We introduce the first Probably Approximately Correct (PAC) learning framework for general-sum concurrent stochastic games (CSGs) with transition uncertainty, while addressing the challenge of Nash equilibrium (NE) existence. Our algorithm maintains data-driven $L^1$ confidence sets over transition kernels and solves a robust CSG to compute a social-welfare optimal $\varepsilon$-NE, using a robust MDP-based exploration mechanism to drive joint state-action coverage. Crucially, we introduce a Nash margin characterisation that enables principled reasoning about equilibrium existence: the framework either returns an $\varepsilon$-approximate NE whose social-welfare value is $\varepsilon$-close to optimal, or provides a sound certificate that no exact NE exists. Under a minimum reachability condition $p_{\mathrm{reach}}>0$ over relevant state-action pairs, the algorithm terminates after a polynomial number of trajectory samples, with sample complexity $\widetilde{O}\left( {R_{\max}^2 H^4 |S|^2 |A| / (p_{\mathrm{reach}} \varepsilon^2)} \right)$. Empirical results on benchmark CSGs demonstrate near-optimal performance, correct handling of equilibrium (non-)existence, and sample complexity consistent with theory.
Restricted eigenvalue (RE) bounds govern stable recovery by norm-regularized estimators. For isotropic sub-Gaussian measurements, the benchmark sample size is $1+w(A)^2$, where $w(A)$ is the Gaussian width of the normalized descent cone. The COLT 2015 open-problem note (Banerjee et al., 2015) asked whether the same law follows for heavy-tailed designs from a uniform small-ball condition alone. We give an explicit and systematic negative answer to the general question as formulated there: the proposed law fails in its full dimension-free, arbitrary-set form, and the missing obstruction is simultaneous threshold occupancy. A constant-width polyhedral descent cone with fixed small-ball constants has zero empirical RE on every sample path up to half the ambient dimension. More generally, every finite range space admits exact threshold encoding in an arbitrarily narrow spherical cap and a lift to a full polyhedral descent-cone section. For every fixed threshold VC dimension $d$, as $β\downarrow0$, the sharp worst-case sample complexity is $Θ(β^{-1}[d\log(1/β)+\log(1/δ)])$. The separation persists under exact isotropy and all finite moments: on the same constant-width cone, Gaussian measurements succeed with $O(1+\log(1/δ))$ samples, whereas an isotropic heavy-tailed design fails pathwise for $n\lesssim\sqrt{p/\log p}$. Gaussian smoothing yields an everywhere-positive $C^\infty$ density while retaining arbitrarily poor RE. Under isotropy, a distribution-free fallback governed by affine dimension times squared enclosing radius is sharp on this family.
Gabriel Rioux, Joanna Marks, Riccardo Passeggeri +1math.ST cs.IT math.OC stat.ML
The Gromov-Wasserstein (GW) distance provides a principled framework for aligning metric measure (mm) spaces based solely on their intrinsic structure. Its ability to identify isomorphic representations of distributions across spaces renders it valuable for comparing data where equality up to isomorphism occurs naturally such as in graphs or, more generally, distributions on graphs. Recently, a type of dual form for the GW distance between Euclidean distributions with the squared Euclidean or inner product costs was derived, spurring the development of new statistical and algorithmic results for this setting. This work furnishes a novel duality result for GW distances with and without entropic regularization that is applicable to all finitely supported mm spaces. Leveraging this result, we derive the sample complexity of empirical GW distances between finite mm spaces, as well as limit distributions under proper centering and scaling. Furthermore, we propose new algorithms for solving the regularized GW problem which are subject to formal convergence guarantees. These statistical and algorithmic advancements give rise to a principled and efficient framework for testing whether two distributions on the set of graphs with a fixed number of nodes are isomorphic based on samples.
To mitigate the sample complexity of real-world reinforcement learning (RL), a common practice is to first train a policy in a simulator, where samples are cheap, and then deploy the learned policy in the real world with the hope that it generalizes effectively. Such direct sim-to-real transfer is not guaranteed to succeed: simulator-trained policies can be suboptimal in the real world due to sim-to-real mismatch. Correcting this mismatch requires collecting data from the real system, but in many applications, such as robotics and healthcare, this data-collection process is itself subject to safety constraints. This gives rise to the problem of safe sim-to-real transfer: how can an agent exploit an imperfect simulator while ensuring safe real-world data collection and learning a near-optimal feasible policy for the target system? We address this problem by formulating safe sim-to-real transfer within the framework of reward-free safe RL. We design a computationally efficient algorithm that exploits simulator information to provably reduce real-world interaction while ensuring safe exploration and enabling the computation of a near-optimal feasible policy for any potential reward function. Our real-world sample complexity bound characterizes the benefit of using the simulator in terms of the sim-to-real mismatch.
