly in high- dimensional settings, making inadequate for sessssens applications where adverse outcomes must be explicitly considered. Moreover, point estimates are only appropriate when the objective function is linear, as the optimization problem in such cases depends solely on the expected value of making variance and higher-order moments irrelevant.
To overcome these limitations, we replace deterministic predictions with a generative model, capturing the full risk distribution. This allows to account for uncertainty in data-driven manner, ensuring that risk-sensitive scenarios are explicitly considered. The optimization problem is then solved using sample-average approximation (SAA) Pagnoncelli et al. (2009); Kim et al. (2055) Emelogu et al. (2016). Formally, we aim to optimize:
w/ (z;; a) == arg min CVaRe-pa(els)) [f w);
(6)
Unlike traditional RO, which requires pre-defined uncertainty set (d:)) - often leading to overly conservative or restrictive formulations our approach treats uncertainty as a learnable distribution. Specifically, we model (c(zzz) using generative model parameterized by 0, allowing it adaptively capture risk-sensitive regions based on empirical data. This approach provides a more nuanced and adaptive approach to uncertainty modeling, ensuring that decisions are informed by the full distribution of possible outcomes rather than rigid, pre-specified constraints.
We emphasize that the proposed Gen-DFL framework is model-agnostic and does not rely on spe- cific generative modeling choice. In this work, we adopt conditional normalizing flows (CNFs) Win kler et al. (2019) to model the conditional distribution p(c|2:) due to their fiexibility and tractable likelihood evaluation. CNFs transform a simple hase distribution pz (e.g., Gaussian) into a complex target distribution via an invertible mapping ge C + Z, where C, Z are the supports of the resulting distribution and the base distribution. This enables the representation of arbitrarily complex, high-dimensional distributions. This transformation follows the change-of -arriaaalle formula Tabak
& Turner (2013); Papamakarios et al. (2021): po(c|z) PZ(g (c; =)) det
This expres-
Bc:
siveness enables our model to generate samples that capture both typical and high-risk scenarios, improving robustness in decision- making under CVaR.
4.2 DRCISION-FOCUSED GENERATIVE LEARNING
We now present the Gen-DFF framework, which provides a decision-focusee solution to the GTO problems in unified end-to-end learning pipeline. For simplicity, we denote the optimal decision
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obtained from model (z; a:) in equation 6 wy. omitting and Similar to other DFL frameworks, Gen-DFL consists of two alternating steps:
1. Ceeiutte-TTeOOOtinniee Generate samples [c using conditional generative model (CGM)
Po(c/z) and solve cquation for the optimal decision via SAA.
2. Model Learning: Given the resulting decision wy. update the generative model parameters by jointly minimizing the generative loss and the decision cost under ug.
A detailed description of the learning procedure is provided in Algorithm iin Appendix we elaborate on key components of our framework.
C.1
Below,
Regret in CVaR. Unlike Pred-DFL, where the decision cost is computed as the regret for a single pair (c, c), in our stochastic optimization problem, the parameter follows distribution, requiring regret to be evaluated over all possible realizations of c. Moreover, in robust derision-making we seek to minimize decision costs based the worst-a% outcomes, rather than the full distribution. To capture this, we define regret using CVaR:
Regrets,pp(; a) = CVaRp(elz) f(c, wā) - f(c, w )a],
(e, where w* :- arg minl. CVaRe=p(da)) (fe,, w); ai] is the optimal decision under the true distribution. The parameter a controls the of risk sensitivity: The lower values emphasize the worst-case outcomes, making decisions more conservative. When = 1, it recovers the standard expected regret across all realizations: Ex-pte/>) (c, wz) - )).
Gen-DFL Loss. In practice, the true data distribution p(c[z) is typically inaccessible, making direct regret evaluation infeasible. To address this challenge, we introduce auxiliary model q(cl2r). trained on available data to approximate p(c]z). Once learned, q(c]ar) remains fixed and serves as a proxy distribution to compute the estimated Regret,, (zz a) and the corresponding surrogate loss function {(0; a, q). This enables regret evaluation even when the true distribution is not directly observable. The training objective for Gen-DFL can then be formulated as the aggregated regret across all inputs æ with an additional regularization term to ensure stability in generative modeling:
Reen-Dru (0: g0:q.00 a)) 3 E.g [Regret,, (x; + (gen((), (gen))
(7)
where Egcn (0) is the generative model loss (e.g., negative log-likelihood, evidence lower bound (ELBO) for variational autoencoders Kingma (2013), or score-matching loss for diffusion models Ho et al. 22200). Here, .9 and are hyperparameters that balance the decision-focused regret loss and the generative model loss. The generative loss term (g) acts as regularization, preventing the learned generative model from deviating excessively from the true data distribution, ensuring reliable sample generation for decision-making
Surrogate Loss Function. When training DFL models with respect the decision loss, it is necessary to backpropagate errors through the decision variable. However, this requires computing the partial derivatives Owž which often involves complex dependency chains. Inspired by Mulambal jet al. (2020), we propose a surrogate contrastive loss in our Gen-DFL. setting to address the challenge of differentiating through the combinatorial optimization mapping:
feen-peL((:00 -
Σ "S
(CVaR If w²) f(cc w ::: + + egen
where w* is the target solution, and the negative samples w" E S C W {w+} are subset of solutions that differ from the target solution practice.
THEORETICAL ANALYSIS
This section provides an analysis of the validity of our sample-based based regret estimation method and compares Gen-DFL and traditional Pred-DFL across different problem settings by examining their regret bounds. Our analysis reveals that as the complexity of the optimization problem increases, whether due to higher dimensionality, greater variance in the data, more nonlinear objective

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function, Gen-DFL's advantage over Pred-DFL becomes more pronounced, leading to improved decision quality in the most challenging practical settings.
We first derive the bound for the loss difference [e(0; a) E(O; q. at), comparing the loss function E(O; p. a) under the ground-truth distribution p(c[z) with the surrogate loss E(0; q. a) computed using the proxy model q(c]ar). The proofs of the theorems can be found in Appendix A.1
Theorem 5.1. Under the assumption that the objective function f(c, w) is g --pLiscchitz continuous with respect to for a fixed decision variable w, the gap between E(6; a) and E(0; q, a) is bounded by
[((0;, a) 2(0;q, S (.
where W(p(el)).g(zz)) is the Wasserstein-1 distance between p(c|z) and q(clz) and Kg is some constant.
The theorem above implies that the surrogate loss provides a valid approximation to the original loss function, provided the proxy model q(c(z) can estimate the ground-truth p(c|z) well. The bound is directly proportional to the W(p(e 2), (c(xx)), which quantifies the discrepancy between these distributions. We now establish the conditions under which Gen-DFL, outperforms Pred-DFL. To facilitate our analysis, first introduce the following two definitions.
Definition 5.2. Let p(c[z) denote the true conditional distribution of c, and let po(e(z) be the generative model. We define Qe to be the "worst a% tail" representative for under p(c|2) based on the target decision w*, Formally,
a= fle w*) VaRa).
