Title: A Gradient Guided Diffusion Framework for Chance Constrained Programming
Venue: NeurIPS 2025
Accepted: yes
Keywords: Chance Constrained Programming

Diffusion Model

Optimization via Sampling

Gradient Guidance

Waveform Design
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

Abstract

Chance constrained programming (CCP) is a powerful framework for addressing optimization problems under uncertainty. In this paper, we introduce a novel G radientG uided D iffusion-based Opt imization 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.

Our code is available at https://github.com/boyangzhang2000/GGDOpt.

1 Introduction

1.1 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 as finance (Bonami and Lejeune [2009]), robot control (Calafiore and Campi [2006]), and wireless communications

Zhiguo Wang ∗

(Wang et al. [2014]). In this paper, we consider a CCP with the following form:

where f : R n → R is a differentiable objective function and X ρ is the chance (or probabilistic) constraint set defined by

In the above, h is a random vector with probability distribution P supported on a set Ξ ⊂ R d , ρ ∈ (0 , 1) , g = ( g 1 , g 2 , . . . , g m ) : R n × Ξ → R m , 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 h { g ( x , h ) ≥ 0 } typically involves a high-dimensional integration, which is computationally intractable. Second, even when g is linear, the feasible set X ρ remains nonconvex, further complicating the optimization.

1.2 Related works

Apart from very special cases where X ρ can be transformed into a convex formulation under strong assumptions (Kataoka [1963], Lagoa et al. [2005], Henrion [2007], Prékopa [2013]), there are two popular approaches to tackling general problem (1), which are Convex Approximation (CA) method and Sample Average Approximation (SAA) method. The CA method seeks to construct a tractable inner approximation of X ρ , but it typically requires the information of the exact distribution P , often assuming that P belongs to specific families such as Gaussian or log-concave distributions (Ben-Tal and Nemirovski [2000], Bertsimas and Sim [2004], Lagoa 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 to ask is: 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 ρ , albeit the explicit formulation of X ρ is unavailable. This motivates us to seek high-quality solutions to the CCP problem (1) from a new perspective via sampling-based methods (Wibisono [2018], Ma et al. [2019], Lee et al. [2021], 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 a 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 a powerful family of generative models, offering high-quality sample generation, stable training dynamics, and scalability to high-dimensional problems (Ho et al. [2020]). 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 a 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]), inverse problems (Chung et al. [2022b], Chung et al. [2022a], Song et al. [2023]), and optimization (Krishnamoorthy et al. [2023], Li et al. [2024], Wu et al. [2024], Kong et al. [2024], Liang et al. [2025]). Recently, Guo et al. [2024] introduced a novel form of gradient guidance to adapt pre-trained diffusion models for user-specified tasks.

Despite their success in 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 of an unknown data distribution (associated with the constraint) and a known Boltzmann distribution (induced by the objective function), but the training data is only available from the unknown component. This makes the application of diffusion models to CCP both novel and nontrivial.

Stage 1: Data Generation

Stage 2:Diffusion and Learning

dxt=a(xt,t)dt+b(t)dBt

min f(x)

(CCP)

x

Intractable

Restriction

Estimation

Restriction Z

Stage 3: Sampling with Guidance

Target p

min f(x)

(RP)

s.t. g(x,h) ≥z

x

Tractabl

dxt = (a(xt,t) - b(t)2(Vxt logpt(xtlp) +Gt))dt + b(t)dBt

Samples

d~()y

U-NetforScoreEstimation

Solutionx

SamplesfromProductDistribution

First-order:

x*~Qβ(x|p) αr(xlp)Bg(x)

Chance

Gt=-βVf(xt)

(→ arg minxex, f(x)

Estimation

Second-order:

Data Set

-2f(x)x+Vf(x)

(x,p) ~r(xlp)<Estimated p

1.3 Our contributions

In this paper, we propose GGDOpt (see Figure 1), a novel G radientG uided D iffusion-based Opt imization framework for solving problem (1), with the following originality:

Applicable to broader problem domains. Built on the basis of diffusion model with classifier-free guidance and optimization via sampling, 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.

