A Gradient Guided Diffusion Framework for Chance Constrained Programming
Authors: Boyang Zhang
Zhiguo Wang
Ya-Feng Liu
Venue: NeurIPS 2025 poster ()
Keywords: Chance Constrained Programming
Diffusion Model
Optimization via Sampling
Gradient Guidance
Waveform Design
Primary area: optimization

Abstract: Chance constrained programming (CCP) is a powerful framework for addressing optimization problems under uncertainty. In this paper, we introduce a novel Gradient-Guided Diffusion-based Optimization framework, termed GGDOpt, which tackles CCP through three key innovations. First, GGDOpt accommodates a broad class of CCP problems without requiring the knowledge of the exact distribution of uncertainty—relying solely on a set of samples. Second, to address the nonconvexity of the chance constraints, it reformulates the CCP as a sampling problem over the product of two distributions: an unknown data distribution supported on a nonconvex set and a Boltzmann distribution defined by the objective function, which fully leverages both first- and second-order gradient information. Third, GGDOpt has theoretical convergence guarantees and provides practical error bounds under mild assumptions. By progressively injecting noise during the forward diffusion process to convexify the nonconvex feasible region, GGDOpt enables guided reverse sampling to generate asymptotically optimal solutions. Experimental results on synthetic datasets and a waveform design task in wireless communications demonstrate that GGDOpt outperforms existing methods in both solution quality and stability with nearly 80\% overhead reduction.

A Gradient Guided Diffusion Framework for Chance Constrained Programming
Boyang Zhang
Zhiguo Wang" Department of Mathematics
School of Advanced Interdisciplinary Sciences University of Chinese Academy of Sciences Beijing 100049, China
Sichuan University Chengdu 610065, China wangzhiguo&sen edu en
zhangboyangg23maal acas.ac.com
Ya-Feng Liu"
Ministry of Education Key Laboratory of Mathematics and Information Networks
School of Mathematical Sciences
Beijing University of Posts and Telecommunications
Beijing 102206, China
yafengllußbupt. edu.cn
Abstract
Chance constrained programming (CCP) is powerful framework for addressing optimization problems under uncertainty. In this paper, we introduce a novel Gradient-Guiie Diffusion-basse Optimization framework, termed GGDOpt, which tackles CCP through three key innovations. First, GGDOpt accommodates broad class of CCP problems without requiring the knowledge of the exact distribution of mncertainty-relyiee the nonconvexity of the chance constraints, it reformulates the CCP sampling problem over the product of two distributions: an unknown data distribution supported nonconvex and Boltzmann distribution defined by the objective function, which fully leverages both first- and second-order gradient information. Third, GGDOpt has theoretical convergence guarantees and provides practical error bounds under mild assumptions. By progressively injecting noise during the forward diffusion process to convexify the nonconvex feasible region, GGDOpt enables guided reverse sampling to generate asymptotically optimal solutions. Experimental results on synthetic datasets and waveform design task in wireless communications demonstrate that GGDOpt outperforms existing methods in both solution quality and stability with nearly 80% overhead reduction.
solely on set of samples. Second, to address
Our code is available at httss//sthabbcooooonnnggguggoogo
1 Introduction
Problem formulation
Chance constrained programming (CCP) is an efficient modeling paradigm for optimization problems with uncertain constraints, which finds wide applications in diverse fields, such finance (Bonami and Lejeune 2009)), ) robot control (Calafiore and Campi (0066), and wireless communications
"Corresponding authors.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

---

Wang et al. (0144). In this paper, we consider a CCP with the following form:
min f(z)
(I)
s.t. x xEXp
where f: IR" + R is a differentiable objective function and Xp is the chance (or probabilistic)
constraint set defined by
E R" Prob, (a, h) 2 0) 2 - o).
In the above, h a random vector with probability distribution P supported on a set C Rd E (0, 1), g (g) 92...... .mm) R" x = + RT and Prob(A) denotes the probability of an event A. Problem ① is generally challenging to solve for the following two reasons. First, evaluating the probability term Prob,, (x, h)) 0} typically involves high-dimensional integration, which is computationally intractable. Second, even when is linear, the feasible set X, remains nonconvex, further complicating the optimization.
1.2 Related works
Apart from very special cases where Xxp can be transformed into a convex formulation under strong assumptions (Kataoka [963]. Lagoa et al.] [2005). Henrion 2007]] Prékopa (2013), there are two popular approaches tackling general problem (T). which are Convex Approximation (CA) method and Sample Average Approximation (SAA) method. The CA method seeks to construct tractable inner approximation of Xx. but typically requires the information of the exacr distribution P. often assuming that P belongs to specific families such as Gaussian or log-concave distributions (Ben-Tal and Nemirovski [2000),Bertsimn, BBetsimas and Sim [2004), [Lago et al. (2005), Nemirovski and Shapiro| [2007]). In contrast, the SAA method approximates P using an empirical distribution based on sampled data, reformulating the CCP as a binary integer program (Ahmed and Shapiro 2008 Pagnoncelli et al. [2009] Adam and Branda (2016). However, this reformulation remains computationally intractable. These restrictive assumptions on the underlying distribution P, along with the high computational cost, significantly limit the practical applicability of CCP.
One important question ask can we design a general framework to efficiently solve CCP when the underlying distribution P is unknown? The answer to the above question is particularly crucial in our interested case where samples can be efficiently drawn from X,, albeit the explicit formulation of Xx is unavailable. This motivates to seek high-quality solutions to the CCP problem (T) from a new perspective via sampling- based methods (Wibisono) [2018], Ma et al.) [2019], Lee et al. [202T] J Chen et al. (2022), (Seyoum and You (2025]). The core idea of applying sampling based methods to solve CCP problems lies in reformulating the original nonconvex CCP with intractable constraints as sampling problem from an unknown distribution. This reformulation leverages probabilistic techniques to handle the challenging constraints through stochastic sampling rather than deterministic evaluation.
Notably, generative models are designed to approximate unknown data distributions based on observed samples, enabling the generation of new data points from the learned approximation. In particular, diffusion models have emerged as powerful family of generative models, offering high-quality sample generation, stable training dynamics, and scalability to -igh-dimensional problems (Ho et al.) 22200). ). The sampling process based on score estimation enables diffusion models to generalize to conditional distributions, thereby generating samples that satisfy requirements through conditional information guidance (Ho and Salimans (2022). ). As powerful generative artificial intelligence (AI)
technology, diffusion model has been successfully deployed across various domains, such as, image generation ( (Yue et al. [2023], Huang et al. [2025] D, inverse problems (Chung et al. [202bb]. Chung ctt al. 2022a Song et al. [2023] ) and optimization (Krishnamoorthy et al. [2033 Li et al. [2024].
Wu et al. 2024], Kong et al. [2024), Liang et al. (2025)) D. Recently, Guo et al. [2024]] introduced a novel form of gradient guidance to adapt pre- tre-trained diffusion models for user-specified tasks.
Despite their success various domains, diffusion models have rarely been explored in the context of CCP. The possible reason behind might be that tackling CCP problems via diffusion models generally requires efficient sampling from a composite distribution, the product an unknown data distribution (associated with the constraint) and a known Boltzmann distribution (induced by the objective function), but the training data only available from the unknown component. This makes the application of diffusion models to CCP both novel and nontrivial.

---

Stage Data Generation
Stuge 2: Diffusion and Learning
a(x,, D)t b()dB,
min f(x)
(CCP) Intractable
s.t. x E x
Restriction
Estimation
Restriction
Stage 3: Sampling with Guidance
U0
Target P
min (e)
(RP) Tractable
s.t. (xx 2
dx,
(a(x,, t) (V., logp, G.) dt b(t)dB
U
Samples A
U-Net Esclmation
Solution
First order:
Samples Distrbhtiion
(g(x(p) (xx) arg minaax, f(x)
Chance Estimation
6, -09((x)
Data Set (r.p) r(x)))
mm
Second-order
Estimated
HoH
6.
(-r1soei + 9700 ao3u
Figure 1: A framework of GGDOpt. (1) Generate training of points satisfying the chance constraint by solving a deterministic restricted problems, (2) Train a diffusion model with classifier- free guidance learn the score of the conditional distribution. (3) Perform the reverse diffusion process with additional gradient guidance to sample from the product of the data distribution and the Boltzmann distribution.
1.3 Our contributions
In this paper, we propose GGDOpt (see Figure 1. novel Gradient-Guider Diffusion based Optimization framework for solving problem T
with the following originality:
Applicable to broader problem domains. Built the basis of diffusion model with classifier-free free guidance and optimization via sampling, GGDOpt accommodates broad class CCP problems without requiring the knowledge of the exact distribution of uncertainty relying solely set of samples.
Problem reformulation with a novel paradigm. GGDOpt reformulates the CCP problem as sampling task over the product of two distributions: unknown data distribution implicitly defined by the constraint and a Boltzmann distribution induced by the objective function with full utilization of first- and second-orde order information of the underlying CCP.
Feasibility data generation and efficient guided sampling. To generate high-quality training data that satisfy the chance constraint, GGDOpt solves a deterministic restricted problems by standard optimization techniques. The solutions are used to guide the training the conditional diffusion model, effectively capturing the geometry of the feasible region. To sample from the product distribution, we develop a gradient-guin guided reverse process derived closed form based on the structure of the product distribution. Compared with Guo et al. 2204 our guidance terms do not require backpropagation through the neural network.
Theoretical convergence and practical evaluation. Regarding the sampling process as a reverse time stochastic differential equation (SDE), GGDOpt is shown generate asymptotically optimal solutions as the time step and inverse temperature go to infinity. A practical error bound is also provided with two components: the limited time length error and limited inverse temperature error.
1.4 Organization
The remainder of the paper is organized as follows. In Section a reformulation of CCP problem 1
provided via sampling, and a gradient guidance-based score estimation schedule provided with both first- and second-order information. A novel GGDOpt framework for solving problem (T) is given Section Theoretical convergence and experimental results are presented in Section and
Section 5 respectively. The conclusion is drawn in Section 6

---

2 Problem reformulation via sampling
(etr( = Ix, (x) denote the indicator function of the chance constraint X Let B3( x e (B(()
represent the Boltzmann distribution associated with the objective function f(x), where 3 >> 0. The
resulting sampling task is draw samples from the following target distribution:
sample æ
Qx(plp) & xr (,(x))
Intuitively, the distribution @s(2)p) p) assigns higher probability density regions where the objective function f()) takes smaller values. Under certain regularity conditions (Kong et al. [2024), ), as - oo, the sampling distribution Q2(x)p)) asymptotically concentrates around the global minimizer of the CCP in (T. Therefore, the CCP (I) admits the following equivalent reformulation:
x² arg min {f(x) + [x, (x:)
sample xx ~ Qs(=lp). B co.
(4)
x
A natural way would be to directly employ Langevin dynamics for sampling from distribution Qs(zlp). However, the unknown nature of component r(x=pp prevents the derivation of an exact expression of the score function. Fortunately, we can obtain a set of feasible samples {x((), p Ji=1, which are drawn from the unknown distribution r(æ)p). p). More details on this will be presented in Subsection This motivates us to leverage diffusion models to directly learn the product distribution Q a p) x r(<lp) Bg where r(x(p) is unknown but Bz(x) is explicitly known.
LV
2.1 Diffusion models
Given observed samples from a distribution of interest, the goal of diffusion model is to learn to model its true data distribution (o(z)). Once learned, we can generate new samples from our approximate model at will. The diffusion model builds a diffusion process by defining forward SDE starting from Po(o)) as follows:
dæ, a(x,, l)dt + b())dB.,
(5)
where E [0, B, is the standard Wiener process (a.k.a., Brownian motion), a(.,t) IR + Rd is a vector valued function called the drift coefficient, and b(-) ER is scalar function known as the diffusion coefficient.
By starting from samples of xT ~ pr(ær) and reversing the process, æg ~ (o())) The reverse a diffusion process is also a diffusion process, running backwards in time and given by the following reverse-time SDE:
obtain samples
dx;- = (a(x,, V logp((z.)) dt b(t)dB,
(6)
where B, is standard Wiener process when the time flows backwards from T to 0. The only unknown term Va, log (xx) is the score function of the marginal density pe (x;). To estimate Va. (It), we can train a time-dependent score-based model se se t) with
0 argmin Ex-u([oTT A₂ A, E
(z., Va. log Poi x., x 1
where Por ( (o) is the transition kernel and can be obtained by the forward process (5). When a(, f) is affine, the transition kernel is always Gaussian distribution, where the mean and variance are often known in closed forms (Särkkä and Solin (019)). With sufficient data and model capacity, score matching ensures that the optimal solution Sg- (x) approximates Va. logp:(Zz)) for almost all De and t.
2.2 Gradient guidance
direct application of diffusion models to CCP ( infeasible, as this requires sampling from the product distribution Qx(())) x (( whereas only samples from r(æ(p) are accessible. Therefore, obtaining precise characterization of the score function of Q2(())p) and its diffused version is crucial.

