In this paper, we study alternating regret in online convex optimization (OCO), motivated by the success of alternating learning dynamics in two-player games. Although previous works have shown that $o(\sqrt{T})$ alternating regret is achievable under various assumptions on the loss functions and feasible domains, the minimax regret rate has remained open even for the expert problem. In this paper, we resolve this question by showing matching lower and upper bounds for both the expert problem and general OCO. Somewhat surprisingly, for the $d$-expert problem, we show that the minimax alternating regret is $Θ(\log d)$, independent of the horizon $T$. This significantly improves upon the best-known $\mathcal{O}(T^{1/3}\log^{2/3} d)$ established by Hait et al. [2025]. We further extend our results to general OCO over a $d$-dimensional compact convex set and prove that the worst-case minimax alternating regret is $Θ\left(d\log \left(1+\frac{T}{d}\right)\right)$, also significantly improving upon the best-known $\mathcal{O}((d\log T)^{2/3}T^{1/3})$ upper bound and resolving the open problem posed by Cevher et al. [2023], Hait et al. [2025]. Technically, our upper bound for the expert problem is achieved by a corrected variant of Hedge, in which carefully designed correction terms cancel the unfavorable curvature arising in the alternating-regret analysis. We extend the same corrected-potential argument to continuous action sets to obtain the optimal alternating-regret rate for OCO. For the lower bounds, the expert construction repeatedly eliminates half of the candidate experts, while the OCO lower bound instance construction replaces this discrete elimination by a more involved multiscale construction on the unit disk.
Yiyang Lu, Hareshkumar Jadav, Mohammad Pedramfar +2cs.LG
We study online convex optimization with dueling (pairwise comparison) feedback, where the learner observes only a binary preference between two queried points. While dueling feedback is well understood in discrete or stochastic settings, the adversarial convex setting has remained unexplored. We propose a simple reduction that converts dueling feedback into approximate gradients, enabling the use of standard first-order methods. We show that regret guarantees transfer under this reduction, yielding the first results for this setting, including $\mathcal{O}(T^{3/4})$ static, adaptive, and dynamic regret. Under additional structure, we obtain improved rates of $\mathcal{O}(T^{2/3})$ for smooth objectives and $\mathcal{O}(\sqrt{T \log T})$ for strongly convex functions.
Adaptive procedures must work without nuisance information an oracle may use, such as a gradient scale or smoothness index, and robust procedures may have to answer queries whose coordinate and inspection time are chosen only after the data are seen. Such comparisons are meaningful only when the oracle advantage and validity contract are stated explicitly. We formalize nuisance adaptation via a slice-normalized minimax ratio retaining the worst-case instance within each nuisance slice, and separately define the robustness cost of expanding from one preannounced Gaussian query to arbitrary post-hoc inspection. Our main result is a finite-horizon composition law for Gaussian certification: from M independent coordinates, a familywise certifier protecting every coordinate and time up to T pays optimal normalized squared half-width of order log(eM) + log log(e^eT), within the sample-mean-centered rectangular class. Epoch stitching gives the upper bound; independent Gaussian block increments across coordinates and geometric time scales give a matching lower bound, already holding on a geometric checkpoint grid, forcing quantiles of the realized maximum width so selection and stopping taxes add. Two benchmark regimes complete the picture: unknown gradient scale in online convex optimization has constant cost, while pointwise adaptation over nested Holder classes costs order (log n / log log n)^(s1/(2s1+1)). Cast as model monitoring, the law lets an analyst inspect any of M slice metrics at any data-dependent time: the naive fixed-query band's selected coverage degrades sharply, to 0.30 at M=1 and to zero for M>=10, while the epoch-stitched certifier holds familywise coverage at an additive iterated-logarithm width cost. Experiments put both sharp predictions at risk of refutation; both survive.
