iZoSGA: Gradient-Free IRS Beamforming

• iZoSGA is a data-driven algorithm that jointly optimizes long-term IRS configuration and short-term AP precoding to maximize weighted sum rate.
• It employs a zeroth-order stochastic quasigradient ascent with two-point Gaussian smoothing to approximate gradients from black-box function evaluations.
• The method tolerates inexact precoding oracles (e.g., truncated WMMSE), ensuring convergence to a neighborhood of a stationary solution despite continuous channel uncertainty.

iZoSGA is a data-driven learning algorithm for joint passive long-term intelligent reflective surface (IRS)-aided beamforming and active short-term precoding in wireless networks. It is based on a zeroth-order stochastic quasigradient ascent methodology designed for tackling two-stage nonconvex stochastic programs with continuous uncertainty and objective functions with “black-box” terms, and where second-stage optimization is inexact. In its canonical formulation, the outer variable is the IRS configuration, the inner variable is the access-point precoder, and the objective is expected weighted sum-rate maximization under continuously varying channel realizations. The defining methodological feature of iZoSGA is that it uses inexact precoding oracles—implemented in the paper with truncated WMMSE—while retaining non-asymptotic convergence to a neighborhood of a stationary solution of the original exact problem under minimal assumptions (Hashmi et al., 20 Sep 2025).

1. Two-stage stochastic program and IRS-aided system model

The algorithm is formulated for a downlink multiuser MISO system aided by a passive IRS. The access point (AP) has $M$ transmit antennas, there are $K$ single-antenna users, and the IRS has $S$ passive reflecting elements. Each IRS element has a tunable complex reflection coefficient parameterized by $\bm{\theta} \in \Theta \subset \mathbb{R}^S$, where $\Theta$ is convex and compact. The operational model is explicitly two-timescale: IRS configuration is optimized in the long term, whereas AP precoding is optimized in the short term for each realized channel state (Hashmi et al., 20 Sep 2025).

The effective channel for user $k$ is denoted

$\bm{h}_k(\bm{\theta},\omega) \in \mathbb{C}^{M},$

where $\omega \in \Omega$ is a random state of nature capturing small-scale fading, geometry, and IRS-related propagation randomness. The stacked channel representation is

$$\bm{H}(\bm{\theta},\omega) \triangleq \mathrm{vec}\big( [\bm{h}_1(\bm{\theta},\omega), \ldots, \bm{h}_K(\bm{\theta},\omega)]\big) \in \mathbb{C}^{M_U},\quad M_U = MK.$$

The AP precoder is similarly vectorized as

$$\bm{W} = \mathrm{vec}\big( [\bm{w}_1, \bm{w}_2, \ldots, \bm{w}_K]\big) \in \mathbb{C}^{M_U},$$

under the total power constraint

$K$0

Performance is measured through weighted sum rate: $K$1 with user weights $K$2 and

$K$3

The resulting design problem is a nonconvex two-stage stochastic program with continuous uncertainty. The short-term subproblem is

$K$4

and the long-term problem is

$K$5

The outer expectation is taken over an unknown continuous distribution of channel states. The inner maximization is itself nonconvex because weighted sum rate is coupled through interference. This combination is the central motivation for a stochastic, model-free, gradient-free outer method.

2. Zeroth-order stochastic quasigradient construction

iZoSGA adopts zeroth-order stochastic quasigradient ascent because neither the channel mapping $K$6 nor the optimizer mapping $K$7 is assumed available in closed form. The method is model-free in the precise sense that it uses observed effective channels and function evaluations rather than analytical derivatives, and it is agnostic to channel models or channel statistics (Hashmi et al., 20 Sep 2025).

The zeroth-order component is based on two-point Gaussian smoothing. For $K$8 and smoothing parameter $K$9, the paper defines

$S$0

Using Wirtinger calculus, this is chained with the co-gradient of $S$1 with respect to the complex channel variable. The resulting compound zeroth-order approximation is written as

$S$2

In implementation, the expectation is not evaluated exactly. Instead, iZoSGA uses the single-sample estimator

$S$3

where $S$4 and $S$5 are the two-point finite-difference evaluations of the real and imaginary channel Jacobian components. Operationally, the method probes the channel at $S$6 and $S$7, approximates directional sensitivity of the effective channel, and composes that estimate with the analytic Wirtinger derivative of the utility.

