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Linear bandits

Found by the topic model: linear bandits, policy gradient, stochastic gradient descent.

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50papers this week−31.5% vs last week396 this month

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Papers

Open problems in Linear bandits

  • Handling non-stationary environments in linear bandits1 paper raises it · 1 addresses it

    Real-world systems often change over time, making static models ineffective for long-term decision-making.

    • Optimal Contextual Pricing in Non-Stationary Environmentsraises · addresses
  • Balancing exploration and regret in pure exploration1 paper raises it · 1 addresses it

    Exploration strategies must account for regret costs relative to a baseline, not just sample efficiency.

    • SHPRR: An Algorithm for Pure Exploration under Constrained Regretraises · addresses
  • Optimizer-dependent precision limits in high-dimensional sensing1 paper raises it · 1 addresses it

    Numerical precision can stall optimization in high-dimensional or ill-conditioned problems, limiting practical performance.

    • The Precision Wall: An Optimizer-Dependent Boundary for Optimization under Power-Law Measurement Spectraraises · addresses
  • Adaptive path selection under mobility and energy constraints1 paper raises it · 1 addresses it

    Mobile sensors must efficiently explore while respecting physical constraints like energy and path feasibility.

    • Adaptive search with path-constrained mobile sensors using multi-arm banditsraises · addresses
  • Domain generalization with adversarial target domains1 paper raises it · 1 addresses it

    Models trained on well-resourced domains often fail in underrepresented or adversarially chosen new domains.

    • Rural Readiness: Domain Generalization with Adversarially Chosen Domainsraises · addresses

Named by an AI from these papers' abstracts.

Rising research topics

  • Spiking neural networks7+40.0%
  • Diffusion priors16+14.3%
  • Algorithmic fairness9+12.5%
  • Channel state information8−20.0%
  • Quantum machine learning14−22.2%
  • Character recognition OCR30−28.6%
  • Linear bandits50−31.5%
  • Molecular property prediction27−32.5%

Papers

  • arXiv cs.LGOct 29

    The Precision Wall: An Optimizer-Dependent Boundary for Optimization under Power-Law Measurement Spectra

    Lngel Kyo

    In low-rank matrix sensing of a rank-$D$ factor $U^\ast \in \mathbb{R}^{V \times D}$, replacing isotropic Gaussian designs with power-law measurement covariance $Σ= \mathrm{diag}(j^{-s})_{j=1}^V$ leaves the Burer--Monteiro landscape free of spurious local minima, yet division-normalised first-order optimizers stall above a critical exponent $s^\ast$. We trace this failure to a numerical precision wall. At the ground truth, the smallest non-gauge expected Hessian eigenvalue scales at leading order as $V^{-(s+1)}D^{-s}$. When this curvature drops below the effective denominator floor $ε_O$ of optimizer $O$, normalisation noise overwhelms rare-direction gradients, yielding the leading-order boundary $s^\ast_O(V, D) = [\log(1/ε_O) - \log V]/\log(VD) + δ_O$, where $δ_O$ is a metric-dependent detection lag. With $δ_{\mathrm{Adam}}$ calibrated on four configurations and frozen, this formula predicts Adam's onset within $0.45$ across all 16 cells of a held-out grid ($V \in \{64,128,256,512\} \times D \in \{4,8,16,32\}$), separates fixed-$VD$ pairs, and distinguishes denominator-precision walls from conditioning and sign-quantisation walls. The approximation fails at $D = 2$, reflects a…

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  • arXiv cs.LGOct 29

    Rural Readiness: Domain Generalization with Adversarially Chosen Domains

    Cynthia Dwork, Andisheh Ghasemi, Lunjia Hu

    Predictive models trained at well-resourced data-rich institutions often fail when deployed in new populations. Motivated by the goal of building clinical diagnosis and decision-making algorithms for data-scarce rural populations using the general knowledge learned from data-rich hospitals, we formulate the fundamental question of \emph{domain generalization with adversarially chosen domains}, which asks a learner to perform on any unseen target domain nearly as well as the worst hypothesis consistent with the evidence from the observed source domains. Assuming only that the hypothesis class contains a "universal" predictor with small error on all $k$ source domains, we characterize the sample complexity of this objective up to logarithmic factors. Letting $d$ denote the VC dimension of the hypothesis class, we find, somewhat surprisingly, that the sample complexity reduces significantly from $Ω(dk)$ to $O(d)$ under a small relaxation of the learning goal in which, roughly speaking, we allow the learned model to compete with hypotheses that perform well on 99% instead of 100% of the source domains. By applying this insight to the meta-distributional setting, in which source…

