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REVIEW 3 major objections 5 minor 2 cited by

Universal priors: solving empirical Bayes via Bayesian inference and pretraining

T0 review · 3 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read The paper proves that a pretrained estimator trained under a simple random prior-on-prior attains near-optimal worst-case empirical Bayes regret uniformly over all test priors, and explains length generalization as Bayesian inference with a

desk verdict A serious conditional theory of universal priors for pretrained empirical Bayes, with the key optimization assumption and test-length universality still gaps. read the letter →

arxiv 2602.15136 v2 pith:XOCHYLDH submitted 2026-02-16 stat.ML cs.LG

Nick Cannella , Anzo Teh , Yanjun Han , Yury Polyanskiy This is my paper · ORCID
classification stat.MLcs.LG MSC 62C1262F1562C2068T07
keywords empiricalBayesuniversalpriorsprior-on-priorPoissonmixtureposteriorcontractionpretrainedtransformerslengthgeneralizationminimaxregret
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper sets out to explain why a transformer pretrained on synthetic data can solve empirical Bayes problems for any test prior. It proves that training under a deliberately simple prior-on-priors—each batch samples O(log n / loglog n) locations uniformly and gives them uniform Dirichlet weights—is universal: the resulting hierarchical Bayes estimator has worst-case regret O(log^3 n / (n (loglog n)^2)) for every prior G0 on [0,A], nearly matching the known minimax rate. The mechanism is posterior contraction: with enough training batches the learned map becomes the posterior mean under the training prior-on-prior, and that posterior concentrates around the unknown test prior. The same analysis shows length generalization is Bayesian inference with an α-posterior, α = n/n_test, which explains both the continued improvement and the saturation observed empirically.

What carries the argument

The central objects are the prior-on-prior Π and the hierarchical Bayes estimator E_Π[θ^n | X^n], whose i-th coordinate equals θ_{G_i}(X_i) with G_i the posterior mean of G given the other n−1 observations. Universality of Π is a coverage property: for every test prior G0 there is a nearby prior G with TV(f_G0, f_G) ≤ n^{-10}, and Π puts mass at least e^{-B_n} on priors G′ whose marginal χ²-divergence from f_G is at most n^{-10}. The proof combines this coverage with a posterior-contraction argument to bound the Hellinger distance between the predictive f_{G_i} and f_{G0}, and a Poisson-specific inequality converts that Hellinger bound into squared-error regret. For length generalization, th

What would settle it

Train the Algorithm 1 transformer at fixed n with a very large number of batches M, then compare its output on test sequences to the exact hierarchical Bayes estimator E_Π[θ^n | X^n] (computed by sampling the prior-on-prior) and record regret against known test priors; if the mean squared output distance does not shrink to a small fraction of the regret as M grows, or if the fitted α at test length n_test deviates systematically from n/n_test, the global-minimizer assumption and the α-posterior explanation are refuted.

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Extended reading notes

Core claim

For Poisson empirical Bayes with an unknown prior G0 supported on [0,A], the paper's central claim is Theorem 1.1: if the transformer is trained by empirical risk minimization on synthetic batches drawn from the simple prior-on-prior of Algorithm 1—k = ⌈c0 log n / loglog n⌉ locations uniform in [0,A] and Dirichlet(1,...,1) weights—and if training finds the global ERM minimizer, then the worst-case regret over all test priors is at most C log^3 n / (n (loglog n)^2). The known minimax regret is Θ(n^{-1}(log n / loglog n)^2), so the pretrained estimator is near-optimal uniformly over the entire class of priors. The proof identifies such priors-on-priors as universal and shows by a minimax theor

Load-bearing premise

The theorems are conditional on the black-box assumption that the training procedure finds the global minimizer of the ERM objective, so the trained transformer exactly computes the hierarchical Bayes posterior mean E_Π[θ^n | X^n]; the numerical evidence for this assumption is limited to small mixtures with m = 2, 5, 10 priors, not the full Algorithm 1.