The algorithmic recursion is correspondingly simple. At iteration $S$8, it samples $S$9 and $\bm{\theta} \in \Theta \subset \mathbb{R}^S$0, obtains an approximate precoder $\bm{\theta} \in \Theta \subset \mathbb{R}^S$1, forms

$\bm{\theta} \in \Theta \subset \mathbb{R}^S$2

and updates the IRS parameter through projection: $\bm{\theta} \in \Theta \subset \mathbb{R}^S$3 After $\bm{\theta} \in \Theta \subset \mathbb{R}^S$4 iterations, it samples $\bm{\theta} \in \Theta \subset \mathbb{R}^S$5 uniformly from $\bm{\theta} \in \Theta \subset \mathbb{R}^S$6 and returns $\bm{\theta} \in \Theta \subset \mathbb{R}^S$7. This randomized output rule is standard in nonconvex stochastic approximation and is the object appearing in the convergence theorem.

3. Inexact precoding oracle and two-stage decomposition

The adjective “inexact” in iZoSGA refers specifically to the second-stage precoding solve. The inner problem is not assumed to be solved exactly at each coherence interval; instead, the method accesses an inexact precoding oracle that returns a feasible approximate solution $\bm{\theta} \in \Theta \subset \mathbb{R}^S$8 (Hashmi et al., 20 Sep 2025).

Formally, for any $\bm{\theta} \in \Theta \subset \mathbb{R}^S$9 and almost every $\Theta$0, the oracle satisfies

$\Theta$1

so that if $\Theta$2 is some maximizer of the second-stage problem, then

$\Theta$3

In the paper’s main instantiation, WMMSE is used as this oracle, and running WMMSE for $\Theta$4 iterations produces an approximation whose error depends on the iteration budget.

A central analytical device is the relationship between functional suboptimality and distance to the optimizer set. The paper states that the function

$\Theta$5

is subanalytic and therefore satisfies a Łojasiewicz inequality with uniform exponent $\Theta$6. Concretely, there exists a positive subanalytic function $\Theta$7 and exponent $\Theta$8 such that

$\Theta$9

Assuming a uniform constant $k$0 that upper-bounds $k$1, any approximate solver with functional gap

$k$2

satisfies

$k$3

This two-level error model is important for interpretation. The outer IRS adaptation is not analyzed against the performance of the approximate inner problem; rather, it is analyzed against the original exact two-stage stochastic program. The inexactness of the short-term oracle therefore appears as a quantifiable perturbation of the exact long-term learning problem, rather than as a reformulation of that problem.

4. Convergence theory and stationarity guarantees

The convergence analysis proceeds under minimal smoothness assumptions on the black-box channel mapping. The paper assumes that for almost every $k$4, the map $k$5 is twice continuously differentiable, uniformly bounded, Lipschitz continuous, and Lipschitz smooth as a function of $k$6, and that the Łojasiewicz constant is uniformly bounded. Under these conditions, prior work implies that the outer objective $k$7 is $k$8-weakly convex (Hashmi et al., 20 Sep 2025).

The analysis therefore uses the Moreau envelope as a stationarity surrogate. With

$k$9

and for $\bm{h}_k(\bm{\theta},\omega) \in \mathbb{C}^{M},$0,

$\bm{h}_k(\bm{\theta},\omega) \in \mathbb{C}^{M},$1

the gradient norm $\bm{h}_k(\bm{\theta},\omega) \in \mathbb{C}^{M},$2 becomes the relevant stationarity measure whenever $\bm{h}_k(\bm{\theta},\omega) \in \mathbb{C}^{M},$3. Small Moreau-envelope gradient implies proximity to a Clarke-stationary point of the original constrained nonconvex problem.

