Regret-bound analysis for gradient sampling

Derive a regret bound for the gradient sampling algorithm for intelligent reflecting surface blind beamforming, which requires addressing the absence of a guarantee that the nonconvex expected-SNR maximization problem reaches a globally optimal solution.

Background

The proposed gradient sampling algorithm is analyzed as a stochastic approximation to gradient ascent on the expected SNR induced by the product-softmax sampling distribution. The paper proves asymptotic first-order stationarity and an O(1/sqrt(T)) finite-time stationarity rate, but the expected-SNR objective is nonconvex, so these results do not establish global optimality. Since the cumulative-regret framework compares performance with an optimal phase-shift configuration, a regret guarantee would require a corresponding global-optimality guarantee for the gradient sampling procedure.

References

Consequently, a regret bound analysis for the gradient sampling algorithm is not available because such an analysis would require a guarantee of global optimality.

— Blind Beamforming for Intelligent Reflecting Surfaces: A Gradient Bandit Approach  (2610.06565 - Lai et al., 5 Oct 2026) in Section 4, “Convergence Analysis,” immediately following Theorem 2 and the discussion of Fig. 4