DIFFRACT: Neuralized Utility Maximization for Wireless Networks by Differentiable Programming Chee Wei Tan∗ , Siya Chen† ∗ Nanyang Technological University, Singapore, Nanyang Ave., Singapore † Department of Computer Science, City University of Hong Kong, Hong Kong
arXiv:2606.07114v1 [cs.NI] 5 Jun 2026
∗ [email protected], † [email protected]
Abstract—Next-generation wireless networks, including satellite-to-Open RAN systems, demand agile and intelligent resource management capable of handling dynamic multi-user interference under stochastic quality of service constraints. This paper introduces DIFFRACT, a neuralized utility maximization framework that leverages differentiable programming to integrate deep learning with optimization in wireless networks. Central to our approach is the exploitation of the mathematical structure of standard interference functions, which are foundational in wireless power control. By developing a duality theory for these functions, we map iterative interference management algorithms into differentiable neural network architectures via algorithm unrolling. This enables distributed, end-to-end gradient-based learning at the network edge, supporting real-time adaptation to interference in both terrestrial and non-terrestrial environments. DIFFRACT allows for scalable and robust utility maximization by modeling complex channel dynamics and leveraging the expressiveness of differentiable models. Experimental results confirm the framework’s theoretical soundness and practical effectiveness for next-generation wireless systems. Index Terms—Wireless networks, Deep learning, Utility maximization, Differentiable programming
I. I NTRODUCTION Lower satellite launch costs and growing global broadband demand have accelerated the commercialization of Low Earth orbit (LEO) satellites, spurring non-terrestrial network initiatives such as Starlink [1], [2]. These satellite-to-cell systems are poised to become integral components of next-generation wireless infrastructure, extending reliable connectivity to remote and underserved areas. At the same time, emerging wireless architectures such as 6G and Open Radio Access Networks (O-RAN) are being designed to support latencysensitive and high-throughput applications by intelligently coordinating edge resources across heterogeneous terrestrial and non-terrestrial links [3]–[5]. The coexistence of diverse wireless interfaces and unpredictable interference poses fundamental challenges to distributed resource management. These complexities often lead to over-provisioned designs or underutilized spectrum, limiting network efficiency. Deep learning has shown great potential for wireless resource optimization, offering the ability to learn adaptive policies directly from data in complex and dynamic environments such as non-terrestrial and satellite-to-cell networks, bypassing the need for explicit modeling of channels and interference [6]–[11]. Traditional optimization methods often
struggle with intractable interference and nonconvex problem structures, whereas neural networks can generalize across a wide range of channel conditions and accelerate computation significantly through parallel processing on GPUs [8]. Yet, deep learning alone often lacks the structure, interpretability, and convergence properties offered by classical theory. This paper proposes DIFFRACT, a framework based on differentiable programming to neuralize utility maximization in uncertain, interference-limited wireless networks. By embedding the mathematical structure of standard interference functions into a differentiable computational graph, DIFFRACT enables gradient-based learning of power control policies that are both scalable and theoretically grounded. This approach is motivated by the emerging trend of differentiable programming, fueled by automatic differentiation in deep learning frameworks such as PyTorch and TensorFlow, which allows complex models to be trained end-to-end using gradient-based methods [12]–[14]. Inevitably, differentiable programming enables end-to-end optimization in AI-native wireless stacks that are GPU-compatible, e.g., NVIDIA’s 6G Sionna [8]. In particular, we present new results rooted in the seminal standard interference function framework [15], uncovering new links between monotone operator and an overlooked differentiability property, which we integrate into a broader differentiable programming paradigm. Our approach is based on advances in contractive interference mappings [16], log-convexity structures [17], and differentiable convex programming frameworks such as LLCPs [14], [18]. These insights lead to DIFFRACT, a differentiable programming framework that embeds the structure of standard interference functions into computational graphs, enabling end-to-end learning through automatic differentiation tools like PyTorch and TensorFlow [12]–[14]. DIFFRACT is useful for satelliteto-cell networks with interference patterns due to atmospheric fading [19]–[21], transforming interference-limited settings into learning systems with differentiable programming. Moreover, the differentiability of these mappings offers programmable flexibility to accurately approximate complex real-world interference patterns, thereby enabling scalable, learning-based solutions for utility maximization in wireless networks. DIFFRACT leverages differentiable programming by integrating deep learning methods–such as algorithm unrolling [22]–[24]–with fixed-point techniques. The duality framework not only guides the optimization but also offers