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  • arXiv cs.ROOct 29

    Adaptive search with path-constrained mobile sensors using multi-arm bandits

    Abraham P. Vinod, Parth K. Thaker, Gautam Dasarathy, Stefano Di Cairano

    We study a constrained sequential decision problem for detecting regions of interest from noisy online measurements collected by a team of energy-constrained mobile sensors. The sensor motion is restricted to a precomputed set of feasible paths that satisfy mobility constraints, limited available energy by requiring return to charging stations, and avoidance of known obstacles. We cast the resulting active path selection as a multi-armed bandit problem and propose an iterative search algorithm that adaptively assigns paths using confidence bounds computed from the collected data. We also propose a graph-based method to construct low-cardinality path covers under the team's constraints. We establish finite-time and anytime probabilistic correctness guarantees for the proposed algorithm, derive non-asymptotic upper bounds on the times required to detect all regions of interest and to label all nodes, and show that these bounds match algorithm-independent lower bounds up to constant factors for a broad class of problem instances. Using numerical simulations, we illustrate the behavior of the proposed approach and the effect of path redundancy and team size on search performance.

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  • arXiv cs.DSOct 29

    Leveraging Recourse and Size Bounds for OnlineKnapsack with Removable Items

    Dimitris Fotakis, Laurent Gourvès, Aris Pagourtzis, Panagiotis Patsilinakos

    In the online unweighted knapsack problem, items arrive sequentially and must be accepted or rejected immediately for a bounded-capacity knapsack, aiming to maximize the total size of accepted items. When decisions are irrevocable, no algorithm achieves a non-trivial competitive ratio. An essential relaxation that circumvents this impossibility is to allow previously accepted items to be removed from the knapsack (a.k.a. removal). In this work, we investigate possible further improvements on the competitive ratio with removal by bringing together two other previously proposed model extensions: knowledge of an upper bound on the item sizes (a.k.a. bound awareness) and ability to reinsert previously rejected items(a.k.a.recourse). Our contributions are twofold. First, we establish close upper and lower bounds on the competitive ratio under bound awareness, removal and recourse. We show that significant improvements on the best known competitive ratio are possible when the upper bound on the item sizes is not too restrictive and only few uses of recourse are available. Second, we introduce the notion of adaptive online algorithms which do not require to know any upper bound on the…

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  • arXiv cs.LGOct 29

    Closing the Log Gap: Minimax-Optimal Regret for Episodic RL

    Guy Zamir, Victor Boone, Matthew Zurek, Yudong Chen

    In this paper, we study the regret minimization problem in episodic tabular MDPs. Our algorithm achieves the first expected regret bound scaling as $\sqrt{H^3 S A K}$ for sufficiently large $K$, where $H$ is the episode length, $K$ is the number of episodes, and $S$ and $A$ are the numbers of states and actions, respectively. Our method is the first to match the minimax lower bound \emph{without extra polylog factors}. We introduce a variant of UCBVI which does not maintain strictly optimistic estimates, and uses MOSS-style exploration bonuses whose logarithmic terms adapt to both visit counts and estimated variances. A key challenge is controlling the propagation of errors caused by under-optimism, which we accomplish in an occupancy-weighted way, avoiding logarithmic overhead that would be incurred by uniform-over-rows concentration. Our analysis develops time-uniform and variance-aware concentration bounds that may be of independent interest. As a corollary we also develop an algorithm for $(\varepsilon, δ)$-PAC best-policy identification, which is the first to match the $Ω\big(H^3 SA\log(\frac{1}δ)\varepsilon^{-2}\big)$ minimax lower bound up to constants in some regimes.

