# Mirror descent algorithms with logarithmic barriers

arXiv: https://arxiv.org/abs/2608.22834 · HTML: https://mlsanity.com/paper/2608.22834/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 26 references that are in the map.
- First posted on arXiv 24 Aug 2026.

## Builds on (most cited first, 4 of 4)

- [Relatively-Smooth Convex Optimization by First-Order Methods, and Applications](https://mlsanity.com/paper/1610.05708/) (2016)
- [Smooth Strongly Convex Interpolation and Exact Worst-case Performance of First-order Methods](https://mlsanity.com/paper/1502.05666/) (2015)
- [Optimal Complexity and Certification of Bregman First-Order Methods](https://mlsanity.com/paper/1911.08510/) (2019)
- [An Interior Point-Proximal Method of Multipliers for Convex Quadratic Programming](https://mlsanity.com/paper/1904.10369/) (2019)

## Abstract (arXiv)

This work derives convergence guarantees for mirror descent and proximal mirror descent algorithms when a logarithmic barrier is used as a distance-generating function. Standard approaches cannot be applied when the solution lies on the boundary, where the Bregman divergence blows up. We show that, in a specific setting, both methods enjoy an O(log k / k) rate, which is also tight. In addition, our contributions include: (i) a new technique for handling the blow-up; (ii) a resolution of a gap in the theory of relative smoothness; and (iii) a comparison of the proposed approach with interior-point methods.