---

For a given data set D - (xx1)) ) we use its empirical pz) to approximate the unknown distribution p(z))p) and denote Po (2o oc ( The diffused distribution then given by the forward process (5). i.e.,
pe(2)(p))= - por(zslaooppd / dwoo
mo
(e(2)ppp= Poc(ass2oo)fnl p)dro x poi (o mt
In order to sample with the reverse process (6) need to characterize the score function of the diffused product distribution Vz, log Pe (xepp)) which is given by the following theorem.
Theorem 1. For any given 3 0, there exists @n(;;) such that the score function of the diffused product distribution can formulated as
V a; log (xx
Vz, Po - af (
gradient guidance G.
where Va, log (xx p) is the score funetion of the diffused data distribution and (xx) satisfies
log (f
f(zo(x)))
Pio p) (20
(10)
3
Theorem demonstrates that sampling from the product distribution can be accomplished by intro- ducing gradient guidance term during the sampling process of the original data distribution, which has a strong connection between the posteriori Pro Pro (Zol p) and the Boltzmann distribution (z
Next, present special case where the gradient guidance terms admit explicit expressions.
Corollary 1, Assume that Pio (2oooaa p) NN Mooe o I), then we have the following results.
First-order guidance: For (R" R), we get
G - Vv
(11)
Second-order guidance: For I E CR R), we get
G: H-1 (- f + ((x))) Boo Moos
+ 10
(12)
Jott
where H
2 f(x)) + Bo I.
worthwhile noting that, for Po(Zo)p) = (Nzollo, I) and the Gaussian transition kernel, the assumption in Corollary holds and the parameters (Mjj) can be expressed explicitly. In practice, we can use Tweedie's 's formula (Efron (201]) to obtain an estimator of μooe, and treat the variance hyper parameter; see Subsection 3.3 for details this. Although the second-order guidance requires computing the inverse of a general Hessian matrix, which may be computationally expensive, it brings faster convergence and better variance reduction.
3 GGDOpt for CCP
In this section, we give our GGDOpt framework for CCP (I. The whole process can be divided into three stages: data generation, diffusion and learning, and sampling with guidance. More specifically, in the data generation stage, a collection of points satisfying the chance constraint is generated to characterize the nonconvex feasible set. The diffusion and learning stage progressively inject noise to convexify the nonconvex feasible region and learn the score function of the conditional distribution order to perform sampling. After learning, the sampling with guidance stage iteratively runs reverse process with an extra gradient guidance to sample from the product distribution, which will asymptotically converge an optimal solution to problem Next, we present the details of the three stages in GGDOpt by one.
5

---

3.1 Stage 1: data generation
First we give an efficient approach to generate high-quality data that satisfy the chance constraint while maintaining lower objective values. Suppose that we have set of samples (h( )ē= denote the empirical mean h I ビ h Notice that in most of cases, it's much easier to solve the following deterministic restricted problem (RP) with fixed h:
min f(x)
(13)
s.t. g( h) Zi.
where 2; 2 is given restriction, i= l.... N. Let (z+) denote the solution to problem L3 for a given 2y. As the smallest clement Emin in 2, increases, the probability of the nonlinear constraint g(x (z)) h) 2 0 also increases. Then, solving problem (13) allows us generate high-quality data that satisfies the chance constraint for arbitrary E (0. 1) while enjoys low objective values.
Since the distribution of the random variable h is unknown, referring SAA method, we approximate the chance constraint using the empirical distribution over samples [/) Then, after getting x( (zz), we have
Proba (((((z)) h) 2 0} ~
En (o)(((((z)),)))))
(14)
1-p(i)
where (g) - 1 2 0 and (g) = O otherwise. By calculating the empirical p(9) an asymptotic approximation of the underlying probability is obtained, requiring no assumption on the underlying distribution P. In the appendix, we give tight lower bound for the probability constraint Prob, (a=az)))) (2z)) ;), f)) 2 0} if the variance and the mean of the random variable h. are known, which helpful to obtain a better approximation p())
Let (+) æ=2)) and repeating the above process, i.e., solving problem 133 and estimating p(), and gradually increasing %;. generate collection of data points D (x (()))))) Ji= which are then used to train our GGDOpt in the next stages.
3.2 Stage 2: diffusion and learning
From Theorem terms, the conditional score Va, log log (xx and the gradient guidance term G, for which explicit forms of first- and second-order guidances have been derived in Corollary 1. Then the challenge reduces to learning the conditional score V log Px<xelp).
observe that the score function of the diffused product distribution has two
In practice, naively conditioning a standard diffusion model by appending the conditioned variable at each step of the sampling process does not work well, as the model often ignores the conditioned information. Related works on conditional score estimation have been studied in (Dhariwal and Nichol (20T]) Dhariwal and Nichol [2021] [Ho and Salimans (022)) ). Here we propose to use the classifier-free guidance (Ho and Salimans [022) give approximation of V log parppp(a) (x|pp).
Instead of training a separate classifier model, classifier-free guidance choose to train an unconditional score estimator to approximate Vx, log Pr(a;) together with the conditional score estimator to approximate Va, log Pt (p))p). Specificity, we train single model se (0 t, p). and the conditioning information p is randomly discarded as empty set 0 with probability Puncond to train unconditionally. Then the conditional score Va, log Pt (x () is estimated by
Va, (x: (1 + (r))
wso
(15)
for a given weight parameter w. Specifically, for the given data set D and network so t, parameterized by 0, the training objective is defined as
Loss 0 Bx- Ee Bey [1s0((())4))) - Va. log (x, | 11 }
(16)
and trained with Adam (Kingma 2014)). The training process of GGDOpt is given in Algorithm I

---

Algorithm 1 Trriniin of GGDOpt
Algorithm Sampling of GGDOpt
Input: {(x11) (((1)) ~ po(x)p))
Input: sg. (x,t, objective f.
Output: sg. (x, p).
Output: zú.
1: repeat
1: ær Pr.
2: Load (To. Po) ~ (ppp).
do 3: Calculate Se (Iz, t. with (18)
fort=
3: Set p 0 with probability Puncond-
4: Sample ~ [].
Calculate G, with (11) or
5: Generate æe ~ Por (Z))
5: Take guided sampling step with 17
6: Take gradient descent step on 16
end for
7: until converged.
7: return Z6 Zo.
3.3 Stage 3: sampling with guidance
Given the forward process ( the corresponding reverse process given by the following reverse-
time SDE with trained so 1. p) and gradient guidance G₁::
dx; = [a(z, t) b(t)) (5o t, p)
G.) dt + b(t)dB,
(17)
where
30 (a t, p) = (1 + s) t, ws (2,, t,
(18)
For the first-order gradient guidance G. in [I we directly use the gradient of the objective scaled by a hyper parameter 3. For the second-orde gradient guidance (12). we need to give the posterior
mean and variance (Moje: ( Here we use Tweedie's formula [Efron [201 get an estimator of the posterior mean follows:
Poj = E (0 0e, p) = Vot (xt + (1 -. (0.,,
(19)
with priori por (a, (r)) = (xx,) VarDo, (1 I) for a specific noising schedule āt.
While Tweedic's formula theoretically provides both the posterior mean and covariance, Eojr (1 (I + (1 ā.) 2 V2 p(::)), computing the covariance requires evaluating the Hessian of ((x). In our framework, the score function se parameterized by a neural network, and computing its second derivatives involves backpropagation through the network's 's Jacobian, which is computationally expensive, especially in high dimensions. To strike a balance between performance and efficiency, we choose to treat the covariance we give a detailed comparison between the fully Tweedie-based method and our approach show that using fixed variance can be practical and robust alternative.
tunable hyper parameter o². In the appendix,
Then the second-order guidance can be calculated by
G. G- - - (v (xx.)
Bo
1 1 r)
(-V V((x))
Bo Moje
1
Male
(20)
and the sampling process of GGDOpt is given in Algorithm
4 Convergence analysis
In this section, give the convergence analysis of the proposed GGDOpt framework in both theoretical and practical aspects. We show that: theoretically, the samples generated by the sampling process will concentrate around the points with the lowest function values within the support of the data distribution; and practically, the gap between the expected function values of generated samples and the optimal value will be bounded by two components.
4.1 Theoretical convergence
As provided by (Pidstrigach) 2022)), under mild assumptions, the sampling distribution of the standard diffusion model will have the exact same support the data distribution. But what if we introduce an
7

---

E D,, < p} as the approximated feasible set of x. The following theorem says that in our settings, as T + oc and B - oo, the samples of GGDOpt will concentrate around the points with the lowest function values within the support of the data distribution Dp for any given p.
extra gradient guidance term? For a given p. denote D, {a(1))
Theorem 2. For any given p E (0, 1), suppose that there exists a constant ö such that the error in the score estimation can be bounded as:
(2 L, p) + G. Va, (og S 6, A ar. For samples Tt ssample ~ Psample (p) generated by the reverse process
(21)
dæ; = (a t) (( tt, p) + G,)] dt b(t)dBr. with prior Pprior = N(0 I), affine drift coefficients a(.,t), t), and
(22)
še (zr, 1, p) (1 + w)sa t, p) wa, t, 0), as T - oc, Psampte(ZalP (Z)lp) will have the same support Po (zolp). Further, as B + 0c, 2. ssample will concentrate around x= = arg min ((x:).
(23)
The assumption in the score estimation error quantifies the approximation accuracy of the trained score network relative to the true score function. It depends on the training quality of the neural network and the expressiveness of the model class. This type of assumption is common in the
theoretical analysis of diffusion models (see, e.g.. ..g..Pidstriiacc) (2022). (2022) De Bortoli et al. [2021] and is used to establish convergence results in generative modeling and sampling.
4.2 Practical error bound
In practice, the forward process cannot reach the stationary distribution and the training is not perfect. This results in the failure of the sample distribution to strictly concentrate on the data points. This will lead to two components of errors: the limited time length error 4 and limited inverse temperature error 12, which given follows:
[ sample)) f(x")] ) [[ [B(((=)))
[z [(x)))
(24)
12
In the above, sampple is sampled from the reverse process 17 x follows the strong solution p" to the Pokker-Plance equation of (17), and x= = arg min D, J (x). Next, we will give practical error bounds of both the two components with finite T and B.
Assumption 1. We assume the following conditions hold:
The forward process is given by dx b(t)ddB,;
The reverse process starts in Pprior N(mr, Er) where my = (l and Ex Cov(Pu(zulp) + T I:
The objective function f(z) satisfies || [x)f(2))l2 & C1 |x||| + Cz.
The first two conditions in Assumption correspond to the VE SDE (Song et al. (000bb) and are primarily used to characterize the discrepancy between the end distribution and the prior distribution. The third assumption is common in the convergence analysis of stochastic optimization and sampling algorithms (sec, c.g.. RRaginsky et al. (2077). In practice, Assumption I holds for broad class of functions, including smooth bounded functions and quadratic objectives, which frequently arise in real-world optimization problems.
Theorem 3. Under Assumption 1, denote (k)) > 1,..., n, the eigenvalues of Ex. For any given P E (0, 1). denote N₂ = [Dpl and x" = arg min f(x). Then for any given T 0 and B > the optimization error can be bounded by
(Taampi)) - f(x)) s CT + (2 + (N, - 1) max [f(x) - f(x")(()))))))) ale 36p,
mED,
(25)
where Cy = log (o and Cr, 5,p are constants.
Theorem 3 provides a non-asymptotic convergence result of GGDOpt with limited time length and inverse temperature. As T > xe and 8 - xc, the optimization error goes to zero and GGDOpt shown to generate asymptotically optimal solutions.
8

