O(T^Β)$ Simultaneous Regret

Convex Optimization with Nested Evolving Feasible Sets (CONES)} was introduced in \cite{CONESVaze} where the objective function $f$ remains fixed but the feasible region evolves over time as a nested sequence $S_1 \supseteq S_2 \supseteq \cdots \supseteq S_T$. The goal of an online algorithm is to simultaneously minimize the regret with respect to hindsight static optimal benchmark and the total movement cost $M_\cA(T)$ while ensuring feasibility at all times. CONES is an optimization-oriented generalization of the well-known \emph{nested convex body chasing} (NCBC). In this paper, we extend CONES to allow for loss functions $f_t'$s to also change over time. When all loss functions are convex, we show that the projected proximal algorithm achieves $O(T^{1-\beta}), O(T^\beta)$ simultaneous regret and movement cost, respectively, for any $\beta \in [0,1)$, over a time horizon of $T$. We also show that any {\it weakly adaptive} online algorithm with $O(T^\beta)$ regret has a movement cost of $\Omega\left(T^{\frac{1-\beta}{2}}\right)$ for any $\beta \in [0,1)$. When all loss functions are strongly convex, we show that the projected proximal algorithm simultaneously achieves $O(1)$ regret and a movement cost of $O(\log T)$. To complement this, we show that any online algorithm with sublinear {\it anytime} regret has a movement cost of $\Omega\left(\log T\right)$.

Learning from correct demonstrations is harder than supervised learning when many answers are correct: after predicting, the learner sees one valid answer but not whether its own answer was valid, nor any reward. Existing reward-hedging guarantees consequently assume a finite reward class. We give the first horizon-free guarantee for continuous classes. The key is to hedge in one shared vote over tolerant optimality tests at every accuracy scale. A target reward has one surviving proxy per scale, and a prediction with gap above that scale doubles the proxy. This yields the simultaneous tail bound $|\{t:\ell_t>2^{-j}\}|\leq \log_2\mathcal N(\mathcal G,2^{-j-1})+j$, where $\mathcal G$ is the class of optimality-gap functions. Integrating the tails gives cumulative hidden gap bounded by a metric-entropy integral, independently of the number of rounds. Polynomial entropy $(A/ε)^d$ gives $O(d\log A)$ total gap and a fast $O(d/m)$ statistical rate. For bounded linear contextual recommendation, the result is $O(d)$ regret for arbitrary compact menus. This is the first polynomial finite bound without structural restrictions on the menus, at the price of improper prediction. Although the general vote can be expensive, it is exactly polynomial-time for one-dimensional Lipschitz parameter curves. Fixed-radius rank-two recommendation takes $O(KT^2)$ time for menus of size $K$. We also prove an $Ω(d)$ lower bound, low-rank and bounded ReLU-network corollaries, and a robust theorem that adds only the demonstrator's cumulative suboptimality. A reproducible adaptive stress test illustrates the predicted scale adaptation. After factorization, an exact MovieLens audit runs in 1.7 CPU seconds across ten users and improves mean latent gap over both a demonstrated-rating policy and a proper online baseline. The learner uses only action demonstrations and never observes a reward or a loss.

This paper answers the one-dimensional local root anti-concentration questions posed by Balcan, Pegden, and Sharma in the context of online optimization of piecewise-Lipschitz functions. For a homogeneous feature curve and coefficients whose density relative to the uniform law on a symmetric convex body $K$ is bounded by $A$, we show that the worst-case interval-hitting constant equals $A$ times a section-averaged projective incidence speed. For cube-supported coefficients, this speed is equivalent, up to universal constants, to the projective Lipschitz constant. This yields a sharp, dimension-free characterization and removes the previous $\sqrt N$ loss. For monic degree-$d$ polynomials under arbitrary coefficient laws, we prove that the interval-hitting constant is finite if and only if the ordered real-root laws have bounded densities, with a factor-$d$ comparison that is sharp. Conditional and joint coefficient-space area formulas, together with a two-chart certificate, make this criterion verifiable for dependent and singular coefficient laws. We also give two graph-learning applications that complete the transition-to-regret chain. A cost-sensitive Gaussian-RBF harmonic classifier uses the projective incidence theorem and achieves expected regret $\widetilde O((An^2D e^{BD}/\ell+1)\sqrt T)$. A common-offset polynomial-kernel model uses rigid translation of the ordered roots and achieves $\widetilde O((qn^2κ+1)\sqrt T)$ regret, even when the induced coefficient law is singular in the ambient coefficient space.

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.