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  • arXiv cs.DSOct 29

    Accumulating Pruning Constraints: New Exact Polynomial-Space Algorithms for Max 2-SAT

    Konstantin Kutzkov

    We present deterministic polynomial-space algorithms that solve Max 2-SAT on $n$ variables exactly in time $\Ostar(2^{(1-γ)n})$, for a fixed $γ>0$, on several classes of instances whose clause density may grow with $n$. Our main tool extends the classic DPLL branching algorithm. When the search gives a variable its less promising value, comparing the two values of the variable yields a pruning constraint that is satisfied by some optimal assignment. Accumulating these constraints along each branch prunes all but an exponentially small fraction of the assignments. The classes are formulas with many biased variables, semirandom 2-CNF formulas, and Max-Cut on regular expander graphs and on graphs generated by a stochastic block model. In a semirandom formula, the pairs of variables that may share a clause are arbitrary, and the clauses on these pairs are random. For semirandom formulas and block-model graphs, the algorithms are still exact on every input, and the time bound holds with high probability over the input. To our knowledge, these are the first polynomial-space algorithms with such a running time for nontrivial classes of dense Max 2-SAT instances. Beyond these classes,…

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  • arXiv cs.LGOct 29

    SHPRR: An Algorithm for Pure Exploration under Constrained Regret

    Georgina Nouli, Flore Sentenac, Maximilian Schiffer

    Best Arm Identification (BAI) conventionally measures the cost of experimentation by the number of samples collected. In many operational settings, however, the cost of exploration relative to a trusted baseline already in place is equally important. We study BAI under both a fixed sampling horizon and a regret budget. Unlike generic resources considered in constrained pure exploration, regret has a distinctive structure: sampling better arms is less costly relative to the baseline. We theoretically show that the regret cost of exploration decreases across the phases of Sequential Halving, a widely used pure-exploration algorithm. This observation yields improved guarantees on the probability of identifying the best arm and motivates a more effective allocation of the regret budget. We propose Sequential Halving with Proportional Regret Rationing (SHPRR), which front-loads the regret budget to allow more exploration when many arms remain. Experiments on synthetic and real-world instances show that SHPRR improves best-arm identification performance over uniform regret allocation and existing baselines.

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  • arXiv cs.LGOct 29

    Optimal Contextual Pricing in Non-Stationary Environments

    Yiding Feng, Alireza Rezaeimoghadam

    We study contextual dynamic pricing in a non-stationary environment. In each round, a seller observes a buyer's context, posts a price, and receives only binary purchase feedback. The buyer's value is linear in the context and an unobserved feature vector that may change over time, subject to a known total variation budget $V$. We measure dynamic regret against the first-best benchmark that extracts each buyer's full value. For identical contexts, we develop a deterministic algorithm that combines binary-search exploration, lower-envelope pricing, and periodic and inconsistency-triggered restarts. It achieves regret $O((\sqrt{TV}+1)\log T)$. With heterogeneous contexts, a new geometric difficulty arises: small feature changes can leave a relaxed feasible set narrow but far from the current feature vector. We address this difficulty through a multiscale shallow-pricing algorithm that maintains relaxed feasible sets and enclosing ellipsoids at several variation scales. Larger scales provide reliable localization certificates, while the smallest active scale determines the exploitation price. The algorithm achieves dynamic regret $O((d^2+d^{3/2}\sqrt{TV})\log^2 T)$, where $d$ is the…

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  • arXiv math.OCOct 29

    Adaptive Federated Optimization Across Smooth and Nonsmooth Regimes

    Zimeng Wang, Ali Dadras, Alp Yurtsever

    AdaGrad-style methods can simultaneously adapt to both the order and constants of smoothness in centralized optimization. However, existing extensions to federated optimization with heterogeneous client objectives and multiple local steps require the smoothness order to be specified in advance. We address this limitation with AdaFed, a federated method that adapts across several regimes under a single choice of parameters. In the deterministic setting, AdaFed achieves an $O(1/T)$ convergence rate for both smooth nonconvex and smooth convex objectives, and an $O(1/\sqrt{T})$ rate for Lipschitz nonsmooth convex objectives up to logarithmic factors. Under a light-tailed stochastic-gradient noise assumption, these guarantees extend to the stochastic setting with additional logarithmic factors and noise-dependent terms. Experiments across the three problem classes further support our theoretical findings.