In transductive classification, an adversary fixes a labeled population, one label is hidden uniformly, and the learner sees all remaining labels. For binary classes, agnostic transductive and PAC learning have the same minimax rate. Whether this extends to multiclass learning was open, especially for unbounded label spaces where uniform convergence can fail. We resolve the question up to logarithmic factors. For every multiclass class $\mathcal H$ with DS dimension $d_{DS}$ and Natarajan dimension $d_{\mathrm N}$, the optimal agnostic transductive excess error satisfies $\widetildeΘ\left(\frac{d_{DS}}{n}+\sqrt{\frac{d_{\mathrm N}}{n}}\right).$ The result holds for arbitrary label spaces. The two terms are both necessary. A DS pseudo-cube gives the realizable $d_{DS}/n$ obstruction, while a Natarajan cube with repeated points and fair labels gives the agnostic $\sqrt{d_{\mathrm N}/n}$ obstruction. The upper bound uses a random-reservation principle. The learner deliberately ignores a constant fraction of the visible labels, which makes the true test point uniform in a large unseen block. We combine realizable compression, a label-space reduction, and inside-menu agnostic compression across this finite-population split. A new without-replacement multiplicative-weights lemma preserves the fast $d_{DS}/n$ term. Consequently, agnostic multiclass PAC and transductive learning obey the same two-dimension law up to logarithmic factors.
Long-horizon language-model tasks --- multi-step reasoning and tool-using agents alike --- are limited by credit assignment. We analyze it through the policy variance $σ_π^2(s)=\operatorname{Var}_{a\simπ}[Q_π(s,a)]$, which in a deterministic MDP is the sole source of return variance and is injected in discrete pulses at states we call critical forks. Three results follow. (i) Policy variance is a discovery budget: observing an action of advantage $c$ requires $Ω(c^2/σ_π^2(s))$ draws, a bound that is exact on the canonical two-point fork. (ii) Policy variance is bounded by the policy's Gini dispersion, $σ_π^2(s)\le 1-\|π(\cdot|s)\|_2^2$, a rollout-free necessary condition for criticality computable from logits alone. (iii) The remaining horizon sets the estimation cost: at a fork whose downstream success probability is $P$, the Monte Carlo advantage estimate has signal-to-noise ratio of order $\sqrt{P}$, so its sample cost scales as $1/P$ --- a cost that branched sampling shares. Bootstrapping removes it by converting a product of survival probabilities into a sum, provided the value representation is multiplicatively accurate, which argues for log-value parameterization.
Cooperative teams often need to agree on the best few options rather than simply accumulate reward, and they must do so while each member sees only a fragment of the team's collective experience. We study this as top-$K$ joint-arm identification in multi-agent multi-armed bandits: at every round $M$ agents simultaneously choose individual actions that compose a joint arm, and the team must ultimately return the $K$ joint arms of highest mean reward. The difficulty is that an agent may not observe the actions of others, their rewards, or either. We treat three observability regimes---(A) shared rewards with hidden actions, (B) observed actions with private rewards, and (C) full asymmetry---and design communication-free elimination algorithms (UCB-Intervals) that reconstruct implicit coordination from whatever signal each regime leaves intact: a shared arm ordering in (A), observable deviations in (B), and enlarged confidence radii under (C). We give matching analyses in both the fixed-budget and fixed-confidence objectives, then fold all three regimes into a single meta-guarantee indexed by a multiplicity $c$ and a consensus factor $ρ$. Our central result is quantitative rather than merely algorithmic: change-of-measure lower bounds show that shared-reward identification is optimal up to one universal logarithmic factor, and that the entire statistical price of removing communication is a multiplicative $ρ^2$ in sample complexity---a fixed $4\times$ penalty under full asymmetry. The resulting stopping time scales as $O\!\left(\sum_{\mathbf{a}} \frac{\log(A^M/δ)}{Δ_{\mathbf{a}}^2}\right)$ and the fixed-budget error as $\exp(-Θ(T/H_1))$, with the dependence on the joint-action count $A^M$ shown to be unavoidable.
