Large language model (LLM) routers improve the efficiency of multi-model systems by directing each query to the most appropriate model while leveraging the diverse strengths of heterogeneous LLMs. Most existing approaches frame routing as a classification problem based solely on the input query. While this reduces overhead by avoiding inference across all models, it overlooks valuable information that could be gleaned from potential outputs and fails to capture implicit intent or contextual nuances that often emerge only during response generation. These limitations can result in suboptimal routing decisions, particularly for complex or ambiguous queries that require deeper semantic understanding. To address this challenge, we propose Lookahead, a routing framework that "foresees" potential model outputs by predicting their latent representations and uses these predictions to guide model selection, thus enabling more informed routing without full inference. Within this framework, we implement two approaches based on causal and masked language models. Empirical evaluations across seven public benchmarks--spanning instruction following, mathematical reasoning, and code generation--show that Lookahead consistently outperforms existing routing baselines, achieving an average performance gain of 7.7% over the state-of-the-art.

Recent advances in computationally efficient non-myopic Bayesian optimization offer improved query efficiency over traditional myopic methods like expected improvement, with only a modest increase in computational cost. These advances have been largely limited to unconstrained BO methods with only a few exceptions which require heavy computation. For instance, one existing multi-step lookahead constrained BO method (Lam & Willcox, 2017) relies on computationally expensive unreliable brute force derivative-free optimization of a Monte Carlo rollout acquisition function. Methods that use the reparameterization trick for more efficient derivative-based optimization of non-myopic acquisition functions in the unconstrained setting, like sample average approximation and infinitesimal perturbation analysis, do not extend: constraints introduce discontinuities in the sampled acquisition function surface. Moreover, we argue here that being non-myopic is even more important in constrained problems because fear of violating constraints pushes myopic methods away from sampling the boundary between feasible and infeasible regions, slowing the discovery of optimal solutions with tight constraints. In this paper, we propose a computationally efficient two-step lookahead constrained Bayesian optimization acquisition function (2-OPT-C) supporting both sequential and batch settings. To enable fast acquisition function optimization, we develop a novel likelihood ratio-based unbiased estimator of the gradient of the two-step optimal acquisition function that does not use the reparameterization trick. In numerical experiments, 2-OPT-C typically improves query efficiency by 2x or more over previous methods, and in some cases by 10x or more.

In reinforcement learning (RL), agents sequentially interact with changing environments while aiming to maximize the obtained rewards.

By replacing the inner optimizer in RAPP we rediscover the family of Lookahead algorithms for which we establish convergence in cohypomonotone problems even when the base optimizer is taken to be gradient descent ascent.

In reinforcement learning (RL), agents sequentially interact with a changing environment, aiming to collect as much reward as possible.

Neural Information Processing Systems http://nips.cc/