a certificate of optimality, enabling distributed wireless network optimization through a “learn to optimize" paradigm. Differentiable programming allows the embedding of iterative algorithms–such as those arising from traditional optimization problems – directly into neural networks, where parameters can be trained through end-to-end gradient descent using automatic differentiation tools [12]–[14], [24], which offers a data-driven alternative to classical optimization-theoretic methods. Furthermore, our study investigates how key design choices–such as the depth of neural networks and the degree of programmability in function approximation and unrolled algorithm iterations–affect the performance and generalization capabilities of DIFFRACT. This helps guide the development of scalable, GPU-compatible learning-based optimization strategies such as amortized optimization [25], [26], advancing new differentiable neural solvers for wireless networks. The main contributions of this paper are: 1) We establish new results in duality and differentiability for the classical standard interference function, a foundational concept in wireless network resource allocation. These results enable differentiable programming for an AI-native optimization stack and guide the design of scalable, principled iterative solvers. 2) We develop a data-driven methodology to learn a broad class of implicit standard interference functions using neural networks. This approach leverages the derived primal-dual iterative algorithms and extends the applicability of standard interference models to a wider range of utility maximization tasks, enabling accurate modeling and fast adaptation in data-centric settings. 3) We introduce DIFFRACT, a deep learning architecture for scalable end-to-end differentiable programming. DIFFRACT efficiently learns interference models built on PyTorch and open-source differentiable proximal solver, supporting real-time deployment in wireless networks. This paper is organized as follows. In Section II, we introduce the wireless network model and the standard interference functions. In Section III, we develop the duality of standard interference functions that can be applied to utility maximization for wireless networks and optimally solved using primal and dual iterative algorithms. In Section VI, we propose a parameterized neural network-based framework for learning interference functions with illustrative examples. Then, we propose DIFFRACT to enable jointly distributed learning of interference functions and utility maximization that enhances the efficiency of training and distributed computation in Section V. Finally, we conclude the paper in Section VI. Notation: Bold uppercase letters denote matrices, bold lowercase letters denote vectors, italics denote scalars, and u ≥ v indicates componentwise vector inequality. Let (Bu)l denote the lth element of Bu. Let diag(u) be the diagonal matrix formed by u. We write B ≥ F if Bij ≥ Fij for all i, j. The Perron-Frobenius eigenvalue of a nonnegative matrix F is denoted as ρ(F), and the right and left eigenvector of F associated with ρ(F) are denoted by x(F) ≥ 0 and y(F) ≥ 0 (or,
simply x and y, when the context is clear) respectively. Recall that the Perron-Frobenius eigenvalue of F is the eigenvalue with the largest absolute value. Assume that F is an irreducible nonnegative matrix.1 Then ρ(F) is simple and positive, and x(F), y(F) > 0 [27]. The super-script (·)⊤ denotes transpose. Denote u/v by the vector [u1 /v1 , . . . , uL /vL ]⊤ and the Schur ⊤ product between x and y by x ◦ y = (x1 y1 , . . . , xL yL ) . For ⊤ L γ̃ γ̃1 any vector γ̃ = [γ̃1 , . . . , γ̃L ] ∈ R , let e = [e , . . . , eγ̃L ]⊤ , ⊤ and log x denotes log x = (log x1 , . . . , log xL ) . II. I NTERFERENCE F RAMEWORK AND S YSTEM M ODEL A. Standard Interference Function We revisit the standard interference function framework from [15], revealing its underlying monotone operator structure and differentiability properties. This enables a new class of data-driven algorithms via differentiable programming. Definition 1 (Standard Interference Function [15]): I(p) is a standard interference function if, for all p ≥ 0, the following properties are satisfied: 1) (Monotonicity) If p1 ≥ p2 , then I(p1 ) ≥ I(p2 ). 2) (Scalability) For any α > 1, αI(p) > I(αp). Lemma 1: [15] If I(p) ≤ p is feasible, I(p) has a unique fixed point p∗ , which can be computed by the iteration p(t + 1) = I(p(t)) with geometric convergence rate. Lemma 2: Positivity and concavity of I(p) implies scalability of I(p). To see this, note that for all α > 1, we have αI(p) = αI(1/α · (αp) + (1 − 1/α) · 0) ≥ α ((1/α) · I(αp) + (1 − 1/α) · I(0)) > α·(1/α)·I(αp)+0 = I(αp) where the last two inequalities are due to concavity and positivity respectively. However, the converse does not hold; that is, positivity and scalability do not imply concavity. A counterexample is Ii (p) = p2i + 3 if 0 ≤ pi ≤ 1 and Ii (p) = 2pi + 2 if pi > 1. We first state a new result for the standard interference function framework in [15] with applications to utility maximization in the next section. Suppose that I(p) is differentiable and concave in p. Denote the derivative of I(p) by: ∂Ii (p) . (1) ∇I(p) = ∂pj L×L Observe that ∇I(p) is an irreducible nonnegative matrix whose entries are nonnegative and continuous in p. The feasibility of I(p) ≤ p can be established by the following necessary and sufficient condition. Theorem 1: If I(p) is differentiable and concave, then I(p) ≤ p is feasible if and only if ρ(∇I(p)) < 1 for some p ≥ 0. Definition 2: I(p) is a log-log convex standard interference function, if it fulfills the condition of standard interference function and, in addition, in the logarithmic domain, i.e., p̃ = log p, I(ep̃ ) is log-convex in p̃. 1 A nonnegative matrix F is said to be irreducible if there exists a positive integer m such that the matrix Fm has all entries positive.
Framework of DIFFRACT Repeat until convergence Algorithm Unrolling
LLCP
p(t)
I(p(t))
p(t + 1)
Solutions to utility maximization
Unrolling p(k)
Sample data of implicit standard interference function
Approximation with maximum of monomials
p(k+1) I(p(t)) with Neural Network
Fixed point of approximate standard interference function
Fig. 1: DIFFRACT as a differentiable programming framework for utility optimization approximates implicit interference functions via differentiable log-convex mapping and uses algorithm unrolling to compute fixed points, enabling end-to-end learning of globally optimal solutions.
We study a broad class of concave and log-log convex standard interference functions that subsume all known cases and underpin wireless utility maximization [15], [28]. Notably, as I(ep̃ ) is convex and monotone, I(ep̃ ) can be treated as a monotone operator established as follows. Theorem 2 (Interference Function as Monotone OperaL tor): Let I : RL ++ → R++ be a standard interference function that is continuously differentiable, positive, monotone and scalable. Define Φ : RL → RL by Φ(p̃) = p̃ − log I(ep̃ ). Then Φ is monotone (e.g., see [29]), i.e., ⊤ (Φ(p̃1 ) − Φ(p̃2 )) (p̃1 − p̃2 ) ≥ 0 for all p̃1 , p̃2 ∈ RL . Let us sketch the proof outline. Define Ψ(p̃) = log I(ep̃ ), so Φ(p̃) = p̃ − Ψ(p̃). The Jacobian is ∇Φ(p̃) = IL − ∇Ψ(p̃), where IL is identity matrix, and by the chain rule with q = ep̃ , ∇Ψ(p̃) = diag(I(q))−1 · ∇I(q) · diag(q).