---

Experimental results
In this section, we perform numerical experiments on both synthetic datasets and a wireless commu-
nications waveform design problem. To generate the data, employ CVX (Grant al. [2008] to solve the restricted problem (3).. In the diffusion and learning stage, we set 7 = 1000 with a linear noise schedule v7(t) ranging from 0.0001 to 0.02, and let a L) - =2()() x and b(t) = (t)) In the sampling with guidance stage, evaluate both first- and second-order gradient guidances via implementing DDIM-based technique (Song et al. (2020al) with a descaled time step T = 100 for accelerated sampling. We employ two variants the U-Net model (Ronneberger et al. [2015) ) as our score estimator: U-Net-ID for the linear chance constrained problem and both for robust waveform design. Additional experimental details are provided in the supplementary materials.
5.1 Linear chance constrained problem
Consider the following linear chance constrained problem:
min EER" 2 x x
(26)
s.t. Probempo (c x d 0) 2 P.
where Pe = (,, and (b, c, d, p) are hyper parameters selected from a test set. The above second-order conic (SOC) program, for which CVX (Grant et al. 22088) is used for solution. To generate training data, we solve the restricted version of problem (6 for N 1000 values of linearly spaced in the interval [0, 0.5]. Then execute the reverse process with first- and second-order gradient guidance to generate samples.
problem be reformulated
We compare our proposed GGDOpt against different types of SAA methods for solving the problem, using the corresponding CVX solutions as performance benchmarks. Each algorithm was executed 100 times (except CVX). The results with 8 are presented in Table [
Table 1:
Comparison results on the linear chance constrained problem (26
Method
Repeat
FvalMean
FvalStd
FvalMedian
Runtime
SOC_ CVX
Grant et al.
2008
-0.6586
-0.6586
0.3214
SAA_ SAA_CVaR
Nemirovski and Shapiro
2007]
100
-0.5893
0.0248
-0.5869
0.3063
SAA_M Pagnoncelli et al.
2009
100
-0.6281
0.0157
-0.6318
15.4502
SAA_PDCA PDCA Wang et al.
2023
100
-0.6389
0.0314
-0.6408
0.6276
SAA_SNSCO
Zhou et al.
2024
100
0.8051
3.4014
-0.6371
0.2793
GGDOpt_ GGDOptt Withourruddddeee
100
0.3481
0.5486
0.2798
0.0465
CGGDOptFFrst-ordde
100
-0.6483
0.0051
-0.6488
0.0486
SGGDOptSSccod-orte
100
-0.6491
0.0056
-0.6503
0.0507
The results in Table demonstrate that, compared to the SOC_ -VC method, which requires explicit knowledge of the underlying distribution, GGDOpt can approximately find the global minimizer with only samples from distribution Pe while simultaneously achieving significant overhead reduction. Compared to SAA methods, GGDOpt achieves superior performance in terms of lower function values and enhanced numerical stability under the effect of gradient guidance.
As expected, the runtime increases with the problem dimension. However, both the first- and second- order versions of GGDOpt remain consistently faster than the bascline SAA_ PDCA method across all dimensions. Moreover, the increase in runtime is moderate, indicating that our approach scales favorably even in nigh-dimensional settings.
Furthermore, as the runtime increases with the problem dimension, both the first- and second-order versions of GGDOpt reduce the computational time by approximately 80% compared with, offering substantial efficiency improvements. More detailed experimental results on larger problem scale and computational costs are listed in the appendix.

---

5.2 Robust waveform design
Consider the following robust waveform design problem (Wang al. (2014)
K
min S SRERMINN
T((S))
(27)
s.t.
ProD (R. 2 r.} 1,, 1,2,..., Si.... SK - 0 K,
where N₂ the number of antennas the base station and K' the total number of users. For each user i, S; = O, h2, RG and ri 2 denote the signal covariance matrix (to be designed), the random channel vector, the achievable rate, and the desired rate target, respectively.
Firstly, we use U-Net-2D as the score estimator. Notice that during the data generation, all the solutions to the restricted problem (13) exhibit rank-one structure (Huang and Zhang (2007). [Chang
et al. (0008) |uang et al. 2020)) Remarkably, the generated samples maintain this rank-one property (with dominant eigenvalue accounting for >99% of the total eigenvalue) after training, suggesting that the solutions to the robust waveform design problem (27) inherently reside on a rank-one manifold with extremely high probability (Wang al. (2014). which GGDOpt successfully captures. This implies that rank-one decomposition can be effectively applied after generation, enabling the use of U-Net-1D score estimator reduce computational costs in both training and sampling process.
Table 2 summarizes the comparison results of GGDOpt and two state-of satte-of-the-ard the-art methods for solving problem (27) with N₂ 16 and K 3, where the worst probabilities that the chance constraints satisfy for users are underlined. Notably, both baseline methods rely on explicit knowledge of the underlying distribution, whereas GGDOOtt operates solely based on samples. The results show that GGDÖpt outperforms existing convex approximation methods, achieving superior feasible solutions outside the convex restriction of the feasible set, while significantly reducing computational overhead. Complete experimental details are provided in the appendix.
Table 2: Optimization methods comparison for robust waveform design
Method
Metrie
p= 0.05
p 0.10
p 0.15
p 0.20
Probability 0.99; 0.99; 0.99
0.99; 0.99; 0.99
0.99; 0.99; 0.99
0.99; 0.99; 0.99
Sphere Bounding
FuncValue
0.1374
0.1366
0.1361
0.1357
Ben-Tal and Nemirovski
2000
Runtime
1.4688
1.4375
1,4113
1.3875
Probability
0.96; 0.95; 0.96
0.93; 0.93; 0.93
0.91; 0.91; 0.92
0.90; 0.90: 0.91
Bernstein-lype type Inequality
Vanue
0.1260
0.1253
0.1248
0.1244
Wang et al. 2014
Runtime
1.2938
1.2813
1,2593
1.2652
Probubility
0.99; 0.95; 0.99
0,92; 0.98: 0.91
0.93; 0.86; 0.94
0.87: 0.81: 0.91
GGDOpt
Funceaalue 0.1279
0.1265
0.1254
0.1247
First-order guidance
Runtime
0.0691
0.0628
0.0603
0.0635
Probubility
0.97; 0.95; 0.96
0.90; 0.94; 0.90
0.88; 0.85; 0.86
0.88; 0.80; 0.87
GGDOpt
FuncValue
0.1260
0.1246
0.1239
0.1237
Second-order guidance
Runtime
0.0788
0.0712
0.0687
0.0682
Conclusion
In this paper, have proposed GGDOpt, a gradient-guided diffusion framework that efficiently solves nonconvex CCP without requiring the exact distribution knowledge. By reformulating CCP as sampling problem over the product of an unknown data distribution and Boltzmann distribu- tion, GGDOpt leverages both first- and second-order gradient information during reverse sampling. Theoretical convergence guarantees and practical error bounds are provided under mild assumptions. Experimental results demonstrate that GGDOpt outperforms existing methods in both solution quality and numerical stability with significant overhead reduction.
10

---

Acknowledgments
The work of Boyang Zhang and Ya-Feng Liu was supported part by the National Natural Science Foundation of China (NSFC) under Grant 12021001 and Grant 12371314. The work of Zhiguo Wang was supported part by The National Key Research and Development Program of China under Grant 2020YTA0714003 and in part by NSFC under Grant 63203313.
References
Lukáš Adam and Martin Branda. Nonlinear chance constrained problems: optimality conditions, regularization and solvers. Journal of Optimization Theory and Applications, 170:419 436, 2016.
Shabbir Ahmed and Alexander Shapiro. Solving chance-constrained stochastic programs via sampling and integer programming. In ofate-of-the-Ari Decision-Making, Tools in the Information- -ntensivve Age, pages 261-269. Informs, 2008.
Aharon Ben-Tal Tal and Arkadi Nemirovski. Robust solutions of linear programming problems contami- nated with uncertain data. Mathematical Programming, 18:411-4244 2000.
Dimitris Bertsimas and Melvyn Sim. The price of robustness. Operations research, 52(1):35-53, 2004.
Pierre Bonami and Miguel Lejeune. An exact solution approach for portfolio optimization problems under stochastic and integer constraints. Operations Research, 57(3):650-670, 2009.
Giuseppe Carlo Calafiore and Marco C Campi. The scenario approach to robust control design. IEEE
Transactions on Automatic Control, 51(5):742-753, 2006.
Tsung-Hui Chang, Zhi-Quan Luo, and Chong-Yune Chi. Approximation bounds for semidefinite relaxation of max-min-fair multicast transmit beamforming problem. IEEE Transactions on Signal Processing, 363-)3393,,3943, 2008.
Yongxin Chen, Sinho Chewi, Adil Salim, and Andre Wibisono. Improved analysis for a proximal algorithm for sampling. Conference on Leurning Theory, pages 2984-3014. PMLR, 2022.
Hyungjin Chung, Jeongsol Kim, Michael T Mccann, Mare Klasky, and Jong Chul Ye. Diffusion posterior sampling for general noisy inverse problems. arXiv preprint 22KV::2209.14687 2022a.
Hyungjin Chung, Byeongsu Sim, Dohoon Ryu, and Jong Chul Ye. Improving diffusion models for inverse problems using manifold constraints. Advances in Neural Information Processing Systems, 35:25683-25699 2022b.
Valentin De Bortoli, James Thornton, Jeremy Heng, and Arnaud Doucet. Diffusion schrödinger bridge with applications to score-based generative modeling. Advances in Neural Information Processing Systems, 34:17695-17709, 2021.
Prafulla Dhariwal and Alexander Nichol. Diffusion models beat gans image synthesis. Advances in Neural Information Processing Systems, 34.8780-8794, 2021.
Bradley Efron. Tweedie's s formula and selection bias. Journal of the American Statistical Association, 106(496):1622-1114 2011.
Michael Grant, Stephen Boyd, and Yinyu Ye. Cvx: Matlab software for disciplined convex program- ming, 2008
Yingqing Guo, Hui Yuan, Yukang Yang, Minshuo Chen, and Mengdi Wang. Gradient guidance for diffusion models: An optimization perspective. arXiv preprint arXiv:2404.1474, 2024.
René Henrion. Structural properties of linear probabilistic constraints. Optimization, 56(4):425-440, 2007.
Jonathan Ho and Tim Salimans. Classifier-free diffusion guidance. arXiv preprint arXiv:2207.12500 2022.

---

Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in Neural Information Processing Systems, 33:6840-6857 2020.
Yi Huang, Jiancheng Huang, Yifan Liu, Mingfu Yan, Jiaxi Lv, Jianzhuang Liu, Wei Xiong, He Zhang, Liangliang Cao, and Shifeng Chen. Diffusion model- based image editing: A survey. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2025.
Yongwei Huang and Shuzhong Zhang. Complex matrix decomposition and quadratic programming. Mathematics of Operations Research, 32(3):758-768, 2007.
Yongwei Huang. Sergiy Vorobyov, and Zhi-Quan Luo. Quadratic matrix inequality approach to robust adaptive beamforming for general-rank signal model. IEEE Transactions Signal Processing, 18:22244-2255, 2020.
Shinji Kataoka. A stochastic programming model. Econometrica: Journal of the Econometric Society, pages 181-196, 1963.
Diederik Kingma. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980. 2014.
Lingkai Kong, Yuanqi Du, Wenhao Mu, Kirill Neklyudov, Valentin De Bortoli, Dongxia Wu, Haorui Wang, Aaron Ferber, Yi-An Ma, Carla P Gomes, et al. Diffusion models as constrained samplers for optimization with unknown constraints. arXiv preprint 240.::2402.18012, 2024.
Siddarth Krishnamoorthy, Satvik Mehul Mashkaria, and Aditya Grover. Diffusion models for black- box optimization. In International Conference on Machine Learning, pages 17842-17857. PMLR, 2023.
Constantino Lagoa, Xiang Li, and Mario Sznaier. Probabilistically constrained linear programs and risk-adjusted controller design. SIAM Journal Optimization, 15(3):938-951, 2005.
Yin Tat Lee, Ruoqi Shen, and Kevin Tian. Structured logconcave sampling with restricted gaussian oracle. In Conference on Learning Theory, pages 2993-3050. PMLR, 2021.
Zihao Li, Hui Yuan, Kaixuan Huang, Chengzhuo Ni, Yinyu Ye, Minshuo Chen, and Mengdi Wang. Diffusion model for data-driven black-box box optimization. arXiv preprint arXiv:2403. 13219, 2024.
Ruihuai Liang, Bo Yang, Pengyu Chen, Xianjin Li, Yifan Xue, Zhiwen Yu, Xuelin Cao, Yan Zhang, Mérouane Debbah, H Vincent Poor, et al. Diffusion models network optimizers: Explorations and analysis. IEEE Internet of Things Journal, 2025.
Yi-An Ma, Yuansi Chen, Chi Jin, Nicolas Flammarion, and Michael Jordan. Sampling can be faster than optimization. Proceedings of the National Academy of Sciences, 116(42)20881-20888 2019.
Arkadi Nemirovski and Alexander Shapiro. Convex approximations of chance constrained programs.
SIAM Journal on Optimization, 17(4)99999996 2007.
Bernardo K Pagnoncelli, Shabbir Ahmed, and Alexander Shapiro. Sample average approximation method for chance constrained programming: theory and applications. Journal of Optimization Theory and Applications, 142(2):399-416, 2009.
Jakiw Pidstrigach. Score- generative models detect manifolds. Advances in Neural Information Processing Systems, 355335522-58866 2022.
András Prékopa. Stochastic programming, volume 324. Springer Science & Business Media, 2013.
Maxim Raginsky, Alexander Rakhlin, and Matus Telgarsky. Non-convex learning via stochastic gradient langevin dynamics: a nonasymptotic analysis. In Conference on Learning Theory, pages 1674-1703. PMLR, 2017.
Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation. In Medical Image Computing and Computer-assisted Intervention-MICCAI 2015: 18th international conference, Munich, Germany, October 5-9, 2015, proceedings, part III 18, pages 234-241. Springer, 2015.
12