The main theorem states that if

$\bm{h}_k(\bm{\theta},\omega) \in \mathbb{C}^{M},$4

is the oracle error at iteration $\bm{h}_k(\bm{\theta},\omega) \in \mathbb{C}^{M},$5, and if the number of outer iterations and smoothing parameter are chosen as

$\bm{h}_k(\bm{\theta},\omega) \in \mathbb{C}^{M},$6

then

$\bm{h}_k(\bm{\theta},\omega) \in \mathbb{C}^{M},$7

where

$\bm{h}_k(\bm{\theta},\omega) \in \mathbb{C}^{M},$8

is the averaged oracle error. Using the functional-gap bound above, this can be refined to

$\bm{h}_k(\bm{\theta},\omega) \in \mathbb{C}^{M},$9

The theoretical message is exact and limited. If the oracle is exact, the previous ZoSGA rate is recovered. If the oracle is inexact, iZoSGA converges to a neighborhood of a stationary solution of the original exact problem, and the neighborhood size is controlled additively by averaged oracle inexactness. This suggests that moderate inner error is tolerable, but large persistent second-stage inaccuracies directly inflate the stationarity residual.

5. Implementation profile and numerical behavior

The algorithm is designed to be practically lightweight at the outer level. Each iteration uses exactly three channel estimations: one at $\omega \in \Omega$0 for precoding, and two at $\omega \in \Omega$1 for the finite-difference estimate. Projection onto $\omega \in \Omega$2 is assumed easy, and the remaining cost is dominated by WMMSE, whose per-iteration complexity is described as typically $\omega \in \Omega$3. The claimed iteration complexity

$\omega \in \Omega$4

grows only like $\omega \in \Omega$5 in IRS dimension $\omega \in \Omega$6, which the paper presents as one reason the method remains viable for large IRSs (Hashmi et al., 20 Sep 2025).

The representative simulation setup uses an AP with $\omega \in \Omega$7 antennas, $\omega \in \Omega$8 single-antenna users, and an IRS with $\omega \in \Omega$9 purely phase-shift elements. Amplitudes are fixed to unity, phases vary in $$\bm{H}(\bm{\theta},\omega) \triangleq \mathrm{vec}\big( [\bm{h}_1(\bm{\theta},\omega), \ldots, \bm{h}_K(\bm{\theta},\omega)]\big) \in \mathbb{C}^{M_U},\quad M_U = MK.$$0, and the effective channel is modeled as

$$\bm{H}(\bm{\theta},\omega) \triangleq \mathrm{vec}\big( [\bm{h}_1(\bm{\theta},\omega), \ldots, \bm{h}_K(\bm{\theta},\omega)]\big) \in \mathbb{C}^{M_U},\quad M_U = MK.$$1

The paper notes that the total number of cascaded links is about 38,192 and that each curve is averaged over 60 independent runs. iZoSGA is explicitly described as oblivious to these structural details because it uses only effective channels in its gradient estimates.

The numerical findings are organized around oracle accuracy. For 10 or more WMMSE iterations, the oracle error is small and iZoSGA achieves similar performance across 10–50 iterations. For 1–3 WMMSE iterations, the oracle is quite inexact and final performance drops significantly. The most practically notable regime is 5 WMMSE iterations: while WMMSE alone with random IRS exhibits a noticeable gap relative to 10 iterations, iZoSGA tuning of IRS parameters closes this gap after about 2500 outer iterations and ultimately matches the higher-accuracy oracles. This is presented as empirical confirmation that sufficiently accurate long-term IRS adaptation can compensate for moderate short-term precoding inexactness.