Problem reformulation with a novel paradigm. GGDOpt reformulates the CCP problem 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 with a full utilization of first- and second-order information of the underlying CCP.

Feasibility-aware 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 of the conditional diffusion model, effectively capturing the geometry of the feasible region. To sample from the product distribution, we develop a gradient-guided reverse process derived in closed form based on the structure of the product distribution. Compared with Guo et al. [2024], 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 to 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 2, a reformulation of CCP problem (1) is provided via sampling, and a gradient guidance-based score estimation schedule is provided with both first- and second-order information. A novel GGDOpt framework for solving problem (1) is given in Section 3. Theoretical convergence and experimental results are presented in Section 4 and Section 5, respectively. The conclusion is drawn in Section 6.

2 Problem reformulation via sampling

Let r ( x | ρ ) = I X ρ ( x ) denote the indicator function of the chance constraint X ρ . Let B β ( x ) ∝ e -βf ( x ) represent the Boltzmann distribution associated with the objective function f ( x ) , where β > 0 . The resulting sampling task is to draw samples from the following target distribution:

Intuitively, the distribution Q β ( x | ρ ) assigns higher probability density to regions where the objective function f ( x ) takes smaller values. Under certain regularity conditions (Kong et al. [2024]), as β →∞ , the sampling distribution Q β ( x | ρ ) asymptotically concentrates around the global minimizer of the CCP in (1). Therefore, the CCP (1) admits the following equivalent reformulation:

A natural way would be to directly employ Langevin dynamics for sampling from distribution Q β ( x | ρ ) . However, the unknown nature of component r ( x | ρ ) prevents the derivation of an exact expression of the score function. Fortunately, we can obtain a set of feasible samples { x ( i ) , ρ ( i ) } N i =1 , which are drawn from the unknown distribution r ( x | ρ ) . More details on this will be presented in Subsection 3.1. This motivates us to leverage diffusion models to directly learn the product distribution Q β ( x | ρ ) ∝ r ( x | ρ ) B β ( x ) , where r ( x | ρ ) is unknown but B β ( x ) is explicitly known.

2.1 Diffusion models

Given observed samples x 0 from a distribution of interest, the goal of a diffusion model is to learn to model its true data distribution p 0 ( x 0 ) . Once learned, we can generate new samples from our approximate model at will. The diffusion model builds a diffusion process by defining a forward SDE starting from p 0 ( x 0 ) as follows:

where t ∈ [0 , T ] , B t is the standard Wiener process (a.k.a., Brownian motion), a ( · , t ) : R d → R d is a vector valued function called the drift coefficient, and b ( · ) : R → R is a scalar function known as the diffusion coefficient.

By starting from samples of x T ∼ p T ( x T ) and reversing the process, we can obtain samples x 0 ∼ p 0 ( x 0 ) . The reverse of a diffusion process is also a diffusion process, running backwards in time and given by the following reverse-time SDE:

where ¯ B t is a standard Wiener process when the time flows backwards from T to 0. The only unknown term ∇ x t log p t ( x t ) is the score function of the marginal density p t ( x t ) .

To estimate ∇ x t log p t ( x t ) , we can train a time-dependent score-based model s θ ( x t , t ) with

where p 0 t ( x t | x 0 ) is the transition kernel and can be obtained by the forward process (5). When a ( · , t ) is affine, the transition kernel is always a Gaussian distribution, where the mean and variance are often known in closed forms (Särkkä and Solin [2019]). With sufficient data and model capacity, score matching ensures that the optimal solution s θ ∗ ( x t , t ) approximates ∇ x t log p t ( x t ) for almost all x t and t .

2.2 Gradient guidance

A direct application of diffusion models to CCP (1) is infeasible, as this requires sampling from the product distribution Q β ( x | ρ ) ∝ r ( x | ρ ) B β ( x ) , whereas only samples from r ( x | ρ ) are accessible. Therefore, obtaining a precise characterization of the score function of Q β ( x | ρ ) and its diffused version is crucial.

