# iPiano: Inertial Proximal Algorithm for Non-Convex Optimization

arXiv: https://arxiv.org/abs/1404.4805 · HTML: https://mlsanity.com/paper/1404.4805/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Apr 2014.

## Built on by (most cited first, 30 of 30)

- [Gradient Step Denoiser for convergent Plug-and-Play](https://mlsanity.com/paper/2110.03220/) (2021)
- [Proximal Denoiser for Convergent Plug-and-Play Optimization with Nonconvex Regularization](https://mlsanity.com/paper/2201.13256/) (2022)
- [Learning Weakly Convex Regularizers for Convergent Image-Reconstruction Algorithms](https://mlsanity.com/paper/2308.10542/) (2023)
- [Restarted Nonconvex Accelerated Gradient Descent: No More Polylogarithmic Factor in the O(ε-7/4) Complexity](https://mlsanity.com/paper/2201.11411/) (2022)
- [Convergence of Random Reshuffling Under The Kurdyka-Lojasiewicz Inequality](https://mlsanity.com/paper/2110.04926/) (2021)
- [Convergent Bregman Plug-and-Play Image Restoration for Poisson Inverse Problems](https://mlsanity.com/paper/2306.03466/) (2023)
- [A relaxed proximal gradient descent algorithm for convergent plug-and-play with proximal denoiser](https://mlsanity.com/paper/2301.13731/) (2023)
- [Optimization-Inspired Cross-Attention Transformer for Compressive Sensing](https://mlsanity.com/paper/2304.13986/) (2023)
- [Escaping Saddle Points Faster with Stochastic Momentum](https://mlsanity.com/paper/2106.02985/) (2021)
- [Training Deep Neural Networks with Adaptive Momentum Inspired by the Quadratic Optimization](https://mlsanity.com/paper/2110.09057/) (2021)
- [The Role of Momentum Parameters in the Optimal Convergence of Adaptive Polyak's Heavy-ball Methods](https://mlsanity.com/paper/2102.07314/) (2021)
- [Learning Sparse Masks for Diffusion-based Image Inpainting](https://mlsanity.com/paper/2110.02636/) (2021)
- [Convergent plug-and-play with proximal denoiser and unconstrained regularization parameter](https://mlsanity.com/paper/2311.01216/) (2023)
- [Accelerated Proximal Alternating Gradient-Descent-Ascent for Nonconvex Minimax Machine Learning](https://mlsanity.com/paper/2112.11663/) (2021)
- [A New Random Reshuffling Method for Nonsmooth Nonconvex Finite-sum Optimization](https://mlsanity.com/paper/2312.01047/) (2023)
- [Inertial Proximal Deep Learning Alternating Minimization for Efficient Neutral Network Training](https://mlsanity.com/paper/2102.00267/) (2021)
- [Polyak's Heavy Ball Method Achieves Accelerated Local Rate of Convergence under Polyak-Lojasiewicz Inequality](https://mlsanity.com/paper/2410.16849/) (2024)
- [Efficient Neural Generation of 4K Masks for Homogeneous Diffusion Inpainting](https://mlsanity.com/paper/2303.10096/) (2023)
- [Compressed Decentralized Momentum Stochastic Gradient Methods for Nonconvex Optimization](https://mlsanity.com/paper/2508.04950/) (2025)
- [Multivariate Fields of Experts for Convergent Image Reconstruction](https://mlsanity.com/paper/2508.06490/) (2025)
- [Block Majorization Minimization with Extrapolation and Application to β-NMF](https://mlsanity.com/paper/2401.06646/) (2024)
- [A Generalization Result for Convergence in Learning-to-Optimize](https://mlsanity.com/paper/2410.07704/) (2024)
- [Adaptive Lipschitz-Free Proximal Gradient Methods for Stochastic Composite Nonconvex Optimization](https://mlsanity.com/paper/2502.21099/) (2025)
- [Stochastic Gradient Descent with Momentum is Algorithmically Stable](https://mlsanity.com/paper/2605.28517/) (2026)
- [On the Generalization of Stochastic Gradient Descent with Momentum](https://mlsanity.com/paper/2102.13653/) (2021)
- [A Refined Inertial DC Algorithm for DC Programming](https://mlsanity.com/paper/2104.14750/) (2021)
- [A theoretical and empirical study of new adaptive algorithms with additional momentum steps and shifted updates for stochastic non-convex optimization](https://mlsanity.com/paper/2110.08531/) (2021)
- [Accelerated Gradient Flow: Risk, Stability, and Implicit Regularization](https://mlsanity.com/paper/2201.08311/) (2022)
- [DRSOM: A Dimension Reduced Second-Order Method](https://mlsanity.com/paper/2208.00208/) (2022)
- [Deep Inertia Lp Half-Quadratic Splitting Unrolling Network for Sparse View CT Reconstruction](https://mlsanity.com/paper/2408.06600/) (2024)

## Abstract (arXiv)

In this paper we study an algorithm for solving a minimization problem composed of a differentiable (possibly non-convex) and a convex (possibly non-differentiable) function. The algorithm iPiano combines forward-backward splitting with an inertial force. It can be seen as a non-smooth split version of the Heavy-ball method from Polyak. A rigorous analysis of the algorithm for the proposed class of problems yields global convergence of the function values and the arguments. This makes the algorithm robust for usage on non-convex problems. The convergence result is obtained based on the KL inequality. This is a very weak restriction, which was used to prove convergence for several other gradient methods. First, an abstract convergence theorem for a generic algorithm is proved, and, then iPiano is shown to satisfy the requirements of this theorem. Furthermore, a convergence rate is established for the general problem class. We demonstrate iPiano on computer vision problems: image denoising with learned priors and diffusion based image compression.