Regret

Region-level retrieval aims to align user-specified image regions with relevant regions or textual descriptions, playing a crucial role in realworld applications such as e-commerce product search and RAG. Although recent Large Multimodal Models (LMMs) have made significant strides in multimodal retrieval, they primarily focus on global-level tasks and struggle to capture effective region-level representations. To bridge this gap, we present RegRet, an LMM-based Region-level Retrieval framework that enhances the regional representations without compromising overall global retrieval performance. At its core, RegRet integrates a Region-Aware Encoder to capture detailed regional features while balancing them with the global background context. To further enhance the fine-grained understanding and discriminability of representations, we design a multi-stage training pipeline that includes detailed localized captioning and regional contrastive learning tasks. In addition, considering the absence of region-level contrastive training data and the limited diversity of evaluation tasks in current benchmarks, we introduce the REGMB benchmark. It comprises 225k contrastive pairs, covering four multimodal retrieval tasks. Extensive experiments validate the effectiveness of our approach. RegRet outperforms strong baselines in the zero-shot setting. Further training with contrastive learning leads to an average improvement of more than 20% on both REGMB and public benchmarks, while achieving comparable or better results on global-level retrieval tasks.

Online gradient descent is usually studied through external regret, where the learner competes with fixed alternatives. Recent work shows that first-order methods control richer action-dependent deviations. We ask for a geometric characterization of the deviations with respect to which online gradient descent, mirror descent, and follow-the-regularized-leader (FTRL) achieve no regret. We identify exactness as the common principle. Exactness means that the relevant displacement field is generated by a scalar potential, or equivalently that the associated one-form is exact in the geometry used by the algorithm. This geometry depends on the algorithm. For gradient descent it is Euclidean geometry, for mirror descent it is the geometry induced by the regularizer, and for FTRL it is the cumulative dual state. Under mild regularity conditions, exactness yields sublinear regret, while nonzero circulation provides the complementary obstruction and leads to linear regret. This gives a unified geometric framework for understanding the deviation classes controlled by these algorithms and reveals that different first-order methods can control genuinely different classes of deviations. These deviation classes have direct consequences for learning, particularly in games. We study the equilibrium notions induced by exact-form deviations and introduce conservative correlated equilibrium, reflecting both the conservative geometry of the underlying displacement fields and the restricted family of deviations available to the players. We characterize its relation to correlated equilibrium, determine when the resulting equilibrium notions coincide and when they separate, and show how these relationships depend on the geometry and the learning algorithm. Overall, this work gives a unified geometric account of what first-order online learning algorithms are no-regret with respect to, beyond fixed comparators.

We introduce an uncoupled learning algorithm which, when employed by all players of an arbitrary $N$-player normal form game with up to $K$ actions per player, guarantees $O(N^3\log^2 K)$ individual regret, uniformly over the horizon of play. The proposed algorithm - which we call higher-order optimism with discounting (HOOD) is a variant of optimistic follow-the-regularized-leader (OptFTRL) that combines a discounted $(N+1)$-th order predictor with entropic regularization over a suitable "lifting" of the game's strategy space. This combination of ingredients is purposefully designed to dampen large oscillations of the induced sequence of play in a controlled manner, removing in this way a key stumbling block of previous attempts to achieve constant regret in general games. Our approach bears several striking similarities to the concurrent - and completely independent - work of Liu, Farina, and Ozdaglar (arXiv:2608.31166), who very recently derived an $O(N^{21}\log^{4} K)$ regret bound through the use of higher-order optimism and an exponential moving average estimator.

We study a variant of the Thompson Sampling (TS) algorithm, called $α$-TS, for solving stochastic generalized linear bandit problems. Existing analyses of TS require inflating the posterior variance to derive near-optimal regret guarantees. We formalize the idea of variance inflation by introducing $α$-TS that uses a fractional or $α$-posterior instead of the standard posterior. Our main contribution is to identify general regularity conditions on the prior and reward distributions that enable a regret analysis of $α$-TS without assuming any tractable approximation of the posterior distribution, unlike previous works. For a specific choice of $α\propto d^{-1}$, our general regret bound yields the best known regret bound of $O(d^{3/2}\sqrt{T}\log T)$ for both the exponential and sub-Gaussian families of reward distributions. We further provide an $α$-dependent lower bound showing that the regret constant depends on the product $αd$, and that when $α\propto d^{-1}$ the regret scales as $Ω(d^{3/2}\sqrt{T})$, explaining the origin of the $d^{3/2}$ factor in the upper bound. Our proof technique adapts and combines recent advancements in the analysis of linear bandit problems with first- and second-order posterior concentration theory from the Bayesian statistics literature.

Uncoupled no-regret dynamics provide a decentralized route to equilibrium, but prior guarantees for individual regret retain a polylogarithmic dependence on the horizon. We remove this dependence for every finite $N$-player normal-form game under full-information feedback. We introduce \emph{ECHO-OFTRL}: optimistic follow-the-regularized-leader (OFTRL) equipped with an EMA cascade for high-order optimism (ECHO), where EMA denotes exponential moving average. The algorithm is deterministic and fully uncoupled. If $m_{\max}$ denotes the largest action-set size, then, simultaneously for every horizon $T\geq1$, it guarantees that each of the $N$ players in the game incurs regret upper bounded by $O(\textrm{poly}(N, \log m_{\max}))$. Our algorithm leverages a new form of optimism inspired by modern filter design.

We determine the minimax cumulative log-loss regret of a finite-alphabet, exogenously driven source with genuinely infinite input memory: independent Rademacher inputs $(U_t)$ are observed sequentially and the next binary mark has logit $\sum_{j\ge1}θ_jU_{t+1-j}$, the unknown coefficients obeying a summable envelope $|θ_j|\le r_j$, $\sum_jr_j\le B$. At horizon $T$, lag $j$ can move the logit by at most $r_j$ and is exercised in only $n_{T,j}=(T-j+1)_+$ rounds, and the two limitations combine into the sum $Γ_T(r)=\sum_{j\le T}\log(1+n_{T,j}r_j^{2})$. One coordinate-localised Bayesian mixture achieves $R_T(r)\le CΓ_T(r)$ for \emph{every} summable envelope with $C$ universal. Our main result is a matching nonasymptotic converse for the canonical exponential and polynomial envelopes; its new ingredients are a modular finite-sample information bound for logistic experiments with an exogenous random design, and a conditioning estimate for the overlapping Toeplitz lag matrix obtained by exhibiting each off-diagonal Gram sum as a sum of independent Rademacher variables indexed by the edges of a forest, needing neither local asymptotic normality nor any spectral theorem for random Toeplitz matrices. So $Γ_T(r)$ is the minimax regret scale here, giving $Θ(α^{-1}\log^{2}T)$ for $r_j=Ae^{-αj}$ and $Θ(T^{1/(2s)})$ for $r_j=Aj^{-s}$, $s>1$ --- the latter without the extra $(\log T)^{1-1/(2s)}$ factor any window-truncation analysis pays. We also show memory decay cannot determine regret, and that a profile-scaled online Newton predictor attains $O_B(Γ_T(r))$ in polynomial time per round.

This paper studies the regret analysis for parallel Gaussian process (GP) bandit optimization. The known regret upper bounds for the widely used GP batched upper confidence bound and GP batched Thompson sampling (GP-BTS) suffer from a multiplicative factor with respect to the batch size $Q$. To avoid this degradation, existing analyses require a polynomial number of uncertainty sampling (US) for $Q$ at the beginning of optimization. However, this initial US phase is often ineffective in practice. This paper shows that the regret upper bound without the multiplicative factor on $Q$ can be achieved without the initial US phase, using GP-BTS as an example. Furthermore, we show much better regret upper bounds in the noiseless setting than in the noisy setting, as in the sequential GP bandit setting.