We study regret minimization for learning CDF-related objectives of the form $$g(x)\cdot\mathbb{P}_{X\sim\mathcal{D}}(X\le x),$$ over $[0,1]^2$, where $g$ is a known Lipschitz function and $\mathcal{D}$ is an unknown distribution. At each round $t$, the learner selects a point $x_t$ and observes the binary feedback $\mathbb{I}(X_t\le x_t)$, where $X_t\sim\mathcal{D}$. We design an algorithm achieving regret $\widetilde{\mathcal{O}}(T^{7/10})$, improving over the previous best-known bound of $\widetilde{\mathcal{O}}(T^{3/4})$ and showing that the curse of dimensionality can be at least partially lifted for this class of objectives, though a gap remains with the $Ω(T^{2/3})$ lower bound. As an application, our techniques yield the same $\widetilde{\mathcal{O}}(T^{7/10})$ regret bound for profit maximization in repeated bilateral trade with fixed prices.

We establish a $\widetildeΩ(d^{5/4}\sqrt T)$ lower bound on the minimax expected regret of stochastic bandit convex optimization of $1$-Lipschitz functions on the Euclidean ball. This presents the first nontrivial regret lower bound that grows faster than $d\sqrt{T}$ for this problem, establishing that stochastic bandit convex optimization is fundamentally harder than linear bandits. The hard class of convex functions we construct takes the following form in dimension $2d$: for an action $a = (a^1,a^2) \in \mathbb{B}^{2d}_2$, each function is the scaled soft maximum of a "tube", $r^{-1} \| W^\star a^1 - \frac{r}{8\varepsilon} a^2 \|_2$ (hyperparameterized by $\varepsilon,r$), and a squared distance function, $\frac12 \| a^1 - u^\star \|_2^2 - \frac12 \| u^\star \|_2^2$. Here, $W^\star \in \mathbb{R}^{d \times d}$ is an unknown linear transformation, and $u^\star \in \mathbb{R}^{d}$ is an unknown vector which must be learned to minimize the function. Observations are informative about $u^\star$ only when the learner's action lies near the tube determined by $W^\star$, satisfying $a^2 \approx \frac{8\varepsilon}{r} W^\star a^1$: thus the learner must either find this tube without knowing $W^\star$, or spend observations learning useful directions of $W^\star$. Formally, our regret analysis exploits this tradeoff by bounding the posterior spread of Fisher information matrices obtained under an adaptive sequence of actions. Together, these ingredients give a sample complexity lower bound of $\widetildeΩ(d^{5/2}/\varepsilon^2)$ to find an $\varepsilon$-optimal action, which translates to an $\widetildeΩ (d^{5/4} \sqrt{T})$ regret lower bound. We also extend this lower bound to the unconstrained setting where the action space is $\mathbb{R}^d$.

Stochastic nonconvex optimization is central to training deep networks and LLMs in modern machine learning. We give a black-box reduction from stochastic nonconvex optimization to ordinary static regret minimization in online convex optimization (OCO), thereby resolving the open problem posed by Chen and Hazan (2024). Our reduction maintains a predictable gradient tracker, while a black-box online learner $\mathcal{A}$ selects a preconditioner that transforms this tracker into the update direction. Given a $β$-smooth function with a range bounded by $M$ and an unbiased gradient oracle with variance bounded by $σ^2$, we bound the expected average squared gradient norm by $O(σ\sqrt{Mβ/T}+\sqrt{Mβ}\mathrm{Reg}_T(\mathcal{A})/T+\frac{Mβ}{T})$, where $\mathrm{Reg}_T(\mathcal{A})$ is the static regret of $\mathcal{A}$. Thus, any OCO oracle with $O(\sqrt{T})$ regret recovers the classical $O(T^{-1/2})$ convergence rate. We further extend the framework to nonsmooth nonconvex objectives, still relying only on ordinary static regret, and attain the optimal convergence rate for Goldstein-type stationarity. Finally, we conduct numerical experiments on nonconvex objectives to illustrate how the reduction exploits online-selected preconditioners while using the same stochastic-oracle budget as stochastic gradient descent.

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.

This paper studies the problem of regret minimization in Markovian bandits with \emph{non-observable states} and possibly \emph{constrained} decision epochs. The focus is restricted to a ``pure'' regret benchmark, that compares the performance of the learning algorithm to the best \emph{pure policy} which -- akin to optimal policies of stochastic bandits -- picks the optimal arm from start to finish without ever switching. We introduce a generalization of rested Markovian bandits, \emph{self-degrading Markovian bandits}, for which pure policies are always asymptotically optimal.We show that without prior knowledge on the underlying bandit, the regret of algorithms that switch arms rarely necessarily scales super-logarithmically for every bandit, i.e., as $ω(\log(T))$, where $T$ is the learning horizon. Despite the unreachability of the logarithmic regime, we design UCB-NOM, an optimistic algorithm inspired by UCB, of which the regret is nearly logarithmic. Lastly, we show that given prior knowledge on the Markovian bandit in the form of a bound on the bias functions of its arm, a proper instantiation of UCB-NOM achieves $O(\log(T))$ regret. We further show that this prior knowledge allows for a $O(\sqrt{T \log(T)})$ worst-case regret bound for UCB-NOM. Notably, our regret bounds do not depend on the number of states of the underlying Markov chains. Our findings suggest that the non-observability of states is a mild inconvenience in self-degrading Markovian bandits.

