Convergence of the distributed stochastic preconditioned forward-backward algorithm with stochastic approximation

Let Assumptions

$$,$$

, and

$$hold. Let the sequence be generated by Algorithm$$

, using the approximation in

$$, for the stochastic generalized Nash equilibrium game in$$

. Convergence conjecture. The generated sequence converges to a v-SGNE of the game. The conjecture concerns the stochastic-approximation (SA) scheme, for which the variable metric does not satisfy the deterministic convergence condition; the authors report numerical evidence of convergence, although possibly very slowly, while a proof is not provided.