# Conic Descent Redux for Memory-Efficient Optimization

arXiv: https://arxiv.org/abs/2308.07343 · HTML: https://mlsanity.com/paper/2308.07343/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 30 references that are in the map.
- First posted on arXiv 13 Aug 2023.

## Builds on (most cited first, 10 of 10)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Dropping Convexity for Faster Semi-definite Optimization](https://mlsanity.com/paper/1509.03917/) (2015)
- [Scalable Semidefinite Programming](https://mlsanity.com/paper/1912.02949/) (2019)
- [Achieving Exact Cluster Recovery Threshold via Semidefinite Programming](https://mlsanity.com/paper/1412.6156/) (2014)
- [Conditional Gradient Algorithms for Norm-Regularized Smooth Convex Optimization](https://mlsanity.com/paper/1302.2325/) (2013)
- [Heavy Ball Momentum for Conditional Gradient](https://mlsanity.com/paper/2110.04243/) (2021)
- [Conic Programming Reformulations of Two-Stage Distributionally Robust Linear Programs over Wasserstein Balls](https://mlsanity.com/paper/1609.07505/) (2016)
- [Greedy Algorithms for Cone Constrained Optimization with Convergence Guarantees](https://mlsanity.com/paper/1705.11041/) (2017)
- [Robust Smart-Grid Powered Cooperative Multipoint Systems](https://mlsanity.com/paper/1506.02336/) (2015)
- [Enhancing Parameter-Free Frank Wolfe with an Extra Subproblem](https://mlsanity.com/paper/2012.05284/) (2020)

## Abstract (arXiv)

Conic programming has well-documented merits in a gamut of signal processing and machine learning tasks. This contribution revisits a recently developed first-order conic descent (CD) solver, and advances it in three aspects: intuition, theory, and algorithmic implementation. It is found that CD can afford an intuitive geometric derivation that originates from the dual problem. This opens the door to novel algorithmic designs, with a momentum variant of CD, momentum conic descent (MOCO) exemplified. Diving deeper into the dual behavior CD and MOCO reveals: i) an analytically justified stopping criterion; and, ii) the potential to design preconditioners to speed up dual convergence. Lastly, to scale semidefinite programming (SDP) especially for low-rank solutions, a memory efficient MOCO variant is developed and numerically validated.