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.

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.

LLM wiki systems compile knowledge into pre-filled KV caches for efficient inference, but assume a static corpus -- an assumption that fails whenever the underlying information landscape evolves. We formalize Streaming Knowledge Compilation: given a document stream, a fixed token budget, and future queries unknown at ingestion time, maintain a compiled wiki that minimizes cumulative regret against an offline oracle with perfect foresight. The enabling insight is a materiality signal $φ_t(k,n)\in[0,1]$ that scores document importance for entity $k$ at time $t$, acting as a query-relevance surrogate for proactive pinning before queries arrive; we prove an $O(\sqrt{T\log K})$ regret bound where $\varepsilon=\mathbb{E}[|φ_t-\hatφ_t|]$ is the only domain-specific quantity. We instantiate in two domains: finance, where $φ_t$ is abnormal stock volatility predicted by frozen Llama 3.1 8B classification head (AUROC = 0.728 on 76K articles, strict temporal split; $1.49\times$ higher realized forward volatility for predicted-material articles); and Wikipedia, where $φ_t$ is the Abnormal Edit Ratio (AER), a cross-sectionally normalized edit velocity -- showing the same algorithm generalizes beyond the finance domain. End-to-end QA evaluation on 173 matched pairs (finance) and 119 (Wikipedia) reveals a pervasive LLM-as-judge confound on post-training knowledge, establishing that regret analysis -- not absolute QA scores -- is the reliable evaluation metric for compiled knowledge systems. Finance cumulative regret converges to -20.0 (-0.12/step); Wikipedia to +16.0 (+0.13/step), with the positive sign confirming that Wikipedia edit content is genuinely post-training -- richer context consistently improves scores (No Wiki 3.80 vs. Oracle 4.74) -- and eliminates this confound. The $O(\sqrt{T\log K})$ guarantee applies to any domain where knowledge gaps can be predicted from streaming signals.

We study the two-action apple-tasting problem with switching costs against an oblivious adversary. In an equivalent normalized formulation, at each round the learner chooses between a revealing action and a blind action: the revealing action gives reward $0$ and reveals the hidden value $x_t\in[-1,1]$ of the blind action; the blind action gives reward $x_t$ but reveals nothing. The learner pays one unit whenever they switches actions, and regret is measured against the best fixed action in hindsight. General feedback-graph algorithms with switching costs give $\widetilde O(T^{2/3})$ regret guarantees for this problem. The two-action apple-tasting graph was the natural candidate for the missing $Ω(T^{2/3})$ obstruction in the switching-cost classification: such a lower bound would have transferred to a large family of still-unclassified feedback graphs. We prove that this obstruction is not there: the oblivious minimax expected regret for this problem satisfies $$\frac{1}{2\sqrt3}\cdot\sqrt T \le R_T^\star \le 2\sqrt{3}\cdot \sqrt{T}.$$

Curvature adaptivity is a classical theme in online optimization: for convex Lipschitz losses, adaptive methods interpolate between the optimal $O(\sqrt{T})$ regret for general convex losses and $O(\log T)$ regret under strong convexity. Recent work has shown that Follow-the-Perturbed-Leader (FTPL) achieves optimal $O(\sqrt{T})$ regret even for online non-convex Lipschitz losses, assuming access to an approximate offline-optimization oracle, but these guarantees do not exploit curvature. We show that FTPL can be made curvature-adaptive in the non-convex setting, without knowing in advance how curvature will accumulate over time. Our algorithm replaces the fixed perturbation scale of standard FTPL with a time-varying scale chosen using only past information. We give a simple follow-the-leader tuning rule for this scale and show that it competes, up to constants, with the best choice in hindsight. The resulting method achieves $O(\sqrt{T})$ regret for arbitrary non-convex Lipschitz losses and improves as cumulative curvature grows; with sufficiently accurate oracle calls, it achieves $O(\log T)$ regret when cumulative curvature grows linearly, which includes the classical strongly convex regime. We complement these upper bounds with matching lower bounds for prescribed cumulative-curvature sequences, already for one-dimensional convex losses, showing that the tradeoff between worst-case non-convex regret and curvature-driven fast rates is intrinsic.