For a given data set D = { ( x ( i ) , ρ ( i ) ) } N i =1 , we use its empirical p 0 ( x 0 | ρ ) to approximate the unknown distribution r ( x 0 | ρ ) and denote ˜ p 0 ( x 0 | ρ ) ∝ p 0 ( x 0 | ρ ) B β ( x 0 ) . The diffused distribution is then given by the forward process (5), i.e.,

In order to sample with the reverse process (6), we need to characterize the score function of the diffused product distribution ∇ x t log ˜ p t ( x t | ρ ) , which is given by the following theorem.

Theorem 1. For any given β > 0 , there exists ˆ x 0 ( x t ) such that the score function of the diffused product distribution can be formulated as

where ∇ x t log p t ( x t | ρ ) is the score function of the diffused data distribution and ˆ x 0 ( x t ) satisfies

Theorem 1 demonstrates that sampling from the product distribution can be accomplished by introducing a gradient guidance term during the sampling process of the original data distribution, which has a strong connection between the posteriori p t 0 ( x 0 | x t , ρ ) and the Boltzmann distribution B β ( x 0 ) .

Next, we present a special case where the gradient guidance terms admit explicit expressions.

Corollary 1. Assume that p t 0 ( x 0 | x t , ρ ) = N ( x 0 | µ 0 | t , σ 2 0 | t I ) , then we have the following results.

First-order guidance: For f ∈ C 1 ( R n , R ) , we get

Second-order guidance: For f ∈ C 2 ( R n , R ) , we get

It is worthwhile noting that, for p 0 ( x 0 | ρ ) = N ( x 0 | µ 0 , σ 2 0 I ) and the Gaussian transition kernel, the assumption in Corollary 1 holds and the parameters ( µ 0 | t , σ 0 | t ) can be expressed explicitly. In practice, we can use Tweedie's formula (Efron [2011]) to obtain an estimator of µ 0 | t , and treat the variance as a hyper parameter; see Subsection 3.3 for details on 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 (1). 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 in order to perform sampling. After learning, the sampling with guidance stage iteratively runs the reverse process with an extra gradient guidance to sample from the product distribution, which will asymptotically converge to an optimal solution to problem (1). Next, we present the details of the three stages in GGDOpt one by one.

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 a set of samples { h ( ℓ ) } L ℓ =1 , denote the empirical mean ¯ h = 1 L ∑ L ℓ =1 h ( ℓ ) . Notice that in most of cases, it's much easier to solve the following deterministic restricted problem (RP) with a fixed ¯ h :

where z i ≥ 0 is a given restriction, i = 1 , . . . , N . Let x ( z i ) denote the solution to problem (13) for a given z i . As the smallest element z min in z i increases, the probability of the nonlinear constraint g ( x ( z i ) , h ) ≥ 0 also increases. Then, solving problem (13) allows us to generate high-quality data that satisfies the chance constraint for arbitrary ρ ∈ (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 { h ( ℓ ) } L ℓ =1 . Then, after getting x ( z i ) , we have

where ℓ 0 / 1 ( g ) = 1 if g ≥ 0 and ℓ 0 / 1 ( g ) = 0 otherwise. By calculating the empirical ρ ( i ) , an asymptotic approximation of the underlying probability is obtained, requiring no assumption on the underlying distribution P . In the appendix, we give a tight lower bound for the probability constraint Prob h { g ( x ( z i ) , h ) ≥ 0 } if the variance and the mean of the random variable h are known, which is helpful to obtain a better approximation ρ ( i ) .

Let x ( i ) := x ( z i ) and repeating the above process, i.e., solving problem (13) and estimating ρ ( i ) , and gradually increasing z i , we can generate a collection of data points D = { x ( i ) , ρ ( i ) } N i =1 , which are then used to train our GGDOpt in the next stages.

3.2 Stage 2: diffusion and learning

From Theorem 1, we observe that the score function of the diffused product distribution has two terms, the conditional score ∇ x t log p t ( x t | ρ ) and the gradient guidance term G t 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 ∇ x t log p t ( x t | ρ ) .