Editorial extensions

If this is right

  • Near-optimal, distribution-free empirical Bayes from one pretrained model: a single transformer trained once on synthetic batches matches, up to a log factor, the best possible worst-case regret for every prior G0 on [0,A].
  • Length generalization is predicted to be Bayesian but not fully adaptive: regret keeps decreasing for n_test > n but plateaus at order 1/n rather than 1/n_test, because the learned map's α-posterior update uses α = n/n_test.
  • The same universal-PoP recipe extends beyond the compact Poisson setting: Gaussian empirical Bayes gets regret O(log^5 n / (n loglog n)), subexponential Poisson priors get O(log^4 n / n), and estimating g(θ) = θ^p gets regret O(log^{p+2} n / (n (loglog n)^{p+1})), each near the corresponding minimax rate.
  • Finite pretraining data suffices in principle: with M ≥ exp(C(log^2 n / loglog n + B_n)) batches, permutation-invariant ERM attains the same regret bound, making precise the practical observation that pretraining needs many batches.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the proof's only quantitative dependence on the transformer is through the ERM minimization error; this suggests the practical bottleneck is not architecture but optimization and data scale, and that other sequence models satisfying the same approximation could inherit the same guarantee.
  • Editorial inference: the α-posterior mechanism gives a testable handle on length generalization—for any permutation-invariant architecture trained at length n, the optimal α at test length n_test should be n/n_test; the paper's small-mixture experiments already show this, and the claim could be probed at larger scale and with positional encodings removed.
  • Editorial inference: the universal-PoP construction is a template for other exponential-family empirical Bayes problems: any mixture family with metric entropy O(log²(1/ε)/loglog(1/ε)) and a usable Hellinger-to-regret inequality should admit the same argument; the Gaussian and subexponential extensions already exercise parts of that template.
  • Editorial inference: the paper leaves the regime A_n ≫ log n as a conjecture; if the missing high-probability bound on −log Π_{X_n | X_{n-1}} can be established, the same pretraining scheme would recover the expected slow-growth regret and would cover unbounded-support priors with slowly growing tails.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper asks why a transformer pretrained on synthetic data can solve Poisson empirical Bayes (EB) problems with arbitrary test priors. It constructs a universal prior-on-prior (PoP): in each training batch, sample k = ceil(c0 log n / log log n) atoms uniformly on [0,A] and weights from Dirichlet(1,...,1); train a transformer by ERM on associated (theta^n, X^n) pairs. The main result (Theorem 1.1) states that, if the training procedure finds the global ERM minimizer with M -> infinity, the resulting estimator has worst-case regret O(log^3 n / (n (log log n)^2)), matching the minimax rate up to a log factor. Theorem 1.3 extends this to any PoP satisfying a universal-prior condition, and Theorem 1.5 analyzes length generalization through fractional (alpha-) posteriors. The proofs combine a minimax identity, posterior-contraction lemmas, metric-entropy bounds, and a volume argument for the random prior. Numerical experiments validate approximate Bayesian inference for small finite-mixture PoPs and exhibit the predicted alpha-posterior behavior for length generalization.

Significance. If the theorems hold, the paper gives a compelling conceptual explanation: a simple, explicitly defined random prior is approximately least favorable, so a single pretrained Bayes estimator can achieve near-minimax EB regret uniformly over all test priors. The technical development is careful and mostly self-contained, with explicit universal constants, no constants fitted to data, and detailed proofs. The identification of fractional posteriors as the mechanism for length generalization is elegant and is supported by targeted experiments. The main caveat is that the central theorems are conditional on an idealized ERM oracle; the extent to which real transformers realize this oracle is only partially validated. This limits the direct practical claim but does not invalidate the conditional mathematical result.