Definition 5.3. Given the target decision w* and the decisions found by Pred-DFL (w (red)) and Gen-DFL (we),, we can define the regret of Pred-DFL and Gen-DFL as:
Rored (x; = f(Qe[a),
w*), Rgon (z; = CVVa [( (f) (o),/,c,))))))
Next, we develop regret bound that quantifies the performance gap between Gen-DFL and Pred- DFL, incorporating data variance and optimization complexity, such as the dimensionality of the parameter space and the risk-sensitive level. The proof can be found Appendix[A.3.
Theorem 5.4. Let g X - C be the predictor in Pred-DFL. Assume the objective function f w) Lipschitz continuous for any c, w. There exists some constants Luw, Ley K1,, K2, Ki, such that the following upper-bound holds for the aggregated regret gap Ez E [R(x)];
2Lw Eg [AR())] S Ex
[R] W(pe, p) + K2 Bias[g]l]] + 22 K3 + 2Le) / /|rlcc æ]]l
2
a
+ CVaRp(ela) [iassia ([]Baas((<)]l:::]) where AR(z) = Rprea(z; (z; a) Rgon (z; at.
The above results reveal how the following three factors affect the performance gap between Gen-DFL and Pred-DFL. (1) Variance of the parameter space [|Var[c|x]]:: Higher variance c conditioned increases uncertainty and amplifies the difficulty of accurately approximating the objective. Pred-DFL, which relies on point estimates from g(x), struggles in high-variance settings. In contrast, Gen-DFF benefits from modeling the full distribution p(c]æ), capturing the variability and structure needed for robust decision paaisiionmmiinngg under uncertainty; (di) Dimensionality of the parameter space, including de and dg As the dimensionality increases, the estimation error of the predictor in Pred-DFL. grows rate of ((v(d, + de)/n//) (see proof Appendix A.5). making it increasingly difficult to obtain reliable point estimates; (iii) Risk level a: The inverse dependence of the estimation error on a implies that smaller values of a make quantile regression more challenging for Pred-DFL, as data in the tail regions of the worst a% outcomes become increas- ingly sparse. This leads to a larger bias in g(z) for smaller a. In contrast, Gen-DFL leverages a generative model to capture the full conditional distribution pl p(c)z). Together, these insights demon- strate that Gen-DFL offers significant advantages over Pred-DFL in complex, nigh-dimensional. and risk-sensitive scenarios.
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Task
Painwis
Fietwioo
NCF
MAP
DiK-DRO
2Singe
Gon-DFF.
Deg-2
1148:00
20.67:(1.11)
6.57:c0-46)
8.88::0.44)
6.92100
8.50::0
Portfolio
Dag-6
Dog-4
11.54+00
11:16+0
30.20-(1.19)
18.57+0087)
869+<061)
843-10.55)
8.5++00.3)
7.01+0022 (0.
1.25-1660
7.1++00677)
8.56+:0.71)
14.00-00633)
16,024-00.30
3.81-10.221
(..1-
10441100.00
21.92100.95)
7.9510040)
8.9010044)
6.981(0.95)
8.65100.50)
16.17.110,00)
3,59_10
Deg-2
94.93-19.30
17.03-1844)
24 3475577.87
15,54-44700
21,00
30,274/0..4
17.00-(3.28)
Knupsack
Deg-
33.85188
(6.30))3.02)
40:33:005633
17.45117.21
16.58.10..00
15-21 (3.75)
17.
Deg-5
33.252500488
20.38::16700
22 23.367778
34.07806.00)
(2.80
21 21.48::00220
21.134/7..40)
(3.29:
Deg-2
6.301:2355)
2.65::00255
12.92:03633
3.231:0:72)
2.91100090)
Path
Deg-
29.63277.20
18911515.00
12.19:11104)
33.15124.600
4287712
52447a16.49)
28.736(11.22)
26.46119.331
23.762/0.21)
(1.78./22.89
22.442/2.84)
38.642/2.3)
3.64::0.417
6.52.:00.11
Deg-8
(18.8)
51 51.65=(13.77)
139.09
178.17=(36.2) 28)
81.78+
39.81=(5446)
45.75+
13.36-
Energe
1.652/0223)
(.67
1.91/0013)
1.59:/0.111
(.561
(491:/0112)
1.912/00222
1.09L100009
COVID Resource
17.912(1.85)
16.33
16.48a/222
(6.59
(7.94)
(6.41:(33))
(8/46//3.2)
(6.96)
Table I: Comparison of decision quality (average relative regret, J, lower is better) across tasks in high-variance settings (o = 20).
EXPERIMENTS
6.1 EXPERIMENTAL SETUP
We evaluate the proposed framework on three synthetic optimization problems including Portfolio Management, Fractional Knapsack, and Shortest-Path. We also consider real-world tasks, namely an Energy Management Problem dataset Ifrim et al. (2012); Simonis et al. (1999) and COVID-19 resource allocation problem adopted from Mandi al. (2022); Kong et al.
2022
Synthetic Data. We evaluate our approaches on synthetic benchmarks (Portfolio, Knapsack, and Shortest-Path), adopting the data-generation process from Elmachtoub & Grigas (2022). We first overview the optimization setup for the Portfolio problem. In the Portfolio problem, optimization parameters represents the asset prices and the dimension of c is the number of assets. Our non-linear, risk-sensitive Portfolio problem is then formulated as:
w² (x; a) == arg min CVaRpte x)) --c c-T T Ew; s.t. w E [0, 1]", 1 S1
(8)
where ===LLT ( is the covariance among the asset prices c, and the quadratic term w Ewe reflects the amount of risk. The configurations of our synthetic experiments include the training size, feature dimension d,, polynomial degree, and the noise scale a that reflects the amount of variance in the parameter space and the non-linearity of the above stochastic optimization, since, by our construction, a would affect the magnitude of the quadratic term w" Ew. The problem setup and model configurations for the Fractional Knapsack and Shortest-Path problem are similar to that of Portfolio. Full details of the data synthesis process and the corresponding optimization formulation for each experiment are provided in Appendix B
Real Data. For the real data experiments, we consider real-world Energy-cost Aware Scheduling problem and a COVID-19 resource allocation problem that we adopt from Mandi et al. (2022): Kong et al. (2022). In the Energy-cost Aware Scheduling problem, we consider a demand response program in which an operator schedules electricity consumption Pt IR21 over time horizon / E Ω2 The objective is to minimize the total cost of electricity while adhering to operational constraints. In the COVID-19 19 resource allocation problem, we focus the problem of allocating the number of hospital beds w E R. for the next seven days based on the forecasted number hospitalized patients E IR? The details of optimization problem in each experiment can be found in Appendix B
Model Configuration. The hyperparameters in our learning algorithm include the decision cost weight 3 and the negative log-likeling likelihood weight in equation which serves regularization. We introduce an additional hyperparameter 3 in our experiment to study how different magnitude DFL loss will affect the model's performance. We set 7 1 across all experiments and study the effect of different / values on Gen-DFL's performance (Figure B) When / = O, the loss reduces to that of a standard generative model, only fitting data without considering decision costs, which results in the worst regret in all risk- sensitie settings. Increasing 3 improves downstream decision quality across all risk levels. Full hyperparameter details provided in Appendix C.2
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200 Published as a conference paper at ICLR 2026
30
Degree 4
Degree 6
0.18
25-
Model Trained by
Average Regret (96
Avcrage Regret 186)
A=0.0
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8=10
15
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B=10.0
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400
Eval
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Nolse
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Figure Decision quality against different risk- sensitive regions vs various 3 in Portfolio problem.