Now, the symmetric part of ∇Φ is 21 (∇Φ + ∇ΦT ) = IL − 21 (∇Ψ + ∇ΨT ). Since I(q) is monotone and differentiable, ∇I(q) is positive semi-definite, and diag(I(q))−1 , diag(q) are positive definite. Thus, ∇Ψ has a positive semi-definite symmetric part, so IL − 21 (∇Ψ + ∇ΨT ) is positive semi-definite, implying that Φ is monotone, i.e., ⊤ (Φ(p̃1 ) − Φ(p̃2 )) (p̃1 − p̃2 ) ≥ 0 for all p̃1 , p̃2 ∈ RL [29]. Theorem 2 enables the use of operator splitting techniques such as proximal point or forward-backward methods in [29], ⋆ [30] to solve equations of the form p̃⋆ = log I(ep̃ ) with guaranteed convergence. An example is Nesterov’s adaptive step-size rule update [30]: p̃(k + 1) = p̃(k) + αk (log I(ep̃(k) ) − p̃(k)), 2 1+ρ(∇I(ep̃(k) )
(2)
with αk = where the distance between one and the Perron-Frobenius eigenvalue of the Jacobian, δ (i.e., 1 − ρ(∇I(ep̃ )) as the Jacobian is nonnegative) governs the convergence rate [29], [30]. The error after k iterations of (2) satisfies the bound [29], [30]:∥p̃(k) − p̃⋆ ∥2 ≤ C(1 − δ)k = Cρ(∇I(ep̃(k) ))k , for some positive constant C determined by the initialization. This is as competitive as the fixedpoint iteration in Lemma 1! This operator perspective enables
differentiable programming [13] to address a broad class of wireless network optimization efficiently, as shown in Fig. 1. B. System Model Consider the system model of a wireless network with L users, each comprising a transmitter-receiver pair, communicating simultaneously over a shared spectrum. The vector p = (p1 , . . . , pL )⊤ denotes the allocation of transmit power, where pl represents the transmit power of the lth user. Let G = [Glj ]L l,j=1 > 0L×L represent the wireless channel gain, where Glj is the channel gain from the jth transmitter to the lth receiver, and n = (n1 , . . . , nL )⊤ > 0, where nl is the noise power at the lth user. We define the Signal-to-Interferenceand-Noise Ratio (SINR) at the lth receiver as the ratio of the received signal power to the combined interference and noise power under frequency-flat fading, given by: SINRl (p) = PL
Gll pl
j=1,j̸=l Glj pj + nl
.
(3)
Next, let us define a nonnegative vector: T nL n1 n2 , ,..., v= , G11 G22 GLL and a nonnegative matrix F with entries: 0, if l = j Flj = . Glj /Gll , if l ̸= j
(4)
Moreover, we assume that F is irreducible, i.e., each link has at least one interferer. Then the SINR of the lth user can be rewritten as: pl . (5) SINRl (p) = PL j=1 Flj pj + vl Suppose each SINRl(p) must exceed a reliability threshold γl , i.e., SINRl(p) ≥ γl . This constraint induces the well-known standard interference function [15]: Il (p) =
L X j=1
γl Flj pj + γl vl ≤ pl
∀, l,
(6)
which is feasible if and only if ρ(diag(γ)F) < 1, as established in Theorem 1. When the wireless channels experience fading (e.g., Rayleigh fading, Rician fading or Nakagami fading), the received power from the jth transmitter at the lth receiver is given by Glj hlj pj where hlj is a random variable reflecting the fading environment. The SINR of the lth user is a random variable in terms of the channel realization: hll pl . (7) SINRl (p) = PL j=1 Flj hlj pj + vl An outage occurs when the received SINR of the lth user falls below γl , a minimum SINR threshold for reliable communication. This means that when SINRl (p) ≥ γl , the transmission at the lth receiver is successful; otherwise, the transmission fails. Denote the outage probability of the lth receiver/transmitter pair for a power vector p by Ol (p) and the outage constraint for the lth user by Ōl . The link reliability function of the lth user is given as the complement of the outage probability: Ol (p) = Prob(SINRl (p) < γl ) ≤ Ōl .
(8)
While a closed-form standard interference function expression for (8) is available under Rayleigh fading (e.g., see [31]), such expressions are generally intractable for other fading conditions, particularly atmospheric fading in space-toground communication scenarios [19]–[21]. This motivates the development of a framework that can algorithmically handle implicit interference functions for utility maximization under these more complex channel conditions.
2) Directional Monotonicity: For any s > 1 and p > 0, ui (sp) > ui (p) for all i. 3) ui (ep̃ ) is concave and monotonically increasing for all i. Now, taking a logarithmic transformation of the variables, (9) can be reformulated as:
A. Utility Maximization and Iterative Algorithm Let u(p) denote the network utility, representing a metric of network quality-of-service. The power vector p = [p1 , . . . , pL ]⊤ is adjusted to maximize this utility, subject to interference constraints I(p) ≤ p. Consider the wireless utility maximization problem: maximize
u(p)
subject to
I(p) ≤ p.