---

Simo Särkkä and Amo Solin. Applied stochastic differential equations, volume 10. Cambridge University Press, 2019.
Nahom Seyoum and Haoxiang You. Beyond smoothness and convexity: Optimization via sampling. arXiv preprint arXiv:2504.0288: 2025.
Bowen Song, Soo Min Kwon, Zecheng Zhang, Xinyu Hu, Qing Qu, and Liyue Shen. Solving inverse problems with latent diffusion models via hard data consistency. arXiv preprint arXiv:2307.08/23, 2023.
Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. arXiv preprint arXiv:2010.02502 2020a.
Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. arXiv preprint arXiv:201 134466, 2020b.
Kun- Yu Wang, Anthony Man-Cho So, Tsung-Hui Chang, Wing-Kin Ma. and Chong- Yung Chi. Outage constrained robust transmit optimization for multiuser miso downlinks: Tractable approx- imations by conic optimization. IEEE Transactions Signal Processing, 62725)55590-5775
2014.
Peng Wang, Rujun Jiang, Qingyuan Kong, and Laura Balzano. A proximal de algorithm for sample average approximation of chance constrained programming. arXiv preprint arXiv:2301.004 000423, 2023.
Andre Wibisono. Sampling as optimization in the space of measures: The langevin dynamics as composite optimization problem. In Conference on Learning Theory, pages 2093-3027. PMLR, 2018.
Dongxia Wu, Nikki Lijing Kuang, Ruijia NIu, Yi-An Ma and Rose Yu. Diff-bbo: Diffusion-based inverse modeling for black-box optimization. arXiv preprint arXiv:2407.0061 2024.
Zongsheng Yue, Jianyi Wang, and Chen Change Loy. Resshift: Efficient diffusion model for image super-resolution by residual shifting. Advances in Neural Information Processing Systems, 36: 13294-13307, 2023.
Shenglong Zhou, Lili Pan, Naihua Xiu, and Geoffrey Ye Li. A 0/1 constrained optimization solving sample average approximation for chance constrained programming. Mathematics of Operations Research, 2024.
13

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: The main results and contributions of this paper are all included in the abstract and introduction clearly.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
It is fine to include aspirational goals as motivation long is clear that these goals are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [Yes]
Justification: We point out all assumptions and discuss the limitations of the work thoroughly in the supplementary material.
Guidelines:
The answer NA means that the paper has limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create a separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (c.g.. independence assumptions, noiseless settings, model sellsspeciiiationn asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect on the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low or images are taken in low lighting. Or speech-to-tex, system might not be used reliably to provide closed captions for online lectures because it fails handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and faimesss.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't 't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community, Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory assumptions and proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
14

---

Answer: [Yes]
Justification: All the assumptions used are included in the main paper, and the proofs are provided in the supplementary material.
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced in the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental result reproducibility
Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that affects the main claims and/or conclusions of the paper (regardless whether the code and data are provided or not)?
Question:
Answer: [Yes]
Justification: The main configuration of experiments is claimed in the Experimental results section, and more details are provided in the supplementary material. We will release the code once the paper published.
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, No answer this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data provided or not.
the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how replicate the results, access a hosted model (e.g.. in the of large language model), releasing of a model checkpoint, or other means that are appropriate the research performed.
While NeurIPS does not require releasing code, the conference does require submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) the contribution is primarily a new algorithm, the paper should make clear how reproduce that algorithm. (b) If the contribution is primarily
model architecture, the paper should describe the architecture clearly and fully.
(c) the contribution is a new model (e.g., a large language model), then there should either be way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).
(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should possible for other researchers to have some path to reproducing verifying the results.
15

---

5. Open access to data and code
Question: Does the paper provide open access to the data and code, with sufficient instruc- tions faithfully reproduce the main experimental results, described in supplemental material?
Answer: [No]
Justification: The data generation algorithm is provided in this paper and can be reproduced easily. The code is a straightforward implementation the proposed framework, and will be released once the paper is published.
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurlPS code and data submission guidelines (https://nips cc/
for more details.
While we encourage the release of code and data, we understand that this might be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https
mtiiss.cc/public//ee
for more details.
The authors should provide instructions data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. only subset of experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information as possible in supplemental material (appended to the paper) recommended, but including URLs to data and code is permitted.
6. Experimental setting/details
Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
Justification: The experimental settings presented in the main paper, and full details provided the supplementary material.
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented in the of the paper to level of detail that is necessary to appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, or as supplemental material.
7. Experiment statistical significance
Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Question:
Answer: [Yes]
Justification: In the experiments, we run multiple times for each method and the stability is shown in the main paper.
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by emoor bars, confi- dence intervals, or statistical significance tests, least for the experiments that support the main claims of the paper.
16

---

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
The assumptions made should be given (e.g., Normally distributed errors).
It should be clear whether the error bar is the standard deviation the standard error of the mean.
It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not to show in tables figures symmetric error bars that would yield results that are out range (e.g. negative error rates).
If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments compute resources
Question:
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed reproduce the experiments?
Answer: [Yes]
Justification: The information of the compute resources is provided in the supplementary material.
Guidelines:
The answer NA means that the paper does not include experiments.
The paper should indicate the type of compute workers CPU GPU, internal cluster, or cloud provider, including relevant memory and storage.
The paper should provide the amount of compute required for each of the individual experimental runs as well estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g.. preliminary or failed experiments that didn't make into the paper).
9. Code of ethics
Question: Does the research conducted the paper conform, in every respect, with the NeurlPS Code of Ethics httpps://nuurpss..... http://pp....pppp..pc/aaccaaeaea
Answer: [Yes]
Justification: The research conducted in the paper conform, in every respect, with the NeurlPS Code of Ethics.
Guidelines:
The answer NA means that the authors have reviewed the NeurIPS Code of Ethics.
If the authors answer No, they should explain the special circumstances that require deviation from the Code of Ethics.
The authors should make to preserve anonymity (e.g.. if there is special consid- eration due to laws or regulations in their jurisdiction).
10. Broader impacts
Question: Does the paper discuss both potential positive socictal impacts and negative socictal impacts of the work performed?
Answer: [NA]
Justification: This paper focus on the theoretical results of Gradient Guidance and frame- work for solving chance constrained problems. There no direct path to any negative applications of this paper.
17

---

Guidelines:
The answer NA means that there no societal impact of the work performed.
If the authors answer NA or No. they should explain why their work has no societal impact why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
If there negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how system learns from feedback over time, improving the efficiency and accessibility of ML).
Safeguards
Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Justification:
This paper poses no such risks.
Guidelines:
The answer NA means that the paper poses no such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
Question:
Answer: [Yes]
Justification: All the original papers of used models and algorithms properly cited in this paper.
Guidelines:
The answer NA means that the paper does not existing assets.
The authors should cite the original paper that produced the code package dataset.
The authors should state which version of the asset used and, possible, include URL.
The name of the license (c.g., CC-BY 4.0) should be included for each asset.
18

---

For scraped data from a particular source (e.g., website). the copyright and terms of service of that souree should be provided.
assets are released, the license, copyright information, and terms use in the package should be provided. For popular datasets, paperssi thood cww datasets has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.
If this information is not available online, the authors encouraged to reach out to the asset's 's creators.
13. New assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [NA]
Justification: This paper does not release new assets, and our code will be released once the paper is published.
Guidelines:
The answer NA means that the paper does not release new assets.
Researchers should communicate the details of the dataset/code/model part of their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent was obtained from people whose asset is used.
At submission time, remember anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
14. Crowdsourcing and research with human subjects
Question:
For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well details about compensation (if any)?
Answer: [NA]
Justification:
This paper does involve crowdsourcing nor research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Including this information in the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then as much detail as possible should be included in the main paper.
According to the NeurlPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional review board (IRB) approvals or equivalent for research with human subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
Answer: [NA]
Justification:
This paper does not involve crowdsourcing nor research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsoureing nor research with human subjects.
19

---

Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if' applicable), such as the institution conducting the review.
16. Declaration of LLM usage
Does the paper describe the usage of LLMs if is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, originality of the research, declaration is not required.
Question:
Answer: [NA]
Justification: The core method development in this research does not involve LLMS as any important, original, non-standard components.
Guidelines:
The answer NA means that the core method development in this research does not involve LLMs as any important, original, non-standard components.
Please refer to our LLM policy (https://neurips /att::///earrrpsscc//oooeerreee/ cc///oooeeeeeecs//22/////.. for what should or should not be described.
20


