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arxiv: 2606.17489 · v1 · pith:UY5BNBQSnew · submitted 2026-06-16 · 💻 cs.LG · cs.AI

Online LLM Selection via Constrained Bandits with Time-Varying Demand

Yin Huang , Qingsong Liu , Jie Xu This is my paper

Pith reviewed 2026-06-27 01:51 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords constrained banditsLLM selectiononline learningtime-varying demandedge-cloud systemsregret boundsresource constraints
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The pith

A constrained stochastic bandit algorithm selects LLMs online with sublinear regret and sublinear constraint violations despite unknown distributions and time-varying demand.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper casts LLM selection for tasks with heterogeneous accuracy, latency, and cost as a constrained stochastic bandit problem that includes both hard packing constraints like monetary budgets and soft covering constraints like latency targets. It introduces an algorithm that combines confidence-bound estimates of unknown model performance with forecasts of changing task demand to make sequential choices from partial feedback only. The work proves that cumulative regret and covering violations remain sublinear relative to an offline benchmark possessing full information. This matters because static selection rules break down once workloads fluctuate and resource limits must be respected over long horizons.

Core claim

We formulate the LLM selection problem as a constrained stochastic bandit learning task with packing-type and covering-type constraints under time-varying task demand. We develop a novel online learning algorithm that leverages confidence-bound estimates and demand predictions to balance reward maximization with long-term constraint satisfaction. We provide theoretical guarantees showing sublinear regret and sublinear covering constraint violations compared to an offline benchmark with full information.

What carries the argument

The online algorithm that uses confidence-bound estimates together with demand predictions to enforce long-term satisfaction of both packing and covering constraints while maximizing cumulative reward in a stochastic bandit with partial feedback.

If this is right

  • The algorithm simultaneously respects hard resource budgets and soft service-level requirements over long sequences of tasks.
  • Performance guarantees hold without knowledge of the underlying reward, cost, or latency distributions.
  • Partial feedback from each selected model is sufficient for the confidence bounds to drive sublinear growth in violations.
  • Demand predictions allow the method to adapt selection policies as task workloads shift over time.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same structure could be tested on other sequential resource-allocation tasks that mix hard budgets with soft quality targets.
  • Accurate short-term demand forecasting becomes a practical lever for tightening the achieved regret and violation bounds.
  • The framework suggests examining whether similar confidence-bound techniques extend to settings with multiple simultaneous covering constraints.

Load-bearing premise

The approach depends on the availability of demand predictions to handle time-varying task demand.

What would settle it

An experiment that measures linear growth in either regret or covering violations when demand predictions are removed or when the algorithm is run on real heterogeneous LLM performance traces would falsify the sublinear guarantees.

Figures

Figures reproduced from arXiv: 2606.17489 by Jie Xu, Qingsong Liu, Yin Huang.

Figure 1
Figure 1. Figure 1: Online LLM selection under Budget Constraint [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Performance with i.i.d model. 0 2000 4000 6000 8000 10000 Time Step 0.2 0.4 0.6 0.8 1.0 |Qt Q| Estimation Error over Time (a) Estimation Error 0 2500 5000 7500 10000 Timeslot 0 10000 20000 Accumulated Regret COPAC-UCB SW-UCB AD-UCB PD-BwK (b) Cumulative Regret 0 2500 5000 7500 10000 Timeslot 2 0 2 Constraint Violation COPAC-UCB SW-UCB AD-UCB PD-BwK (c) Constraint Violation [PITH_FULL_IMAGE:figures/full_fi… view at source ↗
Figure 3
Figure 3. Figure 3: Performance with AR (1) model. constraint violations (Fig. 3c). While COPAC-UCB continues to outperform the baselines in both cumulative reward and SLA satisfaction, its performance advantage narrows in the AR setting, especially during the early rounds. This is ex￾pected: time-correlated demand requires longer adaptation before accurate prediction stabilizes, and the dual update mechanism becomes more con… view at source ↗
read the original abstract