External regret certifies stability only against replacing one's behavior by a fixed alternative. In a quantum game, this misses a natural physical move: a player can apply a local completely positive trace-preserving (CPTP) map to the state it actually received or prepared. We introduce coherent swap regret as the regret benchmark against all such local CPTP deviations, and give an algorithm achieving $O(\sqrt{dT\log d})$ coherent swap regret via entropic mirror ascent on the CPTP Choi slice with a fixed-point play rule. The main result is a three-level deviation-class landscape. Replacement channels recover ordinary external regret at rate $Θ(\sqrt{T\log d})$. Unital channels, including unitary deviations and mixtures of unitaries, have zero minimax regret. Deterministic measurement-and-preparation channels already force $Ω(\sqrt{dT\log d})$ regret in the moderate-horizon regime, and this rate is also sufficient for all CPTP deviations. Thus the hardness comes from non-unital use of the recommendation register, not from quantum coherence alone. As an application, decentralized full-information learning in finite quantum games reaches an $\varepsilon$-approximate separable quantum correlated equilibrium after $T=O(\max_i d_i\log d_i/\varepsilon^2)$ rounds. We identify these equilibria with channel-proofness of mediated quantum recommendation protocols, give an SDP audit for local CPTP exploitability applicable to arbitrary finite-dimensional states, and include a probing-bandit extension with pseudo-regret $O(d^{4/3}T^{2/3}(\log d)^{1/3})$ under Haar-random pure-state probes.

We study stochastic linear bandits under a natural combination of batching and communication constraints: the time horizon is partitioned into batches of equal size $B$, and during each batch the learner sends $B$ requested arm pulls to an agent, who then observes the corresponding $B$ rewards and responds with a single bit of feedback to the learner. For each batch, the learner specifies the 1-bit quantization rule the agent uses, which may depend on all previously received bits but not on any past rewards directly. This setting addresses a significant yet unexplored ``middle ground'' between previous models having per-round quantization only or total bit budgets only. We establish a minimax lower bound showing that $Ω(B\min\{d,\log\lvert \mathcal{A} \rvert\})$ regret is unavoidable due to the 1-bit communication bottleneck, even in the absence of noise. Combined with standard statistical limits, this yields a general lower bound of $\widetildeΩ(B\min\{d,\log\lvert \mathcal{A} \rvert\} + \sqrt{dT \min\{d,\log\lvert \mathcal{A} \rvert\}})$. We develop two phased-elimination algorithms based on $G$-optimal designs and 1-bit mean estimation. The first achieves $\widetilde{O}(dB + d\sqrt{T})$ regret, matching the lower bound up to logarithmic factors when $\lvert \mathcal{A} \rvert = \exp(Ω(d))$, and the second incorporates a safe-arm identification and warm-start procedure to obtain $\widetilde{O}(B\log\lvert \mathcal{A} \rvert + d^{3/2}\sqrt{B} + \sqrt{dT\log\lvert \mathcal{A} \rvert})$ regret, which is near-optimal in broad scaling regimes of $(\lvert \mathcal{A} \rvert, B, d, T)$. Together, our results demonstrate that a single bit of feedback per batch suffices to nearly match the minimax regret of unconstrained linear bandits in broad scaling regimes, even for batch sizes as large as $Θ(\sqrt{T})$.

We study adversarial online learning with hidden-convex losses, i.e., nonconvex losses that become convex after a nonlinear reparameterization. Ghai, Lu and Hazan (2022) proved that, under geometric and smoothness assumptions, online gradient descent (OGD) on such nonconvex losses approximately simulates online mirror descent (OMD) on the underlying convex losses with a suitable regularizer, yielding $\mathcal{O}(T^{2/3})$ regret. They left open whether the optimal $Θ(\sqrt{T})$ regret from online convex optimization can be recovered in this hidden-convex setting. We answer this question affirmatively. More specifically, via a sharper discrete-time algorithmic equivalence argument, we prove that OGD achieves $\mathcal{O}(\sqrt{T})$ regret under the same assumptions, matching the optimal worst-case rate for adversarial online convex optimization. We also address another open question of Ghai, Lu and Hazan (2022) by clarifying the geometry required for this algorithmic equivalence. We replace the diagonal-Jacobian sufficient condition with a necessary-and-sufficient Hessian compatibility condition, thereby expanding the class of admissible reparameterizations. We complement our tight regret bound with a lower bound showing that the Hessian compatibility assumption is essential for OGD; when it fails, we construct a smooth reparameterization and an adversarial sequence of hidden-convex losses for which OGD suffers $Ω(T)$ regret. Finally, we extend our analysis to one-point bandit feedback and prove a $\mathcal{O}(T^{3/4})$ expected regret bound for bandit OGD with spherical smoothing, matching its classical rate on convex losses.