Figure 4: Decision quality in the portfolio prob- lem under different settings (J lower is better).
Baseline Methods. We evaluate the performance of Gen-DFI. against various wate-of-the-ar Pred-DFL. baselines across all tasks. Specifically, we compare against Smart-Predict- Seediic--Then-OPAe "hhen-Optimize (PPO))Elmachtoo & Grigas) (2022), contrastive loss-based Pred-DFL models (NCE, MAP) Mu- lamba el al. (2020), ranking-basse based Pred-DFL models Mandi et al. (0022), and the recently proposed PFed-DFL approach with differentiable Distributionally Robust Optimization layers, which we refer to as Diff-DRO Ma et al.] (2024). These baselines represent a range of decision-focuse learning strategies, differing in their loss formulations and optimization objectives. The main results of our comparison are summarized in Table
We evaluate the decision quality of different models on various tasks in terms of the average relative "-
100%. where lower indicates greater risk sensitivity. For our real data experiment, we will first train proxy model q(clar) given the data, which will then be used evaluate the average relative regret during evaluation. We set a when we compare against the baseline models since the above metric is equivalent to the standard relative regret used in previous Pred-DFL literature, which makes the comparison fair.
regret, E,
ap [f(,w""
6.2 RESULTS
Comparison with Baselines. Table presents the comparative performance of Gen-DFL, Pred-DFL, and the two-stage method across different problem settings. Gen-DFL consis- tently outperforms baseline methods, reducing regret by up to 58.5% compared to Diff-DRO and up to 48.5% compared SPO+ in Portfolio tasks. Gen-DFL's advantage is particularly pro- nounced in high-dimensional tasks like Shortest- Path (Deg-8), where it achieves remarkable 83.7% reduction in regret over SPO+ (13.36 vs. 81.78). This demonstrates Gen-DFL's abil- ity to overcome the curse of dimensionality by effectively capturing the distributional structure of p(c|z) rather than relying on point estimates. Conversely. in Knapsack (De Gen-DFL's improvements over SPO+ and Diff-DRO are more moderate (19.6% and 10.3% respectively), suggesting that the benefits of generative modeling are especially significant problems where uncertainty is highly non-linear or where high-dimensional interactions dominate the optimization landscape. The statistical significance tests of Table can be found in the Appendix
GeNtR Sexte
ERC
Regret
10
Degree
Figure 5: Comparison with conventional RO under varying polynomial degree in Portfolio problem.
Figure 4 Ilustratte the impact of variance Var[e|æ], problem dimensionality, and training size model performance. Gen-DFL demonstrates robustness across all variance levels (a E [40, 60, 80, 100)), effectively capturing the full conditional distribution p(c|æ), unlike Pred-DFL models, which rely less expressive predictors and are more sensitive to variance. As dimensionality increases, baseline methods suffer from the curse of dimensionality, leading to higher regret. In contrast, GF1

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Degree
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Degree
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Model Trained by n=200
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Eval a
Eval a
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(a) Training a vs. risk level a.
(b) Number of generated samples vs. risk level a.
Figure 6: Portfolio problem ablation results.
maintains superior performance by learning the structural complexity of p(c(ær), as predicted in Theorem 5.4 Additionally, while Pred-DFL performance deteriorates with smaller training sizes due increased predictor bias, Gen-DFL remains stable by effectively modeling the underlying distribution. The quadratic term in the objective further amplifies the non-linearity in high digh-variance settings, demonstrating Gen-DFL's 's adaptability to complex optimization problems.
We further compare our Gen-DFL. with conventional data-driven Robust Optimization methods (e.g., LRO Wang et al. (2025), E2E-CRO EEROChenreddy & Delage (2024), E2E- CHEE-CofformalYee Yeh al.) (2024) in Figure 5 Instead of learning fixed-geometry uncertainty sets for a hard min-max problem, Gen-DFL learns a full generative model to directly capture complex uncertainties
minimizes the Conditional Value-at-Rii (CVaR). We also did an ablation study the choice of the generative models which shows that the power our method lies in this core paradigm; while a Conditional Normalizing Flow (CNF) architecture outperforms Variational Autoencoder (VAH) (of which attribute to the advantages of exact likelihood training over the ELBO approximation), the primary contribution remains the development and validation of the Gen-DFL framework itself. We thus leave the integration of other generative models or optimization schemes to future work.
Risk-sensitive Settings. We also evaluate Gen-DFL under various rinsisensi settings (indicated by the "Eval a" on the x-axis, where smaller "Eval a" indicates that we are evaluating under the higher-risk regions) using CVaR, which measures the decision quality (in terms of regret) over the worst-a% of outcomes. Figurel6a that models trained with smaller or (e.g., or = 0.5) outperform those trained with larger a (e.g., 1.0), demonstrating better adaptation to adverse outcomes. The performance gap widens as risk sensitivity increases, confirming that smaller a enhances robustness while larger prioritizes average-case performance. To further assess stability. examine the impact sample size in the aample-average-eee apprraaaapppooimaation step (Figure 6b). Increasing generated samples consistently improves decision quality across all risk levels, reinforcing the importance of uncertainty modeling in Den--FF. These results highlight Gen -DFLss flexibility, making it particularly effective in high-stakes, risk-sensitive environments.
7 CONCLUSION
We presented Gen-DFL, novel decision-focused learning framework that leverages generative modeling to solve robust decision-making problems under various risk-sensitive settings and provided a thorough theoretical analysis demonstrating the performance gain under various high-risk decision- making problems, verified by comprehensive experiments.
ETHICS STATEMENT
This paper presents work whose goal is to advance the field of Machine Learning. There many potential societal consequences of our work, none which we feel must be specifically highlighted
here.
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REPRODUCIBILITY STATEMENT
We include our full implementation at htt.os//wshau.cccm//////ggepeogp
dEl
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A THEOREMS AND PROOFS
A.1 SURROGATE LOSS FUNCTION
In this subsection, present theoretical bound the gap between the loss function E(0; p, a) under the ground-truth distribution p(c æ)) and the surrogate loss E(0; q, o)) under proxy distribution q(c 2) that approximates p(c z).
Theorem A.1. Let p(elz) be the ground-truth distribution and g(c()) be a surrogate distribution that approximates p(c|z))
Under the assumption that the objective function f(,, w) is -Lpischhtitz continuous with respect to e for a fixed decision variable the gap between the loss function E(O: p. a) and the surrogate loss f(0: q, a:) is bounded by
(e(0:p.)) a) ((0;q,a)) Kg- Er (W(o(cl))
where W(p(el)) is the Wasserstein distance between p(c(z) and a(clx) and Kg is some constant.
Proof.
