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.
Context-adaptive Kalman filters calibrate their noise covariance matrices Q and R from innovation residuals via online regression. When the underlying sensor or signal carries periodic structure -- mechanical LiDAR rotation harmonics, engine vibration, ground multipath, weekly and annual demand cycles, dosing-interval rhythms, weekly media-buying cadence -- the regression input is contaminated and the fitted covariance models structural modes rather than genuine state uncertainty. We introduce a four-role FFT pre-filter that solves this problem at $O(N\log N)$ cost and serves three additional roles "for free": (i) it whitens coloured noise before the Kalman update, restoring the optimality assumption; (ii) it cleans innovations before covariance regression, preventing periodic contamination of $\hat{R}$ and $\hat{Q}$; (iii) it generates spectral context features that enrich the downstream bandit's regime-selection state; (iv) it deseasonalises the input feature vector before any supervised regression that produces a sensitivity coefficient (beta, dose offset, bid modifier). We position the algorithm inside the Gated Decoupled Compositional Bandits (GDCB) family, where it acts as a preprocessing layer for the supervised scaler. The single $O(N\log N)$ FFT call thereby serves four downstream consumers, fits in <0.1% of the sensor-fusion or pricing-pipeline compute budget, and is a drop-in addition with no changes to the Kalman filter, bandit, or runtime composition operator. We summarise empirical validation across six independent domains (rocket descent, autonomous-vehicle tracking, short-term rental pricing, clinical drug dosing, airline fare distribution, and ad-operations bid calibration), all returning a PROVES verdict under a pre-registered evaluation protocol.
In fixed-confidence best-arm identification, proofs often use a union bound across the competing arms. From a multiple-testing point of view this can look puzzling: if the best arm is unique, only one hypothesis of the form ``arm $i$ is best'' can be true. Why then should there be a Bonferroni-type factor of $K-1$? The answer is that there are two natural ways to orient the hypotheses. In one orientation, best-arm identification is literally a strong familywise-error-rate (FWER) problem with $K-1$ true nulls. In the opposite orientation, exactly one null is true, but a pairwise implementation can falsely reject that one null through any of $K-1$ comparisons. Thus the multiplicity has not disappeared; it just pops up in different places. This note makes the equivalence explicit in the terminology of both communities.
The Lipschitz bandit problem extends the traditional multi-armed bandit framework to continuous action spaces by assuming that the reward functions satisfy a Lipschitz condition. This work investigates Lipschitz bandits under arbitrary feedback delays, where reward signals are not received immediately upon taking an action but after an arbitrarily chosen delay. We consider both stochastic and adversarial reward settings, proposing an elimination-based algorithm and an EXP3-based algorithm, respectively. For both settings, our algorithms achieve a regret bound of $\tilde{O}\left(T^{\frac{d_z+1}{d_z+2}}+\sqrt{D}\right)$ over a time horizon $T$ with total delay $D$, where the main difference between settings lies in the definition of the zooming dimension $d_z$. Our bounds match existing delay-free regret guarantees for Lipschitz bandits and characterize the additional $\tilde{O}(\sqrt{D})$ impact introduced by feedback delays.
Changshuai Wei, John Bencina, Phuc Nguyen +2cs.LG cs.AI
Large-scale targeting and recommendation systems are typically built around predictive scores fed into heuristic or local allocation. When the business goal is incremental impact, as in marketing campaigns, incentives, and notifications, this paradigm systematically misallocates resources toward users who would have acted anyway. We present a decision-centric framework that instead optimizes causal effects under global constraints, aligning three components under a single objective: a causal neural network with a Transformer backbone for individual treatment-effect estimation, a Bayesian neural-bandit layer for uncertainty-aware exploration, and a dual-based large-scale linear-programming layer for constrained allocation. The framework also supports sequential context and multi-outcome, attribute-conditioned scoring through a Transformer encoder and outcome embeddings. We evaluate it with offline simulations on a public bandit dataset, targeted architectural ablations, and an online A/B test on LinkedIn Feed marketing traffic. We also distill production lessons on causal training-data construction and cost and delivery control, which were critical to successful deployment. The end-to-end treatment policy delivered a statistically significant $+7.20\%$ lift in the primary long-term-value metric, demonstrating the feasibility of production-scale causal optimization under business constraints.
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.