We study constrained online convex optimization with adversarial time-varying constraints. At each round the learner acts before observing the loss and constraint, and is compared with the best fixed action satisfying all constraints in hindsight. The goal is to obtain minimax-optimal regret while controlling cumulative constraint violation (CCV). Prior algorithms achieved $O(\log T)$ regret with $O(\sqrt{T\log T})$ CCV for strongly convex losses, and $O(\sqrt{T})$ regret with $O(\sqrt{T}\log T)$ CCV for convex losses. We propose NP-OGD, an iterated nested-projection algorithm. For strongly convex losses it achieves $O(\log T)$ regret and $O(\log T)$ CCV; for convex losses it achieves $O(\sqrt{T})$ regret and $O(\sqrt{T})$ CCV. The analysis relies on a geometric movement bound: after lifting the nested projected-gradient trajectory to one higher dimension, the lifted path is self-contracted under a nonstandard norm, so a finite-length theorem for self-contracted curves controls the total projection movement. We also prove complementary lower bounds using layered sphere packings. For strongly convex losses, any online algorithm with polynomially sublinear regret can incur CCV at least $Ω((\log T)^{(d-1)/(d+1)}/\log\log T)$. For convex losses, we prove CCV lower bounds $Ω(T^{(d-1)/(2(d+3))})$ for weakly adaptive algorithms and $Ω(T^{(d-1)/(2d)})$ for NP-OGD. Finally, for the constrained experts special case over $N$ experts, an active Hedge algorithm attains $O(\sqrt{T\log N})$ regret and $O(N)$ CCV, with a matching minimax CCV lower bound for sufficiently large horizons.

We study an online market-making problem in which a learner sequentially posts bid and ask prices for a single asset while interacting with traders holding private valuations. Unlike existing online learning formulations that assume fully censored feedback, we introduce an action-dependent feedback model inspired by real limit order books: when a trade occurs, the trader's valuation remains hidden, whereas when no trade occurs, informative feedback about supply and demand is revealed. We show that this additional information fundamentally changes the learnability of the problem. In the stochastic setting with i.i.d. market prices, we propose an elimination-based algorithm that achieves $O(\sqrt T)$ regret with high probability, without requiring any smoothness assumptions on the distribution of trader valuations. We then extend this result to a broad class of mean-reverting price processes by considering both local, autoregressive dynamics and a weaker global drift condition based on cumulative deviations from the mean. Under either assumption, we establish high-probability $O(\sqrt T)$ regret bounds, relying on a new concentration inequality of independent interest. Finally, in the adversarial setting with oblivious prices, we design an explore-then-perturb algorithm that guarantees $O(T^{2/3})$ regret in expectation. Our results quantify the value of observing the order book in online market making and demonstrate that even limited, action-dependent feedback can substantially improve regret guarantees compared to standard bandit feedback models.

This paper addresses the problem of learning to sparsify stochastic linear bandits, where a decision-maker sequentially selects actions from a high-dimensional space subject to a sparsity constraint on the number of nonzero elements in the action vector. The key challenge lies in minimizing cumulative regret while tackling the potential NP-hardness of finding optimal sparse actions due to the inherent combinatorial structure of the problem. We propose an adaptively phased exploration and exploitation algorithmic framework, utilizing ordinary least squares for parameter learning and specialized subroutines for sparse action selection. When the action set is a Euclidean ball, optimal sparse actions can be efficiently computed, enabling us to establish a $\tilde{\mathcal{O}}(d\sqrt{T})$ regret, where $d$ is the dimension of the action vector and $T$ is the time horizon length. For general convex and compact action sets where finding optimal sparse actions is intractable, we employ a greedy subroutine. For general strongly convex action sets, we derive a $\tilde{\mathcal{O}}(d \sqrt{T})$ $α$-regret; for general compact sets lacking strong convexity, we establish a $\tilde{\mathcal{O}}(d T^{2/3})$ $α$-regret, where $α$ pertains to the approximation ratio of the greedy algorithm. Finally, we validate the performance of our algorithms using extensive experiments including an application to recommendation system.

We extend quantum state tomography with minimal cumulative disturbance, first investigated in [arXiv:2406.18370], to arbitrary finite-dimensional pure states. A learner sequentially receives fresh copies of an unknown pure state, chooses a rank-one projector for each copy using the previous outcomes, and performs the corresponding two-outcome projective measurement. The goal is to learn the state while keeping the chosen projectors close to the unknown state in order to minimize disturbance. The qubit solution relies on the special geometry of the Bloch sphere and does not extend directly to qudits, where pure states form a curved manifold. We show that this obstruction can be overcome by working locally on the pure-state manifold. The algorithm proceeds in epochs. In each epoch, it fixes a current estimate, measures pairs of nearby rank-one projectors obtained by moving in opposite tangent directions, and takes differences of the corresponding outcomes. This gives an exact linear observation of the tangent component of the error. The resulting local linear models are combined with a robust variance-adaptive estimator and a hot-start regularization that transfers precision across epochs. For every unknown pure state in dimension $d$, after $T$ measured copies, our protocol achieves cumulative regret $\mathcal{O}(d^3\log^2 T)$, and at each intermediate time $t\leq T$ its current estimate has online infidelity $\mathcal{O}(d^3\log(T)/t)$. Hence, pure-state tomography with essentially no cumulative disturbance is not a peculiarity of qubits but a geometric phenomenon that persists for qudits.