Safety-critical IoT systems such as industrial closed-loop control, V2X coordination, and remote teleoperation require every sensor's peak Age of Information (peak AoI, also abbreviated PAoI) to stay below a hard per-slot deadline, not merely an average bound. Existing approaches meet this requirement only under restrictive assumptions: stochastic channels for Whittle-index AoI, simulator rollouts for deep reinforcement learning, or sublinear cumulative violation for long-term constrained online convex optimization. Under adversarial coefficients, OCO-PAoI-Hard guarantees zero per-slot violation of the modeled AoI state under one-step viability and O(sqrt(T)) regret against any static safe comparator; packet-level safety requires stronger service assumptions. Our key observation is that the fractional peak-AoI deadline collapses exactly to an affine half-space constraint on the resource-allocation vector, turning hard real-time scheduling into time-varying constrained online convex optimization over a polyhedral safe set. A strictly causal proposal-shield-update loop enforces feasibility through one Euclidean projection per slot, the gradient step preserves no-regret behavior, and the classical virtual queue is reduced to an a-posteriori certificate. We establish closed-form static and dynamic regret bounds, a matching Omega(sqrt(T)) minimax lower bound, a margin-safe variant against execution noise, and a deadline-induced competitive ratio. On a four-sensor adversarial fluid-model trap channel, OCO-PAoI-Hard attains zero modeled-state deadline violations across all ten seeds, while four representative baselines miss between 1.65 percent and 64.0 percent of slots, and the empirical normalized regret stays below the theoretical envelope across two orders of magnitude in T.
We study online convex optimization (OCO) in non-stationary environments under heavy-tailed noise, where the stochastic gradient oracle admits only a finite $p$-th central moment for some $p \in (1, 2]$. While static regret is well-understood, achieving universal dynamic regret in a parameter-free manner remains an open challenge. We resolve this by proposing \textbf{HT-PAder}, a parameter-free algorithm combining restarted AdaGrad experts over a geometric pool of block lengths with a pathwise meta-algorithm, \textbf{AdaGrad-Hedge}, which requires no moment conditions on meta-losses. For a domain of diameter $D$, Lipschitz constant $G$, noise level $σ$, and comparator path length $P_T$, HT-PAder achieves an expected universal dynamic regret of \[ \widetilde O\left( GD\sqrt{T(1+P_T/D)} + σD T^{1/p}(1+P_T/D)^{(p-1)/p} \right). \] The algorithm does not require prior knowledge of any of these problem parameters. Even in the special case of finite variance ($p=2$), HT-PAder provides the first parameter-free minimax universal dynamic regret guarantee. We also prove a matching lower bound, establishing the optimality of the path-length exponent.
Constrained online convex optimization requires minimizing regret against adversarial convex costs while satisfying a convex constraint at every round, as needed in safety-critical applications. A computationally efficient method combines online gradient descent with a Polyak feasibility step, using one constraint evaluation and one subgradient per round. Although this method achieves O(sqrt(T)) regret with per-round feasibility, we derive a tighter, data-dependent analysis by retaining two quantities omitted by the standard worst-case argument. First, we replace the gradient envelope G_f^2 T with the observed accumulation G_T = sum_t ||grad f_t(x_t)||^2. Second, we identify a nonnegative Polyak correction P_T that measures the cumulative squared displacement caused by feasibility projections and enters the regret bound with a negative sign. The resulting improvement, Delta_T = (eta/2)(G_f^2 T - G_T) + P_T/(2 eta), is always nonnegative. We further propose AdaOGD-PFS, an adaptive-step-size method that achieves O(sqrt(G_T)) regret while preserving per-round feasibility. Experiments on ball- and halfspace-constrained problems improve the regret bound by 38 to 43 percent, with both data-dependent gradients and Polyak corrections contributing substantially.