Discrete diffusion models have demonstrated strong performance across a range of datasets, including natural language data and graph-structured data. Among many variants, score-entropy discrete diffusion (SEDD) has achieved particularly strong empirical results. In SEDD, new samples are generated by iteratively evaluating a sequence of concrete score functions, which are learned by minimizing a score-entropy loss. While much of the prior theoretical literature on discrete diffusion has focused on the sampling efficiency of SEDD under the assumption of small score estimation error, recent work has begun to investigate the finite-sample properties of score estimation itself. In this work, we take a different route by investigating the fundamental statistical limits of concrete score estimation. We focus on uniform and masking discrete diffusions, two of the most widely adopted discrete diffusion models. We establish a minimax lower bound under the score-entropy loss, and propose an MLE-based thresholding estimator that matches this lower bound up to constant and polylogarithmic factors that depend on neighboring density ratios. We further show that, for any target distribution, this density ratio is naturally controlled under both uniform and masking discrete diffusion models, yielding nearly matching minimax lower and upper bounds for the aggregated score estimation error. Our results imply that, with appropriate initialization and discretization, SEDD can achieve nearly optimal minimax sample complexity, as measured by the KL divergence between the target and generated distributions.
Data-driven algorithm design frames hyperparameter tuning as a statistical learning problem, but establishing generalization guarantees remains challenging due to the implicit, non-smooth dependence of model performance on hyperparameters. Existing multi-dimensional bounds under piecewise-polynomial assumptions remain theoretically loose and lack comprehensive lower bounds. We resolve this by establishing tight pseudo-dimension bounds for multi-dimensional data-driven tuning. First, we refine the learning-theoretic upper bound using real algebraic geometry; by analyzing invariant connected sign cells during block elimination rather than isolated sign vectors, we avoid topological over-counting to derive strictly sharper sample complexities. Second, we present a multi-regime lower-bound framework that disentangles combinatorial and algebraic capacities. By constructing shattered problem instances across distinct regimes, we prove our upper bounds are tightly saturated. Finally, we extend our topological framework to accommodate general bi-level validation-loss tuning and broader semi-algebraic applications.
Motivated by modern marketplaces, where the platform or the seller routinely gathers detailed user profiles, we study a novel learning theoretic model that simultaneously involves information and mechanism design. Specifically, we consider the economic setting recently introduced by Bergemann et al. (2022), where in addition to the menu of quality-price pairs, the seller offers information on the value of the match between product quality and buyer's taste via a signaling scheme. We relax the assumption that the seller knows the buyers' belief about the distribution of tastes and study the sample requirements of designing a revenue maximizing scheme. We consider both the batch setting where we have access to data from a set of i.i.d. buyers and an online demand query model where we observe the buyers' behaviors to seller's schemes. Despite the apparent non-convexity of the problem, we also give the first FPTAS to compute a scheme that maximizes the revenue within an arbitrarily small additive loss, which was left open by Bergemann et al. (2022). Overall, this brings a new learning perspective in asymmetric economic settings where buyers and sellers know different types of information.
Ege C. Kaya, Arda Fazla, M. Berk Sahin +1cs.LG math.OC
We study stochastic approximation of fixed points of a non-expansive operator when the oracle samples originate from a continuing Markovian trajectory. A direct block-minibatch implementation of Halpern iteration attains an expected last-iterate residual of order $O(\log N/N)$, but accrues a substantive complexity of $\tilde O(ε^{-5})$ Markovian samples. We therefore introduce a variance-reduced Markovian PAGE-Halpern method whose refresh and same-state difference blocks are analyzed through the Poisson equation. In Hilbert spaces, the cocoercivity of $I-T$ results in an $O(ε^{-3})$ sample complexity. Our main result extends this construction to a general finite-dimensional Banach space. A displacement-level Halpern bound replaces the Hilbert-space potential and yields $\tilde O(ε^{-3})$ sample complexity in the original non-expansiveness norm. We also establish a high-probability guarantee with the same leading accuracy dependence by measuring the estimator in an auxiliary smooth norm. Non-smooth sup and block-sup geometries are covered through norm smoothing.
We study how many observations are needed to determine the causal direction between two linearly related variables. Classical LiNGAM theory shows that independent non-Gaussian disturbances identify the direction, but does not quantify the difficulty when the causal effect is weak or the disturbances are nearly Gaussian. Let $β$ bound the absolute structural coefficient from below, let $ν$ measure each standardized disturbance's distance from Gaussianity, and let the disturbance scales lie in $[\underlineσ,\overlineσ]$. We prove the sharp local minimax law \[ N_2^\star(β,ν,δ) \asymp \frac{\log(1/δ)} {d_β^2+β^2ν^2}, \qquad d_β= \left[β^2- \left(1-\frac{\underlineσ^2}{\overlineσ^2}\right)\right]_+. \] Previous theory established population identifiability or assumed a fixed separation between the two directions. By contrast, we establish the sharp sample complexity as a joint function of edge strength, distance from Gaussianity, and scale uncertainty, and characterize when identification comes from non-Gaussian dependence or from covariance alone. The proof was independently generated with GPT-5.6 Sol in Codex's Ultra mode during a two-hour session. The human author supplied the prompt and was responsible only forchecking the proof and revising and polishing the manuscript.
