Diffusion-DFL: Decision-focused Diffusion Models for Stochastic Optimization
Authors: Zihao Zhao
Christopher Yeh
Lingkai Kong
Kai Wang
Venue: ICLR 2026 Poster (2026)
Keywords: Decisoin-focused learning
stochastic optimization
diffusion models
Primary area: optimization

Abstract: Decision-focused learning (DFL) integrates predictive modeling and optimization by training predictors to optimize the downstream decision target rather than merely minimizing prediction error. To date, existing DFL methods typically rely on deterministic point predictions, which are often insufficient to capture the intrinsic stochasticity of real-world environments. To address this challenge, we propose the first diffusion-based DFL approach, which trains a diffusion model to represent the distribution of uncertain parameters and optimizes the decision by solving a stochastic optimization with samples drawn from the diffusion model. Our contributions are twofold. First, we formulate diffusion DFL using the reparameterization trick, enabling end-to-end training through diffusion. While effective, it is memory and compute-intensive due to the need to differentiate through the diffusion sampling process. Second, we propose a lightweight score function estimator that uses only several forward diffusion passes and avoids backpropagation through the sampling. This follows from our results that backpropagating through stochastic optimization can be approximated by a weighted score function formulation. We empirically show that our diffusion DFL approach consistently outperforms strong baselines in decision quality. The source code for all experiments is available at https://github.com/GT-KOALA/Diffusion_DFL.

Published as a conference paper at ICLR 2026
DIFFUSION-DFL: DECISION-FOCUSED DIFFUSION MODELS FOR STOCHASTIC OPTIMIZATION
Zihao Zhao
Christopher Yeh
School of Computer Science Georgia Institute of Technology zzhao 62 80 gatech. edu
Department of Computing and Mathematical Sciences California Institute of Technology
cyeh@caltecki edu
Lingkai Kong
Kai Wang
School of Engineering and Applied Sciences Harvard University
School of Computational Science and Engineering Georgia Institute of Technology kwang 920 gat.taachhhh.......ee.cc.cecccae edu
Lingkaikonggg. www.aaaraad.... edu
ABSTRACT
Decision-focused learning (DFL) integrates predictive modeling and optimization by training predictors to optimize the downstream decision target rather than merely minimizing prediction error. To date, existing DFL methods typically rely on de- terministic point predictions, which are often insufficient to capture the intrinsic stochasticity of real-world environments. To address this challenge, we propose the first diffusion-basse based DFL approach, which trains diffusion model to represent the distribution of uncertain parameters and optimizes the decision by solving a stochastic optimization with samples drawn from the diffusion model. Our contri- butions are twofold. First, we formulate diffusion DFL using the reparameterization trick, enabling end-to-end training through diffusion. While effective, it is mem- ory and compute-intensive due to the need to differentiate through the diffusion sampling process. Second, we propose lightweight score function estimator that uses only several forward diffusion passes and avoids backpropagation through the sampling. This follows from our results that backpropagating through stochastic optimization can be approximated by weighted score function formulation. We empirically show that our diffusion DFL. approach consistently outperforms strong baselines in decision quality. The source code for all experiments is available at https: /ttp//ggtt///tt////////////tttgt cww.coocc//coo/ooo GT BAA//GGTKKKLLLAA fusionnDFFL.
INTRODUCTION
Many real-life decision-makinn tasks require selecting actions that minimize a cost function involving unknown, context-dependent parameters. These parameters must often be predicted from observed features. For example, in supply chain management, future product demand must be estimated before deciding how much inventory to order (Tang & Nurmaya Musa, 2011). common approach the predict-then-optine -ootiiiiiee pipeline, where a predictive model is first trained using loss function such as mean squared error (MSE), and the resulting predictions are then passed to an optimization solver to guide decisions. While simple and widely adopted, this two-stage method can be misaligned with the true objective: minimizing decision cost. In particular, lower prediction error does not always lead to higher-quality decisions (Bertsimas & Kallus, 2020; Elmachtoub & Grigas, 2022).
Decision-focusee learning (DFL) addresses this misalignment by integrating the prediction and optimization stages into single end-to-end framework (Donti et al., 2017; Wilder et al., 2019; Mandi et al., 2024). Unlike the two-stage approach, DFL trains the prediction model specifically to improve decision outcomes, often resulting in solutions with lower regret. However, most existing DFL methods rely on point (deterministic) predictions as inputs to the optimization layer, despite the fact that in many real-world scenarios, the underlying parameters may follow complex distributions. Ignoring this uncertainty can lead to overconfident models and degraded decision quality (Kochenderfer et al., 2015).
inherently uncertain and

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In this work, we introduce novel DFL approach that leverages diffusion probabilistic models to capture the environment uncertainty in an end-to-end fashion. Here, we use a conditional diffusion model (Tashiro et al., 2021) to represent the distribution of uncertain parameters given contextual features. The advantage of integrating a diffusion model into DFL. is that, unlike simple distribution predictions (e-g.. Gaussian), diffusion models capture multi-modal or complex distributions. However, the sequential sampling procedure of diffussin models introduces a challenge when training diffusion model end-to-end for stochastic optimization. To address this, we develop two algorithms: reparameterization and score function. First, the reparameterization trick is common approach that expresses a random sample as deterministic function of the model parameters and some noise, and we can backpropagate through sampled prediction to solve the DFL problem.
However, this approach can be very costly in memory and computation because it requires differen- tiating (and therefore tracking gradients) through the diffusion sampling process. To address this, introduce lightweight score function estimator that avoids differentiating through the sampling process. Specifically, we use a score function surrogate to approximate the gradient of the diffusion predictor and plug into the KKT (Karush-Kuhn- Kuhn-Tuccer the total derivative of the decision objective. In addition, we further mitigate the high variance that arises from using only score functions for few steps by employing a tailored importance sampling strategy. Specifically, instead of sampling diffusion timesteps uniformly, we assign sampling proba- bilities proportional to the gradient magnitude each timestep, which keeps the estimator unbiased while substantially reducing its variance.
cuplicit-differentien
approach obtain
We evaluate our proposed methods in various applications, including (synthetie) product allocation, energy scheduling, and stock portfolio optimization. Experimental results show that our diffusion DFL. methods consistently outperform all baselines, with more gains larger-scale problems. Moreover, the score function estimator achieves decision quality comparable to that of the reparameterization method, while significantly reducing GPU memory usage from 60.75 GB to 0.13 GB.
The contributions of this paper are the following:
We introduce the first DFL method that uses diffusion models to capture the downstream uncertainty and employs the reparameterization trick for end-to-end gradient estimation. We propose a lightweight score function estimator that avoids backpropagating the reversing process in the reparameterization method, significantly reducing memory and computation cost.
We evaluate our methods in three real-world optimization tasks and observe consistent improve- ments over strong baselines.
2
RELATED WORKS
Decision-focused learning. DFL is growing and increasingly influential approach that trains models end-to-end to directly optimize decision quality rather than minimize prediction error (Donti al., 2017; Wilder et al., 2019; Mandi al., 2024). Despite the success in aligning learning objectives with decision-making, limitation of most existing DFL. methods is that they typically rely on deterministic point predictions of uncertain parameters (Wilder et al., 2019; Shah et al., 2022). By ignoring distributional uncertainty, deterministic point predictions cannot represent the full outcomes and may lead to lower decision quality (Wang et al., 2025). Empirically, classic DFL was observed to struggle in high-oimennsionl and risk-sensitive real-world settings with significant uncertainty (Mandi et al., 2022).
Therefore, the gap in uncertainty modeling motivates the need for more comprehensive DFL with stochastic predictions, where several works have started integrating uncertainty awareness into the DFL pipeline (Silvestri et al., 2026; Wang et al., 2025; Shariatmadar et al., 2025; Jeon et al., 2025). For instance, Wang et al. (2025) proposes generative DFL approach (Gen-DFL) based on normalizing flow models as the predictor. However, normalizing flows require bijective network architecture, which restricts the expressiveness of the stochastic predictor.
In this paper, we propose using diffusion models (Ho et al., 2020) as a more expressive predictor. By leveraging diffusion models in the DFL paradigm, our approach extends DFL by predicting an accurate full distribution of the unknown parameters, which addresses the overconfidence of deterministic optimization and better aligns with downstream decision-making needs.
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Orthogonally, Silvestri et al. (2026) propose using score function gradient estimation to extend DFL to settings with non-differentiable optimization solvers. However, their approach relies on predictors for which the score admits closed-fon expression. In contrast, the score cannot be computed in closed form for diffusion models. To address this, we approximate the score using the gradient the EBLO as surrogate and provide theoretical bound on the approximation emror.
Diffusion models in optimization. Diffusion probabilistic models have achieved great success in modeling high-dimensional data distributions in recent years (Sohl-Dickstein al., 2015; Song & Ermon, 2019; Dhariwal & Nichol, 2021). Originally popularized for image generation and related structured outputs, its ability capture multi-modal and high-variety distributions has made attractive beyond vision tasks. such as combinatorial optimization (Sun & Yang, 2023; Sanokowski al., 2025) and black-box optimization (Krishnamoorthy et al., 2023; Kong et al., 2025).
To our best knowledge, however, no prior work has integrated diffusion models into a predict-then- optimize learning pipeline for decision tasks. This paper is the first to harness diffusion models in an and-to-end DFL framework. By using a conditional diffusion model, we can learn rich distribution over the uncertain inputs and then propagate this uncertainty through the downstream decision via gradient-basee training (score function and reparameterization). This approach combines the strengths of expressive generative modeling and DFL to improve decision quality under uncertainty.
