PreprintCC BY 4.0arXiv
Mirror descent algorithms with logarithmic barriers
Abstract
We derive convergence guarantees for mirror descent and proximal mirror descent with a logarithmic barrier as the distance-generating function. When the solution lies on the boundary, the Bregman divergence blows up and classical guarantees become vacuous. While existing remedies recover O(log k/k) rates by adapting classical bounds, we propose a more direct route in which the Bregman distance to the solution never appears, and which achieves the same rate with constant stepsize. We also construct an example showing that this rate cannot be improved in general. Finally, we compare the proposed approach with interior-point methods.
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BibTeX
@misc{marchi2026mirror,
title = {{Mirror descent algorithms with logarithmic barriers}},
author = {Alberto De Marchi and Yura Malitsky and Adrien B. Taylor},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.22834},
eprint = {2608.22834},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22834},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mirror descent algorithms with logarithmic barriers
AU - Alberto De Marchi
AU - Yura Malitsky
AU - Adrien B. Taylor
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2608.22834
UR - https://arxiv.org/abs/2608.22834
AB - We derive convergence guarantees for mirror descent and proximal mirror descent with a logarithmic barrier as the distance-generating function. When the solution lies on the boundary, the Bregman divergence blows up and classical guarantees become vacuous. While existing remedies recover O(log k/k) rates by adapting classical bounds, we propose a more direct route in which the Bregman distance to the solution never appears, and which achieves the same rate with constant stepsize. We also construct an example showing that this rate cannot be improved in general. Finally, we compare the proposed approach with interior-point methods.
ER -