First, by linearity expectation and the triangle inequality, we see that
p((0;p,)) a:) E(O: (,)) S E IA B
(9)
where
[CVaR (c, (fe,, For simplicity, we omit inside CVaR for now.
-VVa) (f, w*ll, = (CVaR wž)] -
-CV atd))) w w*)l|
Then, we can begin by examining the gap between the expectations under p(c(ær) and ((c/x)) for fixed context at
By the reverse triangle inequality (||=) ull S 1 y), we have
|A B < CVaRpicle)lf((. w²]] o( (f w |
(C ut)]
CVaR [f w5)1|
Let's define g(c) w*) and h(c) (c, w³). By assumption, f(c, w) is Li-iscchiizz contin- uous with respect to which implies that that ((c) and h(c) are also Lipscchi. Hence, by the Cetturooric--uubsstteee duality for the Wasserstein distance, we have,
(((())
sup [E<ppcla)][((e) S1
(o())]
(M()) Bowgtela) (M [Allas
sup
where the supremum is over all functions 9, that are 1-Lipschitz. By definition of CVaR, we can see that,
(CVaRptele)]](e w²) CVaR CVaRg w")]|= S sup
Eewpte
Again, using the assumption that g(c) and h(c) are also Ly Lipschic,,, we can bound the gap in (2) by
(/) w*)] CVaR w²)) CCVV
wx)] CVaR wE)] s (ll
Finally, taking the expectation over æ on both sides and using equation (1) and set the constant Kg = =2L,, get:
((0:p.a) ((0;g,0) & KG E2 (W(r(c)).
This completes the proof.
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A.2 CVAR/QUANTILE REGRESSION
Theorem A.2 (Finite-Sample Bound for CVaR Estimation). Suppose Y takes values in the interval [m, M. Let CVaR be the empirical estimator derived from
1 (Y; -)++, CVaRa inf Φx. BER
(T( +
1=1
where ()+ max 17, 0} and Yi....,Y, Y, are i...d samples of Then there is a universal
constant C > O such that for all > O, with probability at least 1 5,
CVaRa - CVaRa < (M -m) lm(1/5)
In other words, the estimation error for CVaRa converges on the order of /In(1/8)/n grows. Remark A.3. Here, (Y)
= E Y & VaR. and
VYR.(Y) inf t: Pr(Y t) 2 a}.
The key step in the proof is the Rockafella--Uryasey identity,
(VaRa(Y) inf (1 + : BB((Y- WER
combined with uniform convergence arguments (e.g., Hoeffding or Rademacher complexity bounds).
Proof. Step 1: Rockafellar-Uryase, Representation.
Recall the identity (Rockafellar-rryasey)
CVaR.
min (1 + a nEX
B[(Y n ))+)).
Set
σ(n) + - Bi((Y - 17 )+}
Then (VaR,,Y) mingeR d(1))
Step 2: Empirical Estimator.
Given i.i.d. samples Yı,..., Yn, define the empirical counterpart
1 1 E ( n)+,
Sn(n) 17 a
and let
CVaR,
min )n()).
nE
Similarly, let let ry" e arg min, ((1) and îm E argmin, (n).
Step Uniform Convergence.
Observe that
(m) - -d() (2 Y. 7)+ -A -1 - -1)+) < sup nE
Σ (Y, B[A,0)]
i=1
where fn( (y n)+ is bounded by (M m) if y [m, MJ. By standard Hoeffding (or VC Rademacher) arguments, with probability 2 6,
[I((1/8)
sup yER
1 - n) - E((Y - 7)+1
$ C. (M - -m) <.
n
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for some universal constant C1 0. Hence,
Ci (M m)
[n((1/8) n2
sup
Sn (n) - <(n)l -
: En
VER
a
Step 4: Error Between Minimizers.
By definition of fm and 17"
(fm) I )n(1"))
Also,
((frn)) - o(n') - (f.) [(fin]] + -6(n)) 1 2En.
Thus
((mm)) S p(11)) + DEn = @n(fm)) = =((fm) + [6n(Ph)) - -((m) S 2(11)) + 35.
Similarly, by symmetry, we get p(T)) S (fnn) + +3E, so
16n <(nn)) ) I 3
Since CVaRa(Y) = <((") and CVaR. =ān(fn)) conclude
CVaR. CVaR. < 3
of M-m
((M-m
In(1 (8)
Finally, we absorb constant factors into single C, yielding the stated bound.
Theorem A.4 (Generalization Bound for Conditional CVaR Estimation). Let (X, Y) be distributed X x R. and let G be class of measurable functions g X R. Define the population Rockafellar-Uryasev (RU) risk of any predictor by
R(g) El((X) + a -(Y -x(()))
and let
R = inf R(g), g" E arg min R(g)- gEg
g6g
Given ii.dd samples {(1, y)))= define the empirical RU risk
(g)
E g((,)) (y g(a.))+
n
and let
In E arg min R (g). gEg
Suppose that, with probability at least 1-8,
sup
Rg) R(g)
S
Ens
9EG
where Sn is term that typically of the order of ness, sub-Gausee Gaussian tails, etc.). Then on that event,
In(1/8)
under standard assumptions (boundedd-
R(gn)
R
2 2
Hence the learned predictor 9. achieves CVaR-type risk within e of the best g" E G, with high probability.
Proof. Step 1: Setup & Definitions.
For each E G. define the population RU risk
R(g) = E E(2(X) + (Y (Y- s{
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The empirical counterpart based samples (2), Vb i= is
(g( 1 Σ [g(;;) + a (ui - -((x)))
Let
In arg
aigmin min (g); g" E arg R(g). sEg
DEG
Step 2: Decompose the Excess Risk.
We want R(gn) R(g"). Note that
R(gn) R(g") [R(an) Rh, (en) (R(( (g)
[R((g)) R(g")]
(4)
(B)
(C)
Since In minimizes R,,, the middle term (B) < 0. Hence
R(în) R(g") < (A)+ (C).
But
(A) = R(în) (n(g s supl R(g) (g),
gEg
and similarly
(C) Rn(g") - R(gg) S supl gEG
R, (g) R(g)
Therefore,
R(gn) R(gg) z sup gEg (g) (g) R(o)
Step 3: Uniform Convergence Bound.
By hypothesis (or by a standard Rademacher VC argument), we have
sup
(g) R(o) I Eni
gEg
with probability 2 1-6, 5, where en grows a rate of ( ln(1/8)
Hence that event:
n
R(gn) B(g") S 2
Step 4: Why En Includes Factor of 1/a.
Observe that
d 9. g(x) - g(z))+
Because it scaled by 1, any standard concentration bound (e.g., Hoeffding or Rademacher) for d. incurs extra factor of 1/a. Specifically:
Boundedness: If [g(x)| < Gmax and 1y| < Y maxx then (y (g))4 I 1y g(x) s Y Ymax + Gmax- Hence &a (I, y) < Guax + {Y + Gmax))
Rademacher complexity or Hoeffding: A uniform--oonvergence covering-number argument
yields [m(1/8) factor multiplied by the supremum of |oo/, which is < C for some
constant C.