PROBLEM STATEMENT
3.1 RECISION-FOCUSER LEARNING
We consider general predct-then-optimizz setting (Donti al., 2017: Elmachtoub & Grigas, 2022), where the goal is to make decisions under uncertainty about a key problem parameter. Given a feature vector x E X' and prediction of an unknown parameter yy Y, the decision-make selects = Rd to minimize a decision loss function f y x Rd + R. which measures the cost of applying decision when the true parameter is y². We assume a joint distribution D over (x, y²) pairs.
DFL integrates prediction and optimization into a unified framework. The goal is to learn a decision function 2g X + IRd, parameterized by 0, that minimizes the expected decision loss,
min F(0) := (= [[('" (x))]
The decision z (x) is typically obtained by solving an optimization problem involving a prediction of the uncertainty parameter. Most DFL methods (Mandi et al., 2024) use deterministic point prediction yo (x) of the uncertain parameter
(x) = arg min (1)e(x), 2), s.t. G2 < h, A =
where G E Baxd h E R" A E Rexd b Rº are constraint problem coefficients'.
In contrast, we consider probabilistic model Po(- I)) for the uncertain parameter v" and let (x)
be the solution to a stochastic optimization problem:
z (z) = arg min Py a) (1, s.t. Gz S h, Az
We aim to learn the model parameter such that zσ minimizes the expected decision loss F(O). By the chain rule, the derivative of F is
dF(() d0
Of 0f(v",=(xz)
(z)(x)
= BANDDD
0z
d0
d=j However, computing this gradient (specifically, the dØ term) is challenging because = is implicitly defined by a nested optimization problem. A common solution is to differentiate the KKT system that implicitly defines z w.r.t. 0 (Amos & Kolter, 2017). Another crucial point is the selection of the stochastic predictor in DFL, which in the paper we choose to use diffusion models to represent Pa.
We consider affine constraints in our main paper for simplicity. The extension from affine constraints to general convex constraints follows similar derivation in the linear case.
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3.2 DIFFUSION PROBABILISTIC MODEL
To generate complex multi-modal and high-dimensional distributions, diffusion probabilistic mod- els (Ho et al., 2020) are promising approach. It couples fixed noising chain with a learned reverse denoising chain. Let yo E R denote a sample from the real data distribution (()) and {9g E (0, 1)}ر_ denote the noise schedule. Define 04 = 1 - fr and o = I ds. The forward process q adds Gaussian noise each step y1 through yr:
q(v vt-1) Nue: V1 B. B.T,, t= t=
which guarantees that ((r yo) becomes nearly standard normal distribution T + oc with common schedules (δT - 0). Note that y/8 can equivalently sampled without iterating through intermediate time steps: Yt vāyyo + v âge, where (0, I) is Gaussian noise.
In the reverse process p. the diffusion model predicts the unknown added noise by
poly y/) (yt-;; poly,, 1), of
whose mean (=)+t) parameterized by neural network predictor and variance is either fixed (o? = 33) or learned. The combination of and is equivalent to a hierarchical variational auto- encoder (Vahdat & Kautz, 2020), and thus can be optimized by using the evidence lower bound (ELBO) as the loss function (Hoffmman & Johnson, 2016).
Conditional Diffusion Model. Throughout this paper, denotes contextual features, and every transition probability is conditioned on x (Tashiro et al., 2021). We use Pa(-|z) for the diffusion model's conditional distribution for generated data and po (3t-1 x2) . for its Markov transitions.
4 STOCHASTIC OPTIMIZATION AND REPARAMETERIZATION ESTIMATOR
Real-world decision problems often face significant uncertainty in their parameters. Optimizing with a stochastic predictor (e.g.. diffusion model) yields better results than deterministic optimization, by explicitly modeling the uncertainty and optimizing the expected cost. Figure 1 illustrates a simple example: any deterministic solution ends up at an extreme decision with higher expected cost, while the stochastic solution averages costs across likely outcomes and selects an interior decision with lower expected cost.
(a) Per-sample Decision Cost fly.z)
(b) Expected Decision Cost s(fly,z)
(c) Predicted vs True Distribution
Decision Cost
2.0
Sta
Dei
e
Det
Decision
Densing
3
1.5
19300 C-tiee
1.0
angnin.
0.5
1.5
0.0
0.5
1.0
1.5
2.0
-2
1
y
Figure 1: A comparison of deterministic vs. stochastic optimization with cost function exp(-yz)
as described in Section 6.1. (a) Each curve represents a cost function given sample For any fixed y. the deterministic optimization decision lies one of the boundaries (=* = or = C). (b) When averaging the cost function over many samples of y. the stochastic optimization decision lies in the interior of the feasible region instead of on the boundary. Thus, any deterministic optimization
decision is suboptimal.
(c) A probabilistic (diffusion) model captures distribution over Y that closely resembles the true bimodal distribution.
Solving stochastic DFL. Formally, in the stochastic case, the optimality condition for the decision problem must consider expectation. The stationarity condition for decision problem Eq. 1 becomes
V.L(O, 2
= Ey fvf(y, =)] G + ATD" = 00
where L denotes the Lagrangian. Note that the dependency on 9 in the stationarity condition is in the distribution. Therefore, need to handle this dependency carefully while differentiating the KKT
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system with respect to 8:
i 08 (V_(( X v² x))
② 0e (B (
x ATv 0.
(3)
distribational gradieut
To resolve the dependence of both the predictive distribution Po(y x) and the decision =" on we first adopt the reparameterization trick (Kingma Welling, 2014) for the diffusion model. From Section 3.2, recall that the diffusion sampling process introduces Gaussian noise at each step. Thus, we can reparameterize the reverse process by fixing all the random draws (Gaussian noises). Formally, sample y Po(y 2)) can be expressed transformation y R(e, 0 2)) of a base Gaussian noise sample e P(c). where differentiable in 8. This makes the diffusion sampling a deterministic function of 0. Then have
VelBy~PPo(la))// =)] ELe V [1,,
Next, we incorporate this into the optimization. Following Eq. we can formalize KKT system that contains derivatives of zū, X", n² and VeByy-ro 2)]. Plugging the reparameterized gradient estimator into the KKT system, we can solve for objective F by multiplying dzy
d=g and then obtain the total derivative of the final d8
of (Donti et al., 2017) (proof can be found in Appendix A.1):
H D(NIG A
GT D(G
A
-1
Be- (Vo)) (()) fy 26)]
dF d0
dzy 0
h)
0
0
0
0
where H = By--P((la)) fv, 27]] is the Hessian of the Lagrangian with respect to 2, and D(v) denotes diagonal matrix with v on its diagonal. In practice, one can sample from certain distribution (e.g.. Gaussian) multiple times to estimate the expectation and then obtain the gradient. This gives us reparameterization-bee diffusion DFL using Eq. to run stochastic DFL optimization.
5 SCORE FUNCTION ATTIMATOR
major obstacle to implementing the total gradient (Eq. 5) is the need to backpropagate through the diffusion sampling process. In most cases, the diffusion model's generative process is complex and multi-step (e.g.. 1000 steps), which makes backpropagating through all those steps memory- intensive and prone to instability. To address this, propose diffusion model, which circumvents explicit backpropagation through all sampling steps. The key idea is rewrite the Jacobian Voy in terms of the score Vo log Po æ:), and then approximate the score with the diffusion model's is ELBO training loss.
function' gradient estimator for the
5.1 TRANSFORM THE JACOBIAN INTO SCORE FUNCTION
We begin by rewriting the gradient of expectation as an expectation of a score function using the log-trick (Mohamed et al., 2020). Formally, if Po(-)x)) and is function not dependent on 0, then by the log- derivative trick we have
Vo Ey 2)] = (f 2) Ve log Po x)]-
(6)
Intuitively, instead of differentiating the output through each diffusion step, we only need to compute the gradient for the final log-likeling likelihodoo,, which avoids the need to differentiate through the diffusion sampling process and yields an efficient estimator for the gradient.
Then, one remaining difficulty is that directly computing the exact Va log Po xx) is complicated in practice because Po w) defined as the marginal probability after integrating out the latent diffusion trajectory. To obtain computationally efficient estimator,
the diffusion model's
II this paper, score function refers to the statistical score Vg log Po(y a)) (gradient of log- likelihood w.r.t. model parameters), opposed Stein's 's V. Vyp(y Ve-1, often used the diffusion literature.

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training objective surrogate for the oog-likeihhod. Specifically, diffusion models are typically trained by maximizing an ELBO that lower-bounds the log- log-likelihood:
log Po(uo) log pa (yo (v1) pa (w (12) po(ur- 11xx) (wx) dyier
(Vr-1 116)) (o(
log 1
g (2t-
pa (1h-1 (ee) + logpe(yr))
log
ELBOO 0),
(7)
g(yn 11+-
where the incquality is due to Jensen's. s. To approximate V log Po(vo Ix) conditioned on the gradient of the conditional ELBO loss as a surrogate:
Va log (yvo x)) R Va ELBO x;; 0).
(8)
To motivate this approximation, we provide theoretical justification for the surrogate ELBO gradient. Proposition 5.1 (ELBO gradient approximation error). Let so so(2; yo) Vo log po you 2) denote the joint score function. If sup.. || [2;; vo) & B(0, yo) for some finite upper bound ( yo) > 0, then
Vo log Pa(yo) Ve (y: 0) s v2B(0, vo) KL(g(VIT yo) (po vo)).
