{"id":"8d1e24e2-6f4d-4f3c-a818-99ea446bcc94","arxiv_id":"2412.14297","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A doubly robust estimator and a policy learner achieve minimax optimal regret κ(Π)/√n under concept drift with only conditional Y|X shifts.","lead":"This paper studies robust policy learning when only the conditional outcome distribution changes under a KL budget. It derives a doubly robust estimator with root-n normality and a policy learner with minimax optimal regret, avoiding the conservativeness of joint distributional shift models.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Minimax optimality claim rests on a κ(Π)-to-Ndim(Π) relation that would not match the stated lower bound; the correct entropy bound is not stated or proved.","rationale":"The paper is a serious theory contribution: the debiased estimator, the root-n asymptotic normality, and the regret upper/lower bounds are all plausible and largely correctly argued. The reader's weakest-assumption choice (Assumption 3.4) is legitimate: the debiasing proof collapses if the nuisance rates are not met, and the paper imports these rates partly from self-cited work. However, I found a more specific problem with the central claim of minimax optimality. The upper bound is stated in terms of κ(Π), while the lower bound is stated in terms of Ndim(Π). To conclude that the algorithm attains the minimax rate, the paper must show κ(Π) = O(√Ndim) up to acceptable log factors. Instead, Remark 4.5 and Table 1 state relations that are linear in Ndim (with at most a √(log d) factor). Under that stated relation, O(κ/√n) is O(Ndim/√n), which does not match Ω(√Ndim/√n); the displayed reasoning in Remark 4.7 therefore does not establish the matching. Standard VC-type metric entropy bounds do imply κ = O(√Ndim · poly(log d, log M)), so the conclusion is likely repairable, but as written the proof of optimality has a gap. This is a proof-support issue, not a demonstrated falsehood of the theorem, so it does not by itself require rejection; it strengthens the conditional verdict and points to a specific missing argument. The reader's factor-2 observation about Eq. (17)–(18) is accurate but only affects constants, not rates.","tokens_in":44370,"tokens_out":32797,"duration_ms":298682,"concrete_test":"Compute the Hamming entropy integral κ(Π) for a canonical policy class with known Natarajan dimension, e.g., axis-aligned decision stumps or linear separators in R^d with M=2 and Ndim = d+1. Evaluate covering numbers NH(ε,Π) on adversarially chosen point sets of size n=10^6 for ε ∈ {2^-1, ..., 2^-10}, then numerically integrate κ = ∫_0^1 √(log NH(ε^2,Π)) dε. If κ grows like √Ndim (up to log factors), the stated relation κ=O(√(log d)·Ndim) in Remark 4.5 is not the right bound and the optimality claim needs re-derivation with the correct entropy bound; if κ grows like Ndim, the upper bound does not match the lower bound and the claimed minimax optimality fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 4.3 gives Rδ(π̂^LN) ≤ C0(65 + 8κ(Π) + √log(1/β))/√n, while Theorem 4.6 gives Ω(√(Ndim(Π)/(nε))). The only bridge to the paper's central claim of a matching lower bound and minimax optimality is Remark 4.5 and Table 1, which state κ(Π) = O(√(log d)·Ndim(Π)) (Table 1 states O(log(d)·Ndim(Π))). If that linear-in-Ndim relation were the operative one, the upper bound would be O(Ndim/√n), which is strictly worse than the lower bound Ω(√Ndim/√n) for Ndim > 1; the claimed optimality in policy-class complexity would not follow from the displayed theorems. A standard metric-entropy bound for classes of finite Natarajan dimension is log NH(ε,Π) ≤ C·Ndim·log(1/ε) + O(Ndim·log M), which gives κ(Π) = O(√Ndim) up to log factors and would restore the matching; the paper neither states nor proves this stronger relation, so the minimax optimality conclusion is under-supported as written. The factor-2 constant slip in Lemma D.2 and Eq. (17)–(18) is real but secondary.