The paper also studies dynamic WMMSE schedules. One schedule, $$\bm{H}(\bm{\theta},\omega) \triangleq \mathrm{vec}\big( [\bm{h}_1(\bm{\theta},\omega), \ldots, \bm{h}_K(\bm{\theta},\omega)]\big) \in \mathbb{C}^{M_U},\quad M_U = MK.$$2 iterations, converges smoothly to the best achievable weighted sum rate; another, $$\bm{H}(\bm{\theta},\omega) \triangleq \mathrm{vec}\big( [\bm{h}_1(\bm{\theta},\omega), \ldots, \bm{h}_K(\bm{\theta},\omega)]\big) \in \mathbb{C}^{M_U},\quad M_U = MK.$$3, degrades once the WMMSE iteration budget falls below a threshold around 5 iterations. A plausible implication is that the averaged oracle error $$\bm{H}(\bm{\theta},\omega) \triangleq \mathrm{vec}\big( [\bm{h}_1(\bm{\theta},\omega), \ldots, \bm{h}_K(\bm{\theta},\omega)]\big) \in \mathbb{C}^{M_U},\quad M_U = MK.$$4 admits an operational “safe regime” below which outer adaptation remains effective and above which the long-term learning process becomes visibly impaired.

A second family of experiments replaces the idealized IRS parameterization with a physical electromagnetic model based on varactor diodes, where each element is controlled through a tunable coupling capacitance $$\bm{H}(\bm{\theta},\omega) \triangleq \mathrm{vec}\big( [\bm{h}_1(\bm{\theta},\omega), \ldots, \bm{h}_K(\bm{\theta},\omega)]\big) \in \mathbb{C}^{M_U},\quad M_U = MK.$$5 and amplitude-phase coupling reduces the effective design degrees of freedom. The same qualitative trend is observed: weak performance for 1–3 WMMSE iterations, strong performance for 5–50 iterations, and lower absolute sum rate than the ideal phase-only model because of hardware constraints. This suggests robustness of the method across both abstract and hardware-constrained IRS descriptions.

6. Conceptual position, novelty, and acronym disambiguation

Within the algorithmic landscape, iZoSGA sits at the intersection of stochastic approximation, zeroth-order optimization, and two-stage stochastic programming. It is a stochastic method because it uses random channel realizations and random perturbation directions; it is zeroth-order because it relies on finite-difference probing rather than analytic differentiation through the channel law or the inner optimizer; and it is a two-stage method because the outer objective is the expectation of an inner optimal-value function (Hashmi et al., 20 Sep 2025).

The main novelty claimed for iZoSGA is not merely gradient-free IRS tuning, but gradient-free IRS tuning with explicit accommodation of inexact second-stage optimization. The paper states that the method is truly model-free for IRS-aided beamforming, applies to arbitrary IRS/network configurations, and proves non-asymptotic convergence under inexactness with a clean additive oracle-error term. It also emphasizes practical deployability: three channel estimations per outer iteration, black-box compatibility with WMMSE, and strong empirical performance for moderate inner accuracy.

The principal limitations are equally specific. The convergence neighborhood depends on the average oracle error $$\bm{H}(\bm{\theta},\omega) \triangleq \mathrm{vec}\big( [\bm{h}_1(\bm{\theta},\omega), \ldots, \bm{h}_K(\bm{\theta},\omega)]\big) \in \mathbb{C}^{M_U},\quad M_U = MK.$$6; the analysis is carried out for sum-rate maximization; and the theoretical rates are worst-case and conservative. The paper identifies extensions to other metrics such as fairness, energy efficiency, and secrecy, as well as alternative inexact inner solvers, as open directions.

The acronym can also invite confusion. A separate paper on zero-sum partially observable stochastic games remarks that the query “iZoSGA” sounds like “iterative zero-sum game algorithm” for zs-POSGs or “improved zs-POSG algorithm,” but that paper’s actual named algorithms are SeqPBVI and SimPBVI, not iZoSGA (Dibangoye et al., 27 Feb 2026). In the established sense documented here, iZoSGA denotes “inexact Zeroth-order Stochastic Quasigradient Ascent” for two-stage IRS-aided sum-rate maximization rather than a zero-sum game solver.

In summary, iZoSGA is best understood as a black-box outer-loop optimizer for long-term IRS control in a two-timescale architecture, with the short-term beamforming stage delegated to an approximate oracle. Its theoretical contribution is the coupling of zeroth-order quasigradient learning with a quantified inexact-oracle model; its practical contribution is the demonstration that passive IRS tuning can remain effective even when inner WMMSE beamforming is solved only approximately.