See Appendix A.4 for proof and more details. Consequently, whenever the variational posterior (diffusion reverse process) is a good approximation of the target distribution (diffusion forward process), so that the KL divergence term small, the ELBO gradient is guaranteed to be close to the true score. This justifies our use of V ELBO (yu: 0) as a surrogate for V log Pe (vo).
Cosine similarity between reparam and score function
In practice, we first sample a final output y from the diffusion model given contextual features We then sample a subset timesteps (41, 12,,2,,,f } (k < T)
Cosina similanty
and forward noising process q to generate the tra- jectory (Ve, Ut....., Ut.
0.8
). As in DDPM (Ho et al., 2020), we adopt the simplified form of ELBO Be..22].wo....llle
0.7
(vr, ]. We evaluate the ELBO on the sampled trajectories and compute its gradient w.r.t. as an estimation of the true score.
Epoch
Figure 2: Cosine similarity between the reparameterization and score function gra- dient across different dimensions.
Empirical evidence suggests that the ELBO gradient closely tracks the true score, as shown in Figure 2, mak- ing Eq. 8 reliable proxy in practice.
5.2 OVERALL GRADIENT FOR SCORE FUNCTION
By plugging the ELBO gradient approximation from Eq. 8 into Eq. 6, we can express the KKT conditions without using reparameterization and thus obtain the score function-based derivative:
dF d0
dzg
dF
H
GT
A
E (a))) Vel (V,
D(*)G
D(G2)
h)
0
0
Practical algorithm weighted ELBO gradient. To compute the score surrogate in practice, found it convenient to treat the total gradient as an aepoortacce-eiightee form:
d.F' d
E Po(la)] detach detach[wo(v)
(1/
(10)
dB do
importance weight, no grad
1-slep forward
where ww(y) is the importance weight simplified from Eq. 9 (see Appendix A.3 for complete form). This yields a weighted-ELBo gradient estimator: we treat wy)) as stop-gradient weight and only differentiate the ELBO w.r.t. 8, greatly reducing computations. We implement the entire gradient computation as a user-friendly PyTorch autograd module: the forward pass returns the optimal decision 2" (and X", v²), and the backward pass computes the gradient d' derived above.

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Variance-reduction strategy. While the score-function estimator is effective, naive implementa- tion of the weighted ELBO loss in Eq. 10 can suffer from high variance, leading to unstable training. In practice, we found that carefully designing the sampling strategy for the ELBO loss is crucial to obtaining low-variance and stable gradients. To reduce the variance, we utilize the method from Improved DDPM (Nichol & Dhariwal, 2021) ) for choosing diffusion steps. Specifically, instead of uniform sampling, we use importance sampling over timesteps with probability and weights 1/p::
Vo (o(EL)O Pt
V ELBOO Beepe
where Pt
B V (ELBO:) and Σ Pt =
This method remains unbiased, but the variance is minimized. In essence, this approach gives less weight to the early timesteps that have large gradients and more weight to later timesteps.
EXPERIMENTS
We evaluate the performance of our diffusion-based DFL approaches variety of tasks, comparing against several baseline methods. Specifically, we consider:
Two-stage predict-then-pptimzzz baselines: deterministic MLP, Gaussian probabilistic model, and diffusion model trained to minimize prediction error (Elmachtoub Grigas, 2022). Deterministic DFL: deterministic MLP model with end-to-end DFL training (Donti al., 2017).
Gaussian DFL (both reparameterization and score function); Gaussian probabilistic model with end-to-end stochastic DFL training; see details in Appendix A.6.
Offline Contextual Bandits: solving the full-information contextual bandit using policy-based learning (Brandfonbrener et al., 2021). Note that this method ignores the constraints of the optimization. See details in Appendix A.7.
Diffusion DFL (ours): our diffusion model predictor, trained with either reparameterizatiion or cooreffunction gradient estimators.
Architectural and training details for all methods are summarized Appendix A.8.
6.1 SYNTHETIC PRODUCT ALLOCATION TASK
In this example, we consider factory that decides how much to manufacture for each of d E N products. The parameter E IRd represents the profit margin for each product, i.e., Y, the profit per unit of product i; due to uncertainty in market conditions, Y is uncertain. The factory's decision E [0,C] Cl represents how much of each product to manufacture, where the maximum capacity for each product. For simplicity, do not consider any contextual features in this example. That means DFL learns a distribution that generates y that can minimize the decision objective.
Suppose that the factory has a lisk-everse cost function f(y, 2) the factory wants put a larger weight on the product with higher profit² Y, Under uncertainty, the decision-make seeks to minimize the expected cost by solving a stochastic optimization problem:
exp( -y- 2), which indicates that
arg min Po(-xx) /exp(-x) lexp- sE[0,C]
2)]-
2stn
In this stochastic case, the optimal investment 2 typically lies in the interior of the feasible region, which balances the potential high reward of investing against the risk losses.
Experimental setup. We simulate the uncertain parameter Y drawn from a mixture of Gaussians,
Y
idd
~ - Nia ( (1(-p)))(((b0)))
Specifically, we set = 0.8, = 1,b 3, 0.15, C= We train each model (deterministie, Gaussian, diffusion) on this distribution in a docision-focused manner (for DFL methods) or on pure regression (for two-stage), and evaluate the expected cost achieved by the resulting decision = (z).. We present the results of one product = 1) Figure/ and 10 products (d 10) in Table
'eee, we have ignored the degenerate y = 0. To deal with the degenerate case, one could add zero-centered bump function c(y) the objective f 2).
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6.2 POWER SCHEDULING TASK
In this experiment, we evaluate our method on a real-world energy scheduling problem from Donti al. (2017). This task involves 24-hour generation-scheduligg chooses E 22t (hourly generation). Given realization of demand, the decision loss penalizes shortage and excess with asymmetric linear costs (% and %e) plus a quadratic tracking term; the decision must also satisfy a ramping bound Gr. Let [t]+ == max(u, 0). We have the decision loss as the quadratic problem:
problem in which the operator
1 min By-ro(1 Σ E Bo xiI+ Velas - uil+ 2((= -4,)%,
z1-1 s.t. 2 21-1 S Or for all
Experimental setup. We use more than 8 years of historical data from regional power grid (PJM Interconnection, 2025). Feature includes the previous day's hourly load, temperature, next-day temperature forecasts, non-linear transforms (lags and rolling statisties), calendar indicators, and yearly sinusoidal features. Given z, the prediction model P&( (z) outputs distribution over y E 1224. We report the test decision cost Table and a held-out horizon in Figure
6.3 STOCK MARKET PORTFOLIO TASK
In this experiment, we apply our diffusion DFL approach to financial portfolio optimization problem under uncertain stock returns. Here, the random vector IR" represents the returns for the assets on the next day, and the decision z E IR" represents the portfolio weights allocated to those assets. We consider a mean-variance trade-off decision loss: maximize expected return while keeping the risk (variance) low. This can be written minimizing a loss that is negative expected return plus quadratic penalty on variance:
a yy -- 2). s.t. 11
min By--oo-la)lffy 2)] Eg
0
2
where > risk parameter and constraints enforce that is a valid portfolio. In practice, the deterministic solution may concentrate heavily few assets and yield low average return, whereas stochastic approach can achieve higher returns by accounting for variance.
Experimental setup. We have daily prices and volumes spanning 2004-2017 and evaluate on the S&P 500 index constituents (Quandl WIKI dataset, 2025). The features x R22 include recent historical return, trading volume windows, and rolling averages. The immediate-retum predictor Po(-Iz) is to predict the next day's price. We report the performance of different DFL baselines with 50 portfolios in Table and other sizes of portfolios in Section 7.2.
Table I: Results for different optimization tasks. Our two diffusion DFL methods achieve the best and second-best decision quality in all 3 tasks, significantly better than other baselines. Bolded values are the best in test task losses; underlined values are the 2nd-best. Mean + standard eтror across 10 runs.
Synthetic Example
Power Schedule
Stock Portfolio
Label/ / Method
RMSEJ
Task]
RMSEJ
Task
RMSEJ Task (%)
Two-stage (TS)
Deterministic TS
0.639 20.00
1.987
10.00
0.120
10.00
41,239
13.18
0.027
10.00
0.04% 1:0.4
Gaussian TS
0.720 2.000
1.272 00.23
0.117
10.00
5,580
20.45
0.188
10.03
0.10% 1.004
Diffusion TS
0,905
0.393
0.000
0.147
10.00
7.901
20.78
0.455
1
0.13%
0.0.3
Offline Contextual Bandits
Policy- based Learning
Policy-based
0.873
0.568
4.712
4,440 17
0.064
1.74%
Decisson-focused learning (DFL)
Deterministic
0.640
1.987+p..0
4.997
4.324
0.032
0.07%
Gaussian Reparameterization
0.707
1.1694p.os
4.525 20.12
3.724
20.24
0.189
0.08% 40.08
Gaussian Score Function
0.708
1.132+t..00
4.713
4.087 =0.06
0.187++o.03
0.14% 40.00
Diffusion Reparameterization
0.852
0.365
0.000
3.141 10.00
3.152
0..03
0.06331..00
4.17 5 1:0.24
Diffusion Score Function
0.849
0.362
1.0.00
2.893
3.171
00.02
0.067
1:0.00
3.98%
1.031
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7 DISCUSSION OF EXPERIMENTAL RESULTS AND ABLATION STUDY
7.1 DISCUSSION OF RESULTS IN TABLE
Two-stage vs. DFL. As shown in Table I, across all three experiment tasks, we find that end-to-end DFL leads to better downstream decisions than the conventional two-stage approach. Conventional two-stage methods minimize RMSE during training, but this often leads to poor downstream decisions. In contrast, all variants of DFL directly minimize the decision cost during training and thus achieve lower decision costs.