Reviews:
[{"review_id": "k9lctOR7Rw", "reviewer": "Reviewer_2s5s", "summary": "Chance constrained programming (CCP) is an optimization framework that handles uncertainty by ensuring that constraints are satisfied with a specified probability level. This work presents GGDOpt, a novel Gradient-Guided Diffusion-based Optimization framework designed to solve chance constrained programming problems under uncertainty. This work aims to propose a framework to solve the CCP when the underlying distribution is unknown. Specifically, without requiring the exact distribution of the uncertainty, GGDOpt operates using only sampled data. It reformulates CCP as a sampling problem over the product of a data distribution and a Boltzmann distribution to effectively handle nonconvex constraints. Leveraging both first- and second-order gradient information, the method progressively injects noise through forward diffusion to convexify the feasible region and applies guided reverse sampling for optimization. This work provides some theoretical analyses for the proposed framework, such as convergence and error bounds. However, the experimental results are limited, the writing and presentation of this work need to be enhanced, and more analyses and discussions are required to be added.", "questions": "My questions are based on the weakness:\n\nThe third assumption in Assumption 1 seems uncommon in the existing optimization literature. Could the authors clarify whether all the assumptions used in this paper, particularly the third one, are standard and practical? Would it be possible to provide further discussion or justification for their applicability?\n\nCould the authors include more experiments on additional machine learning applications? The current experimental settings appear limited to relatively simple scenarios, and broader empirical validation may enhance the generality of the proposed method.\n\nThe readability of Figure 1 is somewhat limited. Would the authors consider improving the caption by adding a more detailed explanation of the framework and highlighting its key components, to help readers better grasp the main insights?\n\nThe manuscript appears to lack sufficient discussion of related work. Could the authors provide a more comprehensive review and comparison with existing approaches in the field? This may help position the proposed method more clearly within the current literature.", "limitations": "Yes, this work discusses the limitations in Appendix A.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. The key problem studied in this work is important, that is, how to  efficiently solve CCP when the underlying distribution P is unknown.\n\n2. This work provides theoretical guarantees for the proposed framework, such as the convergence in Theorem 2 and the optimization error bound in Theorem 3.\n\nWeaknesses:\n\n1. The third assumption in Assumption 1 seems not to be widely used. Can you provide more discussions about whether all the used assumptions are widely used and practical?\n\n2. In the experiment, can you conduct more experiments on additional machine learning applications? Because the author just carry out the experiments on some simple scenarios. \n\n3. The readability of Figure 1 is poor. The authors are suggested to offer more explanation about this framework in the caption to help the readers better learn the key insights of this framework.\n\n4. The writing and organization of this work are poor. For instance, the manuscript lacks sufficient discussion of related work. It is recommended to add a dedicated section for this purpose. Moreover, the authors should compare the proposed method with existing approaches (maybe in the related work section) to clarify the contributions of this work.\n\n5. In Theorem 2, the convergence analyses are provided, which is appreciated by the reviewer. However, the non-asymptotic convergence rate is unknown. Can you provide some theoretical analyses or the discussions about the non-asymptotic convergence?\n\n6. In the introduction, the discussion of the contributions is ambiguous. For instance, it is unclear why the claim “applicable to broader problem domains” is considered a **key** contribution. Since this is a theoretical work, a more appropriate key contribution might be the development of a new framework to address a novel problem.", "was_revised": true, "final_justification": "My concerns have been solved.", "rebuttal": "**Reply**\n\nWe sincerely thank the reviewer for the valuable comments and constructive feedback, which have greatly helped us clarify our contributions and improve our work. We hope the following point-by-point responses address your concerns and offer a clearer understanding of our work.\n\n**Reply to Weakness 1 and Question 1:**\n\nWe appreciate your feedback regarding Assumption 1. The first and second assumptions in Assumption 1 describe the forward and reverse processes of an Ornstein–Uhlenbeck (OU) process, respectively; such formulations are standard in the diffusion literature for analyzing the impact of the finite diffusion time on the convergence (e.g., [Song2020]). \n\nThe third assumption imposes **a growth bound on the gradient of the objective function**. This type of regularity condition is common in the convergence analysis of stochastic optimization and sampling algorithms, particularly when studying stability and convergence under Langevin dynamics or diffusion-based methods (see, e.g., [Raginsky et al., 2017]). In practice, this assumption holds for a broad class of functions, including smooth bounded functions and quadratic objectives, which frequently arise in real-world optimization problems.\n\n**Reply to Weakness 2 and Question 2:**\n\nFollowing your suggestion, we have added experiments on a VaR-constrained mean–variance portfolio selection problem, which aims to minimize the risk while pursuing a targeted level of returns with probability at least $1-\\rho$. Let $\\boldsymbol{\\mu}\\in\\mathbb{R}^n$ and $\\boldsymbol{\\Sigma}\\in\\mathbb{R}^{n\\times n}$ denote the expectation and covariance matrix of the returns of $n$ risky assets, and $\\gamma\\in\\mathbb{R}_+$ denote the risk aversion factor. Let $\\boldsymbol{x}\\in\\mathbb{R}^n$ denote the allocation vector. Then this problem is formulated as follows:\n\n$$\\min_{\\boldsymbol{x}\\in\\mathbb{R}^n}\\quad\\gamma \\boldsymbol{x}^{\\top}\\boldsymbol{\\Sigma}\\boldsymbol{x}-\\boldsymbol{\\mu}^{\\top}\\boldsymbol{x}\\quad\\text{s.t.}\\quad\\text{Prob}_{\\boldsymbol{\\xi}}(\\boldsymbol{\\xi}^{\\top}\\boldsymbol{x}\\geq R)\\geq1-\\rho,$$\n\nwhere $R\\in\\mathbb{R}_+$ is a prespecified level on the return. We use 2523 daily return data of 435 stocks included in Standard \\& Poor’s 500 Index between March 2006 and March 2016 and set $R=0.02\\%$ and $\\gamma=2$. Some results are shown below:\n\n**Table:Comparison results of the VaR-constrained mean-variance portfolio selection problem**\n|($\\rho$,$n$)|Metric|MIP|ALDM|PDCA|LAG|GGDOpt(First)|GGDOpt(Second)|\n|---|---|---|---|---|---|---|---|\n|(0.05,100)|fval|-0.0951|-0.0723|-0.0917|-0.0936|-0.0904|-0.0946|\n||time|15.58|2.418|4.602|0.9433|0.3768|0.4071|\n||prob|0.8600|0.8666|0.9700|0.8467|0.9200|0.8933|\n|(0.05,400)|fval|-0.0874|-0.0750|-0.0814|-0.0859|-0.0827|-0.0867|\n||time|204.2|66.68|93.42|2.7570|1.2732|1.3559|\n||prob|0.9066|0.8308|0.9891|0.8933|0.9533|0.9267|\n|(0.1,100)|fval|-0.0951|-0.0721|-0.0856|-0.0927|-0.0915|-0.0936|\n||time|13.31|2.388|6.258|0.9365|0.3420|0.4218|\n||prob|0.8600|0.7633|0.9233|0.8533|0.9067|0.8667|\n|(0.1,400)|fval|-0.0874|-0.0713|-0.0826|-0.0864|-0.0829|-0.0870|\n||time|148.6|67.95|81.95|2.7323|1.2546|1.2818|\n||prob|0.9058|0.8158|0.9266|0.8800|0.9267|0.9133|\n\nIn the above experiments, we compare our algorithm with several classical methods, including the mixed‐integer program (MIP), the augmented Lagrangian decomposition method (ALDM), the proximal difference‐of‐convex algorithm (PDCA), and the diffusion‐based Look‐Ahead Guidance (LAG) method [Guo2024]. We set $\\rho=0.05,0.1$ and $n=100,400$, reporting the final‐iteration objective function value (fval), total runtime (time), and the empirical probability of the chance constraint computed over randomly sampled daily returns (prob).\n\nTable 1 shows that MIP achieves the lowest objective values but incurs the highest computational cost, as it fully exploits the data by formulating CCP as mixed integer program. LAG attains competitive objectives but requires additional back‐propagation steps for guidance. In contrast, GGDOpt well balances solution quality and efficiency, significantly reducing runtime while maintaining comparable objective values and constraint satisfaction.\n\n**Reply to Weakness 3 and Question 3:**\n\nWe appreciate your suggestion on Figure 1. To help readers better understand the framework, we provide an additional explanation in the caption of Figure 1 as follows:\n\n>**Figure 1:**Overview of the GGDOpt framework.  \n>(1)Generate a training set of points satisfying the chance constraint by solving a deterministic restricted problem.  (2)Train a diffusion model with classifier-free guidance to learn the score of the conditional distribution.  (3)Perform the reverse diffusion process with additional gradient guidance to sample from the product of the data distribution and the Boltzmann distribution.\n\n**Reply to Weakness 4 and Question 4**\n\nThank you for pointing out the absence of a dedicated related‑work section. In the revised manuscript, we plan to add a standalone subsection to position our work more clearly within the literature. \n\nCompared with [Guo2024] and related methods, our work is the first, to the best of our knowledge, to use diffusion models to solve the general chance constrained problems. The key challenge here is the **lack of direct training data** corresponding to the product distribution of the objective and constraints. We address this through a dedicated data generation stage, followed by conditional training of the score. In contrast, [Guo2024] assumes access to a pre-trained unconditional diffusion model and focuses on a restricted linear-Gaussian setting. Unlike classical convex approximation approaches for CCP, our method does not require prior knowledge of the underlying distribution. Instead, we only assume access to samples from it, which makes our approach applicable to broader and more realistic settings.\n\nFor clarity, we summarize the main contributions below.\n\n- **Problem reformulation with a novel paradigm**  \nWe reformulate the original CCP as a sampling task over the product of two distributions: an unknown data distribution (implicitly defined by the constraint) and a Boltzmann distribution induced by the objective function. This perspective enables us to transform the challenging chance constrained optimization problem into a generative modeling problem amenable to diffusion-based techniques.\n- **Feasibility-aware data generation**  \nTo generate high-quality training data that approximately satisfies the chance constraint, we design a deterministic restricted problem by relaxing the probability constraint and solving it using standard optimization techniques. The solutions are used to guide the training of the conditional diffusion model, effectively capturing the geometry of the feasible region without requiring explicit knowledge of the underlying uncertainty distribution. \n- **Efficient gradient guidance**  \nTo sample from the product distribution, we develop a gradient-guided reverse process. Our guidance terms are derived in closed form based on the structure of the product distribution and do not require backpropagation through the neural network. This is in sharp contrast to [Guo2024], where the guidance involves differentiating the posterior mean estimate with respect to the input, resulting in significantly higher computational cost.\n- **Theoretical convergence and practical evaluation**  \nRegarding the sampling process as a reverse time stochastic differential equation, GGDOpt is shown to generate asymptotically optimal solutions. A practical error bound is also provided with two components: the limited time length error and limited inverse temperature error.\n\n**Reply to Weakness 5:**\n\nThanks for your recognition of Theorem 2 and its convergence guarantee. As you correctly noted, Theorem 3 provides finite-time and finite-temperature error bounds between the samples from the reverse process and the minimizer $\\boldsymbol{x}^*$ of the objective $f(\\boldsymbol{x})$ within an approximated feasible region $D_{\\rho}$. Here we give a corollary of Theorem 3 to present some non-asymptotic upper bounds.\n\nCorollary (Informal): For sufficiently large $\\beta>0$, with the assumption in Theorem 3, we have that $\\mathbb{E}[f(\\tilde{\\boldsymbol{x}}_T)]-f(\\boldsymbol{x}^{*})]\\leq O(\\frac{1}{\\sqrt{T}})$.\n\nIt should be noted that the above corollary is a preliminary convergence result requiring no additional assumptions and similar to the Langevin algorithm for solving nonconvex problems [Ma2019].\n\n**Reply to Weakness 6:**\n\nWe agree that the primary contribution of our work is the GGDOpt framework for solving chance constrained problems under unknown distributions. In the revised manuscript, we will refine the contribution statements—particularly in the introduction—to make this emphasis clearer and to guide readers better understanding our concerns and contributions.\n\nOnce again, we thank the reviewer for the thorough evaluation and thoughtful suggestions. We hope this detailed response clarifies the motivation, methodology, and contributions of our work. Your feedback has been invaluable in strengthening both the presentation and the scope of the paper.  \n\nThe corresponding references are listed in order:\t\n\n1.Song, Yang, et al. Score-based generative modeling through stochastic differential equations. arXiv preprint arXiv:2011.13456, 2020.\n\n2.[Raginsky2017]Raginsky, Maxim, et al. Non-convex learning via stochastic gradient langevin dynamics: a nonasymptotic analysis. Conference on Learning Theory. PMLR, 2017.\n\n3.[Ma2019]Ma, Yi-An, et al. Sampling can be faster than optimization. Proceedings of the National Academy of Sciences 116.42, 2019.\n\n4.[Guo2024]Guo, Yingqing, et al. Gradient guidance for diffusion models: An optimization perspective. Advances in Neural Information Processing Systems, 2024."