A Theoretical Framework for Statistical Evaluability of Generative Models

Statistical evaluation aims to estimate the generalization performance of a model using held-out i.i.d. test data sampled from the ground-truth distribution. In supervised learning settings such as classification, performance metrics such as error rate are well-defined, and test error reliably approximates population error given sufficiently large datasets. In contrast, evaluation is more challenging for generative models due to their open-ended nature: it is unclear which metrics are appropriate and whether such metrics can be reliably evaluated from finite samples. In this work, we introduce a theoretical framework for evaluating language models and establish evaluability results for commonly used metrics. We study two categories of metrics: test-based metrics, including integral probability metrics (IPMs), and similarity-based metrics, including Rényi and KL divergences. We show that IPMs with respect to any bounded test class can be evaluated from finite samples up to multiplicative and additive approximation errors. Moreover, when the test class has finite fat-shattering dimension, IPMs can be evaluated with arbitrary precision. In contrast, similarity-based metrics, including Rényi and KL divergences, are not evaluable from finite samples, as their values can be critically determined by rare events. We also analyze the potential and limitations of perplexity as an evaluation method.

Consider a model trained to generate images or short musical tunes from a dataset. When it produces new samples, are these genuinely novel, or merely artifacts of the training data? Given only sample access to the true distribution, can we reliably determine whether the generator generalizes well? Can indistinguishability between generated and real samples be tested from finite data? In practice, such questions are addressed via evaluation metrics that quantify the discrepancy between the generated and target distributions. However, not every natural metric can be evaluated from finite samples. Our paper introduces a general theoretical framework for determining when such evaluation is possible. In particular, we show that for a broad class of test-based metrics, called integral probability metrics, evaluability is governed by the statistical complexity of the class of tests. We show that likelihood-based measures such as KL and Rényi divergence can fail to be evaluable because they may depend on rare events that are unlikely to appear in any finite test set. Finally, we use our framework to discuss the limitations of perplexity, a widely used score for evaluating models.