We study \emph{multi-armed bandits} (MABs) augmented with \emph{best-action queries}, in which the learner may additionally query an oracle that reveals the best arm in the current round. This setting was recently characterized by Russo et al. [2024] in the \emph{full-feedback} model, where the learner observes the rewards of all arms after each round. They show that, in both \emph{stochastic} and \emph{adversarial} environments, $k$ best-action queries reduce the optimal $\widetilde{\mathcal{O}}(\sqrt{T})$ regret to $\widetilde{\mathcal{O}}(\min\{T/k,\sqrt{T}\})$. Whether this improvement extends to the more realistic \emph{bandit-feedback} model -- where the learner observes only the reward of the played arm -- was left as an open problem. We fully resolve this question. When rewards are stochastic but correlated among arms, we show that the full-feedback result does not carry over: any algorithm must incur regret at least $Ω(\sqrt{T-k})$. This lower bound directly extends to adversarial environments. On the positive side, we show that $\widetilde{\mathcal{O}}(\min\{T/k,\sqrt{T-k}\})$ regret is still achievable when rewards are stochastic and i.i.d., and establish a matching lower bound, up to logarithmic factors. Together, these results provide a complete characterization of the benefits of \emph{best-action queries} in the \emph{bandit-feedback} model.

Convex Optimization with Nested Evolving Feasible Sets (CONES)} is considered where the objective function $f$ remains fixed but the feasible region evolves over time as a nested sequence $S_1 \supseteq S_2 \supseteq \cdots \supseteq S_T$. The goal of an online algorithm is to simultaneously minimize the regret with respect to hindsight static optimal benchmark and the total movement cost while ensuring feasibility at all times. CONES is an optimization-oriented generalization of the well-known nested convex body chasing problem. When the loss function is convex, we propose a lazy-algorithm and show that it achieves $O(T^{1-β}), O(T^β)$ simultaneous regret and movement cost for any $β\in (0,1]$, over a time horizon of $T$. When the loss function is strongly convex or $α$-sharp, we propose an algorithm Frugal that simultaneously achieves zero regret and a movement cost of $O(\log T)$. To complement this, we show that any online algorithm with $o(T)$ regret has a movement cost of $Ω(\log{T})$ for both cases, proving optimality of Frugal.

Reinforcement Learning from Human Feedback (RLHF) has become a cornerstone technique for post-training large language models. While most existing approaches rely on the reverse KL-regularization, recent empirical studies have begun exploring alternative divergences (e.g., forward KL, chi-squared) as regularizers in RLHF. However, a unified theoretical understanding of general $f$-divergence regularization remains under-explored. To fill this gap, this work develops a comprehensive theoretical framework for online RLHF with a general $f$-divergence regularized objective. Rather than treating each possible divergence function individually, we adopt a holistic perspective across the entire function class and propose two algorithms based on distinct sampling principles. The first extends the classical optimism principle with a carefully designed exploration bonus, while the second introduces a new method that exploits the sensitivity of the optimal policy to reward perturbations under $f$-divergence regularization. Theoretical analysis shows that $O(\log T)$ regret and $O(1/T)$ sub-optimality gap are achievable, establishing provable efficiency of both algorithms and, to the best of our knowledge, the first performance bounds for online RLHF under general $f$-divergence regularization.

We propose a new framework for meritocratic fairness in budgeted combinatorial multi-armed bandits with full-bandit feedback (BCMAB-FBF). Unlike semi-bandit feedback, the contribution of individual arms is not received in full-bandit feedback, making the setting significantly more challenging. To compute arm contributions in BCMAB-FBF, we first extend the Shapley value, a classical solution concept from cooperative game theory, to the $K$-Shapley value, which captures the marginal contribution of an agent restricted to a set of size at most $K$. We show that $K$-Shapley value is a unique solution concept that satisfies Symmetry, Linearity, Null player, and efficiency properties. We next propose K-SVFair-FBF, a fairness-aware bandit algorithm that adaptively estimates $K$-Shapley value with unknown valuation function. Unlike standard bandit literature on full bandit feedback, K-SVFair-FBF not only learns the valuation function under full feedback setting but also mitigates the noise arising from Monte Carlo approximations. Theoretically, we prove that K-SVFair-FBF achieves $O(T^{3/4})$ regret bound on fairness regret. Through experiments on federated learning and social influence maximization datasets, we demonstrate that our approach achieves fairness and performs more effectively than existing baselines.

