  Diffusion models have recently been recognised as efficient inverse problem
solvers due to their ability to produce high-quality reconstruction results
without relying on pairwise data training. Existing diffusion-based solvers
utilize Gradient Descent strategy to get a optimal sample solution. However,
these solvers only calculate the current gradient and have not utilized any
history information of sampling process, thus resulting in unstable
optimization progresses and suboptimal solutions. To address this issue, we
propose to utilize the history information of the diffusion-based inverse
solvers. In this paper, we first prove that, in previous work, using the
gradient descent method to optimize the data fidelity term is convergent.
Building on this, we introduce the incorporation of historical gradients into
this optimization process, termed History Gradient Update (HGU). We also
provide theoretical evidence that HGU ensures the convergence of the entire
algorithm. It's worth noting that HGU is applicable to both pixel-based and
latent-based diffusion model solvers. Experimental results demonstrate that,
compared to previous sampling algorithms, sampling algorithms with HGU achieves
state-of-the-art results in medical image reconstruction, surpassing even
supervised learning methods. Additionally, it achieves competitive results on
natural images.
