Large language model (LLM) routing platforms provide users with access to multiple LLMs that differ in cost and quality. We study the problem of learning contracts in such settings. A principal repeatedly offers a scalar linear contract to an agent, who selects one of several candidate LLMs to answer the query. The qualities of the LLMs are initially unknown, while their costs are known. Under mild assumptions, an optimal contract exists in the classical setting when the LLM qualities are known.

Since the qualities are unknown in our setting and are observed by the user only through the realized responses, we formulate this interaction as an online principal--agent bandit problem. We propose the Three-Phase Successive Contract Elimination (TSCE) algorithm, which induces LLMs through contracts.

We provide finite-sample query-complexity guarantees for TSCE to identify an ε-optimal contract. We further demonstrate empirically that the proposed method reliably identifies near-optimal contracts across a wide range of problem instances.

Fairness-aware federated learning seeks to reduce disparities across sensitive groups while preserving the decentralized nature of training. However, estimating global fairness often requires clients to communicate statistics derived from sensitive attributes and model predictions, introducing an additional privacy concern.

We propose PriFedRényi, a differentially private framework for federated fairness that protects this fairness-statistics communication channel. Each client privatizes a normalized joint histogram of predictions and sensitive attributes, from which all shared fairness statistics are derived. We establish record-level privacy guarantees under repeated communication and analyze how privacy perturbations propagate to the global Rényi matrix and its second singular direction used for fairness regularization

Experiments on Adult and Drug show that PriFedRényi maintains accuracy and group-fairness performance close to the nonprivate baseline across a broad range of privacy budgets. The results further show that the practical effect of privacy depends not only on the noise magnitude, but also on local sample size and the stability of the global fairness direction.