Adaptive learning needs both a state that preserves what observations imply and opportunities to act on that state. We study this width--depth tradeoff in stochastic Lipschitz bandits. After each pull, the learner retains at most $W$ bits of live reward-dependent state and organizes its pulls into at most $B$ committed batches. For $W\gtrsim_d\log(eT)$, we characterize minimax expected pseudo-regret up to logarithmic factors; the lower bounds hold for every $W$. Besides the classical sequential and unrestricted-memory batch costs, the frontier contains the new penalty \[ T^{\frac{d+2}{d+3}} \bigl(1+(B-1)W\bigr)^{-\frac1{d(d+3)}}, \] proving that state width and update depth are not interchangeable. The interaction is an information-routing constraint: at regional scale $s$, low regret forces the committed action transcript to encode $Θ_d(s^{-d})$ regional decisions, while the collected boundary states carry at most $(B-1)W$ bits of entropy. Matching policies stream and erase verification statistics while retaining a mask of a safe active set, either in memory or fragment by fragment. The theorem recovers the full-dimensional worst-case batch-only frontier and logarithmic-memory achievability in the fully sequential specialization; static batch boundaries match predictable adaptive ones.
This paper studies generalized low-rank matrix bandits with multiple prioritized objectives. At each round, the learner selects a matrix-valued arm and observes a vector-valued reward, whose components correspond to multiple objectives with different priority levels. Each objective is governed by an objective-specific generalized low-rank matrix model, and the learner evaluates arms according to a lexicographic preference order, prioritizing higher-level objectives before lower-level ones. We propose \textsc{Lexi-LowGLM}, an efficient online algorithm that first estimates objective-specific low-rank subspaces and then performs lexicographic learning in the reduced feature spaces. Unlike existing single-objective algorithms that repeatedly solve a batch generalized linear estimator using all historical observations, \textsc{Lexi-LowGLM} updates each objective-specific estimator via an online Newton step, reducing the estimator-update complexity over $T$ rounds from $O(T^2)$ to $O(T)$. We establish a regret bound of $\widetilde O\left(W_i^{\rm lex}\sqrt{m}\,(d_1+d_2)r\sqrt{T}\right)$ for each objective $i\in[m]$, where $r$ is an upper bound on the ranks of the objective-specific parameter matrices and $W_i^{\rm lex}$ characterizes the lexicographic trade-off effect. This bound depends on the effective low-rank dimension $(d_1+d_2)r$ rather than the ambient dimension $d_1d_2$. Numerical experiments further validate the effectiveness and computational efficiency of the proposed method.
Heavy-tailed distributions arise naturally in sequential decision-making problems such as financial investment, online advertising, and network management, where rare but extreme outcomes can dominate performance. Heavy-tailed bandits model online decision-making in these settings by assuming only that rewards $X$ satisfy $\mathbb{E}[|X|^{1+ε}]\leq u$, for some tail exponent $ε\in(0,1]$ and moment bound $u<+\infty$. However, most existing regret minimization algorithms require these parameters to be known. This assumption is particularly restrictive in practice: $ε$ and $u$ govern the frequency and magnitude of rare events and are therefore precisely the quantities that are hardest to infer reliably from limited observations. Motivated by an open problem posed by Genalti and Metelli at COLT 2025, we resolve the assumption-free adaptation problem for heavy-tailed bandits and characterize the price in the regret of not knowing the tail parameters. We first study adaptation to the moment bound $u$ for a fixed tail exponent $ε$. We prove that every algorithm unaware of $u$, or of any upper bound on it, must obey a sharp trade-off between its distribution-dependent and distribution-free regret guarantees. We then introduce a scheduled-exploration algorithm that requires no knowledge of $u$ and matches the resulting adaptation frontier up to logarithmic factors. Finally, we show that the same algorithm can be instanced without knowing $ε$ by calibrating its exploration schedule to the endpoint $ε=1$. It achieves sublinear regret for every fixed $ε>0$, while no algorithm can guarantee sublinear regret uniformly over all $ε\in(0,1]$. Altogether, our results resolve the COLT open problem without additional distributional assumptions and provide a sharp characterization of the statistical cost of adapting to unknown heavy tails.