u(ep̃ )
subject to
log(I(ep̃ )) ≤ p̃,
(10)
which reduces to a convex optimization problem, thus allowing (9) to be solved optimally. Suppose the constraints in (9) are feasible and there exists an p̃ that is strictly feasible, i.e., Slater’s condition [33] is satisfied. Let us introduce the dual variable λ ∈ RL + and form the partial Lagrangian of (10) as: L(p̃, λ) = −u(ep̃ ) + λ⊤ (log(I(ep̃ )) − p̃). Applying the Karush-Kuhn-Tucker (KKT) stationarity condition [33] and transforming back to the original variables, we obtain the following key result. Theorem 3: The optimal solution p∗ of (9) and the optimal dual solution λ∗ of (10) satisfy: L ∂u(p∗ ) X λ∗j ∂Ij (p∗ ) λ∗l = − + ∀ l. p∗l ∂pl p∗ ∂pl j=1 j
(11)
p(t + 1) = I(p(t)),
(12)
x(t + 1) = −∇u(p(t)) + ∇I(p(t))x(t),
(13)
Furthermore, the following iterations compute p∗ in (9) and λ∗ in (10) from any positive initial points p(0) and x(0):
III. U TILITY M AXIMIZATION WITH LOG - LOG CONVEX I NTERFERENCE F UNCTION In this section, we study a utility maximization problem with log-log convex differentiable interference function constraints. We adopt an operator perspective by exploiting the problem’s duality, which yields fast fixed-point algorithms. Differentiability of these operators then enables a differentiable programming framework to learning implicit models and paving the way for neuralized utility maximization.
maximize
and λ(t + 1) = p(t + 1) ◦ x(t + 1) if and only if ρ(∇I(p)) < 1 for some p ≥ 0. Remark 1: The fact that the spectral radius of the Jacobian matrix ∇I(p(t)) is strictly less than one for any p(t) (as shown in Theorem 1), has been established in [34] from a control-theoretic perspective based on studying the existence of a fixed point and its convergence behaviour, which are established using the contractive interference function findings in [16], [35]. Theorem 3 offers an alternative and practical interpretation of this result by linking Theorem 1 to the use of iterative algorithms for solving optimization problems. B. Convex Relaxation to Utility Maximization
(9)
Letting p̃ = log p, we assume that the utility function u(ep̃ ) is concave and monotonically increasing in terms of p̃. One example is the minimum of a series of competitive utility functions that satisfy the following assumptions [32]. Definition 3 (Competitive Utility Functions [32]): 1) Competitiveness: For all i, the utility ui (p) is strictly increasing with respect to pi and strictly decreasing with respect to pj for all j ̸= i, whenever pi > 0.
When the problem (9) is infeasible, we consider an approximate power solution that relaxes the original constraints. Specifically, we introduce the following relaxation of (9): maximize subject to
u(p) L Y l=1
Il (p) pl
wl
≤ 1,
(14)
where w = [w1 , · · · , wL ]⊤ is a positive vector that is a convex combination of the individual constraints in (9) and satisfying
PL
l=1 wl = 1 and wl ≥ 0. By letting p̃ = log p, (14) can be solved by the following convex optimization problem:
ω1i1
p̃
u(e )
subject to
L X l=1
p̃
wl log Il (e ) − p̃l ≤ 0.
(15)
pL
i
Furthermore, x⋆ in Theorem 3 satisfies x⋆ = ∇I(p⋆ ) x⋆ − ∇u(p⋆ ), or equivalently, −1 x⋆ = IL − ∇I(p⋆ ) −∇u(p⋆ ) .
Moreover, since I(·) is concave, the Jacobian mapping p 7→ ∇I(p) is order-preserving and concave, and the induced operator x 7→ ∇I(p⋆ )x belongs to the class of concave Perron– Frobenius operators. Consequently, x⋆ lies in the Perron– Frobenius cone of ∇I(p⋆ ). The special case where ∇u(p⋆ ) = −wT p and I(p) = Fp + v leads to a solution p⋆ and x⋆ that recovers the uplink-downlink duality and yields a PageRank-like structure, as shown in [40]. Also, the Collatz-Wielandt-type objective function is a restatement of scalability in Definition 1 [41]. Remark 3: Since I(ep̃ ) is convex in p̃, its Jacobian matrix has nonnegative entries and satisfies J(p̃) := ∂I∂ip̃(p̃) j −1 J(p̃) = diag I(ep̃ ) ∇I(ep̃ ) diag ep̃ ,
where ∇I(·), characterized in Theorems 1 and 2, enables efficient automatic differentiation techniques studied next.
=
pL
Power
aKi1
Power
aKiL
ωIi ωIi bKi = ωaKi1 ωbKi aKi1 Prob
ωKiL
→
z1i ωIi = ωz1i
ωIi ωIi = c1i ωb1i ωz1i
!
1, if 1 = kl→ 0, else
Ii
Max
ωIi bKi lnp1 ωbKi
…
p>0
Ii (p) Ii (p) = sup min = 1. pi pi p>0 i
ωIi
ωKi1 ωεKi1
C. Relationship to Nonlinear Perron-Frobenius Theory We remark that an interesting duality arises between the algorithms for computing p and x, which can be understood via nonlinear Perron–Frobenius theory [32], [36]–[39]: the vectors p and x can be viewed as Perron–Frobenius eigenvectors of suitably constructed nonlinear operators. Theorem 4 (Perron–Frobenius Theorem Characterization): L Let I : RL + → R+ be a differentiable concave standard interference mapping and let u : RL + → R be a differentiable utility function. Suppose p⋆ satisfies p⋆ = I(p⋆ ) and ρ(∇I(p⋆ )) < 1. By nonlinear Perron–Frobenius theory, such a map is nonexpansive in Hilbert’s projective metric i pi /qi dH (p, q) := log max mini pi /qi , and admits a unique fixed point ⋆ p up to scaling, which minimizes the projective diameter of the orbit of I(·). The fixed point p⋆ = I(p⋆ ) plays the role of the nonlinear Perron–Frobenius eigenvector, characterized by a Collatz–Wielandt–type formula
b1i
Prob
a1iL
ωIi ωIi b1i = ωa1iL ωb1i a1iL ω1iL ωIi = ωIi b1i lnpL ωε1iL ωb1i
p1
c1i
1i1
Power
…
w QL Remark 2: The term l=1 Il (p) l in (14) is concave as it is the weighted geometric mean of concave nonnegative functions, which preserves concavity under vector composition [33]. If (9) is indeed feasible, then by Theorem 1 and Theorem 3, setting w = p⋆ ◦ x⋆ in (14) recovers the solution p⋆ of (9), and the relaxation is tight.
inf max
Power
…
maximize
p1
ωIi ωIi = b1i lnp1 ωε1i1 ωb1i ωIi ωIi b1i = ωa1i1 ωb1i a1i1 a
bKi
ωIi ωIi bKi = ωaKiL ωbKi aKiL
ωIi ωIi = cKi ωbKi ωzKi
→
ωIi = ωzKi
!