}, {"review_id": "e1HjvNfht7", "reviewer": "Reviewer_AaBr", "summary": "This paper proposes a novel approach for solving chance-constrained optimization (CCP) problems by integrating diffusion models into the solution process. The core challenge lies in sampling from a product distribution that combines the constraint-related randomness with the Boltzmann distribution of the objective function. To address this, the authors develop a three-stage algorithm: (1) a data generation phase that produces high-quality design points using a deterministic surrogate of the chance constraint, (2) a diffusion-based learning stage that estimates the score of the product distribution, and (3) a guided sampling stage based on the learned score. Numerical experiments on CCP problems demonstrate that the proposed method outperforms existing algorithms in terms of both objective value and feasibility.", "questions": "1. Why does the second-order guidance incur no additional cost compared to first-order? Is this due to problem dimension  is small?\n\n2. In what settings is assumption (21) practically valid? It seems somewhat strong.\n\n3. In line 150, the authors write “convexify the nonconvex feasible region.” However, Problem (13) can still be nonconvex. Could the authors clarify what is meant here—perhaps it refers to tightening the constraint conservatively, rather than true convexification?", "limitations": "N/A", "rating": 4, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\nThe paper introduces a new framework that incorporates gradient-guided diffusion models into CCP. Integrating gradient-guided diffusion into CCP is, to my knowledge, new and potentially influential for stochastic optimization. The experimental section demonstrates consistent improvements over competitive baselines, suggesting that diffusion-based sampling can be worthwhile to study further.\n\n\nWeakness:\n\n1. The discussion of gradient guidance in Section 2.2 is somewhat brief. For readers unfamiliar with the topic like me, it would be helpful to provide more explanation or intuition. In particular, it is unclear how much of the guidance mechanism is novel in this work, since similar techniques—such as classifier guidance or energy-based guidance—have been proposed in prior literature (e.g., Guo et al., 2024). A more explicit comparison or positioning relative to these works would clarify the contribution.\n\n2. In section 3.1, the data generation procedure in Section 3.1 is not fully transparent. For instance, the estimation of  $\\rho^{(i)}$, using Equation (14) could be explained more clearly. The notation uses a subscript without explicitly defining the indicator function used to approximate the chance constraint.\n\n3. The assumption in Equation (21) appears weak or hard to verify in practice. It would be helpful to explain in what types of applications it tends to hold, or whether the method is robust when it is only approximately satisfied.\n\nReferences:\nGuo, Y., Yuan, H., Yang, Y., Chen, M., & Wang, M. Gradient Guidance for Diffusion Models: An Optimization Perspective. In The Thirty-eighth Annual Conference on Neural Information Processing Systems.\n\n\nMinor:\nLine190: parametered should be parameterized", "was_revised": true, "final_justification": "My concerns have been addressed, and I have raised my score accordingly.", "rebuttal": "**Reply**\n\nWe sincerely thank you for your thoughtful and constructive feedback. We truly appreciate your effort in identifying the key weaknesses and raising important questions, which helped us better clarify the scope, contributions, and implementation details of our work. Below we provide a point-by-point response to each of your comments.\n\n**Reply to Weakness 1:**\n\nWe appreciate your insightful feedback on our gradient guidance mechanism in Section 2.2. The core intuition of the gradient guidance in our work is to steer the diffusion sampling trajectory toward regions that yield desirable properties (e.g., lower objective values or improved feasibility), by modifying the score function in a principled way. While guidance techniques have been widely applied in areas such as image generation, black-box optimization, and inverse problems, our contribution does not lie in introducing guidance, but rather in deriving **a novel class of guidance terms based on product distributions**, specifically tailored to chance-constrained optimization problems with unknown distributions.\n\nMore specifically, compared to prior work such as [Guo2024], our approach introduces two main innovations:\n\n- **Conditional Training and Applicability Beyond Linear-Gaussian Settings:**  Unlike [Guo2024], which applies guidance to pre-trained unconditional diffusion models and assumes a linear objective with Gaussian data, our framework involves a dedicated data generation process followed by conditional score training. This enables us to address nonlinear and structurally complex chance-constrained problems, where directly sampling from the feasible region is nontrivial.\n\n- **A New Class of Guidance Derived from Product Distributions:** Most existing guided diffusion frameworks follow the general SDE form as follows:\n\n$$d\\boldsymbol{x}_t=[\\boldsymbol{a}(\\boldsymbol{x}_t,t)-b(t)^2(\\boldsymbol{s}(\\boldsymbol{x}_t,t)+\\boldsymbol{G}_t )]dt+b(t)d\\boldsymbol{\\bar{B}}_t.$$\n\nIn our work, we derive two types of guidance terms directly from the product distribution formulation of the target density:\n\n1. a first-order guidance \n\n$$\\boldsymbol{G}_t^{(1)} = - \\beta \\nabla f(\\boldsymbol{x}_t),$$\n\n2. a second-order guidance\n\n$$\\boldsymbol{G}_t^{(2)} = -\\frac{1}{\\sigma^2}[\\boldsymbol{H}^{-1}[(\\nabla^2 f(\\boldsymbol{x}_t) \\boldsymbol{x}_t+\\nabla f(\\boldsymbol{x}_t)) -\\frac{1}{\\beta \\sigma^2}\\boldsymbol{\\mu}]+\\boldsymbol{\\mu}],$$\n\nwhere the terms are computed based on a learned surrogate for the chance constraint and the posterior mean $\\boldsymbol{\\mu}_{0|t}$. \n\nIn contrast, [Guo2024] introduces a Look-Ahead Guidance term designed for linear objectives:\n\n$$\\boldsymbol{G}_{t}^{(3)} = -\\beta(t)\\nabla(y-\\boldsymbol{g}^\\top \\hat{\\mathbb{E}}[\\boldsymbol{x}_0|\\boldsymbol{x}_t])^2,$$\n\nwhere $\\beta(t)$ and $y$ are tuning parameters, $\\boldsymbol{g}$ is the gradient of the linear objective, and $\\hat{\\mathbb{E}}[\\boldsymbol{x}_0|\\boldsymbol{x}_t]$ is an approximation of the posterior mean that can be calculated by the score network, i.e., $\\hat{\\mathbb{E}}[\\boldsymbol{x}_0|\\boldsymbol{x}_t] = \\alpha^{-1}(t)(\\boldsymbol{x}_t + h(t)\\boldsymbol{s}(\\boldsymbol{x}_t,t))$. This approach is effective when the data distribution is Gaussian and the objective is linear, but may degrade under nonlinear or non-Gaussian scenarios. \n\nOur experimental results further demonstrate that for the nonlinear chance constrained programming, the proposed GGDOpt consistently outperforms the Look-Ahead Guidance from [Guo2024] in terms of both objective value (fval) and computational efficiency (sampling time).\n\n**Table:Comparison results with Look-Ahead Guidance [Guo2024].**\n|Method($\\rho=0.1$)| |$n=2$||$n=4$||$n=8$||$n=16$||\n|---|---|---|---|---|---|---|---|---|---|\n| | |fval|time|fval|time|fval|time|fval|time|\n|SOC_CVX| |-0.4558|0.2148|-0.5630|0.2415|-0.6586|0.3214|-0.7394|0.4067|\n|GGDOpt(First-Order)| |-0.4552|0.0156|-0.5615|0.0238|-0.6483|0.0486|-0.7281|0.0614|\n|GGDOpt(Second-Order)| |-0.4558|0.0157|-0.5624|0.0242|-0.6491|0.0507|-0.7372|0.0653|\n|Look-Ahead Guidance[Guo2024]| |-0.4460|0.0329|-0.5181|0.0738|-0.5783|0.1127|-0.6584|0.1436|\n\nAs shown in the above table, our proposed GGDOpt consistently achieves lower objective values and the performance gap between GGDOpt and Look-Ahead Guidance increases with the problem dimension $n$. In terms of computational efficiency, GGDOpt is approximately $2\\times$ faster than the Look-Ahead Guidance across all problem sizes. This performance gain stems from the computational overhead of [Guo2024], where computing the guidance term $\\boldsymbol{G}_{t}^{(3)}$ requires backpropagation through the score network to obtain the gradient of the posterior mean $\\mathbb{E}[\\boldsymbol{x}_0|\\boldsymbol{x}_t]$ with respect to $\\boldsymbol{x}_t$. In contrast, our first- and second-order guidance terms are derived analytically and thus do **not require any additional gradient computations through the network**, making our method more efficient and scalable.\n\n**Reply to Weakness 2:**\n\nThank you for this insightful comment regarding the transparency of the data generation procedure and the definition of $\\rho^{(i)}$ in Section 3.1. In general, the quantity $\\text{Prob}_{\\boldsymbol{h}}(\\boldsymbol{g}(\\boldsymbol{x}(\\boldsymbol{z}_i),\\boldsymbol{h})\\geq\\boldsymbol{0})$ is hard to compute since it requires a multidimensional\tintegration over the distribution of the random variable $\\boldsymbol{h}$. Inspired by the Sample Average Approximation (SAA) approach, we estimate this probability using an empirical average based on $L$ i.i.d. realizations of $\\boldsymbol{h}$:\n\n$$\\text{Prob}_{\\boldsymbol{h}}(\\boldsymbol{g}(\\boldsymbol{x}(\\boldsymbol{z}_i),\\boldsymbol{h})\\geq\\boldsymbol{0})\\approx \\frac{1}{L}\\sum\\ell^{0|1}(\\boldsymbol{g}(\\boldsymbol{x}(\\boldsymbol{z}_i),\\boldsymbol{h}^{(\\ell)})):=1-\\rho^{(i)} ~~~~(1),$$\n\nwhere $\\ell^{0/1}$ is the element-wise indicator function that returns 1 if all components of the argument vector are positive, and 0 otherwise. To compute $\\rho^{(i)}$, we proceed as follows:\n\n1. For each sampled restriction vector $\\boldsymbol{z}_i \\geq \\boldsymbol{0}$, we solve the corresponding restricted problem, which yields a candidate solution $\\boldsymbol{x}(\\boldsymbol{z}_i)$.\n\n2. We then draw $L$ independent realizations $\\boldsymbol{h}^{(\\ell)}$ from the underlying distribution and evaluate  $\\rho^{(i)}$.\n\nAccording to equation (1), we see that $\\boldsymbol{x}(\\boldsymbol{z}_i)$ may satisfy the chance constraint with parameter $\\rho^{(i)}$. Thus, we take $\\boldsymbol{x}(\\boldsymbol{z}_i)$ and the corresponding $\\rho^{(i)}$ generated in this way as the data set of the next training stage. \n\n**Reply to Weakness 3 and Question 2:**\n\nThank you for highlighting the concern regarding the practicality of the assumption in Equation (21). We provide further clarification below.\n\nThe assumption in Equation (21) quantifies the approximation accuracy of the trained score network relative to the true score function. We acknowledge that this assumption can be difficult to verify directly in practice, as it depends on the training quality of the neural network and the expressiveness of the model class. However, this type of assumption is common in the theoretical analysis of diffusion models (see, e.g., [Pidstrigach2022]; [De2021]) and is used to establish convergence results in generative modeling and sampling. \n\n**Reply to Question 1:**\n\nWe fully agree that in high-dimensional settings, computing the inverse of a general Hessian matrix can be computationally expensive.\n\nHowever, in our current experiments, the objective function is quadratic, which has a diagonal Hessian matrix. As a result, the matrix inversion involved in the second-order guidance becomes trivial and introduces negligible overhead compared to that in the first-order method. This is why, in our reported results, the computational cost of the second-order guidance is almost identical to that of the first-order guidance.\n\n**Reply to Question 3:**\n\nThank you for your careful reading. You are correct that the restricted optimization problem (13) can indeed be nonconvex. As noted, this subproblem is used in Stage 1 to generate high-quality training samples for the diffusion model, which is not intended as a convex relaxation of the original chance-constrained problem.\n\nOur phrasing \"convexify the nonconvex feasible region\" refers to the forward diffusion process used in Stage 2, where adding the Gaussian noise to the indicator function of the feasible set gradually transforms it into a log-concave density. The evolving $\\epsilon$-support of this smoothed distribution tends to exhibit an increasingly convex geometry, in an approximate sense.\n\nWe once again thank you for your insightful feedback. We believe your comments have helped us greatly improve the clarity and rigor of our work, and we hope our revisions and responses address your concerns satisfactorily. \n\nThe corresponding references are listed in order. \n\n1.[Guo2024] Guo, Yingqing, et al. Gradient guidance for diffusion models: An optimization perspective. Advances in Neural Information Processing Systems, 2024.\n\n2.[Pidstrigach2022] Pidstrigach, Jakiw. Score-based generative models detect manifolds. Advances in Neural Information Processing Systems, 2022.\n\n3.[De2021] De Bortoli, Valentin, et al. Diffusion schrödinger bridge with applications to score-based generative modeling. Advances in Neural Information Processing Systems, 2021."}, {"review_id": "k9j4KId5YH", "reviewer": "Reviewer_fiiv", "summary": "This paper addresses a broad and important class of optimization problems—those governed by probabilistic constraints. By reformulating these problems as sampling tasks (nonconvex CCP with intractable constraints <-> sampling from an unknown distribution), the authors enable the use of powerful modern generative models from machine learning, such as diffusion models, which can learn to sample effectively from the feasible or high-quality regions of complex optimization landscapes. The approach is supported by both theoretical guarantees and empirical results.", "questions": "First, it would be helpful for the paper to engage more directly with the existing literature on diffusion-guided optimization, particularly in relation to the setup described in lines 62–67. This setting appears to be fairly common in recent work (e.g.,  https://arxiv.org/abs/2404.14743 or https://proceedings.mlr.press/v202/krishnamoorthy23a.html). A discussion of this growing body of literature would provide useful context and help clarify the novelty and significance of the current contribution. Additionally, in lines 141–144, the paper states that it uses Tweedie’s formula for the mean and introduces a free parameter for the variance. This seems like a substantial simplification. Could the authors clarify this choice, especially since the variance can also be derived from Tweedie’s formula? If the Hessian is already being computed for the guidance term, it’s unclear why the variance is treated as an independent parameter. Apologies if I missed this discussion elsewhere—the implementation details in Section 3.3 don’t appear to address it.