We continue to study the learning-theoretic foundations of generation by extending the results from Kleinberg and Mullainathan [2024] and Li et al. [2024] to account for noisy example streams. In the noiseless setting of Kleinberg and Mullainathan [2024] and Li et al. [2024], an adversary picks a hypothesis from a binary hypothesis class and provides a generator with a sequence of its positive examples. The goal of the generator is to eventually output new, unseen positive examples. In the noisy setting, an adversary still picks a hypothesis and a sequence of its positive examples. But, before presenting the stream to the generator, the adversary inserts a finite number of negative examples. Unaware of which examples are noisy, the goal of the generator is to still eventually output new, unseen positive examples. In this paper, we provide necessary and sufficient conditions for when a binary hypothesis class can be noisily generatable. We provide such conditions with respect to various constraints on the number of distinct examples that need to be seen before perfect generation of positive examples. Interestingly, for finite and countable classes we show that generatability is largely unaffected by the presence of a finite number of noisy examples.

Over the past 50 years, predictive machine learning has been a cornerstone for both theorists and practitioners. Predictive tasks like classification and regression have been extensively studied, in both theory and practice, due to their applications to face recognition, autonomous vehicles, fraud detection, recommendation systems, etc. Recently, however, a new paradigm of machine learning has emerged: generation. Unlike predictive models, which focus on making accurate predictions of the true label given examples, generative models aim to create new examples based on observed data. For example, in language modeling, the goal might be to generate coherent text in response to a prompt, while in drug development, one might want to generate candidate molecules. In fact, generative models have already been applied to these tasks and others [Zhao et al., 2023, Jumper et al., 2021]. The vast potential of generative machine learning has spurred a surge of research across diverse fields like natural language processing [Wolf et al., 2020], computer vision [Khan et al., 2022], and computational chemistry/biology [Vanhaelen et al., 2020]. Despite this widespread adoption, the theoretical foundations of generative machine learning lags far behind its predictive counterpart. While prediction has been extensively studied by learning theorists through frameworks like PAC and online learning [Shalev-Shwartz and Ben-David, 2014, Mohri et al., 2012, Cesa-Bianchi and Lugosi, 2006], generative machine learning has, for the most, part

The Minimum Description Length (MDL) principle selects the model that has the shortest code for data plus model. We show that for a countable class of models, MDL predictions are close to the true distribution in a strong sense. The result is completely general. No independence, ergodicity, stationarity, identifiability, or other assumption on the model class need to be made. More formally, we show that for any countable class of models, the distributions selected by MDL (or MAP) asymptotically predict (merge with) the true measure in the class in total variation distance.

The Minimum Description Length (MDL) principle selects the model that has the shortest code for data plus model. We show that for a countable class of models, MDL predictions are close to the true distribution in a strong sense. The result is completely general. No independence, ergodicity, stationarity, identifiability, or other assumption on the model class need to be made. More formally, we show that for any countable class of models, the distributions selected by MDL (or MAP) asymptotically predict (merge with) the true measure in the class in total variation distance.

The Minimum Description Length (MDL) principle selects the model that has the shortest code for data plus model. We show that for a countable class of models, MDL predictions are close to the true distribution in a strong sense. The result is completely general. No independence, ergodicity, stationarity, identifiability, or other assumption on the model class need to be made. More formally, we show that for any countable class of models, the distributions selected by MDL (or MAP) asymptotically predict (merge with) the true measure in the class in total variation distance.

The Minimum Description Length (MDL) principle selects the model that has the shortest code for data plus model. We show that for a countable class of models, MDL predictions are close to the true distribution in a strong sense. The result is completely general. No independence, ergodicity, stationarity, identifiability, or other assumption on the model class need to be made. More formally, we show that for any countable class of models, the distributions selected by MDL (or MAP) asymptotically predict (merge with) the true measure in the class in total variation distance. Implications for non-i.i.d. domains like time-series forecasting, discriminative learning, and reinforcement learning are discussed.