PROTEUS: Lagrangian RL Optimizes Multi-LLM Routing for SLAs

As organizations deploy multiple large language models to handle diverse workloads, a critical challenge emerges: how do you intelligently route requests to maximize efficiency while guaranteeing service level agreements? New research introduces PROTEUS, a sophisticated system that applies Lagrangian reinforcement learning to solve this multi-objective optimization problem.

The Multi-LLM Routing Challenge

Modern AI deployments increasingly rely on multiple LLMs with different capabilities, costs, and latency profiles. A high-capacity model like GPT-4 might handle complex reasoning tasks, while a faster, cheaper model processes simpler queries. The routing decision—which model handles which request—directly impacts both operational costs and user experience.

Traditional approaches to this problem often treat it as a simple load balancing exercise or use static routing rules. However, these methods fail to capture the dynamic nature of real-world workloads where request patterns shift, model availability fluctuates, and SLA requirements vary across different query types.

PROTEUS addresses this gap by formulating multi-LLM routing as a constrained Markov Decision Process (CMDP), where the objective is to minimize serving costs while maintaining SLA compliance as hard constraints rather than soft objectives to be traded off.

Lagrangian Reinforcement Learning Architecture

The core innovation in PROTEUS lies in its application of Lagrangian relaxation to the constrained RL problem. Rather than manually tuning weights between competing objectives (cost minimization vs. latency guarantees), the system learns optimal Lagrange multipliers that automatically balance these concerns.

The mathematical framework transforms the constrained optimization problem into an unconstrained one through the introduction of dual variables. For each SLA constraint—whether latency percentiles, throughput guarantees, or error rate limits—the system maintains a corresponding Lagrange multiplier that adaptively tightens or relaxes based on current constraint satisfaction levels.

This approach offers several technical advantages over alternative methods:

Adaptive Constraint Handling

When SLA violations occur, the corresponding Lagrange multipliers increase automatically, causing the routing policy to prioritize constraint satisfaction over cost savings. Conversely, when constraints are comfortably met, the system can afford to route more aggressively toward cheaper options.

Principled Multi-Objective Optimization

Unlike weighted-sum approaches that require careful manual tuning, Lagrangian methods provide theoretical guarantees about constraint satisfaction at optimality. The dual variables effectively learn the "price" of each constraint, enabling economically interpretable routing decisions.

System Design and Implementation

PROTEUS operates as a middleware layer between incoming request streams and the underlying LLM serving infrastructure. The routing policy network observes several input features:

• Request characteristics: Input length, estimated complexity, required capabilities
• System state: Current queue depths, recent latencies, model availability
• SLA context: Applicable constraints for the request class, current violation margins

The policy network outputs routing probabilities across available LLM backends, with the Lagrangian framework ensuring that exploration during training respects constraint boundaries.

Training employs a two-timescale update scheme common in constrained RL: the policy parameters update on a faster timescale to maximize the Lagrangian objective, while the dual variables update more slowly to track constraint satisfaction levels. This separation prevents oscillatory behavior that can occur when both components adapt at similar rates.

Implications for AI Infrastructure

The PROTEUS architecture has significant implications for the broader AI serving ecosystem. As organizations deploy increasingly heterogeneous model portfolios—mixing open-source models, API-based services, and fine-tuned variants—intelligent routing becomes essential for cost management.

For video generation and synthetic media applications, where inference costs can be substantial and latency requirements vary dramatically between real-time and batch processing, such routing frameworks could enable more efficient resource utilization. A video generation platform might route simple editing tasks to lightweight models while reserving expensive diffusion models for complex creative requests.

The SLA-aware aspect is particularly relevant for enterprise deployments where contractual guarantees around response times and availability translate directly to business outcomes. PROTEUS demonstrates how modern RL techniques can provide these guarantees without sacrificing the flexibility needed to optimize costs.

Technical Considerations

Several practical considerations affect PROTEUS deployment. The system requires accurate latency prediction for candidate models, which itself can be challenging given the variable nature of LLM inference times. The research addresses this through learned latency estimators trained alongside the routing policy.

Additionally, the Lagrangian approach assumes constraints are satisfiable—if SLA targets are set unrealistically given available model capabilities, the dual variables can grow unboundedly. Production deployments would need monitoring to detect such misconfiguration scenarios.

The work contributes to the growing body of research on LLM systems optimization, joining efforts around speculative decoding, KV cache management, and inference scheduling. As foundation models become infrastructure, such systems-level innovations become as important as model architecture advances.

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