1, if K = kl→ 0, else
zKi
cKi
ωIi ωIi = bKi lnpL ωεKiL ωbKi
Fig. 2: Computational graph of the DIFFRACT framework, implementable in PyTorch with automatic differentiation, for modeling standard interference functions (potentially non-smooth) by using Lemma 3 and Theorem 1.
IV. DIFFRACT: LLCP M ODELING AND A LGORITHM U NROLLING FOR U TILITY M AXIMIZATION While the standard interference function framework enables fast-converging iterative algorithms, this is limited to deterministic closed-form expressions. To overcome this limitation, we must extend the framework to accommodate implicit standard interference functions—those that cannot be expressed in deterministic form, such as in the case of general fading in (8). To address this, we propose a data-driven framework called DIFFRACT. Specifically, we approximate a log-log convex but implicit standard interference function using the LLCP model in [14], converting it into explicit expressions with monomials. We then employ algorithm unrolling [22], [23] to transform the fixed-point iterations into a fixed-depth neural network, whose parameters are then data-optimized to maximize computational efficiency. The architecture of DIFFRACT is shown in Fig. 1. A. Modeling Standard Interference Functions with LLCP for Stochastic Outage Probability To approximate the implicit standard interference function in (8), we use a data-driven method in [14] to learn a log-log convex function from data, leveraging a monomial basis for its approximation as given by the following lemma. The approximation accuracy is then demonstrated through illustrative examples with validation against ground truth. Lemma 3: Consider the standard interference function I(p)
(16) L L Y θ Y θ = max ck1 pj k1j , · · · , max ckL pj kLj k=1,··· ,K k=1,··· ,K j=1
j=1
PL
where ckl ≥ 0, θklj ≥ 0. Suppose j=1 θklj + ϵ ≤ 1 for all l, k for some positive ϵ, I(p) is feasible and log-log convex. We learn the parameters ckl and θklj in (16) from training data via the LLCP model in [14]: ŷ(p) = argminy
1⊤ (z/y + y/z)
L Y
(17)
interference signals are subject to Rayleigh fading, then hlj are independent and exponentially distributed with unit mean, i.e., E [Flj hlj pj ] = Flj pj [44]. The outage probability of the ith user can be written in closed-form as [44]:
where ckl and θklj are the learning parameters, and pj denotes the input of the training data to (17). Suppose we have a training dataset D consisting of |D| input-output pairs {p(i), y(i)} ∈ D. We evaluate the performance of (17) on this dataset using the following mean squared error loss:
Ol (p) = Prob(SINRl (p) < γl ) −1 L Y −γl vl γl Flj pj ≤ Ōl . (19) = 1 − e pl 1+ pl
subject to zl = max
k=1,··· ,K
variables : y, z ∈ RL +,
E(c, θ) =M
K X L X
k=1 l=1
1 + |D|
ϕ(
L X j=1
ckl
j=1
θ
pj klj
,∀l
θklj − 1)
X
{p(i),y(i)}∈D
∥y − ŷ(p(i))∥22 ,
(18)
where c = (ckl )K×L , θ = (θklj )K×L×L , M is a large positive PL number and ϕ(·) is a penalty function, e.g., ϕ( j=1 θklj − n P o2 L 1) = max 0, j=1 θklj − 1 . The first term of (18) is differentiable (due to the square and despite max) and ensures PL that j=1 θklj < 1, i.e., I(p) ≤ p is feasible. We minimize the training loss E(c, θ) via gradient descent, with gradients obtained through automatic differentiation (e.g., PyTorch autograd); Fig. 2 illustrates the resulting computational graph. This is summarized in Algorithm 1. Algorithm 1: Monomial approximation via automatic differentiation. Input: Training dataset D with input-output pairs (p(i), y(i)) ∈ D. Initialize: Learning parameters c(0) and θ(0), learning rate ηc and ηθ . for i = 1, . . . , |D| do Obtain ŷ(p(i)) by solving (17) with c(i), θ(i) and p(i). Use automatic differentiation (e.g., Pytorch autograd) to compute the gradients ∇c E(c(i), θ(i)) and ∇θ E(c(i), θ(i)). Update the parameters c(i + 1) = c(i) − ηc ∇c E(c(i), θ(i)) and θ(i + 1) = θ(i) − ηθ ∇θ E(c(i), θ(i)). end Output: Monomial approximation I(p) in (16) with learned c and θ. We evaluate the effectiveness of the LLCP model in (17) in approximating the implicit function I(p) in (8) with explicit expressions under Rayleigh and Rician fading models relevant to satellite-terrestrial networks [19], [20], [42], [43]. Interestingly, Rayleigh fading allows closed-form deterministic functions, hence enabling direct validation of the LLCP model in (17). In contrast, Ricean fading lacks exact expressions, but can still be effectively approximated using the same model. 1) Modeling Interference Function for Outage Constraints under Rayleigh fading: Suppose both the desired signals and
j=1,j̸=l
Taking the logarithm of (19), we have: PL γ F p vl + pl j=1,j̸=l log 1 + l pljl j Il (p) = ≤ pl , log( 1−1Ōl )
(20)