Thus E. necessarily scales like a / In(1/8) (up to constants and possibly adding a (g) term if
large).
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Theorem A.5 Diige--Dimesiinall Conditional CVaR Generalization Bound), Let (X,Y) be random pair taking values in IR x IR',, and let E (0,1) ) be fixed. Suppose we have:
scalar loss e: R x RR R.
A hypothesis class g of measurable functions g Rdd. + R,
and define the Rockafellar-Uryasee (RU) risk of any predictor E G by
R(g) 18 o(x)
1 (l(g(X)) - (M((()))))ssxx))
+
Let R gEG R(g), and choose g such that R(g") = R Given n i..d samples ((ri, ye) C
Rdd= Rdx,, define the empirical RU risk
1 Σ g(1))
+ 1 (elg(T)), ud) -
(g) =
and let In E arg mingeg (g). Assume that with probability at least 1 8, we have a uniform- convergence bound
Rg(g) R(g) S En
sup NEG
where En scales as
= tu
under suitable Saapadddddeesssb---aassiii assumptions on (X,Y) and l. Then on that event,
R(gh) R S 2
Hence the learned predictor In. achieves CVaR-type risk within 2sn of the best g E G, with high probability.
A.3 GEN-DFL VS PRED-DFL
Definition A.6. Let p(c(ar denote the true conditional distribution of and let po(c))) be the generative model. We define Qu to be the "worst a% tail" representative for under p(c[z) based on the target decision w². Formally,
Q. = E f(c, w*) VaR.].
Definition A.7. Given the target decision w", the decision Wored found by Pred-DFL and the decision w/ found by Gen-DFL, can define the regret of Pred-DFL formally as:
Rpred(r; [ f(Qe[a), w wprea) - (Qee(a), w*).
and the regret of Gen-DFLas as:
Ro(x; a) = CVaRplelz) f(c wy) ((,,
Theorem A.8. Let g:::: X - be the predictor in Pred-DFL. Assume the objective function f(c, w) Lipschitz continuous for any c, Then, there exists some constants Lews Ley K1, 12, Rig such that the following upper-bound holds for the aggregated regret gap Ez| Ez|AR(z)]:
22
2L.
F. [R())) F
[Mi
W(po, + 82 [ias[g]]]] +
Kg + 2T.)) Varlc x|||
CVaRp(ola) [|] [[[Bias[g()]]ll
where A AR(z)= = Rprea a) Rgen (x; a).
Remark A.9. Let de and d denote the dimension of C and X, respectively. The bias term ||Bias[g]]] of the predictor grows at a rate of O 1 / (dg + de)/n). This suggests that the smaller the is, the harder for the predictor in the Pred-DFL to get an accurate estimation of Qe[a].
Remark A. We may write E + where Ep(c|z]cc]. Under some mild assumptions, such being Gaussian, the variance term of the order ( Vdo).
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Proof. Step 1: Decomposition of the Regret
A ( p(el))))))) [[(, wz) f(c,w"); a] CVaR "prea) f(c, w w);
CVaRp [[c, f(e [)
((ela) [f(((z). wx) f(g(x), (pread] [f(c, wy) f(g(x),
[c, ( f(g(()) w al
< CVaRp [f(g(z), ug) f(g(a), W )prea):: a|
+ 2L, g(x)]l:a]
p(ea)) ((/(RR)))))/(((()),...)))))))))) w²) [a(x), Wpread; a|
+ 2LL,
Qa(a] Qa[a] a()))laa]
pcea) ([((()) w)) [(((), w uprea) [
(Vaa Qc
+ [| 10 f(((x,, w wpreadi all + 2d x) + CVaR CVCRp(c()) [| ||:
a]]
where we used the fact fie, wx) = f(g(a), wx) + [f(c, wg) - f(g(x), will and f(e, upeea) (g(x), upred + [f(c, w - f(g(z), w uprea)ll
Step 2: Bounding A CVaR p(e|m) (g())) wg) - f(g(z), w wiree):all wt) [g((x), (i ail term.
Now, we need to bound the Arorm
By assumption, for any fixed co. the map w + f(co. w) is L Lipscciii in w. Equivalently. w2)) w2)) I Lw wi w2ll-
[(e, wi) - f(co,
Applying this specifically co g(z). we get:
[((()), wp) [g((), (prea)) S Lw wg w upreall:
Since CVa is merely expectation that does not affect the integrand here (it does not depend on e anymore), we have
! Term
S
L
wg w wpreall:
Step 3: Bounding [|ul - wfoal
First, we define the following auxiliary (aggregate objectives) functions for both Gen-DFL, and Pred-DFI.,
Gen-DFL: Jgen (w) = CVaRp, [S w), Pred-DFL: Spred (w) = (g(x), w).
So
w6 arg min Jgen (w), worod arg min Sprea (w).
Next, let's define
A(w) ggen( (w) - J Jpred (w) CVaRpu [[(,, - [((()), w).
We take a uniform bound over w:
T sup A(w)).
We will then show that,
T Ki |po pll K2 / [|Var(e [=]] + ng Bias|]]
Step 4: Bounding T
By definition,
T sup
CVaRp, (f(c, [o(x),
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To relate this to the true distribution p and Qc[a] = Eewpl cl. we can do the following decomposition:
[fc, f(g(a), w) (f (VaR,I((c,w))
(CVaR,lf(c,u)) -(Q.aa), w)) (f(Qo(a),,)) - [(((x), w))
Hence, we set
T = (B) (C),,
then by triangle inequality:
T (A)
+ (B)
+ (C)
T2
We now bound each piece T2, T3 separately. First, see that
Ti = CVaR, ([ a) CVaR,l/(e,w.)
S W(po,)),
where depends on the Lipschitz constant of f in c. Next, by taking the Taylor expansion, have f(e, w) f(g(=), (((x),
After taking the CVaR expectation, we see that
To = sup CVaR, w);a] - flg(x), S //2|| Varlc x]]|.
where K2 incorporates the Lipschitz constant.
Finally, for T3, assuming that f(, w) is Lipschitz in c, then
T3 suplf((e[aj w) - f(g(x), w) < Le 1Q[la] s(x))l < Le |[Bias[g]]]
Hence,
T3 S K3 |Bass
Combining all the steps.
Collecting Tı,T2, T3:
T= sup CVaR (f(, f(g(()((())))))))))))(<))))))f wi)
1 Ti + T2 + Ts
< W(po.p) + / ////er x]] + K3 [Baasgg]]
Thus,
T Kn W(pop) p) n2 Varlc x]]
Kg |ias [g]l]
Strong Convexity in Yields Solution Stability. Assume (-en(() and (.rea(.) are a-strongly convex w. Then,
2 T.
| wored
< sup sup/A(w))
a
a
Therefore,
[ wireall s (R1 W(po.p) + N2 VI xi] + 63 |Baas(g]])..