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 [2021], Dhariwal and Nichol [2021], Ho and Salimans [2022]). Here we propose to use the classifier-free guidance (Ho and Salimans [2022]) to give an approximation of ∇ x log p t ( x | ρ ) .

Instead of training a separate classifier model, classifier-free guidance choose to train an unconditional score estimator to approximate ∇ x t log p t ( x t ) together with the conditional score estimator to approximate ∇ x t log p t ( x t | ρ ) . Specificity, we train a single model s θ ( x t , t, ρ ) , and the conditioning information ρ is randomly discarded as empty set ∅ with probability p uncond to train unconditionally. Then the conditional score ∇ x t log p t ( x t | ρ ) is estimated by

for a given weight parameter w . Specifically, for the given data set D and network s θ ( x t , t, ρ ) parameterized by θ , the training objective is defined as

and trained with Adam (Kingma [2014]). The training process of GGDOpt is given in Algorithm 1.

3.3 Stage 3: sampling with guidance

Given the forward process (5), the corresponding reverse process is given by the following reversetime SDE with trained s θ ( x t , t, ρ ) and gradient guidance G t :

where

For the first-order gradient guidance G t in (11), we directly use the gradient of the objective scaled by a hyper parameter β . For the second-order gradient guidance (12), we need to give the posterior mean and variance ( µ 0 | t , σ 2 0 | t ) . Here we use Tweedie's formula (Efron [2011]) to get an estimator of the posterior mean as follows:

with priori p 0 t ( x t | x 0 ) = N ( x t | √ ¯ α t x 0 , (1 -¯ α t ) I ) for a specific noising schedule ¯ α t .

While Tweedie's formula theoretically provides both the posterior mean and covariance, Σ 0 | t = (1 -¯ α t )( I + (1 -¯ α t ) ∇ 2 log p ( x t )) , computing the covariance requires evaluating the Hessian of log p ( x ) . In our framework, the score function s θ 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. To strike a balance between performance and efficiency, we choose to treat the covariance as a tunable hyper parameter σ 2 . In the appendix, we give a detailed comparison between the fully Tweedie-based method and our approach to show that using a fixed variance can be a practical and robust alternative.

Then the second-order guidance can be calculated by

and the sampling process of GGDOpt is given in Algorithm 2.

4 Convergence analysis

In this section, we 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 as the data distribution. But what if we introduce an

extra gradient guidance term? For a given ρ , denote D ρ = { x ( i ) | ( x ( i ) , ρ ( i ) ) ∈ D , ρ ( i ) ≤ ρ } as the approximated feasible set of X ρ . The following theorem says that in our settings, as T →∞ and β →∞ , the samples of GGDOpt will concentrate around the points with the lowest function values within the support of the data distribution D ρ for any given ρ .

Theorem 2. For any given ρ ∈ (0 , 1) , suppose that there exists a constant δ such that the error in the score estimation can be bounded as:

For samples ˜ x sample ∼ p sample ( x 0 | ρ ) generated by the reverse process

with prior p prior = N ( 0 , I ) , affine drift coefficients a ( · , t ) , and

as T → ∞ , p sample ( x 0 | ρ ) will have the same support as ˜ p 0 ( x 0 | ρ ) . Further, as β → ∞ , ˜ x sample will concentrate around x ∗ = arg min x ∈D ρ f ( x ) .

The assumption in the score estimation error (21) 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., Pidstrigach [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 I 1 and limited inverse temperature error I 2 , which are given as follows:

In the above, ˜ x sample is sampled from the reverse process (17), x π follows the strong solution p π to the Fokker-Planck equation of (17), and x ∗ = arg min x ∈D ρ f ( x ) . Next, we will give practical error bounds of both the two components with finite T and β .

Assumption 1. We assume the following conditions hold:

The forward process is given by d x = b ( t ) d B t ;

The reverse process starts in p prior = N ( m T , Σ T ) where m T = E [˜ p 0 ( x 0 | ρ )] and Σ T = Cov (˜ p 0 ( x 0 | ρ )) + T · I ;

The objective function f ( x ) satisfies ∥∇ x f ( x ) ∥ 2 ≤ C 1 ∥ x ∥ 2 + C 2 .

