Adaptive Risk Control (ARC) is an online calibration strategy based on set prediction that offers worst-case deterministic long-term risk control, as well as statistical marginal coverage guarantees. ARC adjusts the size of the prediction set by varying a single scalar threshold based on feedback from past decisions. In this work, we introduce Localized Adaptive Risk Control (L-ARC), an online calibration scheme that targets statistical localized risk guarantees ranging from conditional risk to marginal risk, while preserving the worst-case performance of ARC.

Despite the impressive capabilities of large language models (LLMs) across diverse applications, they still suffer from trustworthiness issues, such as hallucinations and misalignments. Retrieval-augmented language models (RAG) have been proposed to enhance the credibility of generations by grounding external knowledge, but the theoretical understandings of their generation risks remains unexplored. In this paper, we answer: 1) whether RAG can indeed lead to low generation risks, 2) how to provide provable guarantees on the generation risks of RAG and vanilla LLMs, and 3) what sufficient conditions enable RAG models to reduce generation risks. We propose C-RAG, the first framework to certify generation risks for RAG models. Specifically, we provide conformal risk analysis for RAG models and certify an upper confidence bound of generation risks, which we refer to as conformal generation risk. We also provide theoretical guarantees on conformal generation risks for general bounded risk functions under test distribution shifts. We prove that RAG achieves a lower conformal generation risk than that of a single LLM when the quality of the retrieval model and transformer is non-trivial. Our intensive empirical results demonstrate the soundness and tightness of our conformal generation risk guarantees across four widely-used NLP datasets on four state-of-the-art retrieval models.

Prediction models are often employed in estimating parameters of optimization models. Despite the fact that in an end-to-end view, the real goal is to achieve good optimization performance, the prediction performance is measured on its own. While it is usually believed that good prediction performance in estimating the parameters will result in good subsequent optimization performance, formal theoretical guarantees on this are notably lacking. In this paper, we explore conditions that allow us to explicitly describe how the prediction performance governs the optimization performance. Our weaker condition allows for an asymptotic convergence result, while our stronger condition allows for exact quantification of the optimization performance in terms of the prediction performance. In general, verification of these conditions is a non-trivial task. Nevertheless, we show that our weaker condition is equivalent to the well-known Fisher consistency concept from the learning theory literature. This then allows us to easily check our weaker condition for several loss functions. We also establish that the squared error loss function satisfies our stronger condition. Consequently, we derive the exact theoretical relationship between prediction performance measured with the squared loss, as well as a class of symmetric loss functions, and the subsequent optimization performance. In a computational study on portfolio optimization, fractional knapsack and multiclass classification problems, we compare the optimization performance of using of several prediction loss functions (some that are Fisher consistent and some that are not) and demonstrate that lack of consistency of the loss function can indeed have a detrimental effect on performance.