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies distributionally robust policy learning under concept drift, where only the conditional reward distribution Y|X is allowed to shift within a KL ball, while the covariate distribution is fixed. The authors derive a pointwise dual representation of the robust policy value, propose a cross-fitted, doubly robust estimator for V_delta(pi), prove root-n asymptotic normality under slower-than-root-n nuisance estimation rates, and then learn a policy by maximizing this estimated value over a class Pi. They state an upper bound of order kappa(Pi)/sqrt(n) on the sub-optimality gap and a lower bound of order sqrt(Ndim(Pi)/(n epsilon)), claiming minimax optimality. Numerical experiments compare the proposed Algorithm 2 with a joint-shift DRO benchmark on simulated and real data.","tokens_in":44604,"tokens_out":16240,"duration_ms":154010,"significance":"If the results are correct, this is a valuable contribution: it extends concept-drift-robust policy learning from the finite-covariate setting of Mu et al. (2022) to general covariate spaces, gives a doubly robust policy-value estimator with a clean rate condition, and provides both upper and lower regret bounds. The paper is also commendably explicit: the appendices contain the strong-duality derivation, the full regret decomposition, the Rademacher chaining argument, and a lower-bound construction, and the authors provide code for the main algorithm. However, the central minimax optimality claim is not actually supported by the theorems as written, because the stated relationship between kappa(Pi) and Ndim(Pi) does not imply a matching bound; this and several proof-level gaps require correction before the central claims are established.","major_comments":[{"comment":"The claimed matching lower bound is not established by the displayed theorems. Theorem 4.3 gives R_delta(pi_hat^LN) = O((65 + 8 kappa(Pi) + sqrt(log(1/beta)))/sqrt(n)), while Theorem 4.6 gives Omega(sqrt(Ndim(Pi)/(n epsilon))). To compare these, one needs an upper bound on kappa(Pi) in terms of Ndim(Pi). Remark 4.5 and Table 1 state kappa(Pi) = O(sqrt(log d) Ndim(Pi)) and O(log(d) Ndim(Pi)), respectively; either relation would imply an upper bound O(Ndim(Pi)/sqrt(n)), which is strictly worse than the lower bound for Ndim(Pi) > 1. The standard metric-entropy bound log N_H(epsilon, Pi) <= C Ndim(Pi) log(1/epsilon) + O(Ndim(Pi) log M) would yield kappa(Pi) = O(sqrt(Ndim(Pi))) and would restore the matching claim, but this bound is neither stated nor proved in the paper. Please add the entropy lemma and use it explicitly in Remark 4.7.","section":"Section 4.3, Remark 4.7, Table 1"},{"comment":"The KL uncertainty set is not mathematically well-defined as written. Definition 2.2 defines KL divergence for distributions, but D_KL(Q_Y|X || P_Y|X) for conditional distributions is not the same object; it is either a pointwise quantity for each x or an averaged conditional KL. The proof of Lemma 2.3 in Eq. (11) uses the pointwise constraint D_KL(Q_{Y|X=x} || P_{Y|X=x}) <= delta with x-dependent multipliers alpha*(x), eta*(x). If the averaged conditional KL is intended, the dual with x-dependent alpha and eta is not valid. The paper should explicitly define the uncertainty set as {Q_Y|X : D_KL(Q_{Y|X=x} || P_{Y|X=x}) <= delta for P_X-almost every x}, since that is the formulation used in all subsequent derivations.","section":"Section 2.1, Lemma 2.3"},{"comment":"The lower-bound construction does not satisfy Assumption 2.1 with the epsilon appearing in the theorem statement. The proof sets pi_0(f_1(x_j) | x_j) = pi_0(f_{-1}(x_j) | x_j) = epsilon/2, so the minimum propensity in the constructed instance is epsilon/2, not epsilon. If epsilon is the overlap lower bound in Assumption 2.1, these hard instances are outside the class P over which the supremum is taken. The construction must either use propensities at least epsilon and adjust the KL calculation, or the theorem must state the lower bound in terms of a separate parameter for the overlap of the hard instances. This is a load-bearing gap in the proof of the main lower bound.","section":"Appendix D.4, Theorem 4.6"},{"comment":"There is an arithmetic inconsistency in the Rademacher bound constants. Lemma D.2's statement gives E_epsilon sup <= (sqrt(sum c_i^2)/n)(32 + 4 kappa(Pi)), but its proof concludes with (sqrt(sum c_i^2)/n)(64 + 8 kappa(Pi)) because Lambda = 2 sqrt(sum c_i^2). In the effective-term argument, Eq. (17) introduces an outer factor 2, then the text writes E_epsilon sup <= 2 C_0 (32 + 4 kappa)/sqrt(n), and Eq. (18) concludes with C_0(64 + 8 kappa + sqrt(log(1/beta)))/sqrt(n). These constants do not follow from one another. The rate is unaffected, but the proof of the constant 65 in Theorem 4.3 is not correct as written and should be repaired.","section":"Appendix D.3.2, Lemma D.2, Eq. (17)-(18)"},{"comment":"The proofs of Terms (II) and (III) in the proof of Theorem 3.5 require a lower bound on the estimated propensity score. For example, the text says 'due to the lower bound on pi_0 and bpi_0' when bounding Term (II), and Term (III) uses factors of epsilon^{-4} that assume bpi_0 is bounded below by epsilon. Assumption 3.4 only imposes L2 convergence of bpi_0 to pi_0 and does not state any lower bound on bpi_0. Please add a clipping condition such as bpi_0 >= epsilon/2 (or an equivalent bounded-inverse-moment condition) to Assumption 3.4, or modify the variance bounds so that they do not require pointwise lower bounds on the estimator.","section":"Assumption 3.4, Appendix D.2"}],"minor_comments":[{"comment":"There is a typo in 'the real-wrold dataset' that should read 'real-world dataset'.","section":"Section 5"},{"comment":"In the lower-bound construction, the expression (1 - epsilon)/(K - 2) should use the number of actions M, not the number of folds K; K is not defined in that proof.","section":"Appendix D.4"},{"comment":"The statement of Theorem E.3 appears to be missing a division by sqrt(n); it should read C(kappa(Pi) + sqrt(log(1/beta)))/sqrt(n).","section":"Appendix E, Theorem E.3"},{"comment":"The two stated relationships between kappa(Pi) and Ndim(Pi) are inconsistent with each other (O(log(d) Ndim) versus O(sqrt(log d) Ndim)). These should be unified after the correct entropy bound is added.","section":"Table 1, Remark 4.5"},{"comment":"The notation P_Y|X and Q_Y|X should be defined explicitly as Markov kernels, and the quantifier 'for P_X-almost every x' should be added to the KL constraint to remove the ambiguity noted in the major comments.","section":"Section 2.1, Eq. (1)"}],"recommendation":"major_revision","confidential_remarks":"The paper has a strong and coherent core: the doubly robust estimator, the strong-duality reduction, and the chaining argument are all in the right direction. The main concern is that the headline minimax optimality claim is not supported by the stated kappa-to-Ndim relation, even though the standard entropy bound would repair it. The lower-bound construction also has an overlap mismatch that makes the current proof invalid as written. These issues are fixable within the scope of the manuscript, so I recommend major revision rather than rejection. I do not see a circularity problem in the use of Proposition 2.5 or the sieve rates from prior work; the dependence on those results is appropriately cited. The acknowledgment of an earlier error in the manuscript is consistent with the current state of the proofs, but the remaining constant and overlap issues should be corrected before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper is a serious piece of work in DRO-based offline policy learning. The main technical achievement is real: it extends concept-drift robust policy learning to general covariate spaces, constructs a cross-fitted doubly-robust policy value estimator, and proves root-n asymptotic normality even with slower nuisance estimation. The regret upper bound and the lower bound are each nontrivial, and the empirical section is honest and reasonably thorough.\n\nThe soft spots are mostly in the packaging. The paper claims a matching lower bound and minimax optimality, but that does not follow from the displayed theorems. Theorem 4.3 gives regret O(κ(Π)/√n) with κ defined as a Hamming entropy integral. Theorem 4.6 gives Ω(√(Ndim(Π)/(nε))). The bridge is Remark 4.5 and Table 1, which state κ(Π) = O(log(d)·Ndim(Π)) (Remark says O(√(log d)·Ndim)). If that linear-in-Ndim bound were the operative one, the upper bound would be O(Ndim/√n), which is strictly worse than the lower bound Ω(√Ndim/√n) for Ndim > 1. The standard metric-entropy bound for Natarajan classes gives κ = O(√Ndim) up to log factors, which would restore a match, but the paper neither states nor proves that stronger relation. So the central optimality claim is under-supported as written. This is a fixable error, but it is load-bearing for the abstract and the discussion.