# On exploration of an interior mirror descent flow for stochastic nonconvex constrained problem arXiv: https://arxiv.org/abs/2507.15264 ยท HTML: https://mlsanity.com/paper/2507.15264/ ## Key facts - Cited by 3 papers in the map (data through 14 Oct 2026). - Builds on 32 of its 69 references that are in the map. - First posted on arXiv 21 Jul 2025. ## Builds on (most cited first, 30 of 32) - [Stochastic subgradient method converges on tame functions](https://mlsanity.com/paper/1804.07795/) (2018) - [Relatively-Smooth Convex Optimization by First-Order Methods, and Applications](https://mlsanity.com/paper/1610.05708/) (2016) - [Conservative set valued fields, automatic differentiation, stochastic gradient method and deep learning](https://mlsanity.com/paper/1909.10300/) (2019) - [Stochastic Methods for Composite and Weakly Convex Optimization Problems](https://mlsanity.com/paper/1703.08570/) (2017) - [Implicit Bias of Gradient Descent on Reparametrized Models: On Equivalence to Mirror Descent](https://mlsanity.com/paper/2207.04036/) (2022) - [First Order Methods beyond Convexity and Lipschitz Gradient Continuity with Applications to Quadratic Inverse Problems](https://mlsanity.com/paper/1706.06461/) (2017) - [First-order methods almost always avoid saddle points: the case of vanishing step-sizes](https://mlsanity.com/paper/1906.07772/) (2019) - [Gradient Descent Only Converges to Minimizers: Non-Isolated Critical Points and Invariant Regions](https://mlsanity.com/paper/1605.00405/) (2016) - [Hessian Riemannian gradient flows in convex programming](https://mlsanity.com/paper/1811.10331/) (2018) - [Adam-family Methods for Nonsmooth Optimization with Convergence Guarantees](https://mlsanity.com/paper/2305.03938/) (2023) - [The effect of smooth parametrizations on nonconvex optimization landscapes](https://mlsanity.com/paper/2207.03512/) (2022) - [Fast Stochastic Bregman Gradient Methods: Sharp Analysis and Variance Reduction](https://mlsanity.com/paper/2104.09813/) (2021) - [Optimal Complexity and Certification of Bregman First-Order Methods](https://mlsanity.com/paper/1911.08510/) (2019) - [An Inertial Newton Algorithm for Deep Learning](https://mlsanity.com/paper/1905.12278/) (2019) - [Hessian barrier algorithms for linearly constrained optimization problems](https://mlsanity.com/paper/1809.09449/) (2018) - [Bregman Proximal Point Algorithm Revisited: A New Inexact Version and its Inertial Variant](https://mlsanity.com/paper/2105.10370/) (2021) - [Nonsmooth nonconvex stochastic heavy ball](https://mlsanity.com/paper/2304.13328/) (2023) - [Riemannian stochastic optimization methods avoid strict saddle points](https://mlsanity.com/paper/2311.02374/) (2023) - [An exact penalty approach for optimization with nonnegative orthogonality constraints](https://mlsanity.com/paper/1907.12424/) (2019) - [Nonconvex Stochastic Bregman Proximal Gradient Method with Application to Deep Learning](https://mlsanity.com/paper/2306.14522/) (2023) - [Global stability of first-order methods for coercive tame functions](https://mlsanity.com/paper/2308.00899/) (2023) - [Stochastic Bregman Proximal Gradient Method Revisited: Kernel Conditioning and Painless Variance Reduction](https://mlsanity.com/paper/2401.03155/) (2024) - [An efficient implementable inexact entropic proximal point algorithm for a class of linear programming problems](https://mlsanity.com/paper/2011.14312/) (2020) - [Hessian barrier algorithms for non-convex conic optimization](https://mlsanity.com/paper/2111.00100/) (2021) - [Adam-family Methods with Decoupled Weight Decay in Deep Learning](https://mlsanity.com/paper/2310.08858/) (2023) - [Stochastic Bregman Subgradient Methods for Nonsmooth Nonconvex Optimization Problems](https://mlsanity.com/paper/2404.17386/) (2024) - [An Exact Penalty Approach for Equality Constrained Optimization over a Convex Set](https://mlsanity.com/paper/2505.02495/) (2025) - [On Squared-Variable Formulations](https://mlsanity.com/paper/2310.01784/) (2023) - [Bregman Proximal Framework for Deep Linear Neural Networks](https://mlsanity.com/paper/1910.03638/) (2019) - [Developing Lagrangian-based Methods for Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2404.09438/) (2024) ## Built on by (most cited first, 3 of 3) - [Non-KKT Accumulation in Entropic Mirror Descent](https://mlsanity.com/paper/2608.01658/) (2026) - [A Support-Set Algorithm for Optimization Problems with Nonnegative and Orthogonal Constraints](https://mlsanity.com/paper/2511.03443/) (2025) - [Establishing Boundary KKT Convergence of Mirror Descent through Reparameterization](https://mlsanity.com/paper/2608.07248/) (2026) ## Abstract (arXiv) We study a nonsmooth nonconvex optimization problem defined over nonconvex constraints, where the feasible set is given by the intersection of the closure of an open set and a smooth manifold. By endowing the open set with a Riemannian metric induced by a barrier function, we obtain a Riemannian subgradient flow formulated as a differential inclusion, which remains strictly within the interior of the feasible set. This continuous dynamical system unifies two classes of iterative optimization methods, namely the Hessian barrier method and mirror descent scheme, by revealing that these methods can be interpreted as discrete approximations of the continuous flow. We explore the long-term behavior of the trajectories generated by this dynamical system and show that the existing deficient convergence properties of the Hessian barrier and mirror descent scheme can be unifily and more insightfully interpreted through these of the continuous trajectory. For instance, the notorious spurious stationary points citechen2024spurious observed in Hessian barrier method and mirror descent scheme are interpreted as stable equilibria of the dynamical system that do not correspond to real stationary points of the original optimization problem. We provide two sufficient condition such that these spurious stationary points can be avoided if the strict complementarity conditions holds. In the absence of these regularity condition, we propose a random perturbation strategy that ensures the trajectory converges (subsequentially) to an approximate stationary point. Building on these insights, we introduce two iterative Riemannian subgradient methods, form of interior point methods, that generalizes the existing Hessian barrier method and mirror descent scheme for solving nonsmooth nonconvex optimization problems.