Motivated by decentralized applications, we study cooperative multi-agent bandits in continuous (Lipschitz) action spaces when the Lipschitz constant is unknown. We consider three information structures: (A)~unobserved actions with common rewards, (B)~observed actions with independent rewards, and (C)~unobserved actions with independent rewards. In each case we design and analyze an algorithm that estimates the Lipschitz constant, chooses a discretization of the joint action space, and applies a cooperative bandit method to the induced discrete problem. Players never communicate once learning starts, so the central difficulty is that they must reach the \emph{same} discretization from their own data. We prove regret guarantees showing that common rewards and observable actions each supply this agreement for free, and that in their absence agreement can still be bought, through a dithered quantization of the estimate, at no cost in the leading order of the regret.

This note aims to serve as an entry point to the literature on learning in games, a topic with significant theoretical appeal and a wide range of applications -- from machine learning and data science to economics and beyond. Our presentation is structured around two complementary viewpoints: We first consider a single agent -- the learner -- engaged in a sequential decision process in an unknown, non-stationary, and possibly adversarial environment. We then examine what happens when the environment is shaped by the decisions of several interacting agents, not necessarily aware of each other's actions or goals, and all seeking to improve their individual rewards. In this general context, we examine a family of regularized learning policies based on best-responding to the past history of play, up to a regularization penalty intended to encourage exploration and prevent over-commitment to suboptimal choices. In the single-agent setting, we present some basic regret bounds for regularized learning in adversarial multi-armed bandits; in the multi-agent setting, we describe an ergodic equilibrium convergence result for zero-sum games in the spirit of classical results on fictitious play, as well as a "folk theorem" linking strategic and dynamic notions of stability -- Nash equilibria and attracting points of regularized learning, respectively. We pay special attention to the information available to the players and, through a unified analysis framework, we study both oracle- and payoff-based (bandit) methods. Our goal is to provide a coherent and comprehensible -- albeit, by necessity, not comprehensive -- account of some recent ideas in the field, and to discuss their implications for the study of rationality.

Bandit algorithms generate data for downstream inference, but adaptive sampling biases post-bandit sample means. We analyze this bias for stable index algorithms, including UCB1 and its generalizations, and derive sharp leading-order expressions for the sample-mean bias and expected $Z$-statistic, in bandit experiments of fixed horizon $T$. Our characterization reveals the algorithmic origin of bias through a key index-function-dependent quantity, which we term effective exploration rate. For example, under UCB1, the effective exploration rate is of order $\sqrt{\log T}$, and the standardized bias of any arm (that is not uniquely optimal) decays at the extremely slow rate $1/\sqrt{\log T}$. We also show how the choice of the index function affects both regret and bias, which reveals a regret-bias trade-off: more exploratory algorithm reduces bias but increases regret. We further show how bias most severely distorts confidence intervals and hypothesis tests when the tested arm is one of the tied-optimal arms. Our sharp characterization for bias uses a novel empirical fluid approximation of the algorithm's sampling dynamics, which may be of independent interest.

This paper studies the policy gradient update for a multi-arm bandit problem in diffusion environment that is described by a stochastic differential equation (SDE) under the continuous-time reinforcement learning framework by Wang et al. (2020), Jia and Zhou (2022b). With the logit parameterization for the stochastic policy, we show that it converges almost surely to the optimal arm under an arbitrary constant learning rate. Furthermore, we derive the non-asymptotic regret upper bound when the constant learning rate is below a time-invariant threshold; and the regret bound has order $O(\log T)$. We improve the analysis in Lattimore (2026a) for the same SDE by constructing a novel Lyapunov function and demonstrate the transparency of analyzing policy gradient using the tools in SDEs. In addition, the same Lyapunov function is also helpful in analyzing the discrete-time policy gradient algorithm.

We study expected improvement (EI) for minimizing a deterministic function $f$ in the RKHS $\mathcal H_k$ of a continuous positive-semidefinite kernel $k$ on a nonempty compact set $\mathcal X\subset\mathbb R^d$. Function values are observed exactly, and EI is computed from a fixed zero-mean Gaussian-process model with covariance $σ^2k$, $σ>0$. A weak-EI policy queries a point whose EI is at least a fixed positive fraction of its maximum. We introduce a notion of sequential separation radius relating ranked selected-point innovation norms to Kolmogorov widths, drawing on greedy approximation. Standard power-function estimates from scattered-data approximation and a finite-budget regret argument yield the rates. After $N$ post-initial queries, every weak-EI policy has simple regret $O(N^{-ν/d})$ for isotropic Matérn kernels of smoothness $ν>0$ and $O(\exp[-c_1\min\{N,N^{1/d}\log(eN)\}])$ for the isotropic squared-exponential kernel, with $c_1>0$. For $d=1$, the sharper bound $O(\exp[-c_2N\log(eN)])$ holds for exact EI, with $c_2>0$. These bounds are uniform over each fixed RKHS ball. If $\mathcal X$ has nonempty interior and $B>0$, the exact EI policy is minimax-rate optimal over the RKHS ball of radius $B$ for Matérn kernels, even among randomized strategies whose final recommendation need not be a query point. For the squared-exponential kernel, it is minimax-rate optimal up to constants in the exponent among deterministic methods whose final recommendation may be any point of $\mathcal X$.

We propose a self-adaptive online learning for control method for tracking unknown target dynamics. The target dynamics can exhibit switching behavior, particularly, a mixture of structured, random, and/or adversarial motion. Such challenging target tracking scenarios arise in applications of dynamic mapping, traffic control, and pursuit evasion, where robots need to track, pursue, or avoid collision with moving landmarks, objects, humans, etc., whose dynamics are unknown. Our method simultaneously learns multiple predictors from scratch, via self-supervised, one-shot, and computationally efficient learning, and adaptively selects the best one to match the observed target behavior. The method enjoys finite-time near-optimality guarantees in expectation, characterized as a function of the learning error of the target dynamics and the frequency that the target dynamics switch. In the absence of both error and switching, the method asymptotically matches the optimal non-causal control policy that knows a priori the target dynamics, i.e., the method enjoys no regret in expectation. In the presence of learning errors and switching, the method degrades gracefully, \eg when there are errors and no switching, the average regret is proportional to the average learning error and switching times. To prove these guarantees, a novel technical approach is required compared to the existing works that employ RFF-based online learning. We validate our method in Crazyflie simulations and hardware experiments, across target trajectories that vary from structured to random to adversarial, in comparison to non-stochastic, kernel-based, and neural-network-based methods for online learning.

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.

This paper investigates a hybrid reinforcement learning setting in tabular Markov Decision Processes (MDPs), where an agent aims to learn an optimal policy by combining online interactions with a target environment and offline data from a source environment. A central challenge is that offline data may be collected from outdated environments with shifted transition dynamics, making naive integration of historical data ineffective. To address this, we propose a unified algorithmic framework featuring two algorithms: MIN-UCB-VI for regret minimization and MAX-LCB-VI for best policy identification. Both algorithms leverage fine-grained bias information to more effectively exploit offline data under general transition shifts. We provide theoretical guarantees for our framework, including both instance-dependent and independent upper bounds on regret and sub-optimality gap. Furthermore, we establish matching lower bounds to demonstrate the optimality of our approach and validate our theoretical findings through extensive experiments.