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Combining all the steps
2 T.
A Term
CVaR,Lf(g(2)), wx) f(g(z), u ar S Lu w preall s Lu
a
Therefore,
2 <
! Term
Lu
|s W(po.p) + K2 Varle aill + wa [ias[g]l]]
Finally, we get,
Eg ||R(()) - Ex L [&, W(po. p) K2
Varlc x:] + rig [Bias[g]]]]
+ 2L VallVarle z]l
CVaRp(e Bias [ias[g())]]]:=
Moreover, by Theorem A.5 that we developed earlier, we can see the bias term rate ( V(d, + dn)
|[Baasg]]| grows at
B EXPERIMENTAL SETUPS
B.1 SYNTHETIC: PORTFOLIO OPTIMIZATION
In the Portfolio experiment, we generate the synthetic data as follows:
T3 NO, Id=),
By (0.5)
Lay Uniform Uniform|-0.00220 -000025e, 0.0025a]
~~ No(,
0.05
deg
Gij =
Bx; + 0.1
+ LL + 0.01o6,
v/
where das de are the dimensionality of the input features I and the cost vector c. The polynomial degree reflects the level of non-linearity between the feature and the price vector. In Portfolio, represents the asset prices and the dimension of c, is the number of assets.
The non- non-linear, risk-sensitive optimization problem Portfolio Management then formulated as,
w (xx) = min CVaR (- w + w Wn'ww; o]
ue
(10)
s.t. E [0, 1]", 1 1,
where Σ = I.I.T + (0.01o) the covariance over the asset prices c, and the quadratic term w Ew reflects the amount of risk.
B.2 SYNTHETIC: FRACTIONAL KNAPSACK
In the Knapsack experiment, we generate the synthetic data as follows:
Iy NO, Id2..
By (0r5
Taj Uniform (0.000250, U-ifomm-0.00255,, 0.0025
~~ C~,00, 7dc)
0.05
deg
Br, + 0.1
+ L.f + 0.01os,
vp
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where dz, the dimensionality of the input features and the cost vector The optimization problem in Knapsack is formulated as:
w (a) min CVaR ((l))()))) w; af
(11)
s.t. W E [0, p"" P B
where p R and B represent the capacity and weight vector, respectively.
B.3 SYNTHETIC: SHORTEST- PATHI
In the Chortest-Path experiment, generate the synthetic data follows:
33 NO, Id=),
Bij Berouuli (0.5).
Cij ~ Uniform 0.5, 1.5]
1 Br, +3) +
deg
Cj
3.5deg
where dæs de are the dimensionality the input features I and the cost vector c. The polynomial degree reflects the level of non-linearity between the feature and the price vector.
The optimization problem in Shortest- is formulated as:
w" (x) = min CVaR CV w; ai)
(12)
s.t. w E [0,
where c'ww represents the cost of the selected path, and the vector of is defined as follows:
deg
d= 3.5deg (/2 BB,
1 +3)
+
c
where B random matrix, and e/ is the noise component.
The features E Rdd follow standard multivariate Gaussian distribution, and the uncertain coefficients cf exist only on the objective function, meaning that the weights of the items remain fixed throughout the dataset. The parameters include the dimension of resources k, the number of items m, and the noise width.
B.4 REAL DATASET: ENERGY-COST AWARE SCHEDULING PROBLEM
In this task, we consider a demand response program in which an operator schedules electricity consumption Pt over time horizon & E Ωg The objective is to minimize the total cost of electricity while adhering to operational constraints. The electricity price for each time step is denoted by mes which is not known in advance. However, the operator within a specified lower bound P. and upper bound P,. Additionally, the total consumption for the day, denoted as pech, must remain constant. This assumes flexibility in shifting electricity demand across time steps, provided the total demand met.
schedule the electricity consumption pt
The optimization problem, assuming perfect information about prices me. be formulated as:
min CVaR, Pt
Σ tE
TyPri
subject the constraints:
Pj < P: S P
VI,
Σ Pt
psch
tEfl
te
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Algorithm 1 Learning Algorithm for Gen-DFL
Input:
Dataset D {(r,, 63) G)) CGM po(clæ),, learning rate v,, regularization ratio 7; sampling proxy model ((c|z) trained on D.
size K, risk-level
while not converged do
(cm)k= po(c)z); N& + number of en that satisfy [(f(ck, w) 2 VaR..
w arg min NE
Mck w)]{S(ck; w) 2 VaR.):
00 a.) Σ
Regret (# + .gea(())
0 <- 0 0e/00:
end while
Here, P.S.A S Pt S P. ensures the consumption at each time step within the allowed bounds, while the equality constraint guarantees that the total electricity consumption remains fixed across the time horizon.
This setup reflects the practical challenges of demand-sice electricity management, where prices are uncertain, and demand shifting across time steps provides opportunities for cost reduction while maintaining overall consumption levels. The problem serves testhed for evaluating optimization approaches under uncertain electricity prices and operational constraints.
B.5 COVID RESOURCE ALLOCATION
The COVID-19 19 pandemic has highlighted the challenges policymakers and epidemiologists face in planning for surges in medical resource demand, such as hospital beds. As the number of infected patients increases, accurate forecasts of hospitalizations become critical for effective resource allocation. To achieve this, epidemiological models based on Ordinary Differential Equations (ODEs) are often employed to capture and forecast the dynamics of infectious disease outbreaks. These forecasts are then used as guidance for planning future resource allocation.
In this task, focus the optimization problem of hospital bed preparation during pandemic, a critical task for ensuring adequate medical infrastructure. Specifically, the goal is to decide how many hospital beds R7 allocate for the next hospitalized patients y E R?. The optimization objective combines linear and quadratic costs to account for both over-preparation (la; y))+) and prepaartrnaaationn (ly; - -as]+) of hospital beds, ensuring both efficiency and safety in resource allocation.
days based on the forecasted number of
The optimization problem is formulated as:
min
Ch la; yel+ + Ch [y: ai]- + qo([v; az)+)² + (na - -y]+)
DCRT
where G and Ca represent the linear cost coefficients for over- and under- preparatre,,,,nnn, while q and qr. represent the quadratic penalty coefficients for the same, These terms reflect the trade-off's between allocating too many beds, which leads to wasted resources, and too few beds, which risks inadequate care for patients.
This formulation integrates ODE-based forecasts to guide prcision-making enabling data-driven approach to resource planning under uncertainty. The problem designed to balance competing objectives effectively, ensuring sufficient resources while minimizing waste during critical periods of high demand.
C IMPLEMENTATION
C. ALGORITHM
The details of Gen-DFT. implementation be found in Algorithm T
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Task
Method
Learning Rate
Variance o
Dimension
Training Size
8
Pairwise
10
20
50
320
Listwise
10
20
50
320
NCE
10-3
20
50
320
Portfolio
MAP
10
20
50
320
SPO
10
20
50
320
MSE (PTO)
10-3
20
50
320
Gen-DFL
10
20
50
320
10.0
Pairwise
10
20
50
320
Listwise
10
20
50
320
NCE
10
20
50
320
Knapsack
MAP
10
20
50
320
SPO
10
20
50
320
MSE (PTO)
10
20
50
320
Gen-DFL
10
20
50
320
10.0
Pairwise
10
5
25
320
Listwise
10
25
320
NCE
10
5
25
320
Shortest-Pate
MAP
10
5
25
320
SPO
10
25
320
MSE (PTO)
10-1
5
25
320
Gen-DFL
10
20
50
320
10.0
Table 2: Hyperparameters and Problem Configurations
C.2 HYPERPARAMETER CONFIGURATIONS
Table 2 summarizes the hyperparameter settings and problem configurations across different tasks and baselines. For all methods, maintain consistent number of training samples (n 320) and input dimensionality (d = 50 for Portfolio and Knapsack, d = 25 for Shortest-Path to ensure fair comparison. The learning rates vary across tasks, with higher value (0.1) used for the Shortest-Path problem, reflecting its different optimization landscape. The noise scale a remains fixed at 20 for Portfolio and Knapsack, while lower value (a 5) used for Shortest-Pard to account for its different problem structure.
