Ergodic-Rate Maximization

• Ergodic-rate maximization is a framework that optimizes long-term average system performance by replacing time averages with ensemble expectations under stochastic uncertainty.
• It underpins applications in wireless communications, financial models, and robust control, employing techniques such as WMMSE, water-filling, and distributed learning algorithms.
• Recent advances address robust formulations, learning-based strategies, and interference constraints, achieving near-optimal performance even with imperfect channel state information.

The ergodic-rate maximization problem is a fundamental optimization framework in wireless communications, information theory, and applied probability, centered on maximizing long-term average (ergodic) achievable rates under stochastic system uncertainties and constraints. Its study spans power/resource allocation in fading channels, beamforming under partial channel state information, robust investment growth in stochastic environments, and model-free policy learning. The problem's distinct character emerges from the ergodicity assumption—where long-term time averages equal statistical ensemble averages—enabling performance metrics, e.g., sum-rate, to be maximized with respect to time-invariant strategies under random processes. Recent work includes robust formulations under various uncertainties, weighted fairness/percentile utilities, complex secrecy and interference constraints, and distributed or learning-based algorithmic solutions.

1. Formal Problem Structure and Ergodicity

The canonical ergodic-rate maximization problem, in its most general stochastic control form, seeks to optimize a utility of the time-averaged service (rate, throughput, or growth) over admissible policies or control mappings, subject to resource and/or reliability constraints under uncertain, often time-varying or random system states. For example, in wireless systems with channel process $h\sim\mathcal{M}$ and parametric action set $\mathcal{X}$, consider the measurable policy $\pi:\mathcal{N}\rightarrow\mathcal{X}$ and a service-level mapping $$f:\mathcal{X}\times\mathcal{N}\rightarrow\mathbb{R}^S$$. The ergodic-rate maximization is then

$$\max_{\pi}\quad g^{o}\big(\mathbb{E}_{h}[f(\pi(h),h)]\big)\qquad\text{s.t.}\quad \mathbb{E}_{h}[f(\pi(h),h)]\geq c,\ \pi(h)\in\mathcal{X},$$

where $g^{o}$ is a (typically concave) utility function, and $c$ prescribes component-wise QoS constraints (Kalogerias et al., 2019).

Ergodicity of the underlying process justifies replacing long-term time averages in system operation by ensemble expectations, which is essential for formulating and analyzing these objectives. This applies to mutual information rates in time-stationary, ergodic channels (0711.4406), portfolio growth in ergodic stochastic diffusions (Kardaras et al., 2018, Itkin et al., 2022), and sum-rates in ergodic wireless networks with fading (Saki et al., 2017, Joudeh et al., 2016).

2. Representative Application Domains and System Models

Multiuser Wireless Communications

• Downlink MISO/RSMA Broadcast Models: Ergodic-rate maximization under imperfect CSIT motivates rate-splitting strategies, where transmit messages are decomposed into common and private streams, precoded subject to power and secrecy constraints, and optimized for weighted ergodic sum-rate or secrecy rate (Xia et al., 2022, Joudeh et al., 2016).
• Cognitive Radio/Multiaccess/Spectrum Sharing: Maximizing secondary network ergodic sum-rate under average/peak transmit and interference constraints leads to waterfilling-type solutions, often with user assignment or TDMA structures, especially in the presence of nonconvex constraints and imperfect channel estimates (Saki et al., 2017, Kang et al., 2014, Bae et al., 2012).
• RIS-aided Wideband MIMO: The ergodic achievable rate maximization with statistical-only CSI involves joint transmit covariance and RIS phase coefficient optimization, frequently handled via alternating algorithms decoupling beamforming and phase design (Li et al., 2022).

Growth-Optimal Investment and Robust Stochastic Control

• Robust Growth under Model Uncertainty: Ergodic robust rate maximization for continuous-time portfolio processes with uncertain drift and known invariant density/covariance leads to calculus-of-variations or PDE problems, e.g., minimizing Donsker–Varadhan rate functions or maximizing eigenvalues of differential operators, yielding explicit optimal feedback policies (Kardaras et al., 2018, Itkin et al., 2022).

Distributed and Learning-Based Policies

• Distributed GNN-based Resource Allocation: The sum ergodic spectral efficiency maximization problem in cell-free massive MIMO can be addressed via distributed GNN-based policies, trained centrally but executed locally, that map locally available channel statistics to transmit power allocations, achieving near-centralized performance without global instantaneous CSI (Tung et al., 2024).
• Model-Free Primal–Dual Learning: When the channel/system model is unavailable, ergodic policy optimization can be achieved using smoothed surrogate objectives, with gradients estimated from finite differences, permitting a fully model-free stochastic approximation (primal-dual updates on parameterized policies), under mild regularity (Kalogerias et al., 2019).

Channels with Memory

• Information Rate Bounds: For discrete-time channels with memory, ergodic information-rate maximization appears as the search for the tightest lower and upper bounds via optimization over auxiliary FSMCs, using EM-type algorithms for stationary ergodic channels (0711.4406).

3. Algorithmic Methodologies

Convexification and Alternating Optimization

• WMMSE Transformation: For nonconvex ergodic sum-rate objectives (especially in MISO BC with RSMA), a Rate-WMSE equivalence is exploited: each user’s rate is mapped to a minimization over appropriately chosen equalizers and weights of a weighted MSE cost, rendering joint precoder/equalizer optimization tractable via block-coordinate updates (Xia et al., 2022, Joudeh et al., 2016).
• Successive Convex Approximation (SCA): Nonconvex security or SINR constraints are linearized via first-order Taylor expansion, iteratively yielding convex subproblems (Xia et al., 2022).
• Alternating Optimization: Problems such as RIS-assisted MIMO beamforming (Li et al., 2022) are efficiently solved by alternately optimizing transmit covariance and RIS coefficients.