Generative models are increasingly adopted in distributionally robust optimization (DRO), but existing approaches trade off model compatibility and adversarial structure: methods that accept arbitrary samplers do not restrict worst-case laws to a generator family, while generator-parameterized adversaries rely on model-specific access such as likelihoods, scores, or training data. We propose Generative Distributionally Robust Optimization (GDRO), a principled framework that accepts any sampleable conditional generator as the nominal model and restricts worst-case laws to a chosen conditional generator family. The key is the sampler-Sinkhorn pairing: samplers represent the conditional laws exactly, while Sinkhorn divergence compares their induced distributions without likelihood access and can be estimated from samples alone. The resulting population problem admits a direct finite-sample approximation and differentiable primal-dual implementation at the active decision context. For Lipschitz losses, the population Sinkhorn radius bounds downstream degradation. Across explicit and implicit generators, our method reduces rare-context inventory regret by 60% and SocialGAN navigation collisions by 50% relative to nominal decisions.

Decision-makers in learning environments face a dilemma when their short-term optimal actions may not favor their long-term benefits the most. To understand the fundamental tradeoff behind the dilemma, we study adaptive experimentation with post-commitment reward shifts. During an experiment phase, the decision-maker may adaptively test multiple options; during a subsequent commitment phase, the decision-maker must commit to a single option, whose reward may differ from its pre-commitment reward. We propose the Reserved Arm Eliminations for Commitment (RAEC) algorithm, which reserves a predetermined portion of the experiment phase to identify the best post-shift option while using the remaining rounds to minimize short-run regret. We establish regret upper bounds for RAEC across all parameter regimes and matching minimax lower bounds, providing a tight characterization of the cost of balancing short-term performance and long-term commitment. We also study two extensions. With prior structural knowledge linking pre- and post-shift rewards, we show that correctly identifying the ranking-changing component of the shift is more important than estimating its absolute magnitude. For settings with concave commitment rewards and portfolio choice, we develop the Reserved Online Stochastic Convex Optimization for Commitment (ROSCOC) algorithm, which directly converts its reserved exploration history into a commitment portfolio and achieves tight regret bound. Finally, we also conduct numerical experiments which confirm that our proposed algorithms achieve the desired regret predicted by our theory, and also outperform other baseline algorithms.

We revisit the regret loss framework introduced in Park et al. (2025), which uses decision-theoretic regret as a direct loss function for training models to make better decisions, through the lens of probability-simplex policies. Our first result shows that a single-layer self-attention model trained with regret loss admits a stationary point whose forward-pass exactly matches smoothed fictitious play with the appropriate stepsize that ensures no-regret behavior-i.e., for any given policy input, the model outputs the same update that smoothed fictitious play would produce. In parallel, we also newly introduce a swap-regret loss function, which extends the regret-loss framework beyond external regret and enables models to directly optimize for swap-deviation robustness. We further show that this swap-regret loss admits a stationary point whose forward pass implements the corresponding swap-regret update induced by classical Blum-Mansour no-pass implementation algorithm, with each head implementing an external-regret update via smoothed fictitious play. Together, these results show that regret-trained attention can realize differentiable mechanisms whose deployment induces equilibrium behavior in games: external-regret dynamics lead to coarse correlated equilibrium, while swap-regret dynamics lead to correlated equilibrium. Thus, regret-based objectives steer minimal attention architectures toward online-learning dynamics with game-theoretic guarantees, without supervised traces of those algorithms.

Ambiguous user requests make clarification a sequential decision problem for conversational LLM assistants: they must decide whether to ask, what to ask, when to stop, and when to answer. We introduce RegretBench, a multi-turn benchmark that evaluates clarification as policy behavior rather than isolated question quality. RegretBench provides a hidden-intent formulation of ambiguity, supports free-form interaction grounded in semantic-state tracking, and introduces a regret-based objective that measures how much value a model loses relative to a reference clarification policy. Experiments on open-domain QA and product recommendation scenarios show that final success alone is insufficient, as models with similar accuracy can differ substantially in efficiency, robustness to user behaviors, and stopping decisions. By jointly measuring intent resolution, interaction cost, ineffective clarification, and regret, RegretBench reveals whether models clarify usefully and efficiently. Our results show that effective clarification requires more than plausible questions: models must ask the right question at the right time and stop once the user's intended meaning is clear.

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 study horizon-free regret minimization for finite-horizon time-homogeneous tabular Markov decision processes with $S$ states, $A$ actions, horizon $H$, and per-trajectory total reward bounded by $1$. We propose a new algorithm and prove a regret upper bound $$\tilde O(\sqrt{SAK}+S^8A^3)$$ with failure probability $δ$, where $K$ is the number of episodes and $\tilde O(\cdot)$ hides $\mathsf{poly}\log(S,A,K,1/δ)$. Thus, the regret is $H$-free and asymptotically optimal, matching the contextual-bandit lower bound $Ω(\sqrt{SAK})$ up to logarithmic factors. This completely removes the $\log H$ dependence from the previous $\tilde O(\sqrt{SAK\log H}+S^2A\log H)$ guarantee of Zhang et al. (2021), and drastically improves the prior best horizon-free regret $\tilde O(\sqrt{S^9A^3K})$ of Zhang et al. (2022) asymptotically. The main technical difficulty is that the optimal value functions $\{V_h^*\}_{h=1}^H$ are time-inhomogeneous even though the transition kernel is time-homogeneous. A direct union bound over all value functions typically incurs an additional $\min\{\log H,S\}$ factor. We avoid this factor by (i) exploiting the monotonicity of $V_h^*$ in $h$ and (ii) non-trivially projecting the value functions onto an $S$-dimensional grid. Our analysis relies on three additional ingredients. First, we introduce a horizon-truncation argument that enables reward-based exploration and removes the cost of a separate reward-free exploration phase. Second, we design a cutting bonus that preserves both optimism and the monotonicity needed for planning. Third, we prove a new bound on total deviation for time-homogeneous MDPs, which controls the clipped variance terms in the cutting bonus with adjustable polynomial dependence on $S$ and without any dependence on $H$. Together, these tools yield an asymptotically optimal horizon-free regret guarantee.

We study online optimization for a broad class of structured non-convex non-smooth problems where each loss is a composition of a difference-of-convex function with a smooth mapping, and the feasible region is defined by constraint functions of the same kind. We propose a time-smoothed proximal linear algorithm and a local-regret measure based on a proximal residual mapping. We show that this residual is a proper stationarity measure for the original problem: its fixed-point condition implies first-order stationarity. Our analysis relies on a tangent-cone characterization for a feasible region described by composite difference-of-convex constraints, which is of independent interest and allows each update to be computed via a convex optimization oracle, despite the non-convexity of the problem. We establish a local-regret bound and a bound on the total number of inner convex subproblems. We also derive an error bound connecting the proximal residual to the distance to stationarity, providing a quantitative certificate of approximate stationarity.

Building on the identity that expected regret equals the covariance between costs and decisions, this paper develops a derivative theory of the covariance regret functional. We derive the Gâteaux derivative, showing that the universal steepest-descent direction is the contrarian policy $-(c-\bar c)$, while ascent yields momentum. For linear policies $\hatπ(c)=Ac+b$, the gradient is the cost covariance matrix $Σ_c$, with a zero Hessian implying boundary-optimal solutions such as the minimum-variance portfolio. We extend to constrained optimization, sign-gradient duality between regret minimization and alpha maximization, finite-sample convergence bounds paralleling Thompson Sampling, and gradient-descent algorithms requiring only input observations.

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.