Offline Contextual Bandits vs. DFL. Table shows that the policy-based offline bandit method can outperform two-stage approaches on tasks such as power scheduling and portfolio tasks, since it directly optimizes the downstream task objective rather than training predictor with a surrogate loss. It is also competitive with deterministic and Gaussian DFL variants. One reason is that, in some settings (such as the synthetic task), DFL with deterministic optimization (the Gaussian written in closed-form as a deterministic optimization) fails to represent multi-modal distributions on uncertain parameters and is therefore fundamentally poor policy class that does not include the optimal policy. An expressive enough OCB policy class, on the other hand, can learn strong context-to-action policy.
can be
Nevertheless, Diffusion -FL remains consistently superior to offline contextual bandits: by incorpo- rating the known optimization structure and coupling flexible generative model with the downstream solver, can better capture complex decision distributions and deliver higher-quality decisions.
Deterministic vs. Stochastic Optimization. Our results show that stochastic DFL methods outperform deterministic DFL in terms of decision quality every task. By modeling uncertainty, stochastic predictors enable the decision optimization to account for risk and variability in outcomes. For instance, in the portfolio experiment, the deterministic DFL yields only 0.07% return, whereas Gaussian DFL modestly improves that, and our diffusion DFL achieves nearly 4% average return. These gains come from the stochastic models' ability predict uncertainty: instead of committing to point prediction of y, the stochastic DFL produces decisions for range of possible outcomes.
Benefits of Diffusion DFL. Among the stochastic approaches, including baselines using Gaussian models, our diffusion DFL method consistently delivers the best decision performance. In particular, the diffusion model's strength is the capacity capture complex, multi-modal outcome distributions that a simple parametric Gaussian cannot represent. The Gaussian DFL sometimes falls short of the optimal decision quality. The diffusion model, on the other hand, can represent more intricate distributions of y. leading to decisions that better reflect complex scenarios.
7.2 ABLATION STUDY
Comparison Cost for Reparameterization and Score function. A key finding from our ablation study is the computational advantage score-function approach over the reparameterization. Here, we measure the trade-off between training cost and the final decision performance for different gradient estimators and sampling budgets.
In Figure 3 (a), we see that all variants reach similar final performance on the test set, indicating that even using as few 50 samples is sufficient to optimize the decision quality accurately. Figure 4 plots the GPU memory cost alongside the final test loss. The reparameterization method very computa- tionally expensive, requiring about 60 GB of GPU memory for backpropagating through all diffusion steps. In contrast, the score-function with 50 samples achieves virtually the same test loss as the reparameterization method (difference within 0.02) while using an order of magnitude less memory. Even with 10 samples, though slightly worse in loss, it still outperforms the deterministic baseline and uses tiny fraction of the compute. These results validate that the score-function approach retains the pecision-quality benefits of diffusion DFL while dramatically cutting computational requirements, making diffusion DFL practical even for complex problems.
Gradient variance reduction. As discussed in Section 5.2, using the score function estimator allows us to avoid backpropagating through the entire diffusion sampling process by only sampling a limited number of diffusion timesteps per update. The reason behind this is that a naive implementation, sampling timesteps uniformly random, would yield very high variance in the gradient estimates,

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Published as a conference paper at ICLR 2026
(b) importance sampling (IS)
Compare GPU cost
test loss
10
scoore
GPU cost -0-
Test DEL, loss
60,75
7.00
DRL loss
Test DFL loss
20
0,15
0.35
3.50
10
20
40
30
20
40
15
S7
10
51.50
rp
Epoch
Epoch
Method
Figure 3: Learning curves for (a) score function with 10 and 50 samples (sf /0 and sf 50) and reparameterization (rp), (b) score function and importance-weiighed score function with 10 samples.
Figure 4: Computation cost vs. performance trade-off for diffu- sion DFL training
which then leads to unstable training. Intuitively, early diffusion steps (large noise levels) dominate the ELBO loss and its gradients, so if they happen to be sampled, they contribute disproportionately and noisily. With a small random subset of timesteps, the gradient estimate can thus be highly imbalanced and noisy, which causes training divergence in practice.
To address this, we adopt an importance sampling strategy for choosing diffusion timesteps. Empiri- cally, as shown in Figure: 3 (b), the learning curves with the importance- weighted sampler are much smoother and more stable than with the uniform sampler. The score-function DFL training no longer diverges; instead, it converges cleanly, indicating that our variance reduction strategy successfully stabilizes the training process for diffusion DFL.
Comparison different problem sizes. A key challenge for DFL is scalability: as the de- cision dimension grows, many methods degrade significantly Mandi et al. (2024). In this exper- iment, we investigate the performance of DFL. methods under various decision dimensions in the stock portfolio. Specifically, we set the deci- sion dimension range from 10 to 100 and report the test regrets. As summarized in Figure 5, the regret gap between diffusion-DEF and Gaussian and deterministic methods increases with the growth of dimensionality, which demonstrates that diffusion DFL scales effectively in more complex decision settings.
Stock Portfolio
Performance Regret
Recret
10
100
securties)
Figure 5: Test regret vs. decision dimension d in the stock portfolio task.
CONCLUSION
We propose the first diffusion-based DFL approach for stochastic optimization, which trains diffusion model to capture complex uncertainty in problem parameters. We develop two end-to- end training techniques integrate the diffusion model into seeision-mmaingg:: reparameterization and score function approximation. As demonstrated with empirical evidence, the score function method drastically reduces memory and computation cost while achieving similar performance to reparameterization and being easy to train. Empirically, diffusion DFL achieves state-of satte-of-the-ar results on multiple benchmarks, consistently outperforming both traditional two-stage methods and prior DFL approaches.
REPRODUCIBILITY STATEMENT
We release open-source repository containing all code, configuration files, and scripts needed to reproduce our results, including data generation and figure plotting. All proofs for the main paper are stated in the appendix with explanations and proper assumptions.
ACKNOWLEDGMENTS
This work was supported by NSF grants (IIS 240-244,,4, CCF-2326609, CNS-2146814, CPS- 2136197, CNS-2106403, NGSDI-2100648), Schmidt Sciences A12050 Fellowship; NIH grant R01HL HLHH81149))
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gifts from Amazon, OpenAl, and Latitude AI; Quad Fellowship; and the Caltech Resnick Sustain- ability Institute.
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A APPENDIX
Notation, We let 0f denote the Jacobian matrix where (z))
01 and v
01
denote
i. J0y
the gradient. For vector v, (()) denotes diagonal matrix with on its diagonal. Let P(.) denote a probability distribution and p(-) denote a probability density; in particular, for diffusion models we use Pg for the model's output distribution and pe for transition densities.
In this appendix, we derive the decision optimization problem with general convex constraints rather than merely linear constraints. Assume the optimization problem is
z (x) arg min Po z)], s.t. h(x, 2) S 0, g(x, 0,
where hax, 2) denotes the convex inequalities constraints and g 2) 0 denotes the equality constraints.
A.1 PROOFS FOR SECTION 4
Proposition A.1 (Reparamcterization trick in diffusion models). Let E N+, and suppose the reverse diffusion model defines Gaussian distribution in Eq. 2 with fixed scalars ot 0 and standard normal prior yr No, I). Let {Ge o be i.i.d. N NO0, I). Then the model output y can be expressed as transformation y R(EO:T, x) of base noise distribution
P(e), where R is differentiable in O. Also assume E, Po(-a()) [f (v, z)] is continuously differentiable. Then we have
VO y Pot
Pem)(s)
Σ
II
Ju
Ag
(of(R(G 1
where A ==
(pu(grat.r) 00
Jg =
@uolvest.r)) Oye
and we define
Il Ju :=
Proof. The conditional diffusion reverse process is defined Vt-1 = po (yte, 1, x) 0e6t-1, VT = GT:
where the noise term d+6t-1 9-independent. Differentiating both sides w.r.t gives
dyc-1 3po(Wi,t, (Me, x)
Opo(yr,t, dy.
00
00
Oy. 00
Denote
( (ye, t,
Jr (yo(ye,t,0) txx)
0y:
A:
G. :=
00
Oye
00
Thus, we have
G A₁ J.Gr =
Our final goal is:
0yo
VoR(EOTT 0|z) =
= Go
00
............... A1 + J1Ag + J1J2A + ... + J.... Jr,
T 4-1
Σ
Ju
A.,
where we define [T0=1 Ju := I. Then, we have
VoEy [f 2)] = V, (f
EewP(e) VoHi(E x)) 1x) (
II II A.) ()
EEWP(e)
Σ
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Lemma A.2 (Gradient of Reparameterization method). Assume the model prediction y be expressed as a transformation = T(e, objective F w.E.. 0 can be computed x), P(c). The total derivative of the decision
dF
H D
G D(h(2,2))) 0
Q
Bew (e))( Dg .0]]
dF d0
dez
0
0
where H = Ey-Baata 2e 2')+ 2 h(x, 2²)) is the Hessian of the Lagrangian with
respect to 2, G = V2h( 2²) is the Jacobian of the inequality constraints in z", and Q = V=g(æ, =*) is the Jacobian of the equality constraints in 2".