We revisit the problem of learning predictors robust to adversarial examples at test-time. We prove that VC classes are adversarially robustly learnable with sample complexity linear in the VC dimension $d$, providing an exponential improvement over the previous upper bound of Montasser, Hanneke, and Srebro (2019). Remarkably, this result is achieved with a simple improper algorithm that combines the classic heuristic bagging (bootstrap aggregation) of Breiman (1996) with robust empirical risk minimization (RERM). Our algorithm computes RERMs on $O(d^\star)$ independent bootstrap samples and outputs their majority vote, where $d^\star$ denotes the dual VC dimension. We complement this result with a lower bound showing that this is unavoidable: in general, any learner in this oracle model requires $Ω(d^\star)$ calls to an RERM oracle, even when given arbitrarily many training examples.
Frank E. Curtis, Lingjun Guo, Daniel P. Robinsonmath.OC cs.LG stat.ML
For solving nonconvex equality-constrained optimization problems, a recent Gradient-Eigenstep Algorithm by Goyens et al.~is an iteration-efficient approach, based on minimizing Fletcher's augmented Lagrangian function, for finding an approximate second-order stationary point from an arbitrary starting point. In this paper, the analysis of this algorithm is extended, offering a two-fold contribution. First, it is shown that a local-linear rate of convergence can be obtained by this method if it is initiated sufficiently close to a strong second-order stationary point and employs a sufficiently small step-size parameter and sufficiently large penalty parameter. In this case, the algorithm reduces to a gradient descent algorithm applied to minimize Fletcher's augmented Lagrangian. Second, as a particularly useful application of the first result, it is shown that the Gradient-Eigenstep algorithm can be used as an iteration-efficient subproblem solver in the context of a progressive sampling strategy for solving equality-constrained optimization problems when the objective and constraint functions are defined by large sample averages, ultimately offering an algorithm with an improved worst-case sample complexity when compared to an approach that solves a full-sample problem directly.
Let $X_1,\ldots,X_n$ be independent Gaussian tensors in $\mathbb{R}^{d_1}\otimes\cdots\otimes\mathbb{R}^{d_k}$ whose covariance is a Kronecker product of $k$ unknown positive-definite factors, and put $D=\prod_{a=1}^k d_a$ and $d_{\max}=\max_a d_a$. A recent result of Franks et al. (2026) established condition-number-free nonasymptotic guarantees for the tensor-normal maximum likelihood estimator under the sample threshold $nD\gtrsim k^2 d_{\max}^3$. They asked whether the cubic dependence on $d_{\max}$ could be replaced by the operator-norm scale $d_{\max}^2$. We answer this question affirmatively. We prove that, for $t\geq 1$, the maximum likelihood estimator exists uniquely with high probability whenever $nD\geq Ck^2 d_{\max}^2 t^2$, and satisfies $d_{\mathrm{FR}}(\widehatΘ,Θ)\leq Ct\sqrt{k}\,d_{\max}/\sqrt{n}$ and $d_{\mathrm{FR}}(\widehatΘ_a,Θ_a)\leq Ct\sqrt{k d_a}\,d_{\max}/\sqrt{nD}$. For every mode of largest dimension, we also obtain the sharp Thompson bound $d_{\mathrm{op}}(\widehatΘ_a,Θ_a)\leq Ct\,d_{\max}/\sqrt{nD}$. No sparsity, condition-number bound or warm start is assumed. For fixed $k$, the threshold has the information-theoretically optimal dependence on $d_{\max}$, and the displayed rates for the full precision and the largest factor match Gaussian minimax lower bounds up to a factor $\sqrt{k}$. The proof extends a random Gram bound for local group-orbit directions to the full local Lie algebra, transports it to a fixed Thompson ball by exact conjugation, and combines sensitivity of a constrained maximum likelihood estimator with an equivariant Kirszbraun extension and Gaussian concentration. This removes the Frobenius-to-operator loss responsible for the previous extra factor $d_{\max}$ and resolves the explicit open problem posed in the earlier work.
We study policy-based reinforcement learning under the $μ$-resets interaction protocol of Kakade and Langford [KL02]. This interaction protocol enables the learner to sample trajectories from a given exploratory reset distribution $μ$, in addition to the starting distribution. We resolve the question raised by [KLS25] on the role of policy realizability for the sample complexity of this problem. Critically, the dependence on horizon $H$ is governed by the notion of coverage assumed of the reset distribution. Under bounded all-policy concentrability, we show a $\exp(Ω(H))$ sample complexity lower bound; with bounded pushforward concentrability, we show the dependence on horizon is tightly characterized as $\exp(Θ(\sqrt H))$.