Advertisers delegate bidding to autobidders; users delegate tasks to language-model agents. A person describes what they want to an automated proxy that acts in a mechanism on their behalf. This is the revelation principle in production, and it forces a question classical theory assumes away: when is it optimal to describe yourself honestly to your own proxy? We show the answer turns on one quantity, the proxy's within-range regret. The most a principal can gain by misreporting equals the regret of the proxy's honest-report action against those the principal could have steered it to take. Honest self-description is optimal exactly when the proxy already plays the best action it can reach, that is, when it is loyal (Theorem 1). The identity unifies auction-specific autobidding results and pins down when the faithful-communication assumption behind language-model elicitation proxies (Huang et al.) holds. The identity constrains guardrails placed on proxies, from bid caps to a model's alignment layer. No guardrail can be at once binding (it displaces the truthful action from the proxy's best reachable outcome), truthful (honest reporting stays optimal), and capability-preserving (that outcome stays reachable through some report); any two preclude the third (Theorem 2). A safety constraint that alters what a model does while leaving its best output reachable makes honest description of intent suboptimal, so a sharper report can gain. This is the incentive behind prompt-engineering and jailbreaking. Because within-range regret is #P-hard to compute exactly, we estimate it from samples and maintain it as a model is updated, at a cost set by how far the model drifts, not how often it changes. Running it on production language models from five providers under an alignment-style cap, we find honest reporting leaves surplus unclaimed on every model, recovered by inflating the report.

We study the bandit-feedback version of online principal component analysis (Bandit PCA): in each round $t = 1,\dots,T$, the adversary selects a $d \times d$ symmetric gain matrix $G_t$ with spectrum in $[0,1]$ and rank at most $r$; the learner simultaneously selects a unit vector $w_t \in S^{d-1}$ and receives the reward $w_t^\top G_t w_t$. The learner receives no other feedback, and aims to minimize the regret against the best unit vector in hindsight. This problem was introduced by Kotlowski and Neu (2019), who gave an algorithm with regret $O(d\sqrt{rT \log T})$ and showed the lower bound of $Ω(r\sqrt{T/\log T})$. We improve upon both of these bounds and essentially bridge the gap between them, establishing the minimax regret of order $r\sqrt{dT}$ up to polylogarithmic factors in $d$ and $T$. The upper bound is attained by a novel algorithm, which combines online mirror descent on the spectrahedron of (real) density matrices with a multiscale exploration scheme in which the eigenspaces with different spectral magnitudes are updated at different rates. For the lower bound, we construct an adaptive adversary that refines a hidden large-reward subspace based on the learner's actions, in such a way that low regret is impossible without estimating the subspace; as a result, lower-bounding the regret reduces to studying the arising subspace estimation problem. Finally, we discuss connections of Bandit PCA with adaptive-measurement quantum tomography.

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.

This paper studies an online variant of the assistance games framework, where an informed agent and an uninformed agent repeatedly interact over $T$ timesteps to optimize a common reward function. While the informed agent (the human) observes a latent state of the world, the uninformed agent (the assistant) observes only the human's actions. We provide the first provably efficient learning algorithms for repeated assistance games. We introduce the notion of assistance regret: the gap between the cumulative utility of interactions and that of the optimal joint policies in hindsight, which map latent states to action pairs. We present decentralized algorithms for both the human and the assistant that achieve a $(1-1/e)$-approximate assistance regret rate of $\widetilde{O}(T^{3/4})$, with runtime polynomial in the size of the action and state spaces. These algorithms are general; in particular, they accommodate any no-regret algorithm for the assistant. We prove that achieving a regret approximation factor better than $(1-1/e)$ is computationally intractable. Furthermore, we demonstrate how these generic no-regret algorithms can be tailored to a pseudo-decentralized setting -- using a shared random string -- to achieve a rate of $\widetilde{O}(T^{1/2})$, optimal up to logarithmic factors.

We study a sequential learning problem for stable matchings in two-sided markets where preferences on both sides are initially unknown. We focus on a centralized setting where an algorithm matches agents at each time step and receives noisy rewards that reflect the preferences of the matched agents, following a semi-bandit feedback structure. We adopt a pure exploration perspective, aiming to efficiently identify the optimal stable matching with high probability. Our work extends prior results by handling \emph{two-sided uncertainty} and by exploiting \emph{partial preference} information. A central ingredient is the notion of \textbf{pervasive stable matching}, which enables the identification of optimal stable matchings under partial preferences. We propose elimination-based algorithms whose stopping criteria exploit the structure of the learned partial preferences, and provide a refined sample-complexity analysis. Beyond pure exploration, we extend our approach to regret minimization and establish regret bounds with respect to the \emph{optimal} stable matching that avoid dependence on the minimum reward gap $Δ_{\min}$.

Many online decision-making problems involve both round-specific feasible actions and drifting reward models: eligible ad impressions, feasible prices, and available treatments can change over time, while user preferences, demand curves, and patient responses may evolve. Motivated by these applications, we study non-stationary linear bandits with round-specific feasible decision sets. Existing methods that obtain the optimal $\widetilde O(T^{2/3}P_T^{1/3})$ dependence, where $P_T$ is the path length of the reward-parameter sequence, impose an orthogonal-structure assumption on round-specific decision sets, which can be restrictive in contextual applications. We address this gap through a unified misspecification-reduction viewpoint: after partitioning the horizon into blocks, we relate each block's dynamic regret to regret against a fixed-parameter linear bandit benchmark, with the within-block parameter drift entering as bounded misspecification. Restarting algorithms with misspecification-dependent regret guarantees then yields the optimal $T^{2/3}P_T^{1/3}$ dynamic-regret dependence for both linear bandits with general compact decision sets and $K$-armed contextual linear bandits.

We study online resource allocation when both rewards and consumption sizes may be continuously distributed. Requests arrive sequentially and must be accepted or rejected irrevocably under fixed resource capacities. Each request belongs to one of finitely many observable types; conditional on an observable request type, both the reward and the scalar size are random, and the realized size scales a fixed type-specific resource-consumption vector. The model allows the deterministic fluid relaxation to be degenerate. We show that additive regret is governed by the size-weighted mass of requests whose value-to-size ratios lie near the active acceptance cutoffs. We formalize this quantity through an active weighted-mass exponent p. When p > 1, this cutoff mass is thin, and the problem is genuinely hard: every online policy must incur regret of order at least $T^{1/2 - 1/(2p)}$, and this holds for every p > 1. A sample-path marginal policy matches this lower bound up to polylogarithmic factors; and when p = 1, so that the mass grows linearly near the cutoff, it attains $O((\log T)^2)$ regret. For example, if the size and the value-to-size ratio are independent and uniformly distributed, then p = 1; if instead the size and the reward are independent and uniformly distributed, then p = 2. Thus the policy achieves $o(\sqrt{T})$ regret throughout this regularity class without any fluid non-degeneracy assumption, allowing both primal degeneracy and dual non-uniqueness.

LLM agents increasingly rely on retrieval buffers to store and reuse past experience, yet the cache management policies governing these buffers remain largely ad-hoc. We formalize this as an online semantic cache replacement problem with switching costs, where items are matched by embedding similarity and hit quality is continuous rather than binary. Through experiments on two datasets from MemoryBench-Full (LoCoMo, DialSim) with 8 replacement policies, we reveal a surprising finding: classic heuristics (LRU, LFU) \emph{consistently underperform} the naive FIFO baseline on semantic workloads, due to the absence of temporal locality and frequency concentration. We propose SOLAR, a learning-augmented framework that derives modification timing from regret accumulation (achieving $\sim$17% modification rate) and content selection from Bayesian online learning over implicit retrieval feedback. We prove SOLAR achieves a constant competitive ratio $\leq 3$, independent of cache size and horizon (vs.\ $Ω(K)$ for FIFO), and eviction regret $O(\sqrt{KT\log T})$, matching the $Ω(\sqrt{KT})$ lower bound up to logarithmic factors. Experiments demonstrate 5--75% relative improvement over FIFO at tight cache sizes, with a clearly characterized phase transition at the working set boundary. Synthetic experiments with 5000-item pools further reveal an inverted-U relationship between pool size and retrieval quality, justifying capacity constraints as a retrieval noise phenomenon rather than a storage limitation.