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  • arXiv stat.MLOct 29

    Formalizing the Intrinsic Dimension of Objective Landscapes by a Generalized Notion of Co-dimension

    Ernesto Araya, Gitta Kutyniok, Weilin Li, Johannes Maly, Manjot Singh

    In this work, we formalize the concept of intrinsic dimension of objective landscapes introduced in Li et al. (2018), which measures the structural complexity of learning problems by training along randomly generated subspaces. We define the intrinsic dimension of a set $Θ\subset\mathbb{R}^p$ as the minimal dimension $d_{\mathrm{int}}(Θ)$ such that a random plane of dimension $d_{\mathrm{int}}(Θ)$ in $\mathbb{R}^p$ intersects $Θ$ with probability one. We show that it is a generalized notion of co-dimension that depends on the global geometry of $Θ$. We study mathematical properties of the intrinsic dimension and develop techniques to estimate the intrinsic dimension of sets $Θ$ that arise in learning problems. More specifically, we derive upper and/or lower bounds where $Θ$ is the set of optimal parameters in student-teacher settings with shallow and deep ReLU networks, including finite and infinite sample cases, and with and without regularization. We discuss how our theory leads to novel types of generalization bounds, thereby showing that the intrinsic dimension is a measure of learning complexity. Our results connect techniques from seemingly different areas of machine…

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  • arXiv stat.MLOct 29

    The Cost of Working Memory in Streaming Mean Estimation

    Zeyu Wu, Zihao Wang, Minghua Deng

    A streaming estimator must compute from the evidence that fits in memory. We determine when storing an accurate mean is easier than computing it. For Gaussian mean estimation over the Euclidean unit ball in $\mathbb{R}^d$, we establish the minimax rate under squared error with $K$ persistent states, a known horizon, and a public time index: $\min\{1, K^{-2/d}+\frac{d}{n} (1+\frac{\log(ed)}{1+(\log K)/d})\}$. Unrestricted computation followed by $K$-level encoding has rate $\min\{1,d/n+K^{-2/d}\}$. At a fixed sufficiently small error, statistically optimal sample size therefore requires order $d\log d$ working bits, while order $d$ bits suffice to store the answer. The lower bound tracks repeated state compression through an entropy potential obtained by Gaussian smoothing. A matching construction combines a multiscale modular counter with workspace reused across coordinate groups. In one dimension, it attains the terminal rate $n^{-1}+K^{-2}$. Any fixed confidence requirement that the realized loss not exceed the zero baseline preserves these rates. Increasing confidence creates a further memory requirement: at fixed nontrivial risk in every fixed dimension, failure probability…

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  • arXiv math.OCOct 29

    Conic-Lifting for Learning-rate-free Convex Optimization

    Julien Grand-Clément, Joon Kwon, Francesco Orabona

    A central topic in online and stochastic optimization is to design algorithms that do not require tuning of hyperparameters. Successful strategies in online learning include AdaGrad-style learning rates, Conic Blackwell algorithms, and parameter-free algorithms. However, these classes require different, often difficult, analyses. We develop a unifying family of Online Convex Optimization (OCO) algorithms that require no user-tuned learning rate on convex domains that may be unbounded. The construction lifts the domain to a cone by adding one scalar coordinate, an idea going back to Blackwell (1954). The lifted subgradients are orthogonal to the corresponding lifted decisions, leaving the comparator scale free to optimize in the regret analysis. Applying Follow-the-Regularized-Leader (FTRL) and Online Mirror Descent (OMD) on the cone recovers Conic Blackwell Algorithm (CBA) and Conic Blackwell Algorithm Plus (CBA$^+$) on bounded domains in Hilbert spaces and extends them to finite-dimensional normed spaces with a strongly convex squared norm. For arbitrary nonempty closed convex domains in these spaces, a known gradient bound yields parameter-free regret guarantees. Surprisingly,…

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  • arXiv cs.LGOct 29

    Last-Iterate Convergence for Constrained MDPs: From Dual Descent to Primal-Dual

    Michael Lu, Max Qiushi Lin, Mo Chen, Sharan Vaswani

    We study policy optimization for infinite-horizon, discounted constrained Markov decision processes. While primal-dual ($\texttt{PD}$) methods are widely used in practice, their theoretical guarantees typically hold for an impractical mixture policy. In contrast, dual descent ($\texttt{DD}$) methods output a single policy ("last iterate") with guarantees on its performance. However, such methods require iteratively solving entropy-regularized MDPs in an inner-loop, limiting their practical effectiveness. Moreover, existing analyses rely on strong coverage assumptions that are difficult to satisfy in practice. Consequently, we first provide a new analysis of $\texttt{DD}$ methods that does not require coverage assumptions. We then employ entropy-regularized natural policy gradient and dual regularization to eliminate the inner loop, resulting in an efficient $\texttt{PD}$ algorithm. We develop a novel tracking analysis which shows that, compared to existing methods, the proposed algorithm achieves improved last-iterate convergence guarantees. Furthermore, our theoretical framework allows us to analyze a truncated bilevel method that interpolates between $\texttt{DD}$ and…

