Lyapunov-guided Offloading Optimization (LOO)

• The topic introduces a framework that transforms long-term queue constraints into tractable per-slot optimization using Lyapunov drift-plus-penalty techniques.
• It integrates stochastic queueing and instantaneous system states to derive deterministic surrogates or reinforcement learning-driven Markov games for resource allocation.
• The framework offers strong performance guarantees by balancing delay-cost trade-offs via parameter tuning, ensuring scalable and decentralized decision-making.

Lyapunov-guided Offloading Optimization (LOO) modules are a class of control frameworks applied to dynamic computation offloading and resource allocation in distributed edge/cloud environments. These modules leverage Lyapunov optimization to transform long-term queue stability and performance constraints—such as execution latency, energy, or throughput—into tractable per-slot or per-frame optimization problems. By integrating online queueing information and instantaneous system states, LOO modules convert dynamic and stochastic multi-agent resource management into either deterministic surrogates or reinforcement learning (RL)-driven Markov games. This guarantees strong queue stability and quantifiable bounds on long-run optimality gaps under system dynamics such as time-varying channel conditions, bursty task arrivals, and hardware heterogeneity.

1. Systemic Context and Problem Formulation

LOO modules are deployed in networked computing environments with tiered resources—typically edge servers, cloud servers, intermediate layers (such as UAVs or RSUs), and end devices (IoT, XR, vehicular nodes). The entities interact via digital queues representing either physical task backlogs or virtual constraints (e.g., energy, delay). Tasks are generated stochastically, commonly following Poisson processes, with uncertain sizes and resource requirements. Wireless transmission rates and server availabilities are time-varying, often affected by mobility and fading.

A canonical LOO-enabled problem seeks to minimize the long-run average cost $\bar{\Phi}$ (such as execution time or weighted energy-delay) subject to queue stability: $$\min\ \bar{\Phi} = \lim_{T\to\infty} \frac{1}{T} \sum_{t=1}^T \mathbb{E}[\Phi(t)]$$ subject to

$$\lim_{T\to\infty}\frac{1}{T}\sum_{t=1}^T E[Q_i(t)/t] = 0\ \forall i$$

Queue variables $Q_i(t)$ encompass backlogs for each network actor and may be physical or virtual (for, e.g., power/energy constraints).

2. Lyapunov Formalism: Drift-plus-Penalty Relaxation

The foundation of LOO is the Lyapunov drift-plus-penalty technique. The Lyapunov function is typically quadratic in the queue vector, e.g.,

$L(Q(t)) = \frac{1}{2}\sum_i [Q_i(t)]^2$

For each slot, the one-step conditional Lyapunov drift is

$$\Delta(Q(t)) = \mathbb{E}[L(Q(t+1)) - L(Q(t)) \mid Q(t)]$$

A penalty term (scaled by tuning parameter $V>0$) is added, leading to the drift-plus-penalty surrogate: $$\Delta(Q(t)) + V \cdot \mathbb{E}[\Phi(t) \mid Q(t)]$$ Derivation by squaring queue updates yields, after bounding, an upper envelope of the form

$$\Delta(Q(t)) + V\mathbb{E}[\Phi(t)|Q(t)] \leq D + \sum_i Q_i(t) \mathbb{E}[\cdots] + V\mathbb{E}[\Phi(t)|Q(t)]$$

where $D$ is a constant and the summand captures backlog effects against arrivals and service.

3. Per-Slot Deterministic Optimization and Markov Games

The minimization of the upper-bound surrogate each slot produces a concrete, smaller-scale optimization—often a mixed-integer nonconvex program. In the maritime scenario (You et al., 18 Jun 2025), the per-slot problem is

$$\min_{o,s,f^u,f^v}\ C(t) = V\Phi(t) - \cdots\ (\text{terms linear in queue backlogs and resource allocations})$$

subject to capacity and binary offloading constraints. This generally couples integer offloading variables (e.g. $o_{i,j}$, $s_{j,k}$) and continuous resource assignments ($f_{i,j}^u$, $f_{j,k}^v$). For heterogeneous, multi-role agents, this per-slot problem is often cast as a Markov game, with each agent (UAV, vessel, device) acting with local and shared state observations.

4. Solution Algorithms: Deep Reinforcement Learning Integration

LOO leverages this structure via distributed or centralized RL solvers for high-dimensional, nonconvex or combinatorially hard decision spaces. (You et al., 18 Jun 2025) proposes a Heterogeneous-Agent Soft Actor-Critic (HASAC) algorithm:

• Each slot, agent $i$ receives an observation, samples an action via its local policy $\pi_{\theta_i}(\cdot)$
• The joint action (offloading and resource scheduling) is executed, and the resulting global reward $r(t) = -C(t)$ and new state are observed.
• Transitions are stored; critics are updated by squared Bellman loss, and policy networks are sequentially optimized by soft-policy KL divergence.
• Critic targets and actors are soft-updated (Polyak averaging).

This enables distributed execution (each agent instantiates its policy locally post-training), computational scalability, and adaptation to dynamic conditions. Learning is bootstrapped on-the-fly using only instantaneous system state (queue lengths, channels), with no need for future or statistical knowledge of arrivals or fading.

5. Parameterization and Trade-off Tuning

The trade-off between queue backlog (hence delay) and cost optimality is controlled by the parameter $V$:

• $V \uparrow$ emphasizes short-term cost minimization (e.g., throughput, latency), allowing average queues to grow ($O(V)$ delay).
• $V \downarrow$ reduces queue backlog at the expense of increased long-term cost.

In empirical deployments, $V$ is tuned by incrementally increasing until average cost improvement saturates or queue lengths approach system constraints (You et al., 18 Jun 2025).

6. Theoretical Guarantees: Stability and Performance Gaps

Standard Lyapunov drift analysis (per Neely’s stochastic network optimization) yields the following guarantees:

• Strong stability of all queues (physical and virtual): $$\lim_{T\to\infty}(1/T)\sum_t \mathbb{E}[Q_i(t)] < \infty$$.
• Time-average performance gap: $\bar{\Phi} \leq \Phi_\mathrm{opt}^* + O(1/V)$, where $\Phi_\mathrm{opt}^*$ is the minimum cost achievable by any stabilizing policy.
• The average backlog scales $O(V)$, so the throughput/delay-cost trade-off is explicit and tunable.

These results hold even in the presence of stochastic arrivals, time-varying channels, and limited statistical knowledge.

7. Implementation, Scalability, and Extensions

LOO modules operate online, utilizing only current queue and channel states. Per-slot computational complexity is dominated by neural network forward and backward passes. In the maritime edge context, per-slot action inference and localized batch RL updates are feasible at timescales of Hz–kHz using lightweight DNNs and mini-batches ($\sim 10^3$ per update). After centralized training, policy execution is decentralized—each UAV or vessel runs only its own policy, enabling scalable operation in large, dynamic environments. In practical deployments, periodic retraining can be used to adapt to traffic shifts or hardware upgrades without interrupting service.

LOO frameworks generalize readily to other domains—core principles and guarantees extend to urban vehicular edge (Liu et al., 2024), collaborative edge systems (Yuan et al., 27 Aug 2025), peer-offloading in small-cell networks (Chen et al., 2017), LLM inference in edge-cloud federations (Wu et al., 28 Dec 2025), and hybrid mobile edge–quantum scenarios (Ye et al., 2023), among others.

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References:

(You et al., 18 Jun 2025, Yuan et al., 27 Aug 2025, Ye et al., 2023, Liu et al., 2024, Chen et al., 2017, Wu et al., 28 Dec 2025)