We investigate the contextual slate bandit problem with generalized linear rewards under limited adaptivity. At each round, the learner is presented with $N$ sets of items, where each item is represented by a $d$-dimensional feature vector. The learner then constructs a slate by selecting one item per set; the resulting slate yields a scalar reward sampled from a Generalized Linear Model (GLM). We propose algorithms under two limited-adaptivity settings: (a) Batched and (b) Rarely-Switching. For the batched setting, we introduce B-SlateGLinCB, which partitions the time horizon into $\mathcal{O}(\log\log T)$ batches such that each batch's policy relies only on data from previous batches. For the rarely-switching setting, we propose RS-SlateGLinCB, which adaptively performs only $\mathcal{O}(Nd\log T)$ parameter updates. Under a diversity assumption on the item sequences, we prove that B-SlateGLinCB and RS-SlateGLinCB achieve regret bounds of $\mathcal{O}(Nd^{3/2}\sqrt{T})$ and $\mathcal{O}(Nd\sqrt{T})$, respectively. Notably, both bounds are independent of the non-linearity parameter $κ$ that is typically found to scale the regret of GLM bandit algorithms. Our algorithms are computationally efficient, requiring only $\text{poly}(N)$ time per round despite $2^{Ω(N)}$ possible slates. Simulations show our algorithms outperform existing baselines with limited adaptivity and remain competitive with Slate-GLM-OFU, a fully adaptive state-of-the-art algorithm. Notably, a slightly modified B-SlateGLinCB empirically matches this baseline. Finally, we demonstrate strong performance in a practical in-context example selection task for language models.

We study the problem of forecasting for an arbitrary number of downstream agents with unknown objectives, each of whom best responds to the forecaster's predictions. We seek a single forecaster that guarantees sublinear swap regret for all downstream agents simultaneously. For two-dimensional outcome spaces, we give a polynomial time algorithm that guarantees $\tilde{O}(\sqrt{kT})$ swap regret for any downstream agent with $k$ actions. This improves over the previously known bound of $\tilde{O}(kT^{5/8})$ and avoids the exponential in $T$ runtime of prior algorithms in this setting. Our algorithm extends nicely to other low dimensional environments, retaining $\tilde{O}(\sqrt{T})$ downstream swap regret while the exponent of $k$ in the regret bound and the exponent of $T$ in the running time both grow with dimension. For arbitrary dimension $d$, we give a forecasting algorithm that guarantees $\tilde{O}(d\sqrt{kT})$ swap regret, assuming the forecaster knows an upper bound $k$ on the number of actions available to any downstream agent, albeit with a much longer runtime. This improves upon previous high dimensional guarantees that had $\tilde{O}(T^{2/3})$ dependence and required additional behavioral assumptions.

Many two-player zero-sum games admit not a unique Nash equilibrium but a convex set of them: a polytope of profiles that all share the minimax value V* yet prescribe different behaviour. Standard solvers each converge to some equilibrium and are treated as interchangeable. We ask whether they instead select different members of the Nash set, systematically as a function of the algorithm rather than the seed. Using a tabular, exactly solvable testbed of six games with analytically known Nash sets -- including a two-dimensional Nash polytope and Kuhn poker -- we find that (i) selection is determined by the algorithm, not the seed, but families differ only on asymmetric Nash sets; (ii) regularized last-iterate methods (R-NaD, magnetic mirror descent) select the maximum-entropy member, the information projection of their uniform reference onto the Nash set -- exactly on the 2-D polytope and at 99.7% of maximum entropy in Kuhn -- while regret-averaging methods (CFR, CFR+, fictitious play) drift to a lower-entropy face; we confirm this on a randomized 180-game ensemble, where R-NaD attains the maximum-entropy member in 100% of converged games while CFR+ sits strictly below it in 94% (paired Wilcoxon p < 10^-27); (iii) the selected member has downstream consequences against sub-optimal opponents that scale with sequential/hidden-information structure but stay bounded -- in Kuhn the max-entropy member is a strictly better hedge, whereas on the matrix games the members differ without either dominating. We also report two negative results correcting common intuitions: removing CFR's positive-orthant (max(R,0)) projection does not eliminate boundary drift; and R-NaD's selection is anchor-following, not initialization-independent. We state the maximum-entropy / I-projection characterization as a strongly data-supported conjecture, checked throughout against analytic ground truth.

Gradient equilibrium (GEQ) is a recently introduced online optimization framework that generalizes first-order stationarity from offline optimization and abstracts problems like online conformal prediction. While GEQ has curious similarities with known online learning frameworks, namely regret minimization, prior work has shown that GEQ error and regret are incomparable objectives, leaving open a precise understanding of how GEQ fits into the broader online learning landscape. In this work, we show that GEQ is equivalent to Blackwell approachability in the algorithmic sense. That is, a Blackwell approachability problem can always be solved using queries to a black-box GEQ oracle, with no asymptotic loss in the oracle's error rate, and vice versa. Taken together with known equivalences between approachability, regret minimization, and calibration, these results imply that GEQ is equivalent to these frameworks, as well. Our reductions are efficient and can be used to transfer refined guarantees, such as optimism and strong adaptivity, from regret minimization to GEQ. Along the way, we also identify necessary and sufficient conditions for GEQ, and establish reductions between different notions of GEQ with unconstrained and constrained decision sets.

In many online learning and bandit problems, the actions we consider possess inherent similarities--for instance because they share latent traits, tags, or hierarchical structure. We study online learning with a similarity-structured action set, encoded by a rooted tree whose leaves are the actions and whose levels quantify how closely two actions are related. The loss sequence is assumed tree-compatible: losses of similar actions are constrained to be close. We establish an impossibility result showing that usual one-point bandit feedback cannot, in general, leverage range or tree-induced similarity, even under very strong similarity constraints. We then provide a unified set of algorithms which adapt to a wide range of richer feedback models, from semi-bandit feedback down to multi-point bandit protocols, including the minimal two-point feedback setting. We show these algorithms exhibit best-of-both-worlds guarantees and provably exploit action similarities by replacing the number of actions $K$ by a similarity-aware effective number of actions $K_{\mathrm{eff}}$ in the regret bounds. As an application, we show that under two-point feedback, it is possible to achieve $\sqrt{T}$ regret in Lipschitz bandits when $d \leq 2$.

Language models turn a worded situation into a numeric plan, and the dominant pipelines (NL4Opt, OptiMUS, ORLM, OR-LLM-Agent) commit to a single objective and point-valued coefficients, then solve once. For decisions that allocate real budget, effort, or clinical attention, that confidence is the failure mode: every objectified number is an assumption, and a plan optimal only if the guesses are exactly right is fragile -- mimicry of computation. YUKTI changes the target of autoformulation. Its representation is a typed-proposition graph whose relationships carry shape priors, coefficient uncertainty, and provenance. YUKTI routes each stage to an exact, nonlinear, or evolutionary solver; couples stages by a distributional Pareto hand-off; and introduces Assumption-Robust Pareto Frontiers (ARPF), resampling assumptions (including structural epsilon-contamination) to score how often each action survives (rho). We prove a bound making rho an exact factor of decision regret, add auditable traceability, and synthesize a benchmark-faithful data foundation when none exists (SRJANA). We validate three ways: under controlled misspecification the robust compromise cuts mean and tail regret by over 90% versus a naive point plan; on a regulated commercial decision we optimize inside a lawful action space and price the downside in euros; and on a real public dataset of 41,188 decisions an out-of-sample backtest beats the logged status quo by 34% and a naive point rule by 4% while reducing the optimizer's curse. The solvers are standard; we claim no benchmark-SOTA win. A head-to-head shows an LLM given the correct numbers, and single-objective optimization, both incur about 47x the held-out regret of YUKTI -- an LLM is a formulator, not a solver. Under long-range causal coupling, the forward hand-off becomes unsound, locating where it must become a backward-induction causal policy.

