To accompany the definitions of Sec. The above framework was introduced by (Ilyas et al., 2019, Tsipras et al., 2019), and we have slightly We first derive the expression in Eq. (8) of the paper. We consider the binary and the multiclass case separately. Binary case: Suppose we would like to evaluate a model described by Eq. (7) at the end of training, Since Eq. (11) describes regression models with LSE ( Inspecting Eq. (15), maximal "confusion" of the classification model is achieved by aligning Eq. (15) has been derived for perturbations of the training data. Then, Eq. (14) becomes: f ( X + ϵ) = (Θ( X, X) +)Θ(X, X) This leads to the multi-dimensional analogue of the linear Eq. (10) for We present the two most obvious methods.

These previous methods mainly focus on unconditional generative modeling. While there exist guided-diffusion models (Song and Ermon, 2019; Song et al., 2021c; Dhariwal and Nichol, 2021; Nichol et al., 2022; Ramesh et al., 2022) that target on generating high-resolution photo-realistic images that match the semantic meanings or content of the label, text, or corrupted-images, we focus on studying diffusion-based conditional generative modeling at a more fundamental level.

Lacking closed form expressions for the context vector, we use numerical integration: we prove exponential convergence for both families.

The following theorem justifies the DBbonus under the Bayesian linear regression perspective.

However, such methods are usually sensitive to environmental dynamics-irrelevant information, e.g., white-noise. To handle such dynamics-irrelevant information, we propose a Dynamic Bottleneck (DB) model, which attains a dynamics-relevant representation based on the information-bottleneck principle.

We initiate a formal study of reproducibility in optimization. We define a quantitative measure of reproducibility of optimization procedures in the face of noisy or error-prone operations such as inexact or stochastic gradient computations or inexact initialization.