Online Linear Programming for Multi-Objective Routing in LLM Serving

Large language models (LLMs) achieve impressive performance across multiple domains, but using the most capable model for every query is prohibitive at scale. LLM routing exploits diversity in model capability and cost by assigning each query to a suitable model to balance utility and budget. Current methods have two limitations: (i) they either use heuristics that do not always enforce the budget constraint or impose a fixed per-query budget that cannot adapt across the workload and leads to suboptimal performance; (ii) they require supervised learning on a dense dataset with statistics for every query-model pair, which is expensive to collect. To address these challenges, we formulate LLM routing as a constrained contextual multi-armed bandit problem and introduce WISERouter (WR for short), a framework that supports offline learning from historical interactions as well as online learning with exploration. We further prove that WR-Online achieves a sublinear regret bound of $O(\sqrt{T})$ over a time horizon $T$. Empirical results on RouterBench and SWE-Bench demonstrate that (i) WR-Offline surpasses existing baselines in performance under a fixed budget and adheres more closely to budget constraints, and (ii) WR-Online achieves comparable performance to the baselines, while using substantially less exploration data.

Inference costs for large language model (LLM) applications are rapidly growing, driven by surging demand and rising infrastructure cost. Users expect high-quality responses, and in commercial settings this is formally codified in Service Level Agreements (SLAs), creating a fundamental tension between cost and quality. Recent progress on cost-aware LLM request routing has shown potential to resolve this tension, but existing approaches rely on complete feedback signals, offline training, extensive per-workload tuning, and most lack SLA guarantees or inference-time adaptivity. We introduce SLARouter, an online routing algorithm that learns a cost-optimal policy from the sparse, one-sided user feedback available in production systems. SLARouter provides theoretical guarantees for both cost optimality and strict SLA compliance. Experiments across a wide range of LLM benchmarks show that SLARouter satisfies SLA constraints without the need for per-benchmark tuning, reducing operating cost by up to 2.2x over existing baselines.

A multi-model language service must route each request while preserving workload-level budgets for compute, latency, memory, or monetary cost. Two features make this problem materially harder than static model selection. Prompt representations are high dimensional, so only a small subset of embedding directions may predict the incremental value of a model, and both the request mix and the model frontier drift after launches, fine-tunes, quantization changes, and system updates. We formulate nonstationary sparse contextual routing with multiple knapsack constraints and an optional shadow-audit stream that evaluates a small fraction of prompts on several models. We propose Drift-Aware Sparse Routing (DRS). The policy estimates reward and resource use from a rolling audit window, routes using pessimistic reward and optimistic cost estimates, updates resource shadow prices online, and applies a hard meter before commitment. The analysis separates control from statistics. On any event with uniform prediction radii $\{β_t\}$, regret against a paced dynamic fluid benchmark is bounded by the sum of the radii, a capacity-buffer term, and an $O(\sqrt{T})$ pacing term. Under a sparse linear model and bounded drift $V_T$, rolling estimation gives $$\widetilde O\left( T\sqrt{\frac{s}{ρW}}+WV_T+\sqrt{T} \right),$$ where $s$ is sparsity, $ρ$ is the audit rate, and $W$ is the window length. Optimizing $W$ yields the usual stationary $O(\sqrt{sT/ρ})$ rate when $V_T=0$ and a $O(T^{2/3}(s/ρ)^{1/3}V_T^{1/3})$ adaptation term under drift.