We introduce optimal coarse correlated equilibria for continuous-time mean field games. A coarse correlated equilibrium is a randomized recommendation scheme from which no player can gain by ignoring the recommendation and switching to an alternative strategy. The problem is as follows: a moderator selects, among all mean-field coarse correlated equilibria, one that optimizes a prescribed performance criterion, which may differ from the representative player's objective. After formulating the problem, we develop a linear programming (LP) formulation, prove the existence of optimal LP coarse correlated equilibria, and relate the LP characterization to the original probabilistic setting. Building on this characterization, we design a no-regret primal-dual algorithm, based on an equivalent Lagrangian formulation of the external-regret constraint, for learning such equilibria. We provide explicit convergence rates for the learning algorithm, and numerical examples illustrate the method.

We study adversarial bandit optimization in which the loss functions may be non-convex and non-smooth. In each round, the learner selects an action and observes only the loss incurred at that action. The loss consists of an underlying convex and $β$-smooth component and an adversarial perturbation that may be chosen after observing the learner's action. The perturbations are subject to a global budget controlling their cumulative magnitude over time. This framework extends the globally budgeted, post-action perturbation model from underlying linear losses to general convex and $β$-smooth losses. For this broader class, we establish expected regret guarantees that explicitly characterize the effect of the perturbation budget. To establish these guarantees, we modify a standard bandit optimization algorithm and develop an analysis that controls the additional regret caused by the perturbations. In the absence of perturbations, our results reduce to regret guarantees for the standard bandit convex optimization setting with $β$-smooth losses.

U-calibration studies online forecasting algorithms whose predictions can be consumed by any unknown downstream agent, guaranteeing sublinear regret simultaneously for all proper loss functions. Existing U-calibration algorithms achieve worst-case optimal $O(\sqrt{T})$ regret for every bounded proper loss, but they fail to adapt to easier losses: as we show, even for smooth losses such as squared loss, they incur $Ω(\sqrt{T})$ regret instead of the optimal $O(\log T)$ regret. In this work, we show that this limitation is not inherent. Specifically, we design a single forecast algorithm that simultaneously achieves $\tilde O(\sqrt{T})$ regret for every bounded proper loss and $O(\log T)$ regret for every bounded smooth proper loss. More generally, our algorithm also attains logarithmic regret for losses that are smooth relative to the log-barrier, which include several non-Lipschitz examples. Our approach is based on a novel variant of Follow-the-Perturbed-Leader (FTPL) in which perturbations are applied directly in the prediction space using self-concordant noise. The resulting analysis also departs substantially from prior FTPL analyses due to the complex nature of this noise and may be of independent interest.

Conditional generative models are increasingly used as scenario generators for stochastic optimization, but standard training objectives emphasize uniform distributional fit rather than the downstream decisions induced by generated scenarios. This creates an objective mismatch: errors in statistically common regions may have little effect on decision regret, whereas errors in decision-sensitive regions can substantially change the optimal action. We propose Decision-Weighted Flow Matching (DW-FM), a regret-aligned training framework that preserves the simplicity of standard flow matching while reweighting its velocity-regression objective using decision-sensitive endpoint information. Theoretically, we connect downstream regret to pathwise velocity mismatch through a loss-induced decision discrepancy and an adjoint transport argument, yielding an ideal regret-aligned surrogate and practical endpoint-weighted objectives with regret guarantees. Empirically, we demonstrate the effectiveness of DW-FM on three CVaR-based contextual stochastic optimization benchmarks spanning synthetic portfolio, semi-real financial, and traffic-CVaR tasks, where DW-FM improves downstream regret over standard baselines.

A filtered approximate-nearest-neighbor (ANN) query returns the k nearest vectors among those satisfying an attribute predicate P of selectivity s. The best execution strategy -- pre-filter, post-filter, or in-filter -- changes with s, so a system must estimate s and choose. We model this as an argmax over a landscape with phases (regions where each strategy wins) separated by boundaries, and show that selectivity-estimation error produces plan regret -- recall lost versus the oracle strategy -- only in the critical regions around those boundaries. The regret is a wedge of log-width equal to the multiplicative estimation error epsilon and height equal to the local cliff |V'(s*)| epsilon; the flip-margin 1/|V'(s*)| is the condition number of a sibling cardinality-estimation study reappearing as the local boundary theory. The two phase boundaries follow from independent mathematics: order statistics place the post-filter cliff at s ~ k/K, and site percolation places the in-filter cliff at s_c ~ 0.83/M for graph degree M (corpus-size independent). Criticality exists only under a constrained budget B < sqrt(k n). Under pre-registered decision rules we confirm, on synthetic sweeps and real SIFT1M, that regret concentrates ~290x at the boundary and that the regret curves obey a finite-size scaling collapse onto one universal wedge across two decades of corpus size. A real approximate index does not mis-locate the boundary, but a biased cost model opens a persistent miscalibration band that estimation-error robustness cannot fix. The contribution is a characterization, not a new index. Code and the full pre-registration are public.

Cardinality-estimation (CE) research ranks estimators by q-error, yet it is well known that q-error is an imperfect proxy for query-plan quality. We give a measurement-driven account of when it is a good proxy and when it is not, and why. Modeling plan selection as an argmin over a piecewise-linear cost landscape, we find that plan regret (the cost of the chosen plan relative to the optimal, under true cardinalities) is governed by plan-cost geometry in a regime-dependent way. (i) For small errors, a true-point condition number kappa predicts regret and out-predicts q-error; its predictive power decays to zero as error grows, as a local linearization must. (ii) For large errors -- where deployed learned estimators operate -- an estimator-independent average-case sub-optimality measure ACS-infinity predicts which queries are regret-prone (Spearman rho ~ 0.54 on STATS-CEB), while q-error is nearly uninformative at the query level (rho ~ 0.05). (iii) The worst case is Haritsa's maximum sub-optimality (MSO). The three are one cost-ratio spectrum under three weightings. We prove a limit law ACS-infinity = sum_k r_k pi_k with cardinality-independent combinatorial weights, and validate every claim on STATS-CEB and JOB-light with four released estimators under pre-registered decision rules, and confirm on real PostgreSQL runtime that ACS-infinity predicts regret where q-error does not. The contribution is conceptual and empirical -- an average-case companion to worst-case robust query optimization, and a characterization of when an accuracy metric tracks plan quality -- rather than a new estimator. Code and the full pre-registration are public.

We study repeated bilateral trade from a fairness perspective. At each round, a fresh seller-buyer pair arrives, and the platform posts a price before observing the traders' valuations. Trade occurs only if both agents accept the price. Rather than maximizing only the gain from trade, we consider platforms that seek balanced divisions of the generated surplus. We show that natural fairness desiderata lead to a one-parameter Rawls-to-Nash family of fair-gain objectives, obtained by aggregating the seller's and buyer's net gains through nonpositive Hölder means. Unlike the standard gain-from-trade objective and the Rawlsian fair-gain objective studied in prior work, our proposed objectives induce a new statistical structure in which expected rewards are recovered from threshold feedback through a two-dimensional singular-kernel integral identity. This leads to a nonstandard pure-exploration problem whose natural estimators are rectangular double sums with row-column dependence and singular weights. Assuming independent i.i.d. seller and buyer valuation sequences with arbitrary unknown marginals, we characterize the optimal learning rates for the whole Rawls-to-Nash family of fair-gain objectives, giving matching fixed-confidence sample-complexity and regret bounds up to polylogarithmic factors.