\n\nSecondly, Equation (15) includes a final parameter whose role isn’t clearly explained in the manuscript—I couldn’t find it defined or referenced elsewhere. It would be helpful to clarify what this parameter represents and how it is chosen or tuned. Additionally, Assumption 1 is presented without much accompanying intuition. A brief discussion of its implications would make the theoretical framework more accessible, especially for readers unfamiliar with the nuances of these types of probabilistic constraints. Providing some intuition or examples to illustrate what the assumption means in practice would greatly improve clarity.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The paper presents a compelling approach to an important class of optimization problems with wide-ranging applications. The writing is generally clear and accessible, making the paper enjoyable to read. The method is well-motivated, and the integration of modern generative models like diffusion into constrained optimization is both novel and promising. However, Figure 1—introduced early in the paper—misses an opportunity to clearly explain the method. Its complexity makes it difficult to interpret, and it would benefit from simplification and a more detailed caption that guides the reader on how to read and understand it.\n\nOn the other hand, several concerns relate to computational cost and experimental evaluation. The paper lacks sufficient discussion of the computational overhead associated with the proposed approach—for instance, the practicality of using the Hessian (as in Equation 12) within a diffusion model framework remains unclear. Additionally, the cost of generating training data in Stage 1 may be significant, and it's important to understand how this compares to the total runtime of the optimization pipeline. The experimental section also raises questions: the use of small problem sizes (e.g., n=8n=8) limits the practical relevance of the results, and the reported runtimes are too short to showcase meaningful performance differences. It’s also unclear whether Table 1 reflects performance across the full optimization run or only at the final iteration. The results in Table 2 are similarly underwhelming; the paper would be strengthened by including more demanding experimental setups with larger problem instances and longer runtimes to better demonstrate the method’s capabilities.", "was_revised": true, "final_justification": "Thanks for the rebuttal, especially around the discussion of Hessian. I (and the other reviewers) are still concerned about the lack of adequate related work and also about paper clarity. Therefore I maintain my score.", "rebuttal": "**Reply**\n\nWe sincerely thank you for your thoughtful and constructive feedback, which allows us to clarify and further improve our work. Below we provide a point-by-point response to each of your comments.\n\n**Reply to Weakness 1:**\n\nWe appreciate your suggestion on Figure 1. To help readers better understand the framework, we provide an additional explanation in the caption of Figure 1 as follows:\n\n>**Figure 1:**Overview of the GGDOpt framework.  \n>(1)Generate a training set of points satisfying the chance constraint by solving a deterministic restricted problem.  (2)Train a diffusion model with classifier-free guidance to learn the score of the conditional distribution.  (3)Perform the reverse diffusion process with additional gradient guidance to sample from the product of the data distribution and the Boltzmann distribution.\n\n**Reply to Weakness 2:**\n\nThank you for your valuable comments regarding the computational cost and evaluation. We test the linear chance constrained problem with $n=8$ and repeat 100 times to calculate the empirical mean of the objective value (fmean), the empirical standard deviation (fstd), and the average sample time (time). The results are summarized in the following table.\n\n**Table:Comparison of the proposed methods.**\n|Method|SOC_CVX|GGDOpt(First-Order)|GGDOpt(Second-Order)||||\n|---|---|---|---|---|---|---|\n||||$\\beta=0.1$|$\\beta=1$|$\\beta=10$||\n|fmean|-0.6586|-0.6483|-0.6341|-0.6548|-0.6585|\n|fstd|0|0.0051|5.6726e-3|2.5112e-05|2.2329e-08|\n|time|0.3214|0.0486|0.0569|0.0527|0.0541|\n\nAs observed, the second-order method achieves lower objective values compared to the first-order method and its performance closely matches the optimal solution obtained by SOC\\_CVX. Moreover, the second-order method leads to significantly lower standard deviations, particularly as $\\beta$ increases.\n\nRegarding the computational overhead, we acknowledge that in general high-dimensional settings, inverting a full Hessian matrix can be costly. However, in our current experiments, the objective is quadratic and its Hessian is diagonal which makes the inversion trivial.\n\nTo summarize, when the problem dimension is modest and the Hessian matrix has a simple structure (e.g., diagonal or sparse), the second-order guidance is both practical and beneficial. In contrast, for high-dimensional problems with dense and expensive-to-invert Hessians, the first-order guidance remains a computationally efficient alternative. We will clarify this trade-off and the underlying assumptions in the revised manuscript.\n\n**Reply to Weakness 3:**\n\nYou are absolutely right that the cost of generating training data in Stage 1 is crucial. In our framework, training samples are generated by solving the relaxed problem (RP), which is typically much easier to solve than the original CCP. Here we provide the costs of three stages for the linear chance constraint problem in Section 5.1. For each $n$, we generate 1000 data in the training stage. During sampling, we execute 100 times of reverse process to analyze the stability of GGDOpt. The total time costed in hour is shown below.\n\n**Table:Computational time of three stages(in hours).**\n|Stage||Linear chance constrained problem|||\n|---|---|---|---|---|\n|||$n=8$|$n=16$|$n=128$|\n|Data generating time||0.03|0.06|0.11|\n|Training time||0.53|0.96|11.64|\n|Total sampling time|First-Order|0.0013|0.0017|0.0057|\n||Second-Order|0.0014|0.0018|0.0063|\n\nFurthermore, our experiments indicate that increasing the quantity of training data alone does not guarantee better performance. Instead, high-quality samples closer to the true optimal solutions are the key of effective guided sampling.\n\n**Reply to Weakness 4:**\n\nThank you for highlighting the limitations regarding problem scale and runtime. We acknowledge that Tables 1 and 2 only report the runtime of **the sampling stage**, excluding the computational cost of data generation and model training.\n\nTo better demonstrate the scalability and applicability of GGDOpt, we have conducted additional experiments on much larger problem instances, increasing the problem dimension $n$ up to 1024. The results are summarized in the table below.\n\n**Table:Experimental results with different $n$.**\n|Method($\\rho=0.1$)|$n=8$||$n=16$||$n=128$||$n=1024$||\n|---|---|---|---|---|---|---|---|---|\n| |fval|time|fval|time|fval|time|fval|time|\n|SOC_CVX|-0.6586|0.3214|-0.7394|0.4067|-0.8951|0.8751|-0.9610|1.2483|\n|SAA_PDCA|-0.6301|0.4572|-0.6686|0.6530|-0.8582|1.7348|-0.8956|6.4432|\n|GGDOpt_FirstOrder|-0.6483|0.0486|-0.7281|0.0614|-0.8836|0.2068|-0.9409|0.4258|\n|GGDOpt_SecondOrder|-0.6491|0.0507|-0.7372|0.0653|-0.8941|0.2269|-0.9547|0.4699|\n\nAs expected, the runtime increases with the problem dimension. However, both the first- and second-order versions of GGDOpt remain consistently faster than the baseline SAA\\_PDCA method across all dimensions. Moreover, the increase in runtime is moderate, indicating that our approach scales favorably even in high-dimensional settings. \n\n**Reply to Question 1:**\n\nThanks for recommending these two related papers. In [Krishnamoorthy2023], a conditional diffusion model is trained using loss reweighting to map function values to high-quality samples, and applied to offline black-box optimization. In [Guo2024], a Look-Ahead Guidance mechanism is introduced to preserve the linear structure of the data, enabling applications to regularized and global optimization in settings where the data distribution is Gaussian and the objective is linear.\n\nIn contrast, our work is, to the best of our knowledge, the first to apply diffusion models to general chance constrained problems. The key challenge is the lack of readily available training data corresponding to the product distribution formed by the objective function and the chance constraint. We address this by introducing a dedicated data generation stage, where high-quality feasible samples are obtained, followed by conditional training of a score-based diffusion model.\n\nFurthermore, we propose a gradient-guided reverse process to sample from the product distribution. Our guidance terms are derived in closed form based on the structure of the product distribution, which has a different form in contrast to the guidance term in [Guo2024]. In addition, the proposed guidance term does not require backpropagation through the score network. \n\n**Reply to Question 2:**\n\nWe apologize for the earlier ambiguity. While Tweedie’s formula theoretically provides both the posterior mean and covariance, $\\boldsymbol{\\Sigma}_{0|t}=(1-\\bar{\\alpha}_t)(\\boldsymbol{I}+(1-\\bar{\\alpha}_t)\\nabla^2\\log p(\\boldsymbol{x}_t))$, computing the covariance requires evaluating the Hessian of $\\log p(\\boldsymbol{x})$.\n\nIn our framework, the score function $\\boldsymbol{s}_{\\boldsymbol{\\theta}}$ is parameterized by a neural network, and computing its second derivatives involves backpropagation through the network’s Jacobian, which is computationally expensive, especially in high dimensions.\n\nTo strike a balance between performance and efficiency, we choose to treat the covariance as a tunable constant. We acknowledge that this introduces an approximation, but as shown below, this achieves comparable objective values to the fully Tweedie-based method, while reducing runtime by more than an order of magnitude. These results confirm that using a fixed variance can be a practical and robust alternative.\n\n**Table:Experimental results with different variance schedules.**\n|$n=8,\\rho=0.1$|Tweedie's $\\boldsymbol{\\Sigma}$|GGDOpt(second-order)|||||\n|---|---|---|---|---|---|---|\n| | |$\\sigma=0.01$|$\\sigma=0.02$|$\\sigma=0.1$|$\\sigma=1$|$\\sigma=10$|\n|fval|-0.6571|-0.6471|-0.6457|-0.6545|-0.6320|-0.6049|\n|time|1.0984|0.0491|0.0496|0.0493|0.0492|0.0493|\n\n**Reply to Question 3:**\n\nThank you for highlighting the unclear notation. The final parameter $\\emptyset$ denotes the absence of conditioning information, following the classifier-free guidance scheme. Specifically, during training, we randomly omit the conditioning variable $\\rho$ so that the model learns both conditional and unconditional score estimates. At inference time, this allows us to construct guided samples via a linear interpolation of these two estimates, as originally proposed in [Ho2022].\n\n**Reply to Question 4:**\n\nWe appreciate your feedback regarding Assumption 1. The first and second assumptions in Assumption 1 describe the forward and reverse processes of an Ornstein–Uhlenbeck (OU) process, respectively; such formulations are standard in the diffusion literature for analyzing the impact of the finite diffusion time on the convergence (e.g., [Song2020]). \n\nThe third assumption imposes **a growth bound on the gradient of the objective function**. This type of regularity condition is common in the convergence analysis of stochastic optimization and sampling algorithms, particularly when studying stability and convergence under Langevin dynamics or diffusion-based methods (see, e.g., [Raginsky2017]). In practice, this assumption holds for a broad class of functions, including smooth bounded functions and quadratic objectives, which frequently arise in real-world optimization problems.\n\nThe corresponding references are listed in order. \n\n1.[Krishnamoorthy2023] Krishnamoorthy, Siddarth, et al. Diffusion models for black-box optimization. International Conference on Machine Learning. PMLR, 2023.\n\n2.[Guo2024] Guo, Yingqing, et al. Gradient guidance for diffusion models: An optimization perspective. Advances in Neural Information Processing Systems, 2024.\n\n3.[Ho2022] Ho, Jonathan, and Tim Salimans. Classifier-free diffusion guidance. arXiv preprint arXiv:2207.12598, 2022.\n\n4.[Song2020] Song, Yang, et al. Score-based generative modeling through stochastic differential equations. arXiv preprint arXiv:2011.13456, 2020.\n\n5.[Raginsky2017] Raginsky, Maxim, et al. Non-convex learning via stochastic gradient langevin dynamics: a nonasymptotic analysis. Conference on Learning Theory. PMLR, 2017."}, {"review_id": "lYw7JEjyYL", "reviewer": "Reviewer_u5CA", "summary": "The paper develops a gradient-guided diffusion framework to solve chance-constrained programming problems without knowing the exact distribution. The framework first samples solutions that satisfy the chance constraints approximately. Next, the paper employs classifier-free diffusion model training and model-guided sampling to build the diffusion model. The model guidance is derived through a Boltzmann distribution defined by the objective function. The proposed algorithm is evaluated for a linear chance constrained problem and a robust waveform design. Overall, the paper applies the classifier-free and model-guided diffusion-based method for a traditional optimization problem: $\\max f(x)$, subject to a chance constraint $\\mathbb{P}_h(g(x,h)>0)\\geq 1-\\rho$. The method is kind of standard, but it is first applied in the chance-constrained problem.", "questions": "See Strengths And Weaknesses.", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "The paper has the following strengths:\n+ The paper proposes a reformulation for chance-constrained optimization problems in Eqn. (4) which utilizes Boltzmann distribution and indicator function to transform the chance-constrained optimization to an unconstrained optimization problem. \n+ Assuming that the backward process starts from a Gaussian distribution and the gradient of the objective function is bounded, a convergence analysis and regret analysis are provided to theoretically understand the performance. The expected error between the output and the optimal solution is bounded by some proof techniques.