\n\nThere is also a secondary constant slip: the symmetrization in Eq. (17) has an outer factor 2, but Eq. (18) applies only the inner Rademacher bound, so the constant 65 in Theorem 4.3 is not literally established. That is minor by comparison.\n\nI also want to flag that Assumption 3.4 is strong: it requires the ERM optimizer to converge at n^{-1/4} in L2 and the nuisance estimates to satisfy γ1+γ2 ≥ 1/2. These rates are imported partly from self-cited work. They are standard in the DML literature, but the paper would be more self-contained with a direct discussion of which sieve classes satisfy them.\n\nWho is this for? Researchers active in robust policy learning and distributionally robust optimization will find the estimator and the lower-bound construction useful, and the paper deserves serious referee time. But the current version should not be accepted as-is: the minimax claim needs to be either proved with the correct entropy bound or substantially qualified. I would send it back for a major revision, with the stress-test note attached.","headline":"Solid debiased estimation and a clean lower bound, but the minimax optimality headline does not follow from the paper's own entropy bound; likely repairable but needs a major fix.","tokens_in":45201,"tokens_out":2609,"would_cite":false,"duration_ms":25371,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62C20","68T05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper shows that when distribution shift is restricted to the conditional reward law—concept drift—a debiased cross-fitted estimator of the worst-case policy value is asymptotically normal, and a policy maximizing it achieves the…","keywords":["distributionally robust optimization","concept drift","policy learning","doubly robust estimation","offline contextual bandits","minimax regret","KL divergence","cross-fitting"],"falsifier":"Generate data from the Section 5 simulation with a smooth conditional reward (satisfying Assumption B.1) but choose the sieve dimension so that $\\hat{\\theta}$ converges at exactly the $n^{-1/4}$ $L_2$ boundary; Theorem 3.5 predicts $\\sqrt{n}(\\hat{V}_\\delta - V_\\delta)$ is still asymptotically normal. If coverage of the resulting Wald intervals visibly degrades as $n$ grows, or if the empirical regret of Algorithm 2 acquires an extra logarithmic factor in $n$, the claimed double robustness and the rate would be contradicted.","tokens_in":44112,"feed_emoji":"🎯","tokens_out":10733,"duration_ms":89295,"temperature":0.7,"pith_summary":"The paper studies a sharper version of distributionally robust policy learning: instead of guarding against arbitrary shifts of the joint covariate-outcome distribution, it assumes the drift is confined to the conditional outcome law $P(Y \\mid X)$—the concept-drift case—with the covariate distribution unchanged. Its central claim is that knowing the drift source is not just convenient but statistically optimal: the proposed policy has worst-case regret $O(\\kappa(\\Pi)/\\sqrt{n})$, with $\\kappa(\\Pi)$ the Hamming entropy integral of the policy class, and a matching lower bound $\\Omega(\\sqrt{\\mathrm{Ndim}(\\Pi)/n})$ shows no learner can do better. To reach that rate the paper builds a doubly-robust, cross-fitted estimator of the worst-case policy value that is asymptotically normal even when nuisance functions converge slower than root-$n$, and a computational shortcut that estimates one nuisance function per action rather than per policy, making the search over an infinite policy class tractable. If the paper is right, practitioners facing concept drift can obtain the same sample efficiency as standard policy learning while hedging against worst-case conditional shifts, without the conservatism of joint-shift robust methods.","feed_headline":"Policies that know the