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.

Understanding the minimal assumptions necessary for generalization is the fundamental question in learning theory. Unfortunately, most results rely heavily on independence (or some proxy thereof) of the data-generating process, while results for strongly dependent data are far more limited. Towards addressing this gap, we introduce the framework of simulatable processes, where the learner has access to a simulator that approximates the distribution generating the data (which may be an arbitrarily complex and dependent process). Surprisingly, given access to such a simulator, we show that we can recover the same learning guarantees as in the classical setting with independent data, namely, error bounds that depend on the VC dimension. Further, we use this framework to study the power of conditional sampling and show strict statistical and computational advantages in this setting. As a highlight of our framework, we exhibit a single algorithm that simultaneously learns any given VC class under all processes samplable in bounded polynomial time, with regret controlled by the time-bounded Kolmogorov complexity of the process. This provides a significant conceptual broadening of the classical PAC model.

Reinforcement learning with verifiable rewards (RLVR) has enabled progress on reasoning-intensive tasks by relying on task-specific verifiers that provide automated correctness signals. However, many realistic language tasks are difficult to equip with reliable verifiers, motivating a growing reliance on reinforcement learning from human feedback (RLHF). In this setting, we argue that a closer examination of how human feedback should be interpreted is essential. We introduce Regret-based Preference Optimization $(\textbf{RePO})$, which reframes RLHF through $\textit{regret minimization}$ rather than reward maximization. Human preferences are often shaped by $\textit{prospective}$ anticipation of outcomes and $\textit{counterfactual}$ comparisons to alternative behaviors, rather than by immediate, outcome-independent utility. $\textbf{RePO}$ captures this structure by modeling preferences as behavior-conditioned assessments of relative suboptimality. Experiments on mathematical reasoning benchmarks and human preference datasets demonstrate consistent performance gains, indicating that $\textbf{RePO}$ is an effective and human-aligned approach for training large language models.

In this paper, we study regret minimization in repeated games with \emph{adaptive} opponents who can respond based on histories of play. The standard metric of \emph{external regret} in online learning is known to fail to capture such adaptivity. To account for players' counterfactual reasoning, we introduce {\tt Repeated Policy Regret (RP-Regret)}, a game-theoretic metric that measures the difference between the \emph{realized} and the \emph{best-in-hindsight} accumulated utility when all players can \emph{respond} to the history of play. Compared to existing regret notions in this setting, ours is native to repeated game playing, enabling stronger comparators and opponents with fewer constraints, while maintaining the possibility of finding better equilibria when all players minimize it. We first identify necessary conditions for obtaining {\tt RP-Regret} sublinear in time, on the variation of the player's comparator strategies in the regret definition and on the memories of both the comparator and opponents' strategies. We then study additional conditions and provable algorithms to minimize {\tt RP-Regret}, which is by definition \emph{non-convex} in the strategy space. To address this challenge, we propose three algorithms: (i) one based on an optimization oracle, as assumed in some prior work in online non-convex learning; (ii) one that minimizes a convex and \emph{linearized} surrogate of {\tt RP-Regret} at each iteration; (iii) one that directly minimizes {\tt RP-Regret} when opponents change strategies slowly. Furthermore, when all players can run algorithms to minimize the {\tt RP-Regret} (or its linearized variant), certain subgame perfect equilibria of the repeated game can be learned. We also provide experiments showing that minimizing our regret notions can lead to more cooperative solutions with higher utility in games such as Stag-Hunt.

This paper investigates non-stationary online learning using the metric of interval regret, which requires an online algorithm to perform well over every time interval. We propose the first online learning algorithm that achieves an interval regret bound scaling with gradient variation, a fundamental measure of the cumulative change in online function gradients, which relates to various problem-dependent quantities and is closely connected to stochastic optimization and other problems. Our method employs a simple and efficient two-layer online ensemble structure that achieves strong theoretical guarantees. Specifically, it enjoys a regret bound that simultaneously adapts to various problem-dependent quantities while also preserving the minimax-optimal rate in the worst case. Moreover, recognizing the challenge of hyperparameter tuning, we introduce a Lipschitz- and smoothness-agnostic variant that automatically adapts to these potentially unknown constants. This is primarily enabled by a novel Lipschitz-adaptive meta algorithm, which may be of independent interest. Beyond interval regret, our method also yields broader implications: it provides versatile bounds for interval dynamic regret, a stronger measure that competes with changing comparators over any interval, and yields the first piecewise characterization for stochastic extended adversarial optimization. Theoretical findings are validated by experiments.

Causal language models factorize sequence probabilities using only preceding context, leaving future information unexploited during training despite its availability in the training data. This paper introduces Regret Pre-training, a self-supervised framework grounded in the Learning Using Privileged Information (LUPI) paradigm. The framework employs a dual-view architecture in which a single model generates both a causal Student distribution and a future-conditioned Teacher distribution. The training objective augments standard language modeling with a regret loss that minimizes the KL divergence from teacher to student, transferring future-aware signals to the causal representations. We investigate two teacher configurations on the OLMoE-1B-7B architecture:LocalRegret, which extends attention by one future token, andGlobalRegret, which conditions on bidirectional context with the target position masked. Experiments on nine downstream tasks following 4 billion tokens of training demonstrate that both configurations consistently outperform the baseline. On average,GlobalRegret andLocalRegret achieve 33.9% and 32.2% accuracy respectively, surpassing the baseline's 30.2%. Most notably,GlobalRegret improves BoolQ performance by 18.1 percentage points (61.0% vs 42.9%). The framework introduces no additional parameters and requires only one extra inference-mode forward pass per training step.

We study $N$-armed stochastic dueling bandits under the Condorcet-winner assumption, where three widely adopted objectives are considered: best-arm identification (BAI), weak regret, and strong regret. We propose Tree-Guided Identify-Then-Exploit (TG-ITE), the first unified framework to tackle all these objectives to our knowledge. Without requiring stronger assumptions, we propose a shared tree-guided identification approach to find a high-confidence incumbent within $O(N)$ comparisons. We further propose varied exploitation strategies to utilize this warm-start stage to optimize the specific objectives at hand. This methodology enables our approach to (1) achieve $O(N)$ sample complexity in BAI without commonly adopted stronger assumptions; (2) build the first winner-stays-style algorithm to achieve $O(N)$ weak regret; (3) enjoy the same $O(N \log T)$ guarantee as specialized strong-regret approaches; (4) realize the joint optimization of BAI and weak regret with $O(N)$ guarantees for both, eliminating the sub-optimal gap of $O(\log N)$ in the existing approach. Our results provide evidence that the trade-off between BAI and regret minimization is relatively benign in dueling bandits.

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.