Stochastic Resource Allocation

• Waterfilling-type Solutions: Multi-level or dynamic water-filling arises under average/peak constraints for maximizing ergodic rate, commonly in MAC or OFDMA settings, with Lagrangian dual and KKT-driven updates for power and subcarrier/user assignment (Saki et al., 2017, Kang et al., 2014, Erpek et al., 2019).
• Percentile-based Objectives: The ergodic percentile beamforming problem is addressed via QFT or LFT transforms, facilitating block-convex updates. The solution targets cell-edge throughput, with convergence to stationary points by minorization-maximization (Khan et al., 2024).

Model-Free and Distributed Learning

• Zeroth-Order Oracle Methods: Natural-gradient or finite-difference surrogates for policy gradients enable model-free primal-dual optimization—provably reducing primal-dual gaps under appropriate smoothing and policy parameterization (Kalogerias et al., 2019).
• Distributed Graph Neural Networks: Message-passing GNNs, trained to respect sum-power and interference constraints, can infer transmit powers per AP/UE from only local statistics and partial summary information, greatly reducing computational and backhaul burdens while preserving ergodic-rate optimality (Tung et al., 2024).

Robust Stochastic Control

• PDE and Eigenfunction Characterization: The robust ergodic-rate problem for stochastic diffusions is reduced to solving a nonlinear eigenvalue PDE, with the optimal feedback derived via calculus of variations and Dirichlet form techniques (Kardaras et al., 2018, Itkin et al., 2022).

4. Constraints, Robustness, and Uncertainty Models

Ergodic-rate maximization is typically embedded within nontrivial constraint structures:

• Power, Interference, and Security: Average and per-block power constraints, probabilistic interference (e.g., to protect primary users or eavesdroppers), and secrecy requirements appear in both resource allocation and beamforming (Saki et al., 2017, Xia et al., 2022).
• Imperfect/Delayed CSIT: The availability and quality of CSI at the transmitter critically affect achievable ergodic rates and the structure of optimal policies. Rate-splitting strategies and robust designs are especially potent under partial, noisy, or delayed CSIT (Xia et al., 2022, Joudeh et al., 2016, Bae et al., 2012).
• Statistical CSI and Learning: In scenarios with only statistic-based CSI, as in large-scale MIMO or RIS-enhanced systems, ergodic policies are developed over channel averages, using tractable approximations and alternating optimization for performance guarantees (Li et al., 2022).

5. Representative Numerical and Theoretical Results

• RSMA and Robustness: RSMA-based secure beamforming consistently outperforms linear precoding and ZF-based schemes in ergodic weighted sum-rate, with advantages increasing in overloaded user regimes and high-SNR, and with superior robustness to CSIT errors (Xia et al., 2022, Joudeh et al., 2016).
• Distributed/Model-Free Schemes: Distributed GNNs for cell-free MIMO achieve within 1–3% of centralized sum-rate with drastic reductions in computation and communication load. Model-free primal–dual learners match “clairvoyant” or batch-optimized policies in ergodic sum-rate without gradient or channel model access (Kalogerias et al., 2019, Tung et al., 2024).
• Robust Portfolio Strategies: For diffusion market models with known occupation densities and covariation, the robust asymptotic growth rate equals the Donsker–Varadhan rate function, and the optimal strategy has log-gradient (functionally-generated) form, independent of welch-factor processes such as stochastic volatility (Kardaras et al., 2018, Itkin et al., 2022).
• Complexity: In block-coordinate or alternating optimization algorithms, each iteration typically reduces the objective, with overall computation dominated by QCQP or convex subproblem solvers (cubic or higher polynomial in system dimensions) (Xia et al., 2022, Li et al., 2022, Khan et al., 2024).

6. Connections and Extensions

• Information-Theoretic ERM: Ergodic information-rate maximization forms the basis of channel capacity characterization for stationary ergodic channels, with applications in bounding achievable rates via auxiliary FSMCs and EM-type algorithms (0711.4406).
• Non-Orthogonal/Interference Channels: Interference enforcing and regime switching based on instantaneous channel state can yield NOMA and throughput gains in ergodic sum-rate, outperforming conventional “treat-as-noise” allocations, with generalized waterfilling algorithms for KKT-based power optimization (Erpek et al., 2019).
• Fairness and Percentile Extremal Problems: Extensions target ergodic percentile-maximization, e.g., sum-least-qth-percentile (SLqP) objectives, for cell-edge throughput optimization, leveraging QFT/LFT algorithms and equivalence to WMSE minimization classes (Khan et al., 2024).

7. Concluding Synthesis

The ergodic-rate maximization problem provides a unified, rigorous framework for optimizing long-term average performance metrics in stochastic, uncertain, and often adversarial environments across information and communication systems, robust finance, and learning-based control. Methodological advances include WMMSE/SCA-based block optimization, smoothed zeroth-order and primal–dual gradient algorithms, eigenfunction/PDE solutions for robust stochastic control, and distributed or GNN-based policy learning. The ergodic assumption is critical, as it guarantees asymptotic stationarity of system performance, enabling the translation of complex time-varying decision processes into tractable optimization objects amenable to both analytical and algorithmic resolution.

Principal references: (Xia et al., 2022, Joudeh et al., 2016, Li et al., 2022, Kalogerias et al., 2019, Tung et al., 2024, Saki et al., 2017, Kang et al., 2014, Kardaras et al., 2018, Itkin et al., 2022, 0711.4406, Khan et al., 2024, Bae et al., 2012, Erpek et al., 2019).