\n+ The paper evaluates the methods on different chance-constrained optimization with real applications and compares the proposed method with several baselines.  The applications include linear chance constrained problem and robust waveform design. The baselines include CVX and SAA methods. They show some performance improvement but it is uncertain whether the baselines are compared in the same level of chance constraint violation. \n+ The paper is well-written. \n\n\nThe paper has the following weaknesses:\n- The method cannot guarantee the satisfaction of the chance constraints since the exact distribution is unknown. The authors didn't discuss the constraint violation either theoretically or experimentally. The authors did not show the violation of the constraints in the experiments in Section 5. I'm concerned that the comparison in the experiments may not be fair since the violations of different algorithms can be different.\n- It is unclear why the proposed method is better than traditional optimization methods. Also, why is the diffusion-based method better than learning-to-optimize algorithms with classical neural networks? Why not use other generative models like autoregressive models? The discussions on the motivation of using diffusion models in this problem is not enough. \n- The paper employs the existing methods of the classifier-free diffusion model and gradient-guided sampling on the problem of chance-constrained optimization. The algorithm lacks enough originality.  The algorithms in Algorithm 1 and Algorithm 2 are standard classifier-free and gradient-guided conditional diffusion-based methods.  There is limited novelty in terms of the methods on conditional diffusion models.\n- The complexity of training a diffusion model and sampling from a diffusion model is usually very large. It would be better to have a complexity analysis for the proposed algorithm.", "was_revised": true, "final_justification": "The authors provided additional experimental results and revised some theoretical analysis during the rebuttal process. Although there are some technical flaws in the current version, the authors give a plan to address them. Given that this paper has a potential to become a rigorous and complete one, I remain a borderline rating but lean toward acceptance.", "rebuttal": "**Reply**\n\nMany thanks for your detailed and insightful comments, which help us clarify important points and improve our manuscript. We hope the following point-by-point responses can address your concerns and offer a clearer understanding of our work.\n\n**Reply to Weakness 1:**\n\nThank you for raising this important point regarding the constraint violation. We provide a comprehensive discussion from both theoretical and empirical perspectives to clarify how our method handles violations of the chance constraints.\n\n-**Theoretical Guarantee:**\nGiven a well-trained diffusion model whose score network closely approximates the true score, Theorem 2 guarantees that the samples generated by GGDOpt converge to the optimal solution of the chance constrained program. In particular, in the limit of infinite sampling time and model capacity, the generated solutions satisfy the chance constraints with high probability.\n\n-**Experimental Verification:**\nIn Section 5.1, we focus on linear chance constrained problems under Gaussian distribution, where the chance constraint can be equivalently reformulated as a second-order cone (SOC) constraint. To ensure a fair comparison across all methods, we project the final solutions generated by all algorithms, including GGDOpt and some baseline methods, onto the feasible SOC region. This ensures that all reported solutions satisfy the chance constraints.\n\nIn Section 5.2, the analytical form of the chance constraint is not available. Therefore, we estimate the constraint satisfaction empirically. For each solution, we draw a large number of samples from the uncertainty distribution and compute the empirical probability. These results are reported in the table shown below, which includes multiple uncertainty levels. For example, when $\\rho=0.05$, a solution is considered feasible if its estimated probability of satisfaction exceeds 0.95.\n\n**Table:Empirical feasibility of different algorithms under some uncertainty levels.**\n|Method|$\\rho=0.05$|$\\rho=0.10$|$\\rho=0.15$|$\\rho=0.20$|\n|---|---|---|---|---|\n|Sphere Bounding|0.99|0.99|0.99|0.99|\n|Bernstein-type Inequality|0.96|0.93|0.91|0.90|\n|GGDOpt(First-Order)|0.99|0.92|0.93|0.87|\n|GGDOpt(Second-Order)|0.97|0.90|0.88|0.88|\n\nAs expected, classical convex approximation methods produce highly conservative solutions that strictly satisfy the chance constraints. GGDOpt, on the other hand, generates solutions closer to the boundary of the feasible region. Nevertheless, GGDOpt consistently achieve estimated satisfaction probabilities near the required threshold, demonstrating practical feasibility.\n\n**Reply to Weakness 2:**\n\nThank you for this important comment, which gives us an opportunity to clarify the motivation and contributions of the proposed GGDOpt framework.\n\n-**Unknown distribution and computational efficiency.**\nThere are two traditional optimization methods, which are Convex Approximation (CA) and Sample Average Approximation (SAA) methods. The CA method seeks to construct an inner approximation of the chance constraint, but it typically requires the exact distribution. The SAA method reformulates CCP as a binary integer program, which remains computationally inefficient. Unlike classical convex approximation approaches for CCP, our method does **not require prior knowledge of the underlying distribution**, which makes our approach applicable to broader and more realistic settings where the distribution is unknown.\n\n-**Sampling can be faster.**\nRecent studies (e.g., [Ma2019]) show that sampling-based methods can outperform optimization-based methods in nonconvex landscapes. The chance constraints are often nonconvex and do not admit a closed-form formulation, which poses major challenges for direct optimization. While Learning-to-Optimize (L2O) is a promising line of work, we are not aware of any L2O algorithm specifically designed for general CCPs. A key reason is that L2O methods often require a strong base optimizer for supervision or unfolding. The chance constraints have no explicit analytical form, making them incompatible with existing L2O pipelines.\n\n-**Theoretical convergence.**\nCompared to other generative models such as VAEs and GANs, diffusion models offer stronger theoretical support for convergence, especially in high-dimensional and multimodal scenarios. We reformulate the original CCP as a sampling problem over the product of two distributions. This novel perspective allows us to cast CCP as a generative problem and leverage the reverse-time SDE underlying diffusion models. Under this formulation, we prove that the samples generated by GGDOpt converge to the optimal solution.\n\n**Reply to Weakness 3:**\n\nThe rising prominence of diffusion models has spurred significant research interest in their underlying mathematical foundations and applications. While guidance techniques have been widely applied in areas such as image generation, black-box optimization, and inverse problems, our contribution does not lie in introducing guidance, but rather in deriving **a novel class of guidance terms based on product distributions**, specifically tailored to chance constrained problems with unknown distributions.\n\nMore specifically, compared to prior work such as [Guo2024], our approach introduces two main innovations:\n\n- **Conditional Training and Applicability Beyond Linear-Gaussian Settings:**  Unlike [Guo2024], which applies guidance to pre-trained unconditional diffusion models and assumes a linear objective with Gaussian data, our framework involves a dedicated data generation process followed by conditional score training. This enables us to address nonlinear and structurally complex chance constrained problems, where directly sampling from the feasible region is nontrivial.\n\n- **A New Class of Guidance Derived from Product Distributions:** In our work, we derive two types of guidance terms directly from the product distribution formulation of the target density:\n\n1. a first-order guidance \n\n$$\\boldsymbol{G}_t^{(1)} = - \\beta \\nabla f(\\boldsymbol{x}_t),$$\n\n2. a second-order guidance\n\n$$\\boldsymbol{G}_t^{(2)} = -\\frac{1}{\\sigma^2}[\\boldsymbol{H}^{-1}[(\\nabla^2 f(\\boldsymbol{x}_t) \\boldsymbol{x}_t+\\nabla f(\\boldsymbol{x}_t)) -\\frac{1}{\\beta \\sigma^2}\\boldsymbol{\\mu}]+\\boldsymbol{\\mu}],$$\n\nwhere the terms are computed based on a learned surrogate for the chance constraint and the posterior mean $\\boldsymbol{\\mu}_{0|t}$. \n\nIn contrast, [Guo2024] introduces a Look-Ahead Guidance term designed for linear objectives:\n\n$$\\boldsymbol{G}_{t}^{(3)} = -\\beta(t)\\nabla(y-\\boldsymbol{g}^\\top \\hat{\\mathbb{E}}[\\boldsymbol{x}_0|\\boldsymbol{x}_t])^2,$$\n\nwhere $\\beta(t)$ and $y$ are tuning parameters, $\\boldsymbol{g}$ is the gradient of the linear objective, and $\\hat{\\mathbb{E}}[\\boldsymbol{x}_0|\\boldsymbol{x}_t] = \\alpha^{-1}(t)(\\boldsymbol{x}_t + h(t)\\boldsymbol{s}(\\boldsymbol{x}_t,t))$. This approach is effective when the data distribution is Gaussian and the objective is linear, but may degrade under nonlinear or non-Gaussian scenarios. \n\nOur experimental results further demonstrate that for nonlinear objectives, GGDOpt consistently outperforms the Look-Ahead Guidance from [Guo2024] in terms of both objective value (fval) and computational efficiency (sampling time).\n\n**Table:Comparison results with Look-Ahead Guidance [Guo2024].**\n|Method($\\rho=0.1$)| |$n=2$||$n=4$||$n=8$||$n=16$||\n|---|---|---|---|---|---|---|---|---|---|\n| | |fval|time|fval|time|fval|time|fval|time|\n|SOC_CVX| |-0.4558|0.2148|-0.5630|0.2415|-0.6586|0.3214|-0.7394|0.4067|\n|GGDOpt(First-Order)| |-0.4552|0.0156|-0.5615|0.0238|-0.6483|0.0486|-0.7281|0.0614|\n|GGDOpt(Second-Order)| |-0.4558|0.0157|-0.5624|0.0242|-0.6491|0.0507|-0.7372|0.0653|\n|Look-Ahead Guidance[Guo2024]| |-0.4460|0.0329|-0.5181|0.0738|-0.5783|0.1127|-0.6584|0.1436|\n\nAs shown above, GGDOpt consistently achieves lower objective values and is approximately $2\\times$ faster than the Look-Ahead Guidance across all $n$. This performance gain stems from the computational overhead of [Guo2024], where computing $\\boldsymbol{G}_{t}^{(3)}$ requires backpropagation through the score network. In contrast, our guidance terms are derived analytically and thus do **not require any additional gradient computations through the network**, making our method more efficient and scalable.\n\n**Reply to Weakness 4:**\n\nThank you for raising this valuable point regarding the computational complexity of our method. We agree that diffusion models are generally associated with high training costs. However, in the proposed GGDOpt framework, **the overall complexity is dominated by the training stage**, while the sampling stage is relatively lightweight, especially in the context of large-scale problems. \n\nTo support this, we report the computational time for each stage. As shown below, the training time grows significantly with increasing dimension $n$. However, both data generation and sampling time remain negligible by comparison.\n\n**Table:Computational time of three stages(in hours).**\n|Metric||Linear chance constrained problem|||\n|---|---|---|---|---|\n|||$n=8$|$n=16$|$n=128$|\n|Total generating time||0.03|0.06|0.11|\n|Training time||0.53|0.96|11.64|\n|Total sampling time|First-Order|0.0013|0.0017|0.0057|\n||Second-Order|0.0014|0.0018|0.0063|\n\nThis observation confirms that, although training a diffusion model is costly, once trained, our method enables efficient sampling, making it suitable for large-scale CCPs. \n\nOnce again, we thank you for your thorough evaluation and thoughtful suggestions. Your feedback has been immensely helpful in strengthening the presentation and scope of the paper. \n\nThe corresponding references are listed in order. \n\n1.[Guo2024] Guo, Yingqing, et al. Gradient guidance for diffusion models: An optimization perspective. Advances in Neural Information Processing Systems, 2024.\n\n2.[Ma2019]  Ma, Yi-An, et al. Sampling can be faster than optimization. Proceedings of the National Academy of Sciences 116.42, 2019."}]

Author rebuttal:

Meta-review: This paper studies chance-constrained optimization problems under unknown distributions and proposes a novel optimization framework leveraging diffusion models to address them. The submission generated a constructive and detailed discussion where reviewers acknowledged both strengths and weaknesses. Concerns were raised about poor readability in some sections, missing discussion of related work, and the absence of experiments on machine learning-related applications. Nonetheless, the overall impression was positive. The paper addresses an important problem and provides a sound theoretical analysis. The proposed framework is original in several aspects, as it reformulates the chance-constrained program as a sampling task over a product of two distributions, the unknown data distribution (supported on a nonconvex set) and a Boltzmann distribution defined by the objective function. The numerical results are promising and suggest potential for broader influence. Importantly, Reviewer u5CA noted that certain results from [Pidstrigach2022] were misapplied, but this issue was clarified during the discussion. A more critical remaining concern is the reliance on condition (21). It is important that the authors revise the proofs and presentation to clearly indicate where this strong assumption is required, and to expand the discussion of related literature accordingly.
Decision: Accept (poster)