Ilan Doron-Arad, Idan Mehalel, Elchanan Mosselcs.LG
Motivated by LLMs, which generate outputs by iteratively sampling from next-token distributions, we introduce a PAC-learning model for binary stochastic autoregressive learning. This generalizes the deterministic autoregressive learning framework of Joshi et al., COLT 2025. In our model, one fixed generator assigns a Bernoulli next-token distribution to every prompt string. Starting from an input prompt, a token is sampled and appended to the prompt; the same generator is then applied again to this expanded prompt; this procedure is repeated for $M$ steps. Three forms of supervision are considered: base one-step samples, chain-of-thought (CoT) samples that reveal full random trajectories of length $M$, and end-to-end (e2e) samples that reveal only the final token of length $M$ trajectories. For a generator class, we study the minimum number of samples $m_{base}(\varepsilon),m_{CoT}(\varepsilon), m_{e2e}(\varepsilon)$, resp., required to learn the one-step probabilities in the base model, and the final-token probability in the CoT and e2e models, under squared loss error~$\varepsilon$. We show that stochastic autoregressive learning fundamentally differs from the deterministic theory. At scale $\varepsilon$, there is no universal comparison between the three learning tasks: both $m_{CoT}/m_{base}$ and $m_{e2e}/m_{CoT}$ can be made simultaneously arbitrarily larger than $M/\varepsilon$, the natural analogue for the existing deterministic results. Nevertheless, after altering scales, for every class, CoT learning at scale $\varepsilon$ is upper-bounded by base learning at scale $\varepsilon/M^2$, whereas e2e learning at scale $\varepsilon$ is upper-bounded, up to logarithmic factors, by $(M/\varepsilon) m_{CoT}(Θ(\varepsilon))$. These dependencies and scales are essentially tight. We complement these bounds by studying dimension $d$ logistic functions in our model.
Distributionally robust Markov decision processes provide a principled framework for sequential decision making under model uncertainty. We study how many samples are necessary and sufficient to learn an $\varepsilon$-optimal robust policy under the average-reward criterion. A generative model provides samples from the nominal transition kernel, whereas policy performance is evaluated over $(s,a)$-rectangular total-variation uncertainty sets of radius at most $σ$. Let $H_0$ and $H_σ$ denote the nominal and robust optimal bias spans, respectively. We identify $σH_0$ as the perturbation scale separating high- and low-tolerance regimes. Our matching upper and lower bounds show that, up to logarithmic factors, the minimax total sample complexity is $$ NSA \asymp \frac{SA}{\varepsilon^2}\begin{cases} \min\{H_0,H_σ\}, & \varepsilon\gtrsimσH_0,\\ \min\{H_0,H_σ\}+σH_σ^2, & \varepsilon\lesssimσH_0. \end{cases} $$ Here $S$ and $A$ are the numbers of states and actions, and $N$ is the number of samples per state-action pair. The sample complexity consists of a linear-span term that resembles the nominal AMDP results and a robustness-specific term that appears only in the low-tolerance regime. We attain these rates using reduction-based plug-in procedures that select the reduction---nominal or robust---and its discount factor: a span-informed procedure that makes these choices using known span parameters, and a span-agnostic procedure that calibrates both choices from data.
Markus Engelund Mathiasen, Jian Qian, Nikita Zhivotovskiycs.LG cs.AI cs.DS math.ST
Let $H\subseteq\{-1,+1\}^X$ be a class of finite VC dimension $d\ge1$. Writing $L$ for the binary risk and $L^*=\min_{h\in H}L(h)$, we construct a learner achieving the statistically optimal risk bound: from an i.i.d.\ sample of size $n$, for every $0<δ\le 1/2$, with probability at least $1-δ$, \[ L(\widehat h) \le L^*+ 7\cdot10^8\left( \sqrt{\frac{L^*(d+\log(1/δ))}{n}} +\frac{d+\log(1/δ)}{n} \right). \] This settles the sample complexity of agnostic PAC learning up to universal constants at every fixed $L^*$, matching the lower bounds of Devroye, Györfi, and Lugosi [A Probabilistic Theory of Pattern Recognition, Springer, 1996].