where Il (p) is proved to be a standard interference function in [45] and its concavity established later in [31], [32]. Besides facilitating the algorithm design development of using Il (p) in (20) to solve a broader class of utility maximization problems in (9) [32], [44], [45], the deterministic form of Il (p) in (20) gives the ground truth to evaluate the generalization error of the LLCP in (17) when fitting an unknown standard interference function. Specifically, we fit the standard interference function in (20) using the maximum monomials through (17), which serves as a reference point to validate our approach. Example 1: Consider a wireless network with two users whose transmit powers are subject to Rayleigh fading with SINR thresholds being γ1 = γ2 = 10 and the noise variance v1 = v2 = 0.1 pW. The channel gain matrix is G = [1.19, 2.13; 1.69, 1.64]⊤ . Let the outage constraints be Ō1 = Ō2 = 0.1. We use the LLCP model in(16) to approximate (20). Setting K = 3 in (16), we obtain c1 = [2.1, 1.6, 6.3], c2 = [1.1, 1.5, 2.5], c3 = [0.1, 1.2, 1.3], θ1 = [0.23, 0.12; 0.50, 0.49]⊤ , θ2 = [0.63, 0.16; 0.27, 0.46]⊤ , and θ3 = [0.47, 0.25; 0.57, 0.26]⊤ . Fig. 3(a) and Fig. 3(b) show the effectiveness of LLCP in (16) to model outage probabilities under Rayleigh fading for User 1 and User 2. Blue scatter points show sample data; surfaces depict the approximated standard interference functions. LLCP-based monomials show a strong fit to the observed outage probabilities. 2) Modeling Interference Function for Outage Constraints under Ricean Fading: If the random variables characterizing the fading channel do not follow the independent Rayleigh distribution, but instead adhere to the Ricean or Nakagami distribution [20], [43], obtaining expressions for outage probabilities in a deterministic manner becomes challenging, and thus using Lemma 1 directly to obtain the fixed points of the implicit standard interference functions for the utility maximization problem in (9) becomes impractical. Under Ricean fading, the received signal envelope has a Rayleigh distribution, and the received signal power has a non-central chi-square distribution [20], [42], [43]: F (hlj |K, Ω)
K +1 (K + 1)hlj = exp (−K − )I0 (2 Ω Ω
r
K(K + 1)hlj ), Ω
where I0 (·) is the zeroth-order modified Bessel function of the first kind. As a special case with K = 0, Rician fading reduces to Rayleigh fading, which is known to yield a concave standard interference function [31]. We approximate Il (p) with monomial-based standard interference functions via the LLCP method, enabling the iterative algorithm in Section III to compute approximate solutions. Example 2: Let us consider another wireless network where the transmit powers are subject to Ricean fading. Given the SINR threshold γi for each user, the outage constraint Ōi , the noise variance vi , and the channel gain matrix as detailed in Example 1, the sampled data generated by Il (p) in (20) is depicted in Fig. 3. Using LLCP-based monomials to model the standard interference function Il (p), the parameters in the approximation I(p) in (16) are as follows: c1 = [1.8, 0.9, 1.6], c2 = [1.8, 0.8, 1.4], and c3 = [1.9, 0.8, 1.5], θ1 = [0.13, 0.43; 0.65, 0.29]⊤ , θ2 = [0.12, 0.42; 0.67, 0.30]⊤ , and θ3 = [0.13, 0.40; 0.70, 0.28]⊤ . Fig. 3(c) and Fig. 3(d) show the effectiveness of LLCP in (16) to model the I(p) in (8) under Ricean fading for users 1 and 2. The blue scatter points represent the sample data, while the surfaces illustrate the approximate standard interference functions, also exhibiting an excellent fit.
As an illustrative example, consider the stochastic outage probability problem under fading channel conditions: minimize subject to
− u(p)
Prob(SINRl (p) < γl ) ≤ Ōl ∀ l.
(21)
Applying the LLCP model in (17) from Section IV to the outage probability constraints, we obtain an approximation of problem (21): minimize subject to
− u(p)
L Y θ max ckl pj klj ≤ pl ∀ l. k=1,··· ,K
(22)
j=1
According to Theorem 3, the optimal solution to the problem (22) can be obtained by the following iteration: L Y θ pl (t + 1) = max ckl pj klj (t) , (23) k=1,··· ,K j=1
xl (t + 1) = −
L L Y X ∂u(p(t)) + ckl pθi kli (t) θklj p−1 j (t)xj (t), ∂pl i=1 j=1
(24)
where k = argmaxk̂ ck̂l and 15
6
I 1 ( p)
I 2 ( p)
8
θk̂lj (t) for k̂ = 1, . . . , K, j=1 pj
λl (t + 1) = pl (t + 1)xl (t + 1).
10
4
QL
(25)
5
2
0 80
60
80
60 40
p2 40 20
20 0
0
p1
60
60 40
p2 40 20
(a)
20 0
0
p1
(b)
10 8 6 4 2 0
6
I 1 ( p)
I 2 ( p)
4 2 0 150 125 100
150125 100 75
p2 50 25
0
(c)
0
25
75 50 p 1
We now apply algorithm unrolling to construct a fixeddepth nonlinear feedforward model, trained to approximate the optimal solution of (21). The key idea is to utilize feedforward networks whose structure corresponds to a finite number of iteration steps in (23), (24), and (25). Denote the model depth by T . We let p̃ = log p and c̃ = log c. The architecture can be interpreted as a time-unfolded recurrent neural network over steps t ≤ T , as follows: sk (t+1) = f (ŵ(t+1)(θk p̃(t) + c̃k )) ,
150 100125
150125 100 75
p2 50 25
0
0
25
75 50 p 1
(d)
Fig. 3: Illustration of using LLCP to approximate the outage probability constraints with monomials under Rayleigh fading ((a) and (b)) and Ricean fading ((c) and (d)).
