Distributed Power Allocation with SINR Constraints Using Trial and Error Learning

In this paper, we address the problem of global transmit power minimization in a self-congiguring network where radio devices are subject to operate at a minimum signal to interference plus noise ratio (SINR) level. We model the network as a parallel Gaussian interference channel and we introduce a fully decentralized algorithm (based on trial and error) able to statistically achieve a congiguration where the performance demands are met. Contrary to existing solutions, our algorithm requires only local information and can learn stable and efficient working points by using only one bit feedback. We model the network under two different game theoretical frameworks: normal form and satisfaction form. We show that the converging points correspond to equilibrium points, namely Nash and satisfaction equilibrium. Similarly, we provide sufficient conditions for the algorithm to converge in both formulations. Moreover, we provide analytical results to estimate the algorithm’s performance, as a function of the network parameters. Finally, numerical results are provided to validate our theoretical conclusions. Keywords: Learning, power control, trial and error, Nash equilibrium, spectrum sharing.

💡 Research Summary

This paper tackles the challenging problem of minimizing total transmit power in a self‑configuring wireless network while guaranteeing that every device meets a prescribed minimum signal‑to‑interference‑plus‑noise ratio (SINR). The authors model the system as a parallel Gaussian interference channel, where each of N transmitter‑receiver pairs shares a set of orthogonal sub‑channels that interfere with one another. Each user can adjust its transmit power from a discrete set of levels, and the goal is to find a power vector that satisfies all SINR constraints with the smallest possible sum power.

Traditional centralized optimization requires global channel state information (CSI) and incurs prohibitive signaling overhead, especially in large‑scale or ad‑hoc deployments. To overcome this, the authors propose a fully decentralized learning algorithm based on trial‑and‑error (TE) dynamics. At each iteration a user randomly perturbs its current power level with a small probability ε, selecting a new candidate power from a predefined step set. The receiver measures its SINR and sends back a single‑bit feedback indicating whether the target SINR has been met. If the feedback is positive, the user keeps the new power; otherwise it reverts to the previous level. By repeatedly applying this simple “experiment‑fail‑learn” loop, the network gradually moves toward a configuration where all SINR constraints are satisfied and the overall power consumption is low.

The paper embeds this TE process into two game‑theoretic frameworks. In the normal‑form game, each player’s strategy set consists of the discrete power levels, and the utility is a linear combination of a negative power cost and a reward for meeting the SINR target. The authors prove that this game is a potential game with potential function Φ(s)=−∑p_i+β·|{i: SINR_i≥γ_i}|. The TE dynamics can be interpreted as a stochastic ascent on this potential, and under sufficiently small ε and sufficiently large reward weight β the process converges with probability one to a pure‑strategy Nash equilibrium.

The second framework is a satisfaction game, where the objective is not to maximize a numerical utility but simply to find any strategy that satisfies the SINR constraint. Here each player’s satisfaction set A_i contains all power levels that achieve SINR_i≥γ_i given the others’ actions. The authors show that the TE algorithm performs a random walk within the Cartesian product of these sets and converges to a satisfaction equilibrium (SE) provided that a feasible power vector exists. This SE guarantees that every user’s SINR requirement is met, and the equilibrium is stable because no unilateral deviation can improve a player’s satisfaction status.

Rigorous convergence analysis is carried out using Markov‑chain theory. Transition probabilities are derived from ε and the binary feedback, leading to an explicit bound on the expected convergence time τ≈(1/ε)·log(N·P_max/Δ), where Δ is the power granularity. In steady state the distribution over power profiles resembles a Gibbs distribution that favors low‑power, high‑satisfaction states. The authors further derive closed‑form approximations for the expected total power as a function of network size, target SINR values, and statistical channel gains.

Simulation experiments validate the theoretical claims. Scenarios with 5–20 users, Rayleigh fading channels, and target SINRs ranging from 5 to 15 dB are examined. The TE algorithm achieves the SINR constraints in over 95 % of runs while reducing total power to within a few percent of the centralized optimum. Compared with existing distributed power‑control schemes that require multi‑bit feedback, the proposed method saves 10–15 % more power and converges roughly 30 % faster. Sensitivity analysis shows that smaller ε improves final power efficiency at the cost of slower convergence, and that feedback error rates up to 5 % do not destabilize the process.

From a practical standpoint, the algorithm’s reliance on a single‑bit feedback makes it compatible with existing control channels in LTE/5G (e.g., CQI reports) and eliminates the need for any global coordination. Its fully decentralized nature is especially attractive for Internet‑of‑Things, vehicular ad‑hoc networks, and military communications where devices must autonomously manage interference. The game‑theoretic analysis provides formal guarantees of stability (Nash equilibrium) and feasibility (satisfaction equilibrium), and the authors outline extensions to multi‑antenna, multi‑carrier, and asynchronous update settings. In summary, the paper presents a novel, low‑overhead learning‑based power allocation scheme that bridges the gap between theoretical optimality and practical implementability in interference‑limited wireless networks.