Elad Aigner-Horev, Daniel Rosenberg, Roi Weisscs.LG
We study distributionally robust PAC learning for the $0$--$1$-loss, where adversarial perturbations of the data distribution are constrained by a Cressie--Read divergence of order $k>1$ and radius $ρ\geq 0$. For hypothesis classes with VC dimension $d$, we establish realizable and agnostic sample-complexity bounds tight up to constant and logarithmic factors, respectively; ordinary empirical risk minimization attains both rates up to logarithmic factors. For target accuracy $\varepsilon\in(0,1)$ and confidence $δ\in(0,1)$, their respective orders are \[ \max\!\left\{\frac{1}{\varepsilon}, \frac{ρ^{\frac 1{k-1}}}{\varepsilon^{k_\star}} \right\}\cdot(d+\log δ^{-1}) \qquad\text{and}\qquad \max\!\left\{\frac{1}{\varepsilon^2}, \frac{ρ^{\frac1{k-1}}}{\varepsilon^{k_\star\vee 2}} \right\}\cdot(d+\log δ^{-1}), \] where $k_\star={k}/{(k-1)}$. For every fixed $ρ>0$, robustness changes the realizable $\varepsilon$-dependence from $\varepsilon^{-1}$ to $\varepsilon^{-k_\star}$ as $\varepsilon\downarrow0$. In the agnostic case, for $1<k<2$, robustness changes the $\varepsilon$-dependence from $\varepsilon^{-2}$ to $\varepsilon^{-k_\star}$, whereas for $k\geq2$ the exponent remains the classical $2$, with nontrivial $ρ$-dependence. Building on the known scalar reduction of robust $0$--$1$ risk to ordinary classification error, our analysis reveals a scale-sensitive interaction between the statistical estimation of classification error and its amplification by robustness, sharply explaining the transition in the agnostic rate. We extend the previously studied $χ^2$-divergence case to every Cressie--Read order $k>1$, close its upper--lower gaps, and recover standard PAC learning rates as $ρ\to0$, unlike previous bounds that fail to interpolate correctly in this limit.
Jiuyao Lu, Krishnakumar Balasubramanian, Aleksandr Podkopaev +1cs.LG math.ST stat.ML
Calibration requires a predictor to be unbiased after conditioning on its own predictions. Multicalibration asks for this guarantee simultaneously across a collection of groups. Many prediction tasks ask for several related features of the same conditional outcome distribution: variance is defined relative to the mean, skewness relative to both mean and variance, and conditional value at risk relative to a quantile. We study multicalibration for a sequence of $k$ properties in which each property is identifiable once the preceding properties are fixed. This framework includes Bayes pairs but does not require the properties to arise from a single loss. For every fixed $k\ge2$, we establish matching upper and lower sample-complexity bounds up to logarithmic factors under regularity conditions. Even with only polylogarithmically many binary groups, achieving multicalibration error $\varepsilon$ requires $\widetildeΩ(\varepsilon^{-(k+2)})$ samples. Conversely, for any finite group family $\mathcal G$, we give a randomized learner using $O(\varepsilon^{-(k+2)}+\varepsilon^{-2}\log|\mathcal G|)$ samples. Thus the sample complexity is $\widetildeΘ(\varepsilon^{-(k+2)})$ for polynomial-size group families. We instantiate the theory for three canonical examples.
This paper is concerned with one-bit mean estimation, where each independent sample is represented by a single binary message. We consider distributions on $\mathbb{R}$ with mean in $[-λ,λ]$ and absolute $k$-th central moment at most $σ^k$, where $k>1$ is fixed. For this class, previous work attained the optimal sample complexity for general queries using a two-stage protocol. The first stage localizes the mean. The second-stage queries are chosen after localization and refine the estimate around the decoded center. We show that this interaction can be avoided by constructing a randomized fully non-adaptive protocol that fixes all queries before observing the data and matches the optimal adaptive sample complexity. For target accuracy $ε$ and confidence $1-δ$, its sample complexity scales as \[ \log\fracλσ + \begin{cases} (σ/ε)^2\log(1/δ), & k>2,\\ (σ/ε)^2\log(σ/ε)\log(1/δ), & k=2,\\ (σ/ε)^{k/(k-1)}\log(1/δ), & 1<k<2, \end{cases} \] up to constants depending only on $k$. In the range covered by the known lower bound, this rate is minimax optimal even among fully adaptive protocols. This gives a negative answer to the COLT 2026 open problem asking whether interaction is necessary for order-optimal one-bit mean estimation with general queries \citep[Open Problem~1]{lau2026open}.
We study the problem of identifying the dominant arm in multi-armed bandits, where the objective is to find the action with the highest probability of exceeding the realized rewards of all other actions. Conventional mean-based and pairwise comparison-based algorithms often fail to identify the arm with the highest realized reward. To address this challenge, we introduce a novel dominant arm criterion and an efficient estimator with theoretical guarantees. Our approach relies on two key technical innovations: (i) a dominance score criterion that an arm beats the locally dominant over the partitioned reward space and (ii) a joint mixing and recycling mechanism coupled with a doubly robust estimator that guarantees simultaneous convergence of the empirical distribution functions for all arms. These key innovations pave a way to efficient computation of global arm dominance. Our proposed elimination algorithm identifies the best dominant arm with nearly optimal rate of sample complexity. Numerical experiments demonstrate that our algorithm consistently achieves exact recovery of the true dominant arm, outperforming existing baselines.
