The problem of constrained online convex optimization is considered, where at each round, once a learner commits to an action $x_t \in \mathcal{X} \subset \mathbb{R}^d$, a convex loss function $f_t$ and a convex constraint function $g_t$ that drives the constraint $g_t(x)\le 0$ are revealed. The objective is to simultaneously minimize the static regret and cumulative constraint violation (CCV) compared to the benchmark that knows the loss functions and constraint functions $f_t$ and $g_t$ for all $t$ ahead of time, and chooses a static optimal action that is feasible with respect to all $g_t(x)\le 0$. In recent prior work Sinha and Vaze [2024], algorithms with simultaneous regret of $O(\sqrt{T})$ and CCV of $O(\sqrt{T})$ or (CCV of $O(1)$ in specific cases Vaze and Sinha [2025], e.g. when $d=1$) have been proposed. It is widely believed that CCV is $Ω(\sqrt{T})$ for all algorithms that ensure that regret is $O(\sqrt{T})$ with the worst case input for any $d\ge 2$. In this paper, we refute this and show that the algorithm of Vaze and Sinha [2025] simultaneously achieves regret of $O(\sqrt{T})$ regret and CCV of $O(T^{1/3})$ when $d=2$.

We study Constrained Online Convex Optimization with Memory (COCO-M), where both the loss and the constraints depend on a finite window of past decisions made by the learner. This setting extends the previously studied unconstrained online optimization with memory framework and captures practical problems such as the control of constrained dynamical systems and scheduling with reconfiguration budgets. For this problem, we propose the first algorithms that achieve sublinear regret and sublinear cumulative constraint violation under time-varying constraints, both with and without predictions of future loss and constraint functions. Without predictions, we introduce an adaptive penalty approach that guarantees sublinear regret and constraint violation. When short-horizon and potentially unreliable predictions are available, we reinterpret the problem as online learning with delayed feedback and design an optimistic algorithm whose performance improves as prediction accuracy improves, while remaining robust when predictions are inaccurate. Our results bridge the gap between classical constrained online convex optimization and memory-dependent settings, and provide a versatile learning toolbox with diverse applications.

In this work, we explore online convex optimization (OCO) and introduce a new condition and analysis that provides fast rates by exploiting the curvature of feasible sets. In online linear optimization, it is known that if the average gradient of loss functions exceeds a certain threshold, the curvature of feasible sets can be exploited by the follow-the-leader (FTL) algorithm to achieve a logarithmic regret. This study reveals that algorithms adaptive to the curvature of loss functions can also leverage the curvature of feasible sets. In particular, we first prove that if an optimal decision is on the boundary of a feasible set and the gradient of an underlying loss function is non-zero, then the algorithm achieves a regret bound of O ( ρ ln T) in stochastic environments. Here, ρ > 0 is the radius of the smallest sphere that includes the optimal decision and encloses the feasible set. Our approach, unlike existing ones, can work directly with convex loss functions, exploiting the curvature of loss functions simultaneously, and can achieve the logarithmic regret only with a local property of feasible sets. Additionally, the algorithm achieves an O ( T) regret even in adversarial environments, in which FTL suffers an Ω( T) regret, and achieves an O ( ρ ln T + Cρ ln T) regret in corrupted stochastic environments with corruption level C .

We study online composite optimization under the Stochastically Extended Adversarial (SEA) model.

In many applications, learning systems are required to process continuous non-stationary data streams.

Modern deep learning is driven by first-order stochastic optimization algorithms.

Modern deep learning is driven by first-order stochastic optimization algorithms.

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There, strictly feasible solutions can cancel out the effects of violated constraints.