drift source reach the optimal rate","feed_subtitle":"A debiased estimator hits the statistical limit; a matching lower bound shows no algorithm can beat it.","key_machinery":"The argument rests on three mechanisms. First, strong duality (Lemma 2.3) converts the worst-case conditional expectation under the KL uncertainty set into the minimization over $(\\alpha, \\eta)$ of the loss $\\ell(x, y; \\theta) = \\alpha \\exp(-(y+\\eta)/\\alpha - 1) + \\eta + \\alpha\\delta$, so the robust value becomes the optimum of an empirical risk minimization problem. Second, a doubly-robust cross-fitted estimator—an importance-weighted residual $\\hat{G} - \\hat{g}$ plus a regression correction $\\hat{g}$—removes the first-order bias of plugging in slowly converging nuisance estimates, with the four error terms (I)–(IV) vanishing at $o_P(n^{-1/2})$ under Assumption 3.4. Third, for policy learning, the observation that $\\theta^*_{\\pi(x)}(x) = \\theta^*_a(x)$ for the chosen action $a$ reduces the infinite family of policy-dependent nuisances to $M$ action-dependent ones, allowing a chaining argument over the Hamming entropy integral $\\kappa(\\Pi)$ to bound the uniform estimation error and yield the regret rate, together with the matching Natarajan-dimension lower bound.","core_discovery":"This paper establishes that robust policy learning under concept drift—where only the conditional reward law $P(Y \\mid X)$ may shift while the covariate distribution stays fixed—admits a minimax-optimal learner for general covariate spaces. Concretely, Algorithm 2, which maximizes a debiased cross-fitted estimate of the KL-robust policy value $V_\\delta(\\pi)$, satisfies with probability at least $1 - \\beta$ that $R_\\delta(\\hat{\\pi}_{\\mathrm{LN}}) \\leq C_0(65 + 8\\kappa(\\Pi) + \\sqrt{\\log(1/\\beta)})/\\sqrt{n}$ (Theorem 4.3), where $\\kappa(\\Pi)$ is the Hamming entropy integral of the policy class; and no algorithm can have worst-case regret better than $\\Omega(\\sqrt{\\mathrm{Ndim}(\\Pi)/(n\\epsilon)})$ for $\\delta \\leq 0.2$ (Theorem 4.6), so the rate is tight up to a log factor in the covariate dimension. On the evaluation side, the paper proves $\\sqrt{n}(\\hat{V}_\\delta(\\pi) - V_\\delta(\\pi)) \\to N(0, \\sigma_\\pi^2)$ even when the nuisance parameters are estimated at slower-than-root-$n$ rates, as long as the cross-fitted rates of Assumption 3.4 are met.","pith_inferences":["If the paper's rate is right, the source-aware factorization principle likely extends beyond KL divergence: any divergence whose dual adds an $(\\alpha, \\eta)$ term to the loss should inherit the same debiasing recipe, making the machinery a template for other $f$-divergences.","The per-action shortcut $\\theta^*_{\\pi(x)}(x) = \\theta^*_a(x)$ suggests a natural bridge to continuous action spaces—replacing the sum over $a \\in [M]$ with an action-indexed model $\\theta^*(x, a)$—which the paper leaves as a future direction but its own decomposition makes immediate.","Because the estimator's asymptotic variance $\\sigma_\\pi^2$ is given explicitly, one can construct Wald confidence intervals for $V_\\delta(\\pi)$ and use them for robust policy comparisons; the paper does not develop this inferential tool, but its variance formula makes it a direct byproduct.","Taking $\\delta \\to 0$, the robust value $V_\\delta(\\pi)$ converges to the plain policy value, so the regret bound should recover standard offline learning rates such as $\\sqrt{\\mathrm{VCdim}/n}$; checking this limit against known results would validate the constants."],"forward_implications":["Knowing the drift source shrinks the uncertainty set: the concept-drift worst-case value $V_\\delta(\\pi)$ is no smaller than the joint-shift worst-case value, so policies trained under concept drift provably dominate joint-shift robust policies when no covariate shift is present.","The regret upper bound $O(\\kappa(\\Pi)/\\sqrt{n})$ and lower bound $\\Omega(\\sqrt{\\mathrm{Ndim}(\\Pi)/n})$ are matching in sample size and policy-class complexity, establishing minimax optimality for general covariate spaces.","The policy value estimator is doubly robust: it remains root-$n$ asymptotically normal when either the