Learning discrete distributions from i.i.d. samples is a well-understood problem. However, advances in generative machine learning prompt an interesting new, non-i.i.d. setting: after receiving a certain number of samples, an estimated distribution is fixed, and samples from this estimate are drawn and introduced into the sample corpus, undifferentiated from real samples. Subsequent generations of estimators now face contaminated environments, a scenario referred to in the machine learning literature as self-consumption. Empirically, it has been observed that models in fully synthetic self-consuming loops collapse -- their performance deteriorates with each batch of training -- but accumulating data has been shown to prevent complete degeneration. This, in turn, begs the question: What happens when fresh real samples \textit{are} added at every stage? In this paper, we study the minimax loss of self-consuming discrete distribution estimation in such loops. We show that even when model collapse is consciously averted, the ratios between the minimax losses with and without source information can grow unbounded as the batch size increases. In the data accumulation setting, where all batches of samples are available for estimation, we provide minimax lower bounds and upper bounds that are order-optimal under mild conditions for the expected $\ell_2^2$ and $\ell_1$ losses at every stage. We provide conditions for regimes where there is a strict gap in the convergence rates compared to the corresponding oracle-assisted minimax loss where real and synthetic samples are differentiated, and provide examples where this gap is easily observed. We also provide a lower bound on the minimax loss in the data replacement setting, where only the latest batch of samples is available, and use it to find a lower bound for the worst-case loss for bounded estimate trajectories.

翻译：从独立同分布样本中学习离散分布是一个已被深入理解的问题。然而，生成式机器学习的进展催生了一个有趣且非独立同分布的新场景：在接收到一定数量的样本后，一个估计分布被固定下来，随后从该估计中抽取的样本被引入样本库，并与真实样本不加区分地混合。后续的估计器世代现在面临着被污染的环境，这一场景在机器学习文献中被称为自消耗。经验上观察到，在完全合成的自消耗循环中，模型会发生崩溃——其性能随着每一批训练而恶化——但积累数据已被证明可以防止完全退化。这反过来引出了一个疑问：如果在每个阶段都添加新的真实样本，会发生什么？在本文中，我们研究了此类循环中自消耗离散分布估计的极小极大损失。我们证明，即使有意识地避免模型崩溃，在拥有和不拥有源信息情况下的极小极大损失之比，也可能随着批次大小的增加而无界增长。在数据积累设置中（所有批次的样本都可用于估计），我们针对每个阶段的期望 $\ell_2^2$ 和 $\ell_1$ 损失，给出了在温和条件下达到阶数最优的极小极大下界和上界。我们给出了在某些机制下，其收敛速率与相应的、能区分真实样本与合成样本的Oracle辅助极小极大损失相比存在严格差距的条件，并提供了容易观察到这种差距的示例。我们还给出了数据替换设置（仅最新批次的样本可用）下极小极大损失的一个下界，并利用它找到了有界估计轨迹最坏情况损失的一个下界。