ICML 2025

Constrained Online Convex Optimization with Polyak Feasibility Steps
Spencer Hutchinson 1 Mahnoosh Alizadeh 1
Abstract
In this work, we study online convex optimiza-
tion with a fixed constraint function g : Rd →
R.
Prior work on this problem has shown
O(
√
T) regret and cumulative constraint satisfac-
tion PT
t=1 g(xt) ≤0, while only accessing the
constraint value and subgradient at the played ac-
tions g(xt), ∂g(xt). Using the same constraint
information, we show a stronger guarantee of any-
time constraint satisfaction g(xt) ≤0 ∀t ∈[T],
and matching O(
√
T) regret guarantees. These
contributions are thanks to our approach of using
Polyak feasibility steps to ensure constraint sat-
isfaction, without sacrificing regret. Specifically,
after each step of online gradient descent, our
algorithm applies a subgradient descent step on
the constraint function where the step-size is cho-
sen according to the celebrated Polyak step-size.
We further validate this approach with numerical
experiments.
1. Introduction
We study the problem of online convex optimization (OCO)
where, in each round t = 1, 2, ..., T, a player chooses an ac-
tion xt from an action set X ⊆Rd and then suffers the cost
ft(xt) according to an adversarially-chosen convex function
ft (Zinkevich, 2003). The player’s goal is to minimize the
regret with respect to the best single action in hindsight,
RegT :=
T
X
t=1
ft(xt) −min
x∈X
T
X
t=1
ft(x).
The OCO problem has emerged as a fundamental setting
for various machine learning domains, such as stochastic
optimization (Cesa-Bianchi et al., 2004), non-convex opti-
1Department of Electrical and Computer Engineering, Uni-
versity of California-Santa Barbara, Santa Barbara, Califor-
nia, USA. Correspondence to: Spencer Hutchinson <shutchin-
son@ucsb.edu>, Mahnoosh Alizadeh <alizadeh@ucsb.edu>.
Proceedings of the 42 nd International Conference on Machine
Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025
by the author(s).
mization (Cutkosky et al., 2023), ...