The first two conditions in Assumption 1 correspond to the VE SDE in (Song et al. [2020b]) 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 (see, e.g., Raginsky et al. [2017]). In practice, Assumption 1 holds for a 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 σ ( k ) , k = 1 , . . . , n , the eigenvalues of Σ T . For any given ρ ∈ (0 , 1) , denote N ρ = |D ρ | and x ∗ = arg min x ∈D ρ f ( x ) . Then for any given T > 0 and β > 0 , the optimization error can be bounded by

where C T = 1 2 log (∏ n k =1 ( σ ( k ) /T ) ) and C I , δ ρ are constants.

Theorem 3 provides a non-asymptotic convergence result of GGDOpt with limited time length and inverse temperature. As T →∞ and β →∞ , the optimization error goes to zero and GGDOpt is shown to generate asymptotically optimal solutions.

5 Experimental results

In this section, we perform numerical experiments on both synthetic datasets and a wireless communications waveform design problem. To generate the data, we employ CVX (Grant et al. [2008]) to solve the restricted problem (13). In the diffusion and learning stage, we set T = 1000 with a linear noise schedule η ( t ) ranging from 0.0001 to 0.02, and let a ( x , t ) = -1 2 η ( t ) x and b ( t ) = √ η ( t ) . In the sampling with guidance stage, we evaluate both first- and second-order gradient guidances via implementing a DDIM-based technique (Song et al. [2020a]) with a descaled time step T ′ = 100 for accelerated sampling. We employ two variants of the U-Net model (Ronneberger et al. [2015]) as our score estimator: U-Net-1D 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:

where p c = N ( c ; ¯ c , I ) and ( b , ¯ c , d, ρ ) are hyper parameters selected from a test set. The above problem can be reformulated as a second-order conic (SOC) program, for which CVX (Grant et al. [2008]) is used for solution. To generate training data, we solve the restricted version of problem (26) for N = 1000 values of z linearly spaced in the interval [0 , 0 . 5] . Then we execute the reverse process with first- and second-order gradient guidance to generate samples.

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 n = 8 are presented in Table 1.

The results in Table 1 demonstrate that, compared to the SOC_CVX method, which requires explicit knowledge of the underlying distribution, GGDOpt can approximately find the global minimizer with only samples from distribution p c 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 secondorder 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.

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 et al. [2014])

where N t is the number of antennas at the base station and K is the total number of users. For each user i , S i ⪰ 0 , h i , R i and r i ≥ 0 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 a rank-one structure (Huang and Zhang [2007], Chang et al. [2008], Huang 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 et 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 as a score estimator to reduce computational costs in both training and sampling process.

Table 2 summarizes the comparison results of GGDOpt and two state-of-the-art methods for solving problem (27) with N t = 16 and K = 3 , where the worst probabilities that the chance constraints satisfy for K users are underlined. Notably, both baseline methods rely on explicit knowledge of the underlying distribution, whereas GGDOpt operates solely based on samples. The results show that GGDOpt 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.

6 Conclusion

In this paper, we have proposed GGDOpt, a gradient-guided diffusion framework that efficiently solves nonconvex CCP without requiring the exact distribution knowledge. By reformulating CCP as a sampling problem over the product of an unknown data distribution and a Boltzmann distribution, 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.

Acknowledgments

The work of Boyang Zhang and Ya-Feng Liu was supported in part by the National Natural Science Foundation of China (NSFC) under Grant 12021001 and Grant 12371314. The work of Zhiguo Wang was supported in part by The National Key Research and Development Program of China under Grant 2020YFA0714003 and in part by NSFC under Grant 62203313.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS AND WEAKNESSES:\nThe 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.\n\nQUESTIONS:\nSee Strengths And Weaknesses.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\nThe 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.\n\nQUESTIONS:\nFirst, 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.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\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\n\nQUESTIONS:\n1. 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?\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 1, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nChance 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.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\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.\n\nQUESTIONS:\nMy 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.\n\nLIMITATIONS:\nYes, this work discusses the limitations in Appendix A.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 1, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-06"}