# Online Primal-Dual Mirror Descent under Stochastic Constraints

## TL;DR

This paper introduces a primal-dual mirror descent algorithm for online convex optimization with stochastic constraints, achieving optimal regret and constraint violation bounds without requiring the restrictive Slater condition.

## Contribution

It removes the need for the Slater condition in constrained online learning, allowing for equality constraints and reducing the dependence on decision dimension.

## Key findings

- Achieves $ ilde{O}(
oot{T}
ull)$ regret and constraint violation.
- Removes the Slater condition requirement for convergence.
- Reduces dependence on decision dimension to logarithmic in the simplex case.

## Abstract

We consider online convex optimization with stochastic constraints where the objective functions are arbitrarily time-varying and the constraint functions are independent and identically distributed (i.i.d.) over time. Both the objective and constraint functions are revealed after the decision is made at each time slot. The best known expected regret for solving such a problem is $\mathcal{O}(\sqrt{T})$, with a coefficient that is polynomial in the dimension of the decision variable and relies on the Slater condition (i.e. the existence of interior point assumption), which is restrictive and in particular precludes treating equality constraints. In this paper, we show that such Slater condition is in fact not needed. We propose a new primal-dual mirror descent algorithm and show that one can attain $\mathcal{O}(\sqrt{T})$ regret and constraint violation under a much weaker Lagrange multiplier assumption, allowing general equality constraints and significantly relaxing the previous Slater conditions. Along the way, for the case where decisions are contained in a probability simplex, we reduce the coefficient to have only a logarithmic dependence on the decision variable dimension. Such a dependence has long been known in the literature on mirror descent but seems new in this new constrained online learning scenario.