Minimum-exposure constraints arise in recommendation, content curation, and regulated allocation when each provider, arm, or group must receive guaranteed exposure inside a period rather than only in aggregate. We study stochastic bandits with exact exposure floors and show that the right object is a rounding problem: a fractional fair schedule is realized as integral pulls, and the exposure error is exactly a discrepancy vector. The main contribution is a blockwise model with time-varying floors. BDQ-UCB satisfies every block floor deterministically and has fair regret governed by the nonmandatory budget $R$, not the horizon $T$, with high-probability regret $O(\sqrt{KR\log(KT)})$. A MOSS residual variant attains $O(\sqrt{KR})$, and a matching lower bound gives the minimax rate $Θ(\sqrt{KR})$, even with positive mandatory exposure; a kl-UCB$^{++}$ residual rule adds instance-dependent optimality. The formulation becomes essential for overlapping group floors: per-arm rounding can violate a group constraint by $Ω(s)$ in the group size, whereas Beck--Fiala null-space rounding meets every group floor within the block budget with violation below the arm degree $t$, and composes with UCB at the same $R$-parametrized regret. For learned group plans, we close disjoint systems at $\widetildeΘ(\sqrt{KT})$, give a dual-ledger decomposition explaining why naive index rules fail under overlap, and prove a plan-sampling rule that is pathwise feasible under an initial cover-slack condition and attains a conditional $\widetilde O(\sqrt{KT})$ guarantee, leaving the condition-free overlap rate open. Experiments on synthetic floors, MovieLens-100k genre exposure, and deployment stress tests show exact feasibility without penalty tuning and regret competitive with tuned Lagrangian baselines.
We introduce Stochastic Reset Pathfinding (SRP), an episodic learning problem on a known directed graph with unknown stationary edge success probabilities. In each episode, the agent commits to a source-to-goal path, and any edge failure during execution resets it to the source. SRP captures settings such as entanglement distribution in quantum repeater networks, payment routing on the Lightning Network, and delivery in unreliable mesh networks. We show that the global-reset structure makes the optimal policy open-loop, placing SRP within the combinatorial cascading bandit (CCB) framework. We propose a Log-Dijkstra meta-algorithm with UCB (PathUCB) and Thompson Sampling (PathTS) instantiations. Our main technical result is a path-level regret bound for PathUCB that decomposes regret over suboptimal paths via a per-path complexity C(pi) combining each edge's prefix and suffix reliability. The bound is complementary to the edge-level CCB bound and more informative on structured graphs with polynomially many source-to-goal paths. Experiments on quantum-network, layered-DAG, grid-world, and Erdos-Renyi domains support the theory and show that PathTS typically achieves the best empirical performance among the algorithms tested. We then exhibit an adversarial instance on which PathTS fails to converge, consistent with a known exponential obstruction for combinatorial Thompson Sampling on multiplicative-reward problems. We recommend PathTS as the practical default while cautioning that adversarial instances exist.
We design and analyze \underline{M}echanism-\underline{E}nforced \underline{S}equential \underline{HA}lving (MESHA), an algorithm for Best Arm Identification (BAI) in strategic linear bandits. In this setting, each arm may strategically misreport its feature vector to maximize the probability of being identified as the best arm, when rewards are generated from the arms' true but unobservable features. The design of MESHA applies the naïve uniform sampling rule and an epoch-wise Grim Trigger Condition (GTC): the former reduces the impact of arms' strategic behaviours and the latter eliminates arms whose reported features severely deviate from the ground truth. Considering an arbitrary Nash Equilibrium, we prove that any arm would attempt to pass the GTC check to maximize its identified probability and derive an upper bound on the failure probability of MESHA within a fixed budget $T$. We also show that state-of-the-art linear BAI algorithms with $G$-optimal design would fail in such strategic environment, as the optimal design (OD)-based sampling rule based on strategically reported features may {\it starve} the optimal arm of any sampling budget. Finally, extensive numerical experiments indicate that MESHA outperforms baselines that rely on OD-based sampling rules as well as the feature-agnostic baselines, corroborating the efficacy of MESHA.
Dimitar Chakarov, Lee Cohen, Nathan Srebrocs.GT cs.LG
We extend Incentive Compatible Exploration beyond the Bayesian full-information setting of Kremer et al. [2014]. We consider agents that may possess external information unknown to the principal. We show such settings require new notions of incentivized exploration, as well as going beyond a Bayesian perspective, and we introduce a definition where agents choose any reasonable (undominated) action. Furthermore, our framework provides for a more robust treatment of ties, and extends to settings where agents lack a single common prior and instead only know that reward distributions belong to a collection of potential priors.
We consider Bayesian bandit models and prove that Thompson sampling makes at most twice the expected number of mistakes (selections of a suboptimal arm) as any other policy. Our analysis applies as long as the latent arm processes are independent and each arm evolves only when played. For stochastic bandits with best arm defined via mean reward, this confirms a conjecture of Guha and Munagala from 2014, where the factor $2$ is already best possible. The result holds under any nonincreasing sequence of round weights, including fixed horizon and geometric discounting.