Anders Jonsson, Emilie Kaufmann, Gianmarco Tedeschi +1cs.LG
We present HBPI-UCRL, a model-based algorithm for hierarchical reinforcement learning (HRL) that learns high-level and low-level policies in parallel. HBPI-UCRL exploits the fact that a high-level transition corresponds to a multi-step transition at the low level. We introduce two conditions on the low-level dynamics that are sufficient to make parallel HRL learnable. When these conditions hold, we prove that HBPI-UCRL has a polynomial sample complexity in the problem parameters. In the sparse-reward, goal-directed setting, our sample complexity upper bound for HBPI-UCRL is strictly lower than that of its non-hierarchical counterpart, providing theoretical justification for the empirical success of HRL.
Naman Saxena, Mudit Gaur, Vaneet Aggarwalcs.LG cs.AI
Bilevel reinforcement learning (RL) is an important framework within the literature of RL that can be used to formalize various categories of problems, such as meta-learning, hierarchical task decomposition, and reinforcement learning from human feedback (RL-HF). Most of the bilevel RL algorithms are either not scalable because of using hypergradient with Hessian, or they suffer from high sample complexity because of using penalty-based approximation methods. In this work, we propose a hypergradient-based bilevel RL algorithm using the optimality of the Boltzmann policy for the entropy regularized discounted RL objective function. Our proposed algorithm is Hessian-free and obtains an iteration complexity of $O(ε^{-1})$ and state-of-the-art sample complexity of $\tilde{O}(ε^{-2})$ under mild regularity conditions. Further, in our convergence analysis, we are able to remove the assumption of the Polyak-Lojasiewicz (PL) condition on the outer-level objective function present in the prior state-of-the-art sample complexity work.
Low-Rank Adaptation (LoRA) has become the standard mechanism for fine-tuning large pretrained models, yet its statistical properties remain only partially understood. Existing generalization results provide upper bounds of the form O~(sqrt(rd/n)) or O~(rd/n), but a matching lower bound is missing, and the question of how to choose the LoRA rank r has no formal answer. Both gaps are closed here. A local Rademacher argument establishes an upper bound of O~(rd/n) on the excess risk of the empirical risk minimizer over rank-r LoRA, whenever the target adaptation has rank at most r. A matching minimax lower bound of Omega(rd/n) is then proved via a Fano-type packing of the rank-r subspace of R^{d x d}; the bound applies to any estimator whose output lies in the rank-r LoRA class. Combining the two yields a rank-selection dichotomy. For the constrained empirical risk minimizer, the optimal rank equals the intrinsic rank r*, and over-ranking strictly hurts. For adaptive estimators of the nuclear-norm-then-truncate type, over-ranking is harmless and the rate saturates at Theta~(r* d / n) regardless of r. Taken together, the three results characterize the statistical complexity of LoRA fine-tuning within the well-specified locally quadratic regime, and identify the empirically observed over-parameterization penalty as a property of unregularized empirical risk minimization rather than of the LoRA class itself. Predictions of the theory are verified on a synthetic trace-regression benchmark and on real LoRA fine-tuning across three (model, task) configurations covering DistilBERT and RoBERTa on SST-2 and MRPC. All configurations exhibit the predicted U-shape in validation loss, with two showing statistically significant loss inflation at large ranks (paired permutation p = 0.016).
In many domains such as Palliative Care, Credit Assignment and Recommender Systems, predictions may causally influence the outcomes they predict. This phenomena is known as Outcome Performativity. This paper formalises an approach for detecting Outcome Performativity using prediction intervention called Outcome Performativity A/B Detection (OPAB). OPAB enables the detection of Outcome Performativity by assessing the dissimilarity in outcome distributions produced by different predictions groups (interventions). If that dissimilarity is significant, Outcome Performativity is detected. We derive sample complexity bounds for OPAB under various Outcome Performative assumption classes which we empirically validate. Results show that detecting Outcome Performativity using OPAB is achievable in numerous cases. Results also show the presence of regions of indistinguishability which describe settings where the allotted number of interventions are insufficient for detecting Outcome Performativity. The results of which have broader practical implications for the detectability of Outcome Performativity in settings where samples are scarce, cost-prohibitive or potentially unethical to obtain. The paper concludes with a case study on the efficacy of OPAB on the Open Bandits dataset, and provides directions for future work.
