Rate-Constrained Optimization Scheme

• Rate-Constrained Optimization Scheme is an optimization framework that enforces explicit limits on variable rates, such as information transfer or state derivatives.
• It utilizes methods like Lagrange duality, alternating minimization, and implicit reformulations to tackle nonlinear, nonconvex, and high-dimensional problems.
• This scheme is applied in wireless communications, federated learning, and control systems to meet practical limits like bandwidth, latency, and fairness constraints.

A rate-constrained optimization scheme is an optimization framework in which the objective function is optimized subject to explicit constraints on the rates of certain variables, such as information transfer, transmission rates, state derivatives, or prediction rates. This class of problems arises in a range of fields including information theory, optimal control, wireless communications, federated learning, machine learning with fairness constraints, and decentralized optimization. Rate constraints enforce practical system limits—such as bandwidth, latency, privacy, fidelity, or group parity—while guiding the solution towards a feasible and often globally or locally optimal point. Recent advancements have further classified and unified these schemes, enabling efficient computational techniques across highly nonlinear, nonconvex, and high-dimensional settings.

1. Formal Definitions and Problem Classes

Rate-constrained optimization spans diverse mathematical formulations, unified by the presence of a constraint or penalty on the "rate" of a process, variable, or decision. Prototypical forms include:

• Continuous-time constrained optimization: Rate constraints bound the time-derivative of state or control variables in nonlinear optimal control:

$$\begin{aligned} &\min_{x(\cdot),\,u(\cdot)} \;\Phi(x(t_0), x(t_f)) + \int_{t_0}^{t_f} L(x(t), u(t), t)\,dt \ &\text{subject to}\quad \dot{x}(t) = f(x, u, t), \;\;\dot{u}_L \leq \frac{d u}{dt}(t) \leq \dot{u}_U \end{aligned}$$

(Nie et al., 2019)

• Information-theoretic rate-distortion problems: One seeks minimal communication rate subject to fidelity constraints:

$\min_{Q} R(Q) \quad \text{s.t.}\quad D(Q) \leq D_0$

where $R(Q)$ is a coding or transmission rate and $D(Q)$ is distortion (Hamidi et al., 2024, Pilgrim et al., 2017, Yuan et al., 2023).

• Empirical risk with group rate constraints: Minimize a classification loss under constraints on groupwise decision rates:

$$\min_{w} \ell(w) \quad \text{s.t.}\quad r_j(w) \leq c_j \;\;\forall j$$

for rates $r_j$ describing e.g. demographic parity or equality of odds (Yaghini et al., 28 May 2025).

• Bandit or contextual decision-making under rate-limited channels: Optimize cumulative reward (or minimize regret) subject to a fixed communication rate, sometimes mapped to a rate-distortion-type constraint between the desired and implementable policies (Pase et al., 2022, Saxena et al., 2019).
• Machine learning with non-decomposable metric constraints: Optimize a primary metric (e.g., precision) at a fixed secondary rate (e.g., recall or FPR) (Kumar et al., 2021).

2. Key Methodologies for Rate-Constrained Optimization

2.1. Lagrange Duality and Augmented Lagrangian Methods

The dominant approach is to form the Lagrangian of the rate-constrained problem, introducing multipliers for each constraint:

$$\mathcal{L}(\theta, \lambda) = f(\theta) + \sum_j \lambda_j (r_j(\theta) - c_j)$$

Optimization proceeds via (stochastic) primal-dual methods, e.g., stochastic gradient descent–ascent (SGDA), mirror descent, or alternating minimization (Yaghini et al., 28 May 2025, Rozendaal et al., 2020, Yuan et al., 2023). For machine learning settings, the dual variable update is designed to adaptively enforce the rate constraint, often under stochastic or minibatch settings with privacy-enhancing noise (Yaghini et al., 28 May 2025).

2.2. Alternating Minimization and Lloyd-type Algorithms

In quantization, distributed consensus, or rate-distortion settings, alternating minimization is applied: alternate between optimizing the assignment variables and updating the quantization or policy parameters (Hamidi et al., 2024, Pilgrim et al., 2017, Yuan et al., 2023, Pase et al., 2022). The Lloyd algorithm is adapted by replacing per-cell costs with augmented cost-plus-rate penalties, altering the boundary and centroid update rules to enforce the rate constraint directly.

2.3. Implicit Function and Unconstrained Reformulations

For constraints that couple parameters via a rate-type metric, the constraint can often be solved implicitly for a parameter (e.g., threshold as a function of weights), collapsed into an unconstrained problem via the implicit function theorem, and solved by chain-rule gradient methods (Kumar et al., 2021).

2.4. On-Mesh Discretized Constraints

In control, rate constraints are discretized directly onto the collocation mesh using linear finite-difference approximations, leading to linear constraints in the nonlinear program and eliminating the issue of singular arcs present in classical approaches (Nie et al., 2019). This improves computational efficiency and robustness.