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  • arXiv cs.LGOct 29

    Achieving Minimax-Optimal First-Order and Best-of-Both-Worlds Regret Simultaneously in Multi-Armed Bandits

    Mengxiao Zhang

    We study first-order and best-of-both-worlds regret guarantees in multi-armed bandits. Pogodin and Lattimore (2019) proposed an algorithm that combines minimax-optimal worst-case regret with a first-order guarantee whose loss-dependent term is $\mathcal{O}(\sqrt{K L^\star}\log T)$, where $T$ denotes the horizon, $L^\star$ denotes the smallest expected cumulative loss among fixed arms, and $K$ denotes the number of arms. However, a logarithmic gap remains between this loss-dependent term and the optimal first-order scale. In this paper, we remove this logarithmic factor by proposing an efficient and parameter-free algorithm that achieves the minimax optimal first-order regret guarantee $\mathcal{O}(\sqrt{K L^\star})$. Technically, our algorithm is built upon FTRL with a Tsallis entropy regularizer (Zimmert and Seldin,2021), with a learning rate adapted to the learner's cumulative loss and a sampling distribution determined by a scalar fixed point. This construction controls the combined estimation and stability cost in terms of the learner's loss. Moreover, we show that the same algorithm achieves a best-of-both-worlds guarantee, with $\mathcal{O}(\sum_{i\ne i^\star}\frac{\log…

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  • arXiv cs.LGOct 29

    Safe Multi-Objective Reinforcement Learning: Zero Constraint Violation and Primal-Averaging Convergence

    Chenxi Li, Yao Shu, Siyang Gao, Shuang Qiu, Guangxin Jiang

    Many applications of multi-objective reinforcement learning (MORL) require policies that balance conflicting objectives while satisfying safety constraints. We study safe MORL with zero constraint violation throughout learning, covering both cumulative and stepwise constraints. A key challenge is to explore sufficiently under unknown dynamics and constraint costs without violating the constraints. For a given preference over objectives, we develop an algorithm to learn a safe Pareto optimal policy by combining optimistic reward estimates with pessimistic cost bounds. With high probability, every policy executed over $T$ episodes incurs zero constraint violation. The algorithm returns a safe Markov policy induced by the averaged occupancy measure and achieves an $\widetilde{O}(1/\sqrt{T})$ suboptimality gap under smooth Tchebycheff scalarization. We further develop a preference-free algorithm that first explores safely without a prespecified preference and then computes a safe policy for any given preference without further interaction. Over $T$ exploration episodes and $K$ planning iterations, it retains the same safety guarantee and achieves an…

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  • arXiv stat.MLOct 29

    Satisficing Regret Minimization in Bandits with Knapsacks

    Qing Feng, Tianyi Ma, Ruihao Zhu

    We study satisficing regret minimization in bandits with knapsacks (BwK), where a learner sequentially selects randomized policies, i.e., distributions over arms, to obtain mean rewards above a satisficing threshold as frequently as possible while respecting finite resource budgets. The performance is measured by satisficing regret, the cumulative deficits of the selected policies' mean rewards below this threshold. Resource constraints introduce two difficulties: an algorithm can incur linear satisficing regret even with small standard regret as deficits to the threshold cannot be compensated, and random consumption can cause early resource depletion even when policies are feasible in expectation. To address these challenges, we propose SELECTwK, a general algorithmic template that uses a BwK learning oracle algorithm with sub-linear expected (standard) regret upper bound to search for and test candidate satisficing policies (i.e., policies whose mean rewards exceed satisficing threshold and whose expected resource consumption is feasible over the time horizon). Our regret analysis, supported by an LP-duality argument, establishes that SELECTwK attains constant expected…