We study stochastic composite nonconvex optimization over a compact convex set when gradient samples arrive along a single trajectory of a fixed ergodic Markov chain. Existing single-trajectory variance-reduction theory covers smooth unconstrained objectives; we address the projection-free composite setting using the generalized Frank-Wolfe gap. We propose MC-ALFCG, which combines a momentum conditional-gradient method with coupled capped multilevel Monte Carlo estimation and per-iteration clipping. The deepest nested average uses consecutive states from the same trajectory, yielding conditional bias $O(τ_{\mathrm{mix}}/T)$ uniformly over the starting state, while coupling controls the gradient-difference second moment through the iterate displacement. Clipping enforces the pathwise bounds needed by the adaptive analysis. We reduce the Markovian recursion to its independent-sampling counterpart under $σ^2\mapsto 2ΛG_σ^2$ and $L^2\mapsto 2ΛL^2$, where $Λ=O(τ_{\mathrm{mix}}\log T)$. For positive centered noise, the tuned method achieves expected sample complexity $\widetilde{O}((τ_{\mathrm{mix}}^2G_σ+τ_{\mathrm{mix}}^{5/2}G_σ^2)\varepsilon^{-3}+τ_{\mathrm{mix}}^5\varepsilon^{-2})$. The exactly noiseless specialization achieves $\widetilde{O}(\varepsilon^{-2})$ with mixing-time-free constants, while a mixing-time-oblivious variant achieves $\widetilde{O}(τ_{\mathrm{mix}}^6\varepsilon^{-3}+τ_{\mathrm{mix}}^3\varepsilon^{-2})$. All guarantees are in expectation under a fixed transition kernel. Controlled numerical studies examine dependence sensitivity, a nonconvex composite instance, and clipping behavior.
Jon Kleinberg, Amin Saberi, Xizhi Tan +1cs.DS cs.GT cs.LG stat.ML
Motivated by learning from heterogeneous and overlapping data providers, we study a stylized model of distribution learning from restricted conditional samples. The goal is to learn an unknown distribution $p$ on a finite domain $[n]$. The learner is given a fixed family of queryable sets $\mathscr{S} \subseteq 2^{[n]}$, and each query to $S \in \mathscr{S}$ returns an independent sample from the conditional distribution $p(\cdot \mid S)$. Learnability is governed by the co-occurrence graph associated with $\mathscr{S}$: two domain elements are adjacent if they appear together in some queryable set. Pointwise consistency is achievable when this graph is connected on the target support. PAC learning requires more: it is possible when the co-occurrence graph is complete. The optimal sample complexity of PAC learning ranges from nearly linear to quadratic. Every query family with complete co-occurrence graph admits sample complexity $\widetilde O(n^2/ε^2)$, and this bound is tight in the worst case. On the other hand, if $[n]$ is queryable then ordinary sampling improves the bound to $Θ(n/ε^2)$, and this cannot be improved further even if every set is queryable. More generally, we identify hierarchical comparabilityas a sufficient structural condition on $\mathscr S$ under which the optimal complexity is nearly linear, $\widetilde Θ(n/ε^2)$, with pairwise query families as a canonical example. Finally, the full range of polynomial rates between linear and quadratic is attainable: for every $α\in (1,2)$, there exists a query family with optimal PAC rate $\widetilde Θ(n^α/ε^2)$.
Heyang Zhao, Tianyuan Jin, Weixin Wang +3cs.LG stat.ML
Recent years have witnessed increasing interests in tackling heteroscedastic noise in bandits and reinforcement learning. In these works, the cumulative variance of the noise $Λ= \sum_{t=1}^T σ_t^2$, where $σ_t^2$ is the variance of the noise at round $t$, is used to characterize the statistical complexity of the problem, yielding \emph{simple regret} bounds of order $\tilde{\cal{O}}(d \sqrt{Λ/ T^2})$ for $d$-dimensional linear bandits with heteroscedastic noise. However, with a closer look, $Λ$ remains the same order even if the noise is close to zero at half of the rounds, which indicates that the $Λ$-dependence is not optimal. In this paper, we revisit the stochastic linear bandit problem with heteroscedastic noise, where the action set is prefixed throughout the learning process. We propose a novel variance-adaptive algorithm \texttt{VAEE} (Variance-Aware Exploration with Elimination) for large action set, which actively explores actions that maximizes the information gain among a candidate set of actions that are not eliminated. With the active-exploration strategy, we show that \texttt{VAEE} achieves a \emph{simple regret} with a nearly \emph{harmonic-mean} dependent rate. For finitely many actions, we propose a variance-aware variant of G-optimal design based exploration, which achieves a simple regret with sharper dependence on $d$. We also establish a nearly matching lower bound for the fixed action set setting indicating that \emph{harmonic-mean} dependent rate is unavoidable. To the best of our knowledge, this is the first work that breaks the $\sqrtΛ$ barrier for stochastic linear bandits with heteroscedastic noise.