We study the $K$-armed logistic bandit problem, where at each round, the agent observes $K$ feature vectors associated with $K$ actions. Existing approaches that achieve a rate-optimal $\tilde{\mathcal{O}}(\sqrt{dT})$ regret bound rely heavily on context diversity assumptions, such as strict positivity of the minimum eigenvalue of a context covariance matrix. These assumptions, however, impose strong restrictions on the context process, as they rule out the situation where the context vectors are concentrated in a low-dimensional subspace. In this paper, we propose SupSplitLog, which, to the best of our knowledge, is the first algorithm for logistic bandits that achieves $\tilde{\mathcal{O}}(\sqrt{dT})$ regret without any context diversity assumption. The key idea is to split the collected samples into two disjoint subsets when constructing estimators; one is used to compute an initial-point estimator, while the other is used to apply a Newton-type one-step correction procedure. The splitting rule is carefully designed to balance the accuracy requirements of the initial-point estimator and the one-step correction procedure. Moreover, SupSplitLog strictly improves on the existing algorithms in terms of the dependence on dimension $d$ in the regret upper bound. Furthermore, SupSplitLog can be adapted simply to deduce a regret bound that grows with a data-dependent complexity measure, avoiding a direct dependence on $d$, which is favorable when the context vectors are concentrated in a low-dimensional subspace. We also provide experimental results that demonstrate numerically the superiority of our algorithm, validating the theoretical results.

We study the problem of adaptive control of the stochastic linear quadratic regulator (LQR) with constraints that must be satisfied at every time step. Prior work on the multidimensional problem has shown $\tilde{O}(T^{2/3})$ regret and satisfaction of robust constraints, leaving open the question of whether $\tilde{O}(\sqrt{T})$ regret can be attained in the constrained LQR setting. We contribute to this problem by showing $\tilde{O}(\sqrt{T})$ regret and satisfaction of chance constraints. This type of constraints allow us to handle unbounded noise and also enable analytical techniques not directly applicable to robust constraints. Our proposed algorithm for this problem uses an SDP to select an optimistic policy, and then "scales back" this policy until it is verifiably-safe. Our theoretical analysis establishes regret and constraint guarantees via a key lemma that bounds the system covariance in terms of the chosen policy. This covariance-based analysis is in contrast with the cost-to-go based analysis that is typically used in adaptive LQR.

Consider betting against a sequence of data in $[0,1]$, where one is allowed to make any bet that is fair if the data have a conditional mean $m_0 \in (0,1)$. Cover's universal portfolio algorithm delivers a worst-case regret of $O(\ln n)$ compared to the best constant bet in hindsight, and this bound is unimprovable against adversarially generated data. In this work, we present a novel mixture betting strategy that combines insights from Robbins and Cover, and exhibits a different behavior: it eventually produces a regret of $O(\ln \ln n)$ on almost all paths (a measure-one set of paths if each conditional mean equals $m_0$ and intrinsic variance increases to $\infty$), but has an $O(\log n)$ regret on the complement (a measure zero set of paths). Our paper appears to be the first to point out the value in hedging two very different strategies to achieve a best-of-both-worlds adaptivity to stochastic data and protection against adversarial data. We contrast our results to those in Agrawal and Ramdas [2026] for a sub-Gaussian mixture on unbounded data: their worst-case regret has to be unbounded, but a similar hedging delivers both an optimal betting growth-rate and an almost sure $\ln\ln n$ regret on stochastic data. Finally, our strategy witnesses a sharp game-theoretic upper law of the iterated logarithm, analogous to Shafer and Vovk [2005].

We study \emph{online episodic Constrained Markov Decision Processes} (CMDPs) under both stochastic and adversarial constraints. We provide a novel algorithm whose guarantees greatly improve those of the state-of-the-art best-of-both-worlds algorithm introduced by Stradi et al. (2025). In the stochastic regime, \emph{i.e.}, when the constraints are sampled from fixed but unknown distributions, our method achieves $\widetilde{\mathcal{O}}(\sqrt{T})$ regret and constraint violation without relying on Slater's condition, thereby handling settings where no strictly feasible solution exists. Moreover, we provide guarantees on the stronger notion of \emph{positive} constraint violation, which does not allow to recover from large violation in the early episodes by playing strictly safe policies. In the adversarial regime, \emph{i.e.}, when the constraints may change arbitrarily between episodes, our algorithm ensures sublinear constraint violation without Slater's condition, and achieves sublinear $α$-regret with respect to the \emph{unconstrained} optimum, where $α$ is a suitably defined multiplicative approximation factor. We further validate our results through synthetic experiments, showing the practical effectiveness of our algorithm.

