Risk-Sensitive Diffusion for Perturbation-Robust Optimization

The essence of score-based generative models (SGM) is to optimize a score-based model towards the score function. However, we show that noisy samples incur another objective function, rather than the one with score function, which will wrongly optimize the model. To address this problem, we first consider a new setting where every noisy sample is paired with a risk vector, indicating the data quality (e.g., noise level). This setting is very common in real-world applications, especially for medical and sensor data. Then, we introduce risk-sensitive SDE, a type of stochastic differential equation (SDE) parameterized by the risk vector. With this tool, we aim to minimize a measure called perturbation instability, which we define to quantify the negative impact of noisy samples on optimization. We will prove that zero instability measure is only achievable in the case where noisy samples are caused by Gaussian perturbation. For non-Gaussian cases, we will also provide its optimal coefficients that minimize the misguidance of noisy samples. To apply risk-sensitive SDE in practice, we extend widely used diffusion models to their risk-sensitive versions and derive a risk-free loss that is efficient for computation. We also have conducted numerical experiments to confirm the validity of our theorems and show that they let SGM be robust to noisy samples for optimization.