propensity score or the conditional-mean nuisance is consistent with product rate $n^{-1/2}$, and even when the $(\\alpha, \\eta)$ optimizer converges at $n^{-1/4}$ in $L_2$.","The same machinery extends to identifiable covariate shifts combined with concept drift: with density-ratio weights $r(X) = dQ_X/dP_X$, the estimator keeps asymptotic normality and the regret bound holds under the modified Assumption E.1.","Algorithm 2 is computationally feasible for infinite policy classes because the ERM nuisance is solved once per action rather than per policy, so policy search over $\\Pi$ reduces to maximizing a single score."],"supporting_citations":[{"why":"Defines the joint-shift robust batch contextual bandit benchmark (SNLN) that the paper outperforms, and supplies the KL-based lower-bound lemma adapted for the matching lower bound.","marker":"Si et al., 2023"},{"why":"Provides the doubly robust DRO policy evaluation and learning framework for joint distributional shifts that serves as the baseline and benchmark implementation.","marker":"Kallus et al., 2022"},{"why":"Prior factored-DRO policy learning that separates covariate and conditional shifts but is restricted to finite covariate spaces; this is the work the paper generalizes.","marker":"Mu et al., 2022"},{"why":"Supplies Proposition 2.5 ($\\alpha^* > 0$), the sieve-based ERM estimator and its $L_2$/$L_\\infty$ convergence rates, and the regularity conditions used to justify Assumptions 3.3 and 3.4.","marker":"Jin et al., 2022b"},{"why":"Sets the unconfoundedness, overlap, and bounded-reward assumptions and the doubly robust policy learning template that the estimators build on.","marker":"Athey & Wager, 2021"},{"why":"The double/debiased machine learning cross-fitting framework that motivates Assumption 3.4's product-rate condition and the debiasing structure.","marker":"Chernozhukov et al., 2018"},{"why":"Cited for the strong duality theorem used in Lemma 2.3 to pass from the KL-constrained infimum to the finite-dimensional ERM dual.","marker":"Luenberger, 1997"}],"fun_headline_variants":["Optimal-rate robust policies under concept drift","Concept-drift robust policy learning hits minimax rate","Debiased estimator reaches optimal rate under concept drift","Minimax-optimal policy learning with drift-aware debiasing","No algorithm can beat this concept-drift policy rate"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on Assumption 3.4: the sieve-estimated $(\\alpha, \\eta)$ optimizer must converge to the truth in $L_2$ faster than $n^{-1/4}$ and uniformly in $L_\\infty$, and the product of the propensity-score and conditional-mean estimation errors must be $o_P(n^{-1/2})$; if these rates slip, both the root-$n$ normality of the value estimator and the $\\kappa(\\Pi)/\\sqrt{n}$ regret bound fail.","fun_headline_variants_meta":{"raw":{"variants":["Optimal-rate robust policies under concept drift","Concept-drift robust policy learning hits minimax rate","Debiased estimator reaches optimal rate under concept drift","Minimax-optimal policy learning with drift-aware debiasing","No algorithm can beat this concept-drift policy rate"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000922,"raw_usage":{"total_tokens":4015,"prompt_tokens":1065,"completion_tokens":2950,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":681,"completion_tokens_details":{"reasoning_tokens":2873}},"tokens_in":681,"tokens_out":2950,"duration_ms":19286,"temperature":1.0,"reasoning_tokens":2873,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:21:58.132267+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Generate data from the Section 5 simulation with a smooth conditional reward (satisfying Assumption B.1) but choose the sieve dimension so that $\\hat{\\theta}$ converges at exactly the $n^{-1/4}$ $L_2$ boundary; Theorem 3.5 predicts $\\sqrt{n}(\\hat{V}_\\delta - V_\\delta)$ is still asymptotically normal. If coverage of the resulting Wald intervals visibly degrades as $n$ grows, or if the empirical regret of Algorithm 2 acquires an extra logarithmic factor in $n$, the claimed double robustness and the rate would be contradicted.","supporting_citations":[],"review_version":1}