Large Language Models (LLMs) are increasingly deployed in edge-cloud inference systems to handle diverse user tasks with heterogeneous accuracy, latency, and cost profiles. Selecting the appropriate LLM for each incoming task is critical for ensuring service quality and efficient resource utilization. However, model heterogeneity, stochastic and unknown performance characteristics, and time-varying task demands make static selection strategies inadequate. Real-world deployments often impose hard resource budgets such as monetary expenditure limits, along with soft service-level requirements such as latency guarantees. These constraints introduce additional challenges for online decision-making. We formulate this problem as a constrained stochastic bandit learning task, where the learner sequentially selects models under both packing-type (hard) and covering-type (soft) constraints, while adapting to time-varying task demand. The learner operates without access to the underlying reward, cost, or latency distributions and must rely on partial feedback. We develop a novel online learning algorithm that leverages confidence-bound estimates and demand predictions to balance reward maximization with long-term constraint satisfaction. We provide theoretical guarantees showing sublinear regret and sublinear covering constraint violations compared to an offline benchmark with full information. Experimental results on synthetic workloads demonstrate the effectiveness and robustness of our approach in dynamic, resource-constrained environments.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript formulates LLM selection for edge-cloud inference as a constrained stochastic bandit problem with packing (hard resource budgets) and covering (soft latency) constraints under time-varying task demand. The learner has no access to reward/cost distributions and uses partial feedback. The proposed algorithm combines confidence-bound estimates with demand predictions to achieve sublinear regret and sublinear covering-constraint violations relative to an offline full-information benchmark; synthetic experiments are provided to illustrate performance.

Significance. If the analysis supplies a precise, quantitative assumption on demand-prediction error that justifies the comparison to the true-demand offline optimum, the result would supply a concrete algorithmic template for constrained online allocation in heterogeneous LLM systems, extending classical constrained bandits to time-varying demand with both hard and soft constraints.

major comments (2)
  1. [Abstract / §1] Abstract and §1: the claimed sublinear regret and covering-violation bounds are stated relative to an offline benchmark that knows the true demand process, yet no assumption quantifying the cumulative or per-step error of the supplied demand predictions appears. Without such a condition (e.g., cumulative prediction error o(T) or decaying as O(√T)), the stated comparison is not internally justified and the bounds hold only conditionally on unstated prediction quality.
  2. [§3 / §4] §3 (algorithm description) and §4 (analysis): the regret and violation proofs must explicitly incorporate the prediction-error term; if the analysis treats predictions as exact or absorbs their error without an explicit bound, the sublinear claims relative to the true-demand benchmark do not follow.
minor comments (2)
  1. [§2] Notation for the covering constraint (latency) and its violation measure should be introduced with an explicit mathematical definition before the algorithm statement.
  2. [§5] The experimental section would benefit from an ablation that varies the quality of the demand predictor to illustrate sensitivity to the (unstated) prediction-error assumption.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive feedback. The comments correctly identify that the current manuscript does not explicitly quantify the demand-prediction error when comparing to the true-demand offline benchmark. We will revise the paper to address both points.

read point-by-point responses

  1. Referee: [Abstract / §1] Abstract and §1: the claimed sublinear regret and covering-violation bounds are stated relative to an offline benchmark that knows the true demand process, yet no assumption quantifying the cumulative or per-step error of the supplied demand predictions appears. Without such a condition (e.g., cumulative prediction error o(T) or decaying as O(√T)), the stated comparison is not internally justified and the bounds hold only conditionally on unstated prediction quality.

    Authors: We agree that the manuscript as written does not state a quantitative assumption on prediction error, which is required to justify the comparison to the true-demand benchmark. In the revision we will add an explicit assumption (e.g., cumulative prediction error o(T)) in §1 and the abstract, and state that all bounds are conditional on this assumption. revision: yes

  2. Referee: [§3 / §4] §3 (algorithm description) and §4 (analysis): the regret and violation proofs must explicitly incorporate the prediction-error term; if the analysis treats predictions as exact or absorbs their error without an explicit bound, the sublinear claims relative to the true-demand benchmark do not follow.

    Authors: We will revise the analysis in §4 (and the algorithm description in §3 where relevant) to explicitly carry the prediction-error term through the regret and violation decompositions. The final bounds will be expressed in terms of both the usual bandit terms and the cumulative prediction error, thereby making the sublinear claims conditional on the assumption introduced above. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on external offline benchmark and standard analysis

full rationale

The derivation compares an online algorithm (using confidence bounds and demand predictions) against an offline full-information benchmark. No quoted step reduces a claimed prediction or guarantee to a fitted parameter or self-citation by construction. The central regret and violation bounds are defined relative to an independent external benchmark rather than internally forced quantities. This is the normal case of a self-contained analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, invented entities, or paper-specific axioms; standard stochastic bandit assumptions are implicitly invoked but not detailed.

axioms (1)
  • standard math Standard stochastic bandit assumptions such as bounded rewards and sub-Gaussian noise for confidence bounds.
    Implicit in the use of confidence-bound estimates to obtain sublinear regret.

pith-pipeline@v0.9.1-grok · 5741 in / 1263 out tokens · 46027 ms · 2026-06-27T01:51:13.017895+00:00 · methodology

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