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 $g_t(x)\le 0$. Currently, the best known algorithm is OGD+Projection algorithm of [Vaze and Sinha, 2025] that has simultaneous regret of $O(\sqrt{T})$ and CCV of $O(T^{1/3})$ for $d=2$ [Balasundaram et al., 2026], and simultaneous regret of $O(\sqrt{T})$ and CCV of $O(\sqrt{T})$ for any $d$ [Sarkar and Sinha, 2026]. In this paper, we show that the CCV of the OGD+Projection algorithm is $Ω(T^{\frac{d-1}{2d}})$. This is the first such lower bound result.
Decentralized online convex optimization (D-OCO) is a popular framework for distributed applications with streaming data. To tackle the communication bottleneck, previous studies have investigated D-OCO with compressed communication and proposed several algorithms that are variants of online gradient descent (OGD). However, for D-OCO with exact communication, the best existing algorithms are variants of follow-the-regularized-leader (FTRL). In this paper, for the first time, we propose two FTRL-type algorithms for D-OCO with compressed communication. Compared with OGD-type algorithms, our algorithms are more elegant in both algorithmic design and theoretical analysis. The key insight is that the dual update mechanism of FTRL allows us to make a simple application of the technique for average consensus with communication compression. More specifically, our first algorithm considers the full-information setting, and can match the existing regret bounds. Our second algorithm is designed for the bandit setting, and can significantly improve both the regret bounds and communication costs of existing algorithms.
We study constrained online convex optimization with adversarial losses and stochastic or adversarial constraints. For stochastic constraints, existing algorithms that achieve nearly optimal regret and constraint violation bounds typically rely on regularity assumptions such as Slater's condition, while adversarial-constraint algorithms avoid these assumptions by using a rather restrictive round-wise feasible comparator. We bridge this gap with an anytime primal-dual framework that incorporates an adaptive regularizer into the dual update. The regularizer stabilizes the dual process without relying on the negative drift induced by Slater's condition. For stochastic constraints and convex losses, our algorithm achieves $O(\sqrt{T})$ expected regret and $O(\sqrt{T}\log T)$ expected cumulative constraint violation. Furthermore, we show that our algorithm also admits high-probability bounds of the same order on regret and constraint violation. For strongly convex losses, the regret bound improves to $O(\log T)$ with a violation bound of the same order. With a minor modification, the framework also applies to adversarial constraints and provides guarantees for hard constraint violation.
Anthony Pineci, Yunzong Xucs.LG eess.SY math.OC stat.ML
Online inventory optimization (OIO) is online convex optimization with physical memory: inventory carryover makes the feasible action set depend on the past. A natural principle, used in stochastic inventory learning and recently in OIO under a single linear capacity constraint, is to maintain a hidden target chosen by an online learner and implement its projection onto the currently feasible order-up-to set. We prove that this simple principle is optimal for OIO on arbitrary bounded convex capacity sets. With online gradient descent as the base learner, the method improves the best known regret guarantee for OIO on general convex sets from inverse to inverse-square-root dependence on the common-demand probability, and we prove a matching lower bound. The same principle gives the first polylogarithmic regret guarantee for strongly convex losses and the first dynamic regret guarantee adapting to Euclidean path variation on general convex capacity sets. The analysis introduces a norm alignment principle: the right state variable is the distance from the hidden target to the feasible set, measured in the same norm as the projection. Under norm alignment, this distance evolves pathwise as a scalar queue, with target movement as arrival and common demand as service. This reduction to one-dimensional queue control resolves the state dependence and extends the guarantees to general convex capacity sets, beyond the reach of prior productwise approaches. Experiments on synthetic and real-world inventory data corroborate the theory.