B. Differentiable Programming by Algorithm Unrolling Algorithm unrolling is a deep learning technique that establishes a link between iterative algorithms (e.g., sparse coding) and neural network architectures [22], [23] . In essence, algorithm unrolling maps each iteration to a network layer, enabling a trainable deep neural network that embeds optimization logic within a differentiable programming framework, enhancing both performance and interpretability [13].
p̃(t+1) =
max {sk (t+1)} ,
k=1,··· ,K
(26) (27)
x(t + 1) = f (w̆(t)∇(I(p(t+1)))x(t) + ∇u(p(t+1))) , (28) λ(t + 1) = p(t + 1) ◦ x(t + 1),
(29)
where f (·) is a nonlinear activation function (e.g., ReLU) and ∇(I(p(t)))lj = ckl QL
L Y pθi kli (t)θklj p−1 j (t),
i=1 θ
where k = argmaxk̂ ck̂l j=1 pj k̂lj (t) for k̂ = 1, . . . , K. Denote the architecture of our unrolled model by [p(T ); λ(T )] = F (θ, c, w), where w = {ŵ(t), w̆(t)} for t = 1, . . . , T . Suppose we are given a training set D̄ with |D̄| training samples. For the mth sample {(θm , cm ), (p∗m , λ∗m )} ∈ D̄, (θm , cm ) are the inputs and (p∗ , λ∗ ) are the ground truth of outputs. The loss function is defined as the squared error
between the predicted solution and the optimal solution to the problem (22): X 1 ∥[p∗m ; λ∗m ] − f (θm , cm , w)∥2 . E(w) = |D̄| ∗ ∗
on minimizing total power while meeting outage probability constraints in (21). For each scenario, we generate 1,000 samples of the implicit standard interference function derived from the outage probability constraints. We then apply the {(θm ,cm ),(pm ,λm )}∈D̄ LLCP model to fit this sample data, allowing us to approximate (30) the implicit standard interference functions using a maximum We train the parameters w by minimizing the loss E(w), of monomials for each problem instance. The LLCP model with gradients obtained via automatic differentiation (e.g., for fitting the implicit standard interference functions is set PyTorch autograd). Algorithm 2 presents DIFFRACT as (16), where ckl and θklj for all k, l and j are learning for computing approximate primal and dual solutions to (22). parameters. We set the number of monomials as K = 3, 5, 8 and 10. The loss function is defined as (18), where the penalty scalar M = 1 × 103 . We set the learning rate as 2 × 10−3 . Algorithm 2: DIFFRACT via Algorithm Unrolling To assess the performance of the LLCP model in (17), we Input: Training dataset D̄ with input-output pairs use metrics: utility accuracy and power allocation accuracy. {Dm , (p∗m , λ∗m )} ∈ D̄, where Dm is a dataset with Utility accuracy measures the ratio between the predicted input-output pairs (pm (i), ym (i)) ∈ Dm . utility and the maximum achievable utility, indicating the Initialize: p(0) = 0,x(0) = 0, learning parameters model’s ability to approximate optimal utility values. Power w(0) = (w1 (0), w2 (0), b1 (0), b2 (0)) and learning allocation accuracy measures the closeness of predicted to rate ηw . ground truth allocations, reflecting the model’s effectiveness in for m = 1, . . . , |D̄| do solving the problem. We approximate the standard interference Obtain cm and θm of the |D̄| monomial vectors function using Lemma 3. For Rayleigh fading, ground truth using Algorithm 1 with Dm . values come from the iterative algorithm in [31], while for for t = 1, . . . , T do Rician and Nakagami fading, we use exhaustive search for benchmarking. Tables I and II compare the utility and power sk (t+1) = f (ŵ(t+1)(θk p̃(t) + c̃k )) , allocation accuracy using different numbers of monomials to p̃(t+1) = max {sk (t+1)} , fit implicit standard interference functions under various fading k=1,··· ,K x(t+1) = f (w̆(t)∇(I(p(t+1)))x(t)+∇u(p(t+1))) , conditions. We observe in our numerical evaluations that the LLCP model approximates these functions well at K = 8 and λ(t + 1) = p(t + 1) ◦ x(t + 1). only marginal gains beyond that. end B. Performance of Algorithm Unrolling Use automatic differentiation (e.g., Pytorch We now evaluate the algorithm unrolling method for the autograd) to compute the gradient ∇E(w(t)) approximate standard interference function using the maxiand update w(t + 1) = w(t) −ηw ∇E(w(t)). mum of monomials. We consider the utility maximization end problem involving eight users, with the number of monomials fixed at 8. The unrolled model for computing the maximum Implemented in PyTorch, Algorithm 2 frames the primalover monomials is given by (26) to (29). The model takes dual optimization loop as a differentiable computational graph. as input the vectors θk and ck , and is trained using the loss Each iteration is modeled as a neural layer using learned in (30). We generate 106 samples, splitting them into 80% monomial mappings for both primal and dual updates. Leverfor training and 20% for testing, with 10-fold cross-validation aging PyTorch’s autograd engine in [18], this setup can for robustness. The validation datasets are constructed using enable end-to-end training with gradient-based optimization the approximate standard interference functions defined earlier. and GPU acceleration integrated with NVIDIA’s SONNIA For problem instances subject to Rayleigh fading, ground truth framework [7], [8] by integrating Algorithm 2 with ∇-Prox values are obtained using the iterative algorithm in [31]. For [46], an ML compilation framework for generating memoryscenarios involving Ricean and Nakagami fading, we compute efficient differentiable solvers, thus enabling rapid prototyping the ground truth using a brute force discretization search. and fully differentiable resource allocation under interference We first examine how the number of layers in the unrolled constraints in an AI-native wireless network stack. model affects the approximation of standard interference funcV. P ERFORMANCE E VALUATION tions for utility maximization under different fading environIn this section, we provide numerical examples to demon- ments. Fig. 4 illustrates the maximum utility results obtained strate the effectiveness of the proposed DIFFRACT framework from the unrolled model across varying numbers of layers under Rayleigh fading, Ricean fading, and Nakagami fading, for solving the utility maximization problem in Section III. respectively. Interestingly, the model achieves the highest utilA. Performance of LLCP ities with 4 layers. We also demonstrate the unrolled model’s Consider a wireless network with eight transmitter-receiver superior performance by comparing it against established pairs under Rayleigh, Ricean, and Nakagami fading, focusing baselines: the Adaptive Outage-based Power Control (AOPC)
