Classifier-guided diffusion models generate conditional samples by augmenting the reverse-time score with the gradient of the log-probability predicted by a probabilistic classifier. In practice, this classifier is usually obtained by minimizing an empirical loss function. While existing statistical theory guarantees good generalization performance when the sample size is sufficiently large, it remains unclear whether such training yields an effective guidance mechanism. We study this question in the context of cross-entropy loss, which is widely used for classifier training. Under mild smoothness assumptions on the classifier, we show that controlling the cross-entropy at each diffusion model step is sufficient to control the corresponding guidance error. In particular, probabilistic classifiers achieving conditional KL divergence $\varepsilon^2$ induce guidance vectors with mean squared error $\widetilde O(d \varepsilon )$, up to constant and logarithmic factors. Our result yields an upper bound on the sampling error of classifier-guided diffusion models and bears resemblance to a reverse log-Sobolev--type inequality. To the best of our knowledge, this is the first result that quantitatively links classifier training to guidance alignment in diffusion models, providing both a theoretical explanation for the empirical success of classifier guidance, and principled guidelines for selecting classifiers that induce effective guidance.

翻译：分类器引导的扩散模型通过将概率分类器预测的对数概率梯度与逆时分数相结合，生成条件样本。在实践中，此类分类器通常通过最小化经验损失函数获得。尽管现有统计理论保证在样本量足够大时具有良好的泛化性能，但此类训练是否会产生有效的引导机制仍不明确。本文在交叉熵损失的背景下研究该问题，该损失函数被广泛用于分类器训练。在分类器满足温和平滑性假设的条件下，我们证明控制扩散模型每个步骤的交叉熵足以控制相应的引导误差。具体而言，达到条件KL散度 $\varepsilon^2$ 的概率分类器可诱导均方误差为 $\widetilde O(d \varepsilon )$ 的引导向量（忽略常数项与对数因子）。我们的结果给出了分类器引导扩散模型采样误差的上界，其形式与反向对数Sobolev型不等式具有相似性。据我们所知，这是首个定量关联分类器训练与扩散模型中引导对齐的研究成果，既为分类器引导的经验成功提供了理论解释，也为选择能诱导有效引导的分类器提供了原则性指导。