Simone Di Gregorio, Anupam Gupta, Stefano Leonardi +1cs.LG cs.DS
We study Online Convex Optimization (OCO) over a convex set $K\subseteq \mathbb R^d$, where in each round $t$ the learner selects $x_t\in K$ and then observes a convex loss $f_t:K\to[0,1]$, with the goal of minimizing regret to the best fixed decision in hindsight. We introduce a unified probing model that generalizes two recent lines of work: sublinear best-expert queries in the experts setting, and pairwise (comparison-based) feedback available every round in OCO. In our framework, the learner has a budget of $k\le T$ pairwise probes; on a probed round it may query two points and learn which one has smaller loss. Our main result shows that even a sublinear and noisy probe budget can provably improve worst-case regret in the full feedback OCO regime. With $k$ $δ$-noisy pairwise probes, we obtain: $ \text{Reg}_T \le O\left(\min\left\{\sqrt{dT\ln T},\; \frac{dT\ln T}{k|1-2δ|}\right\}\right) $, which is tight (up to logarithmic factors in $T$) across $T$, $k$ and $δ$. Specifically regarding the noise parameter $δ\in [0,1]$, the regret guarantee smoothly degrades as the oracle response approaches a coin flip, i.e., $δ$ is close to $\frac{1}{2}$. When applying the same techniques to a finite $K$ for the prediction with $d$ experts setting, the resulting rates are instead completely tight in all parameters, including $d$. Our analysis gives a streamlined treatment of pairwise probing in OCO by quantifying the benefit of probing via a variance reduction effect, combined with a second-order (variance-based) analysis of Continuous Exponential Weights.
Alexander Ryabchenko, Idan Attias, Daniel M. Roycs.LG stat.ML
Online learning with delayed feedback typically assumes that the learner can track all pending rounds until their feedback arrives. In practice, tracking resources are finite, and feedback from untracked rounds is permanently lost. In this paper, we study delayed online convex optimization (OCO) under a hard capacity constraint, where at most $C$ pending rounds can be tracked at any time. To model delay information, we introduce a semi-clairvoyant model that refines the clairvoyant assumption from prior work: rather than requiring delays to be known at prediction time, the learner observes delay expirations online, consistent with the classical unconstrained delayed setting. Our approach proceeds via a reduction to a novel ``delayed and weighted'' OCO problem, using a scheduler that randomizes tracking decisions and importance-weights the resulting observations. For this base problem, we propose and analyze Delayed-Weighted FTRL and its bandit analogue, establishing regret bounds that explicitly characterize the interaction between time-varying weights and delayed feedback. Combining these base learners with our schedulers yields the first regret guarantees for capacity-constrained OCO under convex and strongly convex losses, for both first-order and bandit feedback. For first-order feedback, capacity $C = Ω(\log T)$ suffices to recover standard delayed OCO rates up to logarithmic factors. For bandit feedback, the regret rates are modulated by powers of $(1 + σ_{\text{max}}/C)$, where $σ_{\text{max}}$ is the maximum number of pending observations at any time. This allows the regret bound to degrade gracefully when $C < σ_{\text{max}}$, while remaining sublinear.
We consider Constrained Online Convex Optimization (COCO) with adversarially chosen constraints. At each round, the learner chooses an action before observing the loss and constraint function for that round. The goal is to achieve small static regret against the best point satisfying all constraints while also controlling cumulative constraint violation ($\mathsf{CCV}$). For strongly convex losses, state-of-the-art algorithms achieve $O(\log T)$ regret and $O(\sqrt{T \log T})$ $\mathsf{CCV}.$ The corresponding best-known bounds for convex losses is $O(\sqrt{T})$ regret and $O(\sqrt{T} \log T)$ $\mathsf{CCV}$. In this paper, we give a simple projection-based algorithm that simultaneously achieves $O(\log T)$ regret and $O(\log T)$ $\mathsf{CCV}$ for strongly-convex losses, yielding an exponential improvement in the $\mathsf{CCV}$. For the convex losses, our algorithm improves the $\mathsf{CCV}$ to $O(\sqrt{T})$ while maintaining the optimal $O(\sqrt{T})$ regret. The key to our improvement is a recent geometric result for self-contracted curves, which may be of independent interest.