Rayleign fading
Ricean fading
1.50
1.5
Nakagami fading 1.45
1.45
1.40
1.40
layers= 3 layers= 4 layers= 5 layers= 6
1.2
1.30
layers= 3 layers= 4 layers= 5 layers= 6
1.25 1.20
layers= 3 layers= 4 layers= 5 layers= 6
1.25 1.20 1.15
0 20 40 60 80 10 120 140 160 180 200 220 240 260 280 0
1.10
0 20 40 60 80 10 120 140 160 180 200 220 240 260 280 0
1.15
1.30
0 20 40 60 80 10 120 140 160 180 200 220 240 260 280 0
1.1
1.35
1.35
Utility
1.3
Utility
Utility
1.4
(a)
(b)
(c)
Epochs
Epochs
Epochs
Fig. 4: An illustration of the performance of Algorithm 2 in unrolling the standard interference function to compute the optimal utility values of the utility maximization problem under various fading conditions: Rayleigh fading (a), Ricean fading (b), and Nakagami fading (c).
K=3 K=5 K=8 K = 10
Rayleigh Fading 0.9291 0.9391 0.9834 0.9856
Ricean fading 0.9248 0.93497 0.9876 0.9909
Nakagami fading 0.9235 0.9400 0.9793 0.9834
TABLE I: Comparison of utility accuracy across various monomials used to fit implicit standard interference functions for wireless systems in diverse fading environments.
K=3 K=5 K=8 K = 10
Rayleigh Fading 0.9177 0.9259 0.9697 0.9760
Ricean fading 0.9131 0.9302 0.9617 0.9633
Nakagami fading 0.9195 0.9228 0.9645 0.9687
TABLE II: Comparison of power accuracy across various monomials used to fit implicit standard interference functions for wireless systems in diverse fading environments.
algorithm in [31] for Rayleigh fading, the large deviations principle (LDP) algorithm from [47] for Ricean fading, and the convex relaxation (CR) algorithm from [48] for Nakagami fading. TABLE IV summarizes the results of maximal utility under different fading environments. We observe that while our unrolled model yields slightly lower values than the ground truth, it outperforms other approximation methods. Table IV further shows that it achieves the highest efficiency compared to all baselines. TABLE IV presents the results of maximal utility across various fading environments. Although the output from our unrolled model are slightly below the ground truth values, they surpass the performance of other approximate methods. Furthermore, Table IV compares the efficiency of our proposed methods with several baselines, highlighting that the unrolled model achieves the highest efficiency. Environment Algorithm 1 Algorithm 2 AOPC [31] LDP [47] CR [48]
Rayleigh fading 0.9902 0.9704 1.0000 – –
Ricean fading 0.98851 0.9622 – 0.9525 –
Nakagami fading 0.9899 0.9664 – – 0.9434
TABLE III: Comparison of the utility accuracy performance for approximate standard interference functions using LLCP (Algorithm 1) and Algorithm Unrolling (Algorithm 2), benchmarked against different baselines based on existing state-of-the-art algorithms for various wireless fading conditions.
Environment Algorithm 1 Algorithm 2 AOPC [31] LDP [47] CR [48]
Rayleigh fading 10.9086 1.5690 9.5252 – –
Ricean fading 11.2013 1.6889 – 8.6566 –
Nakagami fading 11.3778 1.2701 – – 9.6425
TABLE IV: Comparison of time efficiency (seconds) for approximate standard interference functions using LLCP (Algorithm 1) and Algorithm Unrolling (Algorithm 2), benchmarked against different baselines under various fading channel conditions.
VI. C ONCLUSION This paper presents DIFFRACT, a general differentiable programming framework for utility maximization in wireless networks, particularly effective in managing resources under unpredictable interference, as commonly encountered in satellite-terrestrial systems. Our work reveals new duality results by establishing the differentiability and monotone operator characterization of standard interference functions. This provides a nonlinear Perron-Frobenius operator perspective that enables smooth integration into differentiable programming and modern deep learning frameworks with efficient automatic differentiation. DIFFRACT models complex interference patterns–such as outage in atmospheric fading, which is critical for space-to-ground communication–within an end-to-end trainable, AI-native stack by unrolling a utility maximization algorithm with implicit interference constraints into differentiable primal-dual updates. Experiments show DIFFRACT offers theoretically sound performance and scalability for AI-native resource control in wireless network optimization. Future work will leverage the monotone operator characteristics in Theorems 2 to 4 to design deep implicit layers in the DIFFRACT framework, enabling differentiation through utility maximization via fixed-point strategies [49], [50]. This approach will integrate first-order neural acceleration to improve convergence and enable scalable, distributed learning in AI-native wireless networks. VII. ACKNOWLEDGEMENT The research was supported in part by NTU startup and the Singapore Ministry of Education Academic Research Fund (MOE-T2EP20224-0009).
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