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  • arXiv cs.LGOct 29

    Strongly Adaptive Online Learning Achieves Uniform Discounted Regret

    Sifan Yang, Wenhao Yang, Lijun Zhang

    Discounted regret measures the performance of an online learner by assigning exponentially decaying weights to past losses, reflecting the greater importance of recent losses in non-stationary online learning. While previous work obtains uniform discounted regret bounds over a continuous range of discount factors using a specialized meta-algorithm, its results are limited to convex functions. In this paper, we demonstrate that strongly adaptive regret bounds can directly imply uniform discounted regret guarantees for convex, strongly convex, and exp-concave loss functions. Our key observation is that discounted regret against a fixed comparator admits an exact decomposition into a convex combination of interval regrets. For convex functions, this reduction yields an $O(\sqrt{\log T/(1-λ)})$ discounted regret bound for all discount factors $λ\in(0,1)$, matching the previous rate over a broader range of discount factors. For strongly convex and exp-concave functions, we provide regret bounds of $O\left(\log T\left(1+\log\frac{1}{1-λ}\right)\right)$.

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  • arXiv cs.ITOct 29

    Pair-Dependent Judge Errors Change the Minimax Rate of Semantic Entropy

    Jiamiao Liu, Dewen Qiao, Yu Zhang, Xuetao Chen

    Semantic entropy, the entropy of the meanings among a language model's sampled answers, is estimated from the verdicts of a judge that declares pairs of answers equivalent or not, and automatic judges err in ways that depend on the answers they compare. The existing minimax theory covers only judges whose false-equivalence probability is the same for all pairs. We model a known judge that declares two answers with meanings $a$ and $b$ equivalent with probability $q+ΔM_{ab}$, for a symmetric matrix $M$ and $q$ bounded away from $0$ and $1$, and show that for three meanings the rate is set by tree statistics on four answers, while on an open set of channels the consistency threshold is set by trees on five. For every channel in an explicit open class, the minimax squared risk is of order $N^{-1/3}$ with $N=n^4Δ^6$ on polynomial weak windows on which $n^5Δ^8$ exceeds a multiple of $\log n$, and of order $\min\{1,N^{-1/3}\}$ on every polynomial weak window when the tree functions on at most four vertices are injective. The exponent comes from a cubic degeneracy at the equitable point. On an open set of channels in the class, the four-vertex tree functions take equal values at two…

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  • arXiv cs.LGOct 28

    Planned Restlessness: Delayed Meta Bandits with Budgets

    Sudipto Guha

    We study episodic finite-horizon Bayesian bandits, or meta-bandits, with independent arms. Each episode draws fresh arm qualities from unknown recurring priors. Observations arrive after known delays, and plays obey hard per-episode caps and total budgets. A play earns reward and informs both the current quality and the recurring prior: the learner must explore, use, and generalize. We formulate a common planning optimization over learning trajectories. Its LP relaxation values experiments through current reward, resource consumption and continuation: the value of future decisions from the learning state produced. Dual prices separate the arms into local programs and certify every admissible policy, including hierarchical Thompson sampling. Execution enforces the hard constraints while preserving the promised learning transitions or accounting for their loss. The framework also represents risk-sensitive objectives and prices purchased observations and buyout options. Delayed reports create planned restlessness: an arm's information changes while it is not played, through reports of its own earlier plays arriving on a known calendar. For stationary priors, finite reward alphabets,…

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  • arXiv cs.LGOct 28

    Rennala NSGD: Asynchronous Stochastic Optimization Beyond Euclidean Geometry

    Igor Sokolov, Alexander Tyurin, Peter Richtárik

    The empirical success of matrix-aware optimizers, such as Muon (Jordan et al., 2024) and Shampoo (arXiv:1802.09568), highlights the potential of non-Euclidean geometry for deep learning, including large language model (LLM) training. Despite this progress, the theoretical understanding of stochastic optimization in general normed spaces remains limited, particularly in distributed settings. We analyze minibatch stochastic gradient descent (SGD) for smooth nonconvex optimization with respect to an arbitrary norm on a finite-dimensional space. Using the framework of $κ$-regular normed spaces (arXiv:0809.0813), we derive high-probability oracle complexity bounds under a light-tail condition stated directly in the dual norm induced by the update norm. The bounds make the dependence on geometry explicit through the regularity constant $κ$ of the dual space. In the Euclidean case, our bound recovers the standard minibatch SGD rate up to logarithmic factors. For some non-Euclidean geometries and noise models, it is sharper than bounds obtained by converting a Euclidean noise assumption, reducing a linear dimension dependence to a logarithmic one. Building on this framework, we introduce…

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