Proof. At the primal-dual optimal solution (zg. Ap. 1/6) to Eq. 1, the following KKT conditions must hold:
V Vi0 20, Ao uo; x) = 0,
Ag ② h (z, 2g) O,
g =6) = 0
A 0, pg 2 0,
h(2, 26) 0.
Since h does not depend on here, we can apply Proposition A.1 the KKT conditions get
C
Ov.Coz*
ov vv
80
Oz 00
0x
00
Ov
00
0z
-EL 01x))
V, V ff(z" b))l + (Ey-ra + x²))
00
01*
0X*
+ V2h(E,, x-)
V V=g(z, =²)
00 = 0.
00
0x 3 h(x
Ox @ h(x, =²) 0x* -(X"'V 02*
O2*
+
+ D(h(I,22))
00 0.
Oz
00
0x
00
00
matrix form, we have
H
GT
E.
v2 f="
D(NO Q
D(h(x,2)))
0
0
86
where H = By Pa V f(" 8)] 222 (A* h(x, 2²)) + 22 g(x, 2 2)), G Veh(z, 2n) and Q = V vlt ))) Furthermore, equalities and inequalities are affine (as in main paper), H reduces E fu, 2)] since V v2_h V2.9
By chain rule, we have
dF
dF
0=
d=-
d0
0
28
GT D D(h(z,=*)) Q
Q
E P(e) (VoT(c
d=
0
0
0
0
A.2 PROOFS FOR SECTIONS 5
Proposition A.3. Let Poly | x) be a probability density parameterized by E θ, and let f ) x dd + R be scalar-valued function that does not depend on 9. Fix any E Rd. Suppose that there exists some neighborhood N ((e)) C around Og E θ such that the following assumptions satisfied:
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Published as a conference paper at ICLR 2026
1. For all 0 E ((b)). the function h(y) Poly) 2) f(y.2) 2) is integrable;
2. For all 0 E (6)) and almost all y E Y the gradient VoPoly x) exists: and
3. There exists integrable function g
R that dominates VoPoly z). That is, for all 0 E (f0) and almost all E y, VV x)]l - lg(v))-
Then,
Vo [f =)] E Pa (1) 2) Vo log Poo (y x)).
Proof. We make of the log-derivative trick:
Poo x) V log Poo (y x) = Peo (y x) Peo (y x)
VaPa. VaPa. (y x).
Then
Vo / f(y. Po. 1x)
Vo By Poo (11))
Vo [f 2) Poo (v/ x)]dy
(by Leibniz integral rule)
fly 2) Pa. (v x V. log Poo (y) x) dy
(by log-derivative trick)
Pa (12) [f 2 Vo log Poo (v x))
Lemma A.4 (Gradient of Score Function). The total derivative of the decision objective F w.r.. can be computed as
GT
QT
Ey (Vg- dELBO
dF
d=
d8
D(N) Q
D(h(z,z)))
0
0
d0
0
Proof. Differentiate this KKT system w.r.t. 0 and applying Proposition A.3 yields
OV.C
0V
Oz
8V.L 0x*
VV Ov
dv* Öv
00
Oz
00
OX
00
00
02"
E By-Pa((
Vef(2" 8) (V log Po()( 11 + (By-
+ 22 (axT)("x)))))
00
OX*
Ov
+ Veh(x, )*)
V.g(z, x
08
00
ax~ h(x, 2²)
02"
ax h(x, 2*) 0x*
D(*')V =*) Oz
0x*
+
+ D(M,,, =*))
3z
00
0x
00
00
00
In matrix form, this becomes
H D(NG D(M(z,z)))
GT
Ey V 2 () log Pa x])
Ox
0
(11)
D0
0
dv
08
where H = ( ..8 V (A-
h(xx,))), (ATTA(( G= Veh(z,2").
Applying the chain rule to F now gives
dF
dF d0
d=
20
A9
0
20
dF
H
GT
Q
PurPoelee (V log Poly x))
D(Y)G D(h(z,=*)) 0
0
Q
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Published as a conference paper at ICLR 2026
Then, replace V log (1 x) with the gradient of ELBO score for sample y and have
dF d= 0
H D(X)G
G
Q
(ot))) v)- (Vo ELBO((:))) '1]
dF d0
D(h(z,z*))
Q
0
Remark A.5 (Why we cannot compute the gradient using score-matching). One may attempt to apply the chain rule Vo log (y(az) VayV log Po x)), and then estimate V y log Po( ss Vylog Polyc+1lye. x) = (y t, x) via score-matching (Song al., 2021) (using the learned score (e t, of the diffusion model). However, this approach is invalid in our setting: Under the log-trick, y is treated as a free variable and enters only through Po(u(æ), so the pathwise term Vay does not exist (see Appendix A.2 for derivation). Our ELBO-bol -O--edd surrogate (Eq. (8) avoids this obstacle entirely.
A.3 PROOF FOR EQ. 10
Based on the results in Lemma A.4, we have
H
GT
By By Pa(1)) (2f(= v) (Vo ELBO((:)))
dF d0
d=
D(X)G
D(h(z,=*))
0
Q
0
d.F
T
H
GT
-1
Po(1))))))) "=ff=""01 ELBO
d
D( D(h(z,2)))
d0
Q
0
(B)T
[V v)
d
E Ev-PotHe)lu((()
ELBO]
d
:= (3)
d
dg Ey (w)
A.4 GRADIENT ERROR OF ELBO vs. LOG-LIKELIHOOD
Under mild assumptions, we can prove an upper bound for the error of our ELBO gradient approx- imation. Recall from Eq. 7, we define ELBO as a lower bound for log-likelihood. Here, we can actually write an equality relationship between them:
log (y(o ELBO + (L((())llo(zlln))
where z denotes latent variable and F.LBO (Vo. 0) = Excay llog (o =)] RE (lg d(=)]. Let denote the score function as s0(2; v0) == Vo log po (Bo. 2). We immediately have the following two results:
Ve log Po(yo) V log
po(yo. 2) dz
/ Vope(yo:>)) dz
Po (vo)
/ po(vo= Vo log po (vo, 2) dz
po (yo)
/ Vo log (o 2) polyo, 2)
(o() dz
Enoco(on) Vo log po (yu, 2)]
Epots/po) [so (s:
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Similarly,
Vo ELBO( 0) = Vo (()) lo (vo. =)]
[x(=) V log Po z)]
[=) [sa (2; 80)].
Then we are ready to state the following proposition:
Proposition A.6 (ELBO gradient approximation error). Let (z(;; Mo) == Va log po 2) denote the joint score function. If sup,,. [[ yu) I B(0, yu) for some finite upper bound B(6, yo) 0, then
Vo log Po (vVo) - V ELBO 0) - /2B(0, vo) KL(((vnr yo Bo)lloo yo))-
Proof.
Ve log Po(yo) Va ELBO No: 0)1
E (: (ole())ss)ssss) (2: (=)=) [o)]
// so (00 30) ((=)) de
$ / (1=o vo) a(=))]] de
= / [1 yo) (po vo) - {(=)] de
- B(0 vo) |po(z vo) g(=)) dz
= B(0, yo) (T)((())).o( vo))
S (2B(( yo) (LL Bo)),
where TV denotes the total variation distance of probability measures: TV 9) = 1/2 2 p(x))dr, and the last inequality is due to TV. S V (L((l)p)
[g(x))
In the context of the diffusion model, the latent variable 2 = Vi.r is the diffusion trajectory, q(vi:)) the distribution over the diffusion trajectory y1.7, and ( (yo) the corresponding diffusion reverse process. Thus, whenever the variational approximation is good (small KL), the ELBO gradient provably close to the true score.
A.S EMPIRICAL EVIDENCE FOR ELBO GRADIENT APPROXIMATION
Assume our model is 0- = (A, B, c). and the noise is predicted by eØ 1,x)) = Aeye + B + ct- True data y ~ N( xr I). this way, there is a closed-form solution for true Ve log p(yolz) since Yo is a Gaussian and yt / + VI- Oye.
A.6 DETERMINISTIC OPTIMIZATION AND GAUSSIAN MODEL IN STOCHASTIC OPTIMIZATION
Deterministic Optimization Since deterministic can also be viewed as reparameterization trick without any randomness e, we can reuse our derivation in Section and compute the gradient V F (x) by Eq. 5.
Gaussian Model in Stochastic Optimization Since there are many recent papers that use the Gaussian model predictor for stochastic DFL, also claim that it has the reparameterization and score function form.
Reparameterization with Gaussian Model. Also using the reparameterization trick:
V
x)] Ee- V v, (v, =)].
But with the predictor instantiated as a Gaussian model, i.c., y is computed by y = R(e, (x)) and R is Gaussian model.
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Distribution of Cosine Similarities (True Grad vs Score_ _BBO_
600
Frequency
400
300
100
0.0
0.2
0.4
0.6
0.8
1.0
Simllarity
Figure 6: Cosine similarity between the true gradient and the estimated gradient using linear model.
2. Score Function with Gaussian Model. Recall we need to approximate the term log (o(l)z) for the diffusion model. However, this term has closed-fore formooommmmmm for a Gaussian model. Assume our Gaussian model N (po. Eg), then the negative log- ligelikll is
1 log Pa(v x) -2100 nO) Eg (y (a) log det Eg + d dlog(22))
Then, the gradient for the score function be calculated using Ve log (v x). Gaussian models are powerful tools for many DFL tasks. However, we want to claim that the diffusion model is more general and requires less model tuning and model assumptions.