3. Algorithmic Realizations and Computational Strategies

Domain/Task	Constraint Type	Main Algorithmic Idea
Nonlinear control	$\dot{x},\dot{u}$ bounds	On-mesh finite-difference constraints
Federated/consensus	Bit-rate, distortion	Generalized geometric/Lloyd-type programming
Fair ML	Group prediction rates	Primal-dual SGDA w/ private histogram
Bandits/online learning	Regret s.t. success rate	Constrained Thompson Sampling, LP projection
Deep compression	Expected distortion	Dual ascent, normalized Lagrange penalty
Non-decomposable metrics	FPR/Recall constraints	Implicit threshold update via chain rule

Direct convexification, deflation of large alphabets, and incorporation of side information or predictive coding are key for tractability in high-complexity settings (Yuan et al., 2023, Pilgrim et al., 2017).

4. Theoretical Properties and Convergence Guarantees

• Convexity and Duality: For convex primal and linear/convex rate constraints, Lagrangian dual methods achieve global optimality; for nonconvex problems (e.g., neural compression), dual ascent still yields constraint satisfaction and local minimum under mild regularity (Rozendaal et al., 2020, Yaghini et al., 28 May 2025).
• Convergence Rates: In SGDA or alternating minimization, rates of $O(1/\sqrt{T})$ in the min-max gap can be achieved, with explicit finite-time convergence bounds accounting for privacy-induced noise (Yaghini et al., 28 May 2025, Yuan et al., 2023).
• Elimination of Singular Arcs: On-mesh rate constraints preclude singular arcs in discretized OCPs, restoring stability absent in auxiliary-control discretizations (Nie et al., 2019).
• Information-theoretic Limits: Fundamental rate limits—e.g., minimum $H(A^*)$ rate for zero asymptotic regret in contextual bandits—are achievable using practical quantization codes (Pase et al., 2022). For consensus or federated quantization, the exponential distortion-rate law $D \sim 2^{-2R}$ holds in high-rate regimes (Hamidi et al., 2024).

5. Applications in Modern Communication, Learning, and Control Systems

• Federated Learning: Communication-efficient FL is achieved by quantization schemes that minimize distortion under strict rate caps, yielding continuous rate-distortion tradeoffs and strong experiment-backed test accuracy gains (Hamidi et al., 2024).
• Fairness and Privacy in Machine Learning: Group fairness constraints (e.g., demographic parity) are enforced via rate-constrained empirical risk minimization, with differential privacy realized through histogram mechanisms and private noise addition (Yaghini et al., 28 May 2025).
• Wireless Resource Allocation: Optimal rate selection under latency (success rate) constraints is realized by Constrained Thompson Sampling and LP-based rounding (Saxena et al., 2019). mMTC network rate allocation leverages blocklength-sensitive throughput formulations and sequential convex relaxation (Liesegang et al., 2024).
• Networked Control and Decentralized Optimization: DLMD with differential exchanges under rate and channel-noise constraints enables robust consensus in distributed convex optimization, tuned by power and consensus rates (Saha et al., 2020).
• Non-decomposable Metric Optimization in ML: ICO allows direct optimization of e.g. FNR at prescribed FPR using implicit constraint elimination and gradient computation, outperforming competing Lagrange or surrogate methods (Kumar et al., 2021).

6. Performance Benchmarks and Practical Outcomes

Algorithms designed using rate-constrained optimization principles have demonstrated substantial improvements in scalability, accuracy, and practical feasibility:

• Federated quantization: Up to 50–60% communication savings at equal accuracy over baseline quantizers (Hamidi et al., 2024).
• Fair DP learning: Pareto improvements in both error and fairness disparity, with 10³× runtime gains over prior DP-fairness optimizers (Yaghini et al., 28 May 2025).
• Consensus coding: State-evolution driven rate allocation achieves near-optimal aggregate communication cost for a specified MSE in finite time, significantly outperforming equal-rate or fixed schemes (Pilgrim et al., 2017).
• Wireless networks: Rate-optimized schemes tune energy transfer and blocklength for reliability-delay-constrained communication, attaining near-ideal rates given sufficient channel diversity (López et al., 2019).
• Decentralized learning: Robustness to quantizer and channel noise is achieved via careful “difference-based” quantization and averaging protocols (Saha et al., 2020).

7. Outlook and Extensions

Open directions include extending rate-constrained optimization to multi-resource multi-constraint environments, improved algorithms for rates coupled across agents or time, integration with reinforcement learning and online convex optimization, and further synthesis between information-theoretic and learning-theoretic constraint enforcement. Existing work establishes rate-constrained optimization as both a foundational theoretical paradigm and a practical algorithmic toolkit across the communication, control, and machine learning domains (Yaghini et al., 28 May 2025, Hamidi et al., 2024, Yuan et al., 2023, Pase et al., 2022, Nie et al., 2019, Rozendaal et al., 2020, Kumar et al., 2021).