Motti Goldberger, Nils Rudics.LG stat.AP stat.ML stat.OT
We study the active learning problem of fixed-confidence top-$k$ identification from noisy pairwise comparisons. In this problem, an algorithm sequentially chooses pairs of items to compare, observes the outcomes, and stops when it can return the set of top-$k$ items with error probability at most $δ$. The objective is to design such a $δ$-correct procedure that minimizes the expected number of comparisons (the sample complexity). This problem falls within the broader literature on fixed-confidence pure exploration in bandit models, where a common target is asymptotic optimality: the algorithm's expected sample complexity matches the information theoretic lower bound as $δ\to 0$. Asymptotically optimal procedures have been developed for a range of fixed-confidence pure-exploration problems, however to the best of our knowledge, for top-$1$, or more generally top-$k$ identification from pairwise comparisons under latent utility models an asymptotically optimal algorithm has not been established. In this setting, we develop such an algorithm. We characterize the structure of the lower bound and formulate it as a saddle-point problem. This structure enables a computationally efficient primal-dual procedure that learns the asymptotically optimal comparison allocation online. We then construct an adaptive comparison-allocation algorithm that tracks the allocation learned by the primal-dual procedure and prove it is asymptotically optimal.
Best-arm identification is a canonical model for data-driven decision-making, but in many applications each reward observation is costly. Motivated by the growing availability of cheap predictions from machine learning and large language models, we study fixed-confidence best-arm identification in which each costly reward pull is paired with a cheap but correlated proxy score. The marginal mean of the proxy can be estimated offline and is treated as known, whereas its correlation $ρ$ with the reward, which governs how much the proxy helps, is unknown and must be learned online in pair with real rewards. We show that a control-variate adjustment turns this model into a heteroscedastic identification problem whose oracle sample complexity improves by residual variance $1-ρ^2$. The central difficulty is that the correlation must be learned from the same costly samples that identification consumes online, and that a plug-in estimate of the residual variance is anti-conservative and can compromise correctness. We propose PROBE (PRoxy OLS for Best-arm Exploration), a phase-elimination algorithm that directly maintains an upper certificate on the residual variance with an ordinary least squares fit, whose exact chi-square law keeps the certificate valid regardless of the unknown correlation. We prove that PROBE is $δ$-PAC and attains the known-correlation oracle sample complexity up to a constant multiplicative factor and a constant additive calibration cost. The guarantee extends to the $(ε,δ)$-PAC setting under minimal changes to the algorithm. Numerical experiments on synthetic instances and on an auto-loan pricing replay with large language model and tabular proxies confirm that the sample savings of PROBE scale with the strength of the reward-proxy correlation, exactly as the theory predicts.
Raghav Bongole, Amirreza Zamani, Tobias J. Oechtering +1cs.LG cs.IT
Minimax risk and regret are expectation-based criteria and do not capture rare but consequential failures. To address this concern, we develop a $δ$-explicit minimax-quantile theory for interactive statistical decision making (ISDM). We first provide structural relations between minimax quantiles, lower minimax quantiles, and minimax risk. This includes a quantile-to-expectation conversion and an equivalence between strict and lower minimax quantiles outside a countable set of confidence levels. We then derive two converse tools for ISDM: a high-probability interactive Fano's method and a high-probability interactive Le Cam's method. Then, we show that mutual-information (MI) privacy can be handled in the same framework by restricting the admissible decision class. For coordinatewise Gaussian privatization, we derive a two-point template that isolates the privacy-induced variance inflation. We instantiate this template for Gaussian mean estimation, and use the same two-point strategy directly for two-armed Gaussian bandits. We then derive a minimax quantile lower bound for the $K$-armed Gaussian bandit problem, showing that the interactive Fano method captures the exploration cost over multiple possible best arms. The resulting lower bounds are explicit in the confidence level $δ$ and in the privacy budget for the private problems. They yield $\log(1/δ)/n$ scaling for squared-error Gaussian mean estimation, $\sqrt{T\log(1/δ)}$ scaling for two-armed bounded-mean Gaussian bandits, and $\sqrt{KT\log(1/δ)}$-type scaling for the $K$-armed bandits, with privacy appearing through a Gaussian variance-inflation factor for the private problems.
François Bachoc, Roberto Colomboni, Emilie Kaufmanncs.LG
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.
