Mirror descent algorithms with logarithmic barriers

What the math gives to ML

The paper develops mirror descent with a logarithmic barrier, addressing the singular Bregman divergence that occurs when the optimizer lies on the boundary of a convex feasible set. The transferable asset is a positivity-preserving update that can approach zero coordinates without evaluating the barrier at the boundary, together with a boundary-aware convergence rate of O(log k/k) in the stated convex setting. A practical neural-network use is optimizing simplex-valued routing or gating probabilities directly instead of relying on clipped SGD or unconstrained logits. The most promising first target is sparse mixture-of-experts routing, where the method can be compared against AdamW on router logits and exponentiated-gradient updates.