Preference alignment is central to improving large language models, but standard reward-based formulations can be restrictive when human preferences are cyclic, non-transitive, or otherwise not representable by a scalar reward. Nash Learning from Human Feedback (NLHF) addresses this limitation by modeling alignment as a preference game and targeting a Nash equilibrium rather than a reward maximizer. However, the learning-theoretic foundations of scalable NLHF remain limited. Existing regret guarantees rely on oracle-based methods that estimate a general preference model and solve KL-regularized minimax problems, while iterative NLHF methods directly optimize policy-level preference losses and are easier to implement but lack regret guarantees. We study online iterative NLHF under general preference models and identify exploration as the key obstacle. First, we show that standard iterative NLHF can suffer an exponential dependence on the KL-regularization parameter, revealing that implicit exploration through policy updates is insufficient for controlling regret. Second, we propose an explicitly exploratory iterative NLHF algorithm that combines SFT-based regularization with adversarial policy exploration. The resulting method retains the direct policy optimization structure of iterative NLHF, avoids explicit preference model estimation, and achieves an $O(\sqrt{T})$ regret bound without an exponential dependence on the KL-regularization parameter. We show that the regret can be improved to $O(\log(T))$ with access to a minimax oracle, clarifying the computational-statistical tradeoff in learning general preference games. Finally, we instantiate our method for LLM fine-tuning and evaluate it on \texttt{Llama-3-8B-Instruct} across multiple benchmarks, where explicit exploration yields consistent improvements over existing NLHF baselines.

This paper studies a one-step lookahead Bayesian optimization (BO) method and its theoretical guarantee. Although the empirical effectiveness of one-step lookahead BO methods, such as entropy search, has been studied extensively, they often rely on computationally intractable approximations, and their regret guarantees remain underdeveloped. Thus, this paper proposes a one-step lookahead BO method called optimal-point variance reduction (OVR), which requires only posterior sampling and Monte Carlo approximations. We obtain a uniform error bound over an input domain for the Monte Carlo estimation in OVR. Furthermore, we show that the regularized OVR, with the slight modification to promote exploration, achieves a vanishing Bayesian expected simple regret upper bound. Finally, we demonstrate the effectiveness of OVR through numerical experiments.

We study the Lipschitz bandit problem, where a learner sequentially maximizes an unknown Lipschitz function $f$ over a domain $\mathcal{X} \subset [0,1]^d$ using noisy pointwise evaluations. Existing regret bounds are either worst-case, scaling as $\tildeΘ \left ( T^{d+1/d+2}\right )$, or adaptive via the zooming dimension $d_z$, yielding $\tildeΘ \left ( T^{d_z+1/d_z+2}\right )$. However, such zooming-based guarantees are only partially instance-dependent, as they depend solely on the asymptotic growth of near-optimal level sets and fail to capture finer structural properties of $f$. We provide an analysis and an algorithm that characterizes the regret through integrals of the suboptimality gap of $f$ over its level sets. This yields regret bounds that adapt to the local growth of level sets, rather than only their asymptotic behavior. As a corollary, when the set of maximizers has dimension $d^\star>0$, we obtain improved adaptive rates of order $\tilde{\mathcal{O}} \left ( T^{d_z+1 / \max(d_z,d^\star)+2}\right )$ strictly improving over classical zooming bounds in this regime. Finally, we extend our analysis to the full-information setting (Lipschitz experts) and show how some of the regularity assumptions can be relaxed.

Test-time adaptation (TTA) aims to adapt models to maintain reliable performance on non-stationary test streams without requiring labeled data. Despite its empirical success, the learnability of TTA under non-stationary streams remains unexplored. A key challenge is the lack of a principled theoretical framework that simultaneously aligns with the TTA objective and captures both continuously evolving distribution shifts and intrinsic information constraints. To address this gap, we propose the first theoretical framework for studying the learnability of TTA and introduce $(ε,δ)$-Recovery Complexity and $(ε,ρ)$-TTA Learnability. Recovery complexity measures the post-shift time needed to maintain excess risk below a target level with high probability, and is further extended to TTA learnability, which measures the long-term reliability of TTA. Within this framework, we introduce a novel discrete surrogate for non-stationary test streams, enabling a unified and tractable analysis of both gradual and abrupt shifts. We derive order-wise matching lower and upper bounds on recovery complexity, revealing fundamental limits of TTA and an intrinsic adaptivity-information trade-off. These results provide unified learnability guarantees for TTA that complement regret-based analyses.

We study the adversarial kernel bandit problem, in which the loss at each round is induced by an arbitrary bounded element of a reproducing kernel Hilbert space (RKHS). We propose an exponential-weights algorithm built on a regularized importance-weighted loss estimator, together with an explicit correction term that cancels the bias introduced by the regularization. Our main result bounds the regret by $\widetilde{O}\big(\sqrt{T\, d_*(λ)\,\log|{X}|}\big)$, where $d_*(λ)$ is a widely-adopted notion of effective dimension that captures the complexity of the kernel. Up to logarithmic factors, this matches the known rate achieved in the related stochastic kernel bandit problem. A notable application is the Matérn$(ν,d)$ kernel with smoothness parameter $ν$ on $\mathbb{R}^d$, for which our bound specializes to $\widetilde{O}\big(T^{(ν+d)/(2ν+d)}\big)$, improving over the best-known prior rate of Chatterji et al. [2019] while simultaneously removing the rank-one adversary assumption required by their analysis. Moreover, this rate is the same as the known optimal rate for stochastic kernel bandits, and also matches a lower bound from concurrent work up to a $\log T$ factor.

We study a stochastic multi-armed bandit problem where an agent is granted a free exploration budget before regret accumulates, a setting not captured by the classic regret minimization or pure exploration paradigms. The goal is to design an adaptive policy that strategically explores the bandit instance in the initial free exploration phase and minimizes the cumulative regret in the subsequent phase. We formalize this regret minimization with free exploration problem and identify an interesting regime where the free exploration budget scales logarithmically with the time horizon. To quantify the amount of regret saved with high probability as a result of the availability of the free exploration phase, we introduce a novel set of policies known as $(α,β)$-probably saving policies. We propose a two-phase, probably saving algorithm, UFE-KLUCB-H, which consists of a principled free exploration policy, UFE, and a history-aware regret minimization policy KLUCB-H. Instance-dependent upper bounds on UFE-KLUCB-H are derived, showing that UFE-KLUCB-H accumulates strictly less regret than policies that do not have access to a free exploration phase. Complementarily, we derive instance-dependent lower bounds based on novel multi-instance perturbation arguments tailored to the free-exploration setting, demonstrating the near-optimality of UFE-KLUCB-H for two-valued bandits. Our upper and lower bounds reveal sharp phase transitions in the accumulated regret depending on the amount of available free exploration. Simulations are conducted to demonstrate that forced exploration and adaptivity in the algorithm lead to greater regret savings.

We study adversarial multi-armed bandits with and without delayed feedback under a safety-aware goal: achieving minimax-optimal worst-case regret while keeping nearly constant regret relative to a designated "safe" baseline policy. Existing approaches can balance this trade-off with immediate feedback for smooth comparators, but arbitrary delays can mistime transitions between conservatism and exploration, endangering the safety guarantee. To bridge this gap, we propose Prudent-Banker, a novel algorithm that combines a delay-adapted variant of Online Mirror Descent with a modified phased-aggression mechanism. Its key technical contribution is a delay-calibrated restart threshold that rigorously accounts for the worst-case distortion induced by unobserved feedback and reliably detects comparator suboptimality. We also establish new lower bounds for safety-constrained adversarial delayed bandits, showing that the regret guarantees of Prudent-Banker are unimprovable, up to logarithmic factors, under the baseline-safety requirement. To the best of our knowledge, Prudent-Banker is the first algorithm to achieve the optimal safety--robustness trade-off: pseudo-regret $\widetilde{O}(\sqrt{T}+\sqrt{D})$ together with $\widetilde{O}(1)$ regret against the safe comparator, both with and without delays. Experiments across diverse delay distributions show that, unlike standard delay-robust baselines, Prudent-Banker effectively balances safety and learning.