Abdellah Aznag, Rachel Cummings, Adam N. Elmachtoubcs.LG cs.IT
We study a \emph{max-risk} objective for active learning in a multi-group mean estimation $d$-armed bandits: a learner adaptively allocates a budget of $T$ samples across $d$ groups to minimize the worst-case uncertainty index $\max_{k\in[d]}σ_k^2/n_k$, where $σ_k$ is the standard deviation of the distribution of arm $d$, and $n_k$ is the number of times arm $d$ is sampled. We develop a local minimax framework and prove the first general lower bound for this objective, valid for any finite-variance hypothesis class. The bound separates difficulty into three orthogonal factors: a \emph{budget} term, a \emph{heteroscedasticity} index measuring how unevenly the uncertainty is spread across arms, and a model-dependent complexity measure, the \emph{Variance Local Curvature} ($\mathrm{VLC}$), which captures how much information a local change of variance creates inside the hypothesis class. For smooth classes, the $\mathrm{VLC}$ is a reparametrization of a variance--Fisher information, with closed-form values for common families. Benchmarking against the strongest available upper bound shows near-optimality up to logarithmic factors in broad regimes, and pinpoints a systematic gap in highly heterogeneous instances. Our proof introduces two key ingredients: a loss-induced $\ell_1$ geometry on the decision space, and a representation-based instance generator that reduces hard-instance construction to an explicit random matrix calculation.
Linzhe He, Yu-Jie Zhang, Sifan Yang +1cs.LG stat.ML
This paper studies efficient online algorithms for multinomial logistic bandits (MLogB), where the feedback distribution over $K+1$ outcomes follows a multinomial logistic model of $d$-dimensional action vectors. A representative UCB-type algorithm, OFUL-MLogB, achieves a regret bound of $\tilde{\mathcal{O}}(Kd\sqrt{T})$, but still requires $\mathcal{O}(K^3d^3)$ time and $\mathcal{O}(K^2d^2)$ space per round due to parameter estimation and optimistic reward construction, which is prohibitive in high-dimensional settings. To address this limitation, we propose EOFD-MLogB, which integrates frequent directions matrix sketching into OFUL-MLogB. By maintaining a low-rank SVD sketch of the accumulated Hessian, constrained online Newton updates in parameter estimation and $Kd \times K$ spectral-norm computations in the reward bonus are reduced to one-dimensional root-finding tasks and $K \times K$ eigenvalue computations, respectively. This yields dominant per-round time complexity $\mathcal{O}(Kd(m+K)^2)$ and space complexity $\mathcal{O}(Kd(m+K))$, where $m \ll d$ is the sketch size. We further prove a regret bound of $\tilde{\mathcal{O}}(Δ_T(Kd\lnΔ_T+m)\sqrt{T})$, where the sketching error factor $Δ_T$ is controlled by the $m$-truncated spectral tail of the Hessian. Thus, when the Hessian is approximately low-rank, the regret is close to that of OFUL-MLogB. Experiments validate the computational efficiency and competitive performance.
We prove that $ρ\text{-}\mathrm{NPTS}_{\mathrm{SG}}$, an anchor-free nonparametric Thompson Sampling algorithm for risk-averse bandits, achieves regret matching the instance-dependent lower bound to leading order in $\log n$, establishing it as asymptotically optimal for any continuous risk functional $ρ$ (CVaR, mean-variance, Sharpe ratio, distortion risk measures, and more) on the class of distributions with bounded density and sub-Gaussian tails, including Gaussian arms. Both this result and its bounded-support counterpart require only continuity of $ρ$: strictly weaker than the dominance condition of prior parametric Thompson Sampling results, and strictly weaker than the Lipschitz condition of UCB-type algorithms, yielding the first instance-optimal guarantees for non-Lipschitz functionals such as the Sharpe ratio without parametric reward assumptions. The bounded-support case is developed first as a stepping stone sharing the same proof structure. The key technical contributions are a discretisation lemma (bounded support) and a truncated discretisation lemma (sub-Gaussian tails), each projecting the growing-alphabet Dirichlet posterior onto a fixed grid via the Dirichlet aggregation property, holding all polynomial prefactors at fixed degree independent of sample size and breaking the super-exponential barrier that blocked prior proofs.
We study the two-action apple-tasting problem with switching costs against an oblivious adversary. In an equivalent normalized formulation, at each round the learner chooses between a revealing action and a blind action: the revealing action gives reward $0$ and reveals the hidden value $x_t\in[-1,1]$ of the blind action; the blind action gives reward $x_t$ but reveals nothing. The learner pays one unit whenever they switches actions, and regret is measured against the best fixed action in hindsight. General feedback-graph algorithms with switching costs give $\widetilde O(T^{2/3})$ regret guarantees for this problem. The two-action apple-tasting graph was the natural candidate for the missing $Ω(T^{2/3})$ obstruction in the switching-cost classification: such a lower bound would have transferred to a large family of still-unclassified feedback graphs. We prove that this obstruction is not there: the oblivious minimax expected regret for this problem satisfies \[ \frac{1}{2\sqrt3}\cdot\sqrt T \le R_T^\star \le 2\sqrt{3}\cdot \sqrt{T}. \]