A.7 CONNECTIONS WITH OFFLINE CONTEXTUAL BANDIT METHODS
DFL and offline contextual bandits both involve learning from contextual features make decisions, they differ in key assumptions and objectives. Specifically, in DFL, the decision z solution of solving known optimization problem, with explicit constraints and objective (e.g., quadratic objective with convex constraints). other words, DFL explicitly incorporates the known structure of the decision task into the learning problem. Predict-then- theeiittthen--pimizze methods specifically aim to leverage this structure of the optimization problem to learn better predictions under fewer samples. This incorporation of task structure is the key motivation for predicc-then-optimiie methods.
In an offline bandit view, each decision (action) obtained by a policy that is learned directly without using the fact that comes from solving particular optimization problem. In theory, if the model is expressive enough, an offline bandit algorithm can learn the decision with enough samples, but it may ignore the underlying optimization structure and thus require more samples.
Here, implement offline full-infonmn informatioonnn (the true cost function is revealed) contextual bandit method using policy-based learning methods. Specifically, we first sample a batch from the offline dataset, and then compute the decisions (actions) by policy network Φb Our goal is to train the policy network & to minimize the following loss function:
1
Σ
f(o(z).v).
B
(RN)EB
After training T epochs, we evaluate a test set and report the results.
A.8 TRAINING DETAILS AND HYPERPARAMETERS
We summarize our model settings for the deterministic DFL, Gaussian DFL, and diffusion DFL
in Table 2.
For all experiments, we perform 10 random seeds to evaluate variability.
Implementation Details
Stochastic optimization introduces two main computational bottlenecks:
(i) sampling multiple Monte Carlo instamces from the diffusion model, and (ii) repeatedly solving the downstream optimization problem. We address both in our implementation:
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Parameter
Deterministic MLP
Gaussian MLP
Diffusion Model
Model Architecture
Trunk (layers width)
2 x 1024
2 x 1024
2 x 1024 SiLU
Activation
ReLU / SiLU E Rd.
ReLU / SiLU
Inputs
E Rd
Me. f. x]
Time embedding
Sinusoidal (16-d) -
FC + SiLU
Output head
E Rd
p(a), log (xx)
èg fv),
Uncertainty
None (point)
Gaussian N(u, diag(o"))
Diffusion Po x))
process
Training
Loss
MSE(y)
Gaussian NLL
Weighted
denoising
MSE
DFL gradient
Implicit diff. via KKT
(1) Reparam (Gaussian)
(1) Reparam (Diffusion)
(2) Gaussian score
(2) Weighted- BLieeddEEBBBO score
Sample size M
10/25/50
10/25/50
(eymthetic/oowee/prrrtoo)))
Learning rate
x 10 (typical)
Reparam: x 10-5
Reparam: 1 x 10-5
Score-fn: 8 10-
Score-fn: 8 x 10-6
Inference
Procedure
Use directly
Sample outputs (m))
Reverse diffusion to get M samples (mm)
Table 2: Architectural and training diffcrences among deterministic, Gaussian, and ceffusiionbbaed DFL methods.
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Parallel diffusion sampling. We sample in parallel across many noise realizations, which makes efficient use of hardware and reduces the per-sample overhead. In our supplementary material, for each task (power sched, stock portfolio, synthetic example), the file (task) | /di f ffusion on opt.py .py implements the function sampl lample e e el which accepts batched inputs and runs the diffusion sampler for the entire batch in parallel. This allows us to draw many Monte Carlo samples in one single forward pass.
Parallel optimization solving. The downstream optimization solver is also executed in parallel across different samples. In (task), / cvxpy {task) -py, the function cvxpy- _aaaak/paaaallll___aa
constructs a batched KKT system for the Monte Carlo
samples and solves these batched KKT systems in parallel (across CVXPY worker jobs).
A.9 DETAILS OF SYNTHETIC' TASK
In this example, we consider factory that decides how much to manufacture for each of d E N products. The parameter E R represents the profit margin for each product, i.e. Y, the profit per unit of product i; due to uncertainty in market conditions, Y is uncertain. The factory's decision E [0,C] CJ represents how much of each product to manufacture, where the maximum capacity for each product. For simplicity, we do not consider any contextual features in this example. That means DFL learns a distribution that generates y that can minimize the decision objective.
Suppose that the factory has a risk-averse cost function f(y,)) exp(- sep(-y"am 2), which indicates that the factory wants to put a larger weight on the product with higher profit Y. Intuitively, if the factory knew Y exactly, then the optimal strategy would be all-or-nothing: set = C if Y; > O, or 2; = 0 Y, < 0. Likewise, with respect to a point prediction of Y. the optimal deterministic decision det E {0, C) is attained on the boundary of the feasible set.
Under uncertainty, the decision-maker seeks to minimize the expected cost by solving a stochastic optimization problem:
arg min Eyyr =E(0,
2)].
sst
In this stochastic case, the optimal investment 2 typically lies in the interior of the feasible region, which balances the potential high reward of investing against the risk of losses.
Then, we compute the necessities for diffusion DFL:
V2ch(x, 2²) T H Ey Po((a)) (V ..,2((("v))))))) + (A*) v2
exp(-y" 2)80"
G (2h(,,
exp(-x) z)y-
For reparameterization,, we have
dloss
dooss
05
d=
20
d0
OX
09
dloss
GT D(N* D(h(a,=*))
[ 17 ΣM
(oyi) T V2 1z" (2=,="z''0 vi)
dsa
where 22 222 (0000())))))))) . = exp(- For the score function, we have
2)2 Ia) in this case.
dloss d0
diose d=
H
GT D(M(r==')) GT
Ey V ((("))) dB
JELBO
0
D(X)G
dioss
H
1 M M
Veg(z*
AELBO, v:)(
do
0
D(Y)G
((M(x,²*))
d0
A. DETAILS ON POWER SCHEDULE TASK
This task involves a 24-hour electricity generation scheduling problem with uncertain demand. The decision & E 24 represents the electricity output to schedule for each hour of the next day. The uncertainty y E R²4 represents the actual power demand for each of the 24 hours. The goal is to meet demand as closely as possible at minimum cost. We also consider a decision cost function that penalizes storage, excess generation, and ramping following Donti et al. (2017):
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1. Let "/ and Ta be the per-unit costs of shortage (not meeting demand) and excess (over- generation), respectively. We use " = 50 and " 0.5 in our experiment.
2. Let c, be penalty hour-to-hour changes in generation. We c. 0.4 in appropriate units.
Formally, if ;21....,22211, and y = (y1..... 122)), the loss for single day
min
2)] Σ E Pd 2il+ + Telzs vil+ + 2(2 (2) yx)
s.t. 2 4-1| cr for all i 2 ..... 243.
Then, compute the necessities for diffusion DFL:
H Po(()) V2 (2
(z" y) + (A*) V2 2x) =
1
0 0
0
G V_h(x,2")
:
:.
:
0
-1
For reparameterization, have
dloss
dloss
0="
d=
d0
09
dloss
GT DDN D(h(z,z*)) ]
M Ei (Voy;) T TV V2 2200(" vi)
d=a
where V V2 4) - in this case.
For the score function, have dloss
T =
ddooss d=*
GT D( D(M(I,2'))
H
Ey V ==9""",00 B)
dELBB d8
d0
dioss
H
GT
M 1=
>2(2 ya) dELBO;
2
d=*
D( D(M(I,z*))
M
0.4
0.30
0.25
0.3
Task Loss
RMSE
0.20
0.10
0.1
0.05
0.00
0.0
12
16
20
24
12 horizon
16
20
24
horizon
Hour of Day
Score fn
-
Reparam
+
Gaussian
Deterministic
Figure 7: Results on the hour power grid scheduling task.
Dataset. We use real, public historical electricity load data from the PJM regional grid (PJM Interconnection, 2025). The features for each day include: the previous day's 24-hour load profile, the previous day's temperature profile, calendar features, and seasonal sinusoidal features. In total, d, 150 features for each day were constructed. We normalized all input features for training. The target label is the next day's s 24-hour load vector.
For completeness, we include extended comparison of different sample sizes in Figure 9, which further highlights that additional samples yield diminishing returns in accuracy while linearly in- creasing compute cost. We also find that adding small regularizer during DFL training help the model learning the data distribution and avoid some bad local minima. leading to a stable training process.
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000
7a DR. 1055
DF.
2,9
0
10
20
30
10
20
30
Epoch
Trech
Tpoch
diffusion of
diffusion rp
Figure 8: Train, validation, and test DFL loss for diffusion DFL using function reparame- terization. Solid lines show the mean loss over multiple runs with different random seeds; shaded regions indicate standard deviation across runs.
14
Earch
2000
Figure 9: Comparison between different sample sizes for score function and reparameterization.
A.11 DETAILS STOCK PORTFOLIO TASK
We consider a mean- variance portfolio optimization problem with uncertain returns. Here y I" represents the random next-day returns of n assets (stocks), and E Rh" are the portfolio weights we assign to each asset (the fraction of our capital invested each stock). Our goal maximize expected return while keeping the risk (variance) low. This can be written minimizing loss that is a negative expected return plus a quadratic penalty on variance:
min Ey [/ =)] = Pa(la))
2 yy z v 2]
=
Then, compute the necessities for difffusion DFL:
H Po 2 [(= v)] + V h(x, + 220 g(xx,²))) =oByyPo Poyloolo J.