We study the dynamic allocation of indivisible resources to strategic agents under long-term constraints, where the planner aims to maximize social welfare, satisfy multiple constraints, and elicit near-truthful reports. We find standard primal-dual methods fragile in this setting: agents easily manipulate their reports to distort dual variables, sacrificing social efficiency for individual utility. To address this, we propose the Incentive-Aware Primal-Dual (IAPD) framework. On the primal side, we integrate three components to suppress manipulation: a VCG-based payment neutralizes immediate misreporting benefits, while epoch-based lazy updates and random exploration together ensure potential future gains are outweighed by immediate penalties. On the dual side, to overcome a learning barrier due to lazy updates -- which we call the "price of incentives" -- we design a novel optimistic online learning algorithm, O-FTRL-FP. It utilizes a fixed-point oracle to resolve the circular dependency between optimistic dual variables and the resulting allocations. Ultimately, our mechanism attains $\tilde{\mathcal O}(\sqrt T)$ social welfare regret, satisfies all long-term constraints, and induces a near-truthful equilibrium. It also smoothly generalizes to multi-unit multi-demand allocation problems. Notably, this $\tilde{\mathcal O}(\sqrt T)$ regret near-matches the non-strategic $Ω(\sqrt T)$ lower bound, demonstrating that incentive-awareness can be accommodated at nearly no cost.

The $K$-Max combinatorial multi-armed bandit problem arises in applications such as recommendation and distributed decision making, where the reward is determined by the maximum outcome among $K$ selected arms. When outcomes are continuous and only the maximum value together with the winner's index is observed, this problem introduces unprecedented difficulties including discretization errors, non-deterministic tie-breaking, and severe estimation biases. To overcome these barriers, we introduce DCK-UCB, an efficient algorithm combining adaptive discretization with bias-corrected confidence bounds. We prove that DCK-UCB achieves a $\widetilde{O}(T^{3/4})$ regret bound, the first sublinear guarantee in this setting. Numerical experiments show strong performance over baseline methods. Furthermore, for the specific case of exponential distributions under full-bandit feedback, we propose the MLE-Exp algorithm that attains a near-optimal $\widetilde{O}(\sqrt{T})$ regret bound. This work establishes fundamental theoretical guarantees and provides a powerful algorithmic solution for continuous combinatorial bandits.

Online linear programming (OLP) has gained significant attention from both researchers and practitioners due to its extensive applications such as online auctions, network revenue management, order fulfillment and advertising. Existing OLP algorithms fall into two categories: LP-based algorithms and LP-free algorithms. The former typically guarantees better performance but requires solving a large number of LPs, which could be computationally expensive. In contrast, LP-free algorithms only require first-order computations but induce a worse performance. In this work, we bridge the gap between these two extremes by proposing a well-performing algorithm that solves LPs at a few selected time points and conducts first-order computations at other time points. Specifically, for the case where the inputs are drawn from an unknown finite-support distribution, the proposed algorithm achieves a constant regret (even for the hard "degenerate" case) while solving LPs only $O(\log\log T)$ times over the time horizon $T$. Moreover, when we are allowed to solve LPs only $M$ times, we design the corresponding schedule such that the proposed algorithm can guarantee a nearly $O\left(T^{(1/2)^{M-1}}\right)$ regret. Our work highlights the value of resolving both at the beginning and the end of the selling horizon, and provides a novel framework to prove the performance guarantee of the proposed policy under different infrequent resolving schedules. Numerical experiments are conducted to demonstrate the efficiency of the proposed algorithms.

Stochastic optimization is a widely used approach for optimization under uncertainty, where uncertain input parameters are modeled by random variables. Exact or approximation algorithms have been obtained for several fundamental problems in this area. However, a significant limitation of this approach is that it requires full knowledge of the underlying probability distributions. Can we still get good (approximation) algorithms if these distributions are unknown, and the algorithm needs to learn them through repeated interactions? In this paper, we resolve this question for a large class of ''monotone'' stochastic problems, by providing a generic online learning algorithm with $\sqrt{T\log(T)}$ regret relative to the best approximation algorithm (under known distributions). Importantly, our online algorithm works in a semi-bandit setting, where in each period, the algorithm only observes samples from the random variables that were actually probed. Moreover, our result extends to settings with censored and binary feedback, where the policy only observes truncated or thresholded versions of the probed variables. Our framework applies to several fundamental problems such as prophet inequality, Pandora's box, stochastic knapsack, single-resource revenue management and sequential posted pricing.