For Gen-DFL., we introduce an additional DFL loss weight 3 which controls the balance between the decision-focused objective and the negative log-likelihood (NLL) regularization, so that
nFn (0;g,a) 3 E,, (Regret, (2; a)) + + Egen(()
Unlike baseline Pred-DFL models, which optimize directly over point estimates, Gen-DFL leverages generative modeling and requires careful tuning 3, 7 to ensure stable training. The uniformity in hyperparameter selection across methods helps isolate the impact of different learning paradigms.
D COMPUATIONAL COMPLEXITY
Regarding training cost, our method scales linearly with the number Monte Carlo samples used in the sample average approximation (SAA). Our experiments on consumer-grade -grudde Apple M2 CPU show: with 200 generated samples per decision, Gen-DFL. roughly five times slower to train than Pred-DFL, and with 800 samples it can be up to twenty times slower. In the portfolio experiment, for example, training Pred-DFL takes about 8 minutes, whereas Gen-DFL with 200 generated samples takes roughly 40 minutes. This overhead stems from our desire to model the full conditional distribution p(c[z) expressively; each additional sample improves the approximation of the CVaR but also increases runtime. We emphasize that this computational overhead does not pose a significant drawback since inference typically relies on pre- pre-trained models rather than real- time training.
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E STATISTICAL ANALYSIS
Task
Setting
p-value (Gen- DFL vy 2Stage)
p-value (Gen-DFL vs SPO+)
Portfolio
Deg-2
< 0.00001
0.0000
Portfolio
Deg-4
0.0001
000000
Portfolio
Deg-6
< 0.0001
0.0001
Portfolio
Deg-8
0.0001
< 0.005
Knapsack
Deg-2
0.79
0.61
Knapsack
Deg-4
0.80
0.43
Knapsack
Deg-6
0.59
0.95
Knapsack
Deg-8
0.83
0.77
Shortest Path
Deg-2
0.0001
0.085
Shortest Path
Deg-4
< 0.0001
0.038
Shortest Path
Deg-6
< 0.0001
0.046
Shortest Path Deg-8
0.0001
0.006
Energy
0.003
0.004
Table 3: Comparison: Gen-DFL vs 2Stage (PTO) and SPO+
26


Reviews:
[{"review_id": "45Mch1sSSD", "reviewer": "Reviewer_DNJQ", "summary": "This paper proposes GEN-DFL, a decision-focused learning approach for contextual stochastic optimization problems that uses generative models (e.g. normalizing flows) to generate samples that are fed into sample average approximation model used to optimize a CVAR objective. In particular, in analogy to classical DFL approaches, they propose to train the generative (prediction) model using a surrogate decision-oriented loss function in order to obtain “decision-focused” samples that minimize a CVAR regret. The paper proposes a contrastive surrogate loss function for this setting, and it provides a theoretical analysis that provides an error bound for the surrogate loss function and a quantification of the improvement of the regret obtained by their model in contrast to classical (single-point-forecast-based) DFL approaches.\nIn a set of computational experiments with classical predict-then-optimize, DFL, and a robust DFL approach as baselines, it is shown that the proposed approach yields much better results in terms of a CVAR-based regret.", "questions": "- The most important question is: In your experiments, given a fixed number of samples, how does generate-then-optimizes using a generation model trained in a classical way compare to your approach that uses a surrogate decision loss for training", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- The paper addresses a critical weakness of classical decision-focused learning approaches, namely the fact that the optimization models rely on a single point forecast, and addresses that weakness by using a SAA-based approach to compute a CVAR objective.\n\n- The paper provides a surrogate loss function that is then usable in a classical end-to-end learning pipeline, and quantifies the error introduced by the surrogate loss.", "weaknesses": "- The paper ignores the fact that there exist many papers that deal with contextual stochastic optimization, see e.g. the survey paper Sadaba et. al (2025) (A survey of contextual optimization methods for decision-making under uncertainty, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2024.03.020), many of them using SAA-based approaches that rely on generating samples from conditional distributions that are generated by machine learning models.\n\n- Given those approaches, for me, the big (and obvious) question is: What is the benefit of training the (probabilistic) machine learning models used for generating the sample with a decision-oriented loss function instead of using classical loss functions? In other words: What is the benefit of step 2 (Model learning) in section 4.2 of the paper? This question is not answered in the computational experiments, and this is, in my opinion, the big weakness of the paper. If there is not a big benefit, why should the reader bother with the expensive DFL pipeline? Personally, I dot not think that the task-specific loss will bring a lot of advantage, but I would love to see results from experiments that show the contrary. Also, I would like to emphasize that in classical DFL, it is common to compare “predict-then-optimize” (with classically trained prediction models) to “predict-and-optimize” aka DFL. This is why I find it strange not to compare “generate-then-optimize” (for which there is a lot of literature, see section 4 of the abovementioned survey) to “generate-and-optimize” (which is obtained by using a decision-oriented loss function).\n\n- In my opinion, the experiments are also flawed in a different way as the approaches use different objective functions. This similar to a setting in which one solves a classical stochastic optimization problem with an expected value objective, and another with a CVAR-objective, and then compares their performance in terms of CVAR – it would come to no surprise that the solution of the first perfroms much worse.\n\n- The paper does not mention approaches such as the one Nathan Kallus, Xiaojie Mao (2022) Stochastic Optimization Forests. Management Science 69(4):1975-1994., which also uses a decision-focused approach to solve risk-averse objective functions involving CVAR, also using a sample-based approxiation objectives (this would also be a good benchmark approach)\n\n\nIn its present state, I find that the weaknesses sketched abive namely neglecting whole stream of relevant literature on contextual stochastic optimization and  not comparing  the decision-focused training of the generative model to a classically trained generative model using the same SAA-based CVAR loss function. If this decision-focused training is not significantly better, the main contributions of this paper are not significant enough to justify a publication at ICLR.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "f29mVsdksK", "reviewer": "Reviewer_KwJ3", "summary": "This paper proposes Gen-DFL, a “Decision-Focused Generative Learning” framework for robust decision-making under uncertainty. It extends decision-focused learning (DFL) by replacing deterministic point predictions with a conditional generative model (e.g., conditional normalizing flow) that captures the conditional distribution $p_\\theta(c|x)$ of optimization parameters. The framework combines this with a CVaR-based risk-sensitive optimization objective, forming a generate-then-optimize (GTO) paradigm. Theoretical analysis provides regret bounds suggesting that Gen-DFL performs better than traditional DFL when variance, dimensionality, or nonlinearity increases.", "questions": "- What is the relationship of $p_\\theta(c|x)$ and $q(c|x)$? Are they refering to the same thing?\n- Since $w^\\star$ is not observable in real-world decision-making, how is regret approximated during training and evaluation?