G = V,h(u,,7) Tn
In
V.g(a, ==]==, =
For reparameterization, we have
dlose
0=
dloss d0
d="
dloss
GT
1
(
To
V2 "="00""0000
d=a
M
21=1
DONG D(h(a,z*)) Q
0
where 2 f( v)
Ey (2ay" in this case.
For the score function, we have
dloss db
dloss 0
H
GT
E =0(2" v) dELBO
TT =
D(N) D(h(I,2")) Q
0
ddoss d2
GT D(VGG D(h(z,z')) Q
H
"9(2** 9))(
JELBO, de
M
1=
0
0
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Published as a conference paper at ICLR 2026
where V Vzg(2", (4) anyyy 2" y4-
Dataset. We use daily stock prices from 2004-2017 for constituents of the S&P 500 index (Quandl WIKI dataset, 2025). We obtained this data via QuandI's API (specifically WIKI pricing data; the user will need Quandl API key to replicate. We compute daily returns for each stock (percentage change). To construct features 2, we use a rolling window of recent history for each asset. Specifically, for each day, a data point for predicting next day's returns: we include the past 5 days of returns for each of the assets, past 5 days of trading volume for each asset, plus some aggregate features. To avoid an explosion of dimension with large n, we also include PCA- CCC-coopressed features: we take the top principal components of the last 5-day return matrix to summarize cross-asset trends. In the end, for n = 50 assets, we ended up with d,- = 28 features. All features are normalized and we use time-series split: first 70% of days for training (2004-2013), next 10% for validation (2014), last 20% for test (2015-2017). We evaluate performance on the test set by simulating the portfolio selection every day and computing the average return achieved.
A.12 (ADDITIONAL TASK) DETAILS ON INVENTORY STOCK PROBLEM
We also validate our approaches on toy inventory control problem. In this task, the uncertain demand y drawn from multi-modal distribution (a mixture of Gaussians), where we vary the number of mixture components K to control the distribution complexity:
K
p(x)
Σ Tf ps. Ey),
j=1
where Ty is the probability of choosing component and p(æ; py, Mj) is a multivariate Gaussian density with parameter (py, M.y). The cost function follows the standard newsvendor formulation with piecewise penalties for under-stock and over-stock:
1
1
fw, = co +
2 -0 + csly =]+ + g =1 + ca(== v]+ 3 - y)
3
Our learning objective is to minimize the expected cost over this stochastic demand, i.e., a stochastic optimization problem:
min L( E En
s.t. S
Zmaz.
Regret Distribution
z Distribution
Win-rate (lower cost wins)
1.0
method
Sto
0.8
74.8%
Density
0.2
Det
0.6
0.4
0.1
25.2%
10
20
Thue
Sto
Det
Sto
Det
Regret
Method
Figure 10: Toy decision task comparing deterministic and diffusion DFL. Left: distribution of per- instance regret (lower is better). Middle: distribution of chosen decision = in the lower-level; the stochastic method tracks the true distribution 2* more closely. Right: pairwise win-rate on test set; a large fraction of costs from the stochastic method are lower than the deterministic one, indicating that modeling uncertainty yields better decisions.
We compare a deterministic DFL model against our diffusion DFL model on this toy task. Figure 10 summarizes the results, where the diffusion model (stochastic DFL.) achieves substantially lower regret average than the deterministic model. Besides, we observe that the decision obtained by our diffusion method closely tracks the true optimal decisions =" by capturing the multi-modal demand uncertainty, whereas the deterministic predictor's decisions deviate more. In Figure 10 (c), we directly compare the decision outcomes via a win-rate: the fraction of test instances where one method achieves lower cost than the other. The diffusion DFL method attains win-rate of about 75% against the deterministic baseline, which confirms that modeling uncertainty leads to better downstream decisions
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Generated Data distribution over test set
2.5
Tre data
Stochastic presictions
Deteriniisk predictions
2.0
Goussian predictions
Donzay
1.-
10-
0.5
1
2
3
y Valle
Figure 11: Prediction distribution for inventory stock problem.
A.13 ADDITIONAL RELATED WORKS AND DISCUSSIONS
Stochastic optimization Making decisions under uncertainty a classic topic in operations research and machine learning (Shalev-Shware et al., 2009). Stochastic optimization formulations explicitly consider uncertainty by optimizing the expected objective over distribution of unknown parameters. A common approach is the Sample Average Approximation (SAA) (Kleywegt et al., 2002; Arjevani et al., 2020; Wang et al., 2024b), which draws many samples from the estimated distribution and solves an approximated deterministic problem minimizing the average cost. While SAA can handle arbitrary uncertainty distributions in theory, it becomes very computationally expensive and still does not consider the distribution during optimization (Kim et al., 2015). It will lead to optimizing the sample mean, which may yield a decision that performs poorly if reality often falls into one of several distinct models far from the mean (Kim al., 2015; Elmachtoub & Grigas, 2022).
When Diffusion- DFL useful? Our experiments suggest that Diffusion DFL will be especially useful in -assiinnmmaiiggggao settings where the uncertain parameter in the objective is high-dimensional, has a multimodal distribution, and limited training data is available. We believe that promising directions of future work include accommodating uncertainty in constraints in addition to the objective function, investigating the trade-off between model calibration and decision-making quality, and adapting Diffusion DFL to online decision-making settings.
Offfine Contextual Bandits (OCB) Offline contextual bandit approaches provide an alternative way to learn decisions from contextual features by directly optimizing a policy that maps each context an action (Agarwal al., 2020; Nguyen- -ang et al., 202 1; Brandfonbrener et al., 2021; Gabbianelli et al., 2024; Sakhi et al., 2023). A key difficulty is that feedback is typically partial: the log reveals rewards only for actions taken by a behavior policy, so learning and evaluation rely on off-policy estimators and assumptions such as overlap between the learned and behavior policies. Recent OCB methods address this challenge via optimistic or pessimistic objectives (Nguyen-Tang et al., 2021; Wang et al., 2024a), variance control (Li et al., 2011; 2012; Wang et al., 2017). conservative regularization (Swaminathan & Joachims, 2015), and robust optimization (Yang et al., 2023) to avoid out-of distr--iistiiuution actions. In all these methods, the policy is learned purely from logged data, without using the fact that the decision comes from solving particular optimization problem. This is fundamentally different from DFL in terms of the prior knowledge and optimization structure. In DFL, the decision 2 is solution of solving known optimization problem, with explicit constraints and objectives. In this paper, we empirically show that Diffusion-DFi achieves better performance than the policy-basee OCB method.
26


Reviews:
[{"review_id": "jvsKMKLLON", "reviewer": "Reviewer_Azep", "summary": "This paper studies a paradigm termed decision-focused learning, where the goal is to solve a decision-making problem that depends on uncertain parameters that must be learned. The authors develop a computational pipeline to propagate uncertainty from the learning procedure to the decision procedure using diffusion models, and show that incorporating uncertainty results in improved decision-making performance.", "questions": "Have I missed some critical contribution in the above points that should justify a different conclusion?", "limitations": "", "rating": null, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 1, "strengths": "I note two key strengths:\n* **Writing attention to detail.** This paper is reasonably polished, and it is clear the authors know how to write a paper to an appropriate technical standard. \n* **Reasonably-careful experiments.** The authors not only demonstrate good performance on a key benchmark of interest compared to pure-deterministic methods, they also include various ablation studies to check individual parts of their conclusions.\n \nIn an ordinary review, I would offer many more details on these points. However, in light of a very serious issue which I will bring to light shortly, I will omit these in the interest of time.", "weaknesses": "This paper has a critical weakness because of which I am confident it needs to be rejected: **a very significant portion of this work's findings are known and covered in standard textbooks, but are presented as novel.** In particular, the paper's actual novelty is limited to introducing diffusion models to well-known decision-making paradigms. If re-written completely in a manner that reflects this contribution, I do believe the authors have done sufficient development to warrant a paper, but the gap between the current manuscript and what would be appropriate is, to say it again, sufficient to warrant a complete rewrite. This is unfortunately far too much to address in the rebuttal phase.\n\nAllow me to elaborate:\n* **Decision-focused learning is presented as a novel emerging paradigm, but is not new, because the authors' formulation is equivalent to a (stochastic) contextual bandit.** Let me be completely precise: the authors define their setting via the optimization objective $\\mathbb{E}f(y^\\star, z^\\star(x))$ where $(x,y^\\star) \\sim D$ and $z^\\star$ is parameterized by some vector $\\theta$ which is to be optimized. Let us define the notation $\\phi^\\star(z) = f(y^\\star, z)$, so the objective becomes $\\mathbb{E} \\phi^\\star(z^\\star(x))$ where $(x,\\phi^\\star) \\sim D$. With this notation change, it is clear that the authors' setting is a (stochastic) contextual bandit: we are given a random context vector x which is correlated with some random objective $\\phi^\\star$ (either in a known way in the Bayesian case, or unknown way in the non-Bayesian case), and our goal is to choose the optimal map from context to actions. With this notational link made, there is a substantial literature of prior work the authors could consider and compare with, which is not mentioned in this work. Note, in particular, that if D is assumed known, maximizing the objective is equivalent to minimizing regret, which is the usual performance objective considered because it behaves much better mathematically.\n* **Using stochastic models inside dynamic programming is not new.** Indeed, this is covered in Chapter 1.5 of \"Sequential Decision Analytics and Modeling\" by Warren Powell, to name just one book that includes this topic.\n* **In the contextual bandit literature, the need for uncertainty is well-understood.** This is the basis not only for classical upper confidence bound algorithms, but also methods based on approximate dynamic programming such as expected improvement algorithms from the Bayesian optimization literature.\n* **Even if one is interested purely in practical aspects involving diffusion models, one should benchmark against standard algorithms.** For example, I see no reason why, in light of the above equivalence, any DFL algorithm in the authors' sense cannot directly be benchmarked against contextual kernel UCB.\n* **Section 4 consists entirely of known ideas.** In particular, the application of the reparametrization trick to handle objectives defined as expectations of this kind is, at this point, completely standard and need not be introduced except in code. The discussion on differentiating through the solution of a parameterized optimization problem is also standard, though less than reparametrization tricks - this should be deferred to an appendix.