\n- The regret definition, why is CVar positioned outside the regret difference rather than inside in equation (5)? Note that these two quantities can be different", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The topic is timely and relevant, as robust decision-making under uncertainty is a rapidly growing area at the intersection of optimization, learning, and generative modeling. The proposed framework is general and conceptually interesting—it integrates generative modeling and decision-focused optimization in a unified formulation that, in principle, could be applied across a wide range of uncertain decision-making problems. The problem setup is clearly motivated, and the generate–then–optimize structure is intuitively appealing.", "weaknesses": "- Limited conceptual novelty: The central idea—modeling parameter uncertainty via a generative model and optimizing CVaR—is highly similar to existing work in distributionally robust DFL and end-to-end conditional robust optimization (E2E-CRO). The paper does not clearly articulate how Gen-DFL differs from or improves upon these prior methods.\n- Mathematical presentation is confusing: As noted in the comments, notation is inconsistent ($p_\\theta(c|x)$ vs. $q(c|x)$ in equation (7)); $w^\\star$ is used even though it is inaccessible; and several definitions (e.g., for CVaR-based regret) are unclear. Some derivations are incomplete or informal, lacking well-defined assumptions and proof steps; and several mathematical objects (e.g., $R(x)$ in Theorem 5.4) are never defined.\n- Lack of explanation for the surrogate loss function: A central component of the algorithm—the surrogate loss function used for training—is introduced without any explanation, justification, or derivation. Since the entire algorithm builds upon this surrogate, the omission makes the methodology difficult to follow and gives an impression of a lack of transparency, as if the authors are glossing over key technical details. This severely hurts the paper’s clarity and credibility.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "Jz7ha2VOkb", "reviewer": "Reviewer_eGUr", "summary": "The paper tackles the problem of making decisions via predictive models. Specifically, this paper focuses on the end-to-end regime of decision focused learning (DFL) which has been explored in recent years. For DFL, the model often takes the form of bilevel structure with a predictive model (for the cost vector) and an inner minimization of a downstream expected objective. This paper defines gen-DFL which instead of a predictor uses a generative model that tries to predict a distribution rather than a point prediction of the cost vector. This results in improvements in high dimensional settings over baselines and the authors provide theoretical bounds on performance.", "questions": "Could the computational overhead of gen-DFL be problematic under settings that require retraining under distribution shift, perhaps? Is there an equivalence in solutions between the \"right size\" of an uncertainty set for pred-DFL with robust optimization and the full tail distribution of gen-DFL?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "It has clear empirical improvements and clear theory on generalization bounds and provides an interesting alternative method to the other formulations of DFL.", "weaknesses": "The justification of DFL versus point-wise robust methods is not theoretically clear. First, while they provide gaps between the point-prediction methods of pred-DFL and gen-DFL, first it is unclear what the \\Delta R term really means. The differing definitions of regret for gen-DFL and pred-DFL make sense as one is with a regret realized by a singular cost value and the other is over a distribution of realizations. To then take the difference between these two terms is what seems a little strange. Later it seems that the definition of \\Delta R in appendix A.8 uses the same CvaR regret for both but doesn’t use the defined pred-DFL regret. Thus, the \\Delta R term is confusing and some clarification would help here.\n\nFurthermore, the purpose of this bound is a little unclear. If the motive is to demonstrate theoretical improvements of the gen-DFL method over pred-DFL type methods, why is it an upper-bound on the absolute residual gap? Shouldn’t it instead be a lower-bound type result on the gap without absolute difference such that the regret of pred-DFL methods is at least some quantity larger than the regret of gen-DFL? \nSo while the upper-bound is valid, it doesn’t necessarily distinguish the two methods, and perhaps a corresponding lower-bound would help.\n\nLastly, while there are many solid synthetic experiments, there are only 2 real-world data ones, for which one of them diff-DRO outperforms (admittedly by a small amount). It would be nice to include another such experiment in which the empirical performance of gen-DFL on real data is demonstrated. Additionally the results on these real world experiments seems lacking.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "F6W6M7LpKt", "reviewer": "Reviewer_Uiro", "summary": "The paper combines the advantages of Decision-focused learning (DFL) and robust optimization (RO) to propose a decision-making framework robust to model uncertainty. The authors introduce a method that uses the conditional distribution learned by generative models instead of uncertainty sets to protect the decision-making against tail regions of the distribution. This results in optimizing a Conditional Value-at-Risk objective. The paper provides theoretical results on the loss difference using an approximation of the conditional distribution and regret gap between standard DFL and their method. They validate their methods on an energy-cost aware scheduling problem and the COVID-19 resource allocation problem, along with three synthetic experiments.", "questions": "* Why aren’t the RO baselines included in Table 1?\n* What data sets are used to produce Figure 3-7?\n* Why is the $\\beta$ hyperparameter introduced in the experiments section? Why can’t $\\gamma$ be varied in experiments instead of $\\beta$?\n* What is this “auxiliary model” in line 259? Are there two models in Gen-DFL?", "limitations": "", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper proposes a method that addresses the faults of RO and DFL. I believe the introduction, related works, and preliminaries provide sufficient context for the paper. The method was robustly validated in experiments with a wide range of data sets and ablation studies. The method was compared against many baselines in RO and DFL. The empirical results make a strong and convincing case for using this method.", "weaknesses": "Theorem 5.4, which is one of the two main theoretical contributions of the paper, studies the upper bound of $\\mathbb{E}_x\\left[|\\Delta R(x)|\\right]$ to characterize the factors that influence the performance gap between standard DFL and Gen-DFL. However, the authors analyze these factors to highlight the failure modes of Pred-DFL, which is reflected by $\\mathbb{E}_x\\left[\\Delta R(x)\\right]$ instead of  $\\mathbb{E}_x\\left[|\\Delta R(x)|\\right]$. While the analysis of the performance gap is interesting, it doesn’t strengthen the case made for Gen-DFL. \n\nAdditionally, it seems like the performance gap, as defined by the authors, should also have a distribution distance term between $p(c\\mid x)$ and $q(c\\mid x)$ (similar to the result in Theorem 5.1). It seems odd that the miscalibration of the generative model has no effect on the performance gap. \n\nIf a satisfactory upper bound and lower bound for $\\mathbb{E}_x\\left[\\Delta R(x)\\right]$ is provided, I would be willing to increase my score.\n\nMinor weaknesses\n* The theorem statement of Theorem 5.4 isn’t the same as Theorem A.8. \n* In lines 344-345, the authors argue that the estimation error of the predictor in Pred-DFL grows at a certain rate, but I don’t think this is immediately obvious. It would’ve been nice if there were some explanation around that statement or a reference to the original result.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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:

Decision: Accept (Poster)