\n* **In Section 5, the key ideas involving score estimation are essentially the same as the standard REINFORCE trick from reinforcement learning. Moreover, the algorithmic justification for using ELBO as a surrogate to the score is not technically sound.** The authors' argument, essentially, is to state that $f(x) \\leq g(x)$ on basis of an evidence lower bound, and argue that this implies $\\nabla f(x) \\approx \\nabla g(x)$, which is not necessarily true even if $|f(x) - g(x)| \\leq \\epsilon$ for an arbitrary given $\\epsilon$. Rather than relying on a theoretical explanation that does not follow, I believe the authors should instead lean on their empirical results in Figure 2, which shows these are close, as this avoids a misleading explanation.\n\nSummarizing, I believe this work's only contribution which is novel is the exploration of diffusion models in the context of stochastic dynamic programming. This contribution is a reasonable fit for ICLR and could make for a good paper, but the paper needs to be re-written in a manner that reflects this.", "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": "jd0yD2KP7W", "reviewer": "Reviewer_dJh7", "summary": "This paper presents a Decision Focused Learning method where the output of the estimator is stochastic, rather than being a point estimate, and the downstream problem is also a stochastic optimization one. As a main advantage, the approach can better address decision problems where no deterministic formulation can result in the correct optimal solution under uncertainty, e.g. because any individual realization leads to an either maximizing or minimizing a variables, while the best value under under uncertainty would be somewhat in the middle.\n\nThe authors use a diffusion process to generate their estimate and present two techniques to enable differentiation. The first -- technically more precise, but expensive in terms of computation time and memory -- is based on a reparameterization of each diffusion step and implicit differentiation. The second -- much lighter from a computational standpoint -- is based on Score Function Gradient Estimation and an Evidence Lower Bound. Using a diffusion process enables modeling complex distributions\n\nThe approach is tested on selected benchmarks from the literature, and at least in one realistic (but still far from true practical applicability) problem it achieved a very significant advantage in terms of solution quality.", "questions": "* Can you confirm that the Gaussin DFL method is in fact a stochastic method, i.e. using a stochastic optimization problem with multiple samples downstream?\n* How many samples were used in the empirical evaluation?", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The insight about the limited expressiveness (for lack of a better term) of deterministic DFL is correct and definitely useful, even if not entirely novel.\n\nThe empirical results on the stock portfolio problem provide evidence that a combination of stochastic prediction and a complex distribution can improve the effectiveness of the DFL approach very significantly, beyond what is possible for either technique. Remarkably, this is true even when the two-phase approach is rather expressive and should be capable of approximating well the ground truth parameter distribution (the PFL diffusion method).\n\nThe suggested differentiation method based on SFGE + ELBO appears to be rather good at improving the computational efficienty, with no major drop in terms of solution quality.\n\nThe SFGE/ELBO method also seems to be applicable, with a few minor modifications, to a _much_ more general setting than the one presented in eq. (2). Since the method only requires function evaluation, it seems it could easily account for solution repairs costs (similar to the other SFGE method by Silvestri et al.) and therefore, combinatorial problems, two-stage problems, or decision problems with uncertain parameters in the constraints.", "weaknesses": "It is my opinion that the paper strengths outweigh its weaknesses. That said, I can find some, of which a few are major and a few minor.\n\nThe first major weakness I can find is that the approach requires solving a stochastic optimization problem per example, and per epoch. Stochastic optimization problems can be very challenging to address and the corresponding solution methods are well known for their rather poor scalability. Combined with the already computational cost of DFL training, this significantly reduces the practical appeal of the method. That said, given the improvements reported in one benchmark, I think there are definitely practical settings with relatively lightweight optimization problems that can benefit from the proposed technique.\n\nThe second major weakness is again related to applicability. If enough data is available and a sufficiently expressive (distributional) ML model is used, the approximation error for the ground truth distribution will eventually converge to zero. Under these conditions, a Prediction Focused Learning (i.e. two phase) pipeline should be capable of reaching optimal or close to optimal results, similarly to what happens in a classical DFL setup with non-misspecified ML models. The empirical results corroborate this intuition, with the Diffusion TS method reaching high quality results in the synthetic benchmark. It would seem that the ideal application case for the proposed method is one where a deterministic decision problem is inadequate, a complex distribution is needed, and the problem cost is not well aligned with the likelihood function used in PFL. While situations like this certainly exists, the applicability of the method is also reduced.\n\nHere is a list of some minor weaknesses:\n\n* The fact that deterministic DFL may not be expressive enough for some optimization problems under uncertainty is also established in [1]. Given that this paper is very recent and that the expressiveness claim is not at the center of this work, this issue is definitely minor. That said, it might be worth including a citation\n* The Gaussian method by Silvestri et al. is not stochastic, but deterministic: the Gaussian distribution is used just for smoothing, while the optimization problem works with a single parameter vector (both at training and inference time). By reading Appendix A.5, it seems the authors instead implemented an actual Gaussian stochastic DFL pipeline, based on their own techniques, but using a simpler distribution. This is again a minor issue, since the original method by Silvestri et al. would likely work even worse in this setting (though it would be much more scalable)\n* This might be my fault, but I did not easily find the number of samples used in the empirical evaluation\n\n[1] Schutte, Noah, et al. \"Sufficient Decision Proxies for Decision-Focused Learning.\" arXiv preprint arXiv:2505.03953 (2025).", "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": "gGCbyOCanq", "reviewer": "Reviewer_ZKgV", "summary": "This paper introduces Diffusion-DFL, a first framework that integrates diffusion probabilistic models into decision-focused learning (DFL) — an approach that trains predictive models to directly optimize downstream decision quality rather than prediction accuracy.\nUnlike prior DFL methods, which rely on deterministic or simple probabilistic predictors (e.g., Gaussian), Diffusion-DFL uses a conditional diffusion model to capture potentially complex and multimodal uncertainty in the parameters of a downstream stochastic optimization problem. The decision layer then optimizes the expected cost under this learned distribution.\n\nTo train the model end-to-end, the authors develop two gradient estimators:\n1) Reparameterization estimator: backpropagates through the diffusion sampling process, providing an exact but memory-intensive gradient.\n2) Score-function estimator: avoids backpropagating through diffusion by applying the log-trick, estimating gradients via the model’s ELBO  — drastically reducing memory and compute cost with minimal accuracy loss.\n\nExperiments on synthetic manufacturing, power scheduling, and portfolio optimization tasks show that Diffusion-DFL consistently yields better decision outcomes than both two-stage and deterministic DFL baselines, while the score-function version achieves similar performance to reparameterization with over 400× lower memory usage.", "questions": "The score-function estimator seems to rely on local smoothness between the diffusion model’s likelihood landscape and the downstream decision cost; can the authors discuss or empirically validate when this assumption might fail, e.g., in highly nonconvex or discontinuous decision problems? Have the authors tested Diffusion-DFL in nonconvex or combinatorial settings, where diffusion’s multimodality could be beneficial?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper introduces the first integration of diffusion models into the decision-focused learning framework, allowing richer uncertainty modeling than deterministic approaches.\n\n- The proposed score-function estimator is a practical contribution: it replaces costly pathwise differentiation with an ELBO-based surrogate, maintaining decision performance while reducing memory and compute by several orders of magnitude.\n- The methodology is well-grounded theoretically (via implicit differentiation and the log-trick) and empirically validated on multiple domains, showing consistent improvements in decision quality.\n- Clear ablation studies (memory vs. accuracy, variance reduction, decision dimension scaling) demonstrate that the approach is both effective and scalable.", "weaknesses": "- The score-function approximation relies on an implicit assumption of local smoothness between the model’s likelihood and decision-loss landscapes, which is not formally analyzed or guaranteed, although supported empirically. \n- Experimental settings are limited to convex or low-dimensional decision problems; performance on more complex or nonconvex tasks (where diffusion expressiveness would matter most) remains unclear.", "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": "MEFmx5d4q2", "reviewer": "Reviewer_1MXz", "summary": "The paper introduces a novel DFL approach that leverages diffusion models to represent the distribution of uncertain parameters. To mitigate the issue raised by using diffusion models, i.e., sequential sampling procedure, the authors develop two algorithms: reparametrization and score function. Experiment results demonstrate that using diffusion models outperforms prior baselines.", "questions": "- I'm not familiar with decision-focused learning. Are there any other benchmarks for evaluating the performance of decision-focused learning? It seems that the three optimization tasks provided in the paper are not complex enough.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Utilizing diffusion models in decision-focused learning is sound and novel\n- Authors successfully mitigate issues raised when using diffusion models in decision-focused learning\n- Experiments and further analysis are comprehensive", "weaknesses": "- The authors say one benefit of utilizing diffusion models is capturing a multi-modal distribution. Does the task provided benefit when we capture the multi-modal distribution? It would be nice to persuade readers why we should capture multi-modal distribution in decision-focused learning.", "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)