major comments (3)
  1. [§1 (Theorem 1.1), §4.2] The central claim is conditional on two idealizations: M -> infinity and the training procedure finding the global minimizer of (ERM). The ERM assumption is explicitly stated in §1 as a black-box modeling assumption and is used in the proof to identify the trained estimator with the hierarchical Bayes estimator E_Pi[theta^n|X^n]. The numerical evidence in §4.2, however, only checks closeness to the HB estimator for simple finite-mixture PoPs Pi_m = (1/m) sum_i G_i^{⊗n} with m=2,5,10; it does not test the full Algorithm 1 PoP with k ~ log n / log log n and Dirichlet weights, nor does it compare the transformer-HB distance with the regret of the HB estimator under the test priors of Theorem 1.1. Since gradient-based training of a transformer is not guaranteed to reach a global minimizer, Theorem 1.1 does not as written establish the paper's headline claim for the actual pretrained model. T
  2. [§1.2 (Theorem 1.5), Appendix A.3] Theorem 1.5's length-generalization bound requires the PoP Pi to be universal at the test length n_test: the proof applies Lemma 2.5 with n = n_test, which invokes Assumption 1 with n = n_test. But Algorithm 1 defines the PoP with k depending on the training length n. Lemma 1.4 establishes universality only for the PoP constructed with k ~ log n / log log n; for n_test >> n, the fixed training PoP has too few atoms to guarantee the chi^2 neighborhood at scale n_test^{-10} for every G0, so B_{n_test} need not be finite for that PoP. The theorem statement writes 'Pi is universal ... with rate B_n' but the bound contains B_{n_test}; this mismatch means the conclusion does not follow from the stated assumptions. The authors should either prove universality for all lengths simultaneously (e.g., via a different construction) or restrict the theorem to settings where B_{n_test} is available for
  3. [§2.1, Lemma 2.1 and proof of Lemma 1.2] The minimax identity in Lemma 1.2 is established via Ky Fan's minimax theorem, but the compactness and lower-semicontinuity conditions on the estimator space are not verified in the text. The estimator space of functions from Z_+^n to [0,A]^n with pointwise convergence is compact, but the regret functional is an infinite sum and its continuity/lower-semicontinuity in the estimator is not shown. While this is likely fixable, the current proof omits a technical step that is load-bearing for the identification of the least-favorable PoP.
minor comments (5)
  1. [Abstract] The abstract cites 'Teh et al., 2025' but the reference list uses [TJP25]; please unify the citation style.
  2. [§4.2, Figure 2] The conditioning on 'the event that the HB estimator does not perform uniformly well on G1,...,Gm' is not fully described in the main text; please clarify how the event is constructed and whether it biases the reported distance/regret comparison.
  3. [Theorem 1.5] The displayed regret bound uses a double sup that is redundant with the definition of Regret. Also, the rate B_{n_test} should be made explicit in the theorem statement to match the proof.
  4. [Algorithm 1] For very small n, k = ceil(c0 log n / log log n) may be zero or one; please specify that n is assumed sufficiently large or handle small n separately.
  5. [Appendix A.2] In the proof of Lemma 1.4, the notation L is used both for the number of atoms in the intermediate prior and for the integer in Lemma A.2; this is slightly confusing. Consider disambiguating.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity: universal-prior construction is proved from external moment-matching and posterior-contraction results; the ERM global-minimizer assumption is explicit and not disguised as a prediction.

full rationale

The derivation chain is not circular. Lemma 1.2 is a standard Ky Fan minimax swap (citing [Fan53]); Theorem 1.3 follows from posterior-contraction lemmas built on [GGVDV00], the metric entropy bound Lemma 2.2 from [WY16], and the Hellinger-regret inequality Lemma 2.1 from [JPW25]. Lemma 1.4 verifies Assumption 1 for Algorithm 1's explicit random-PMF prior-on-prior by moment matching and a volume argument; the universality condition (3) is about Π's chi-squared/TV coverage of the prior space, not about regret, so Theorem 1.3 is not Assumption 1 renamed. The main idealizations are explicit: Theorem 1.1 states 'assume that the training procedure finds the global minimizer of (ERM)', and Section 1 says 'we adopt a black-box modeling of transformers under the key assumption that the transformers can approximate the Bayes estimator.' These are unverified modeling assumptions, with Section 4.2 checking them only for small finite-mixture PoPs Π_m (m=2,5,10) rather than the full Algorithm 1 PoP; that is a conditional/empirical gap, not a circular reduction. The paper itself flags a related open point in Section 3.2: 'we find it difficult to rigorously establish such a bound, and therefore leave it as a conjecture.' Self-citations ([TJP25], [KPT26], [PW21]) supply context, expressiveness theorems, and benchmark lower bounds, but the proof of the regret bound does not reduce to those citations; no fitted constant is renamed as a prediction, and no uniqueness theorem is imported to force the construction.

Assumptions & free parameters 1 free parameters · 7 assumptions · 1 invented entities

The paper's results are conditional on (i) the transformer attaining the global ERM minimizer, and (ii) a specific structural model for length generalization. The main mathematical tools are standard results from posterior contraction, moment matching, and minimax theory, which the paper uses carefully. No parameters are fitted to data; the only hand-chosen constants are hyperparameters like c0.

free parameters (1)
  • c0 (number of atoms hyperparameter in Algorithm 1) = unspecified; chosen 'large enough' so that k ≥ L ≈ A log n / loglog n
    Algorithm 1 samples k = ⌈c0 logn/loglogn⌉ atoms. Lemma 1.4 requires k ≥ L to approximate any prior within TV ≤ n^{-10}; c0 is a hand-chosen constant that the proof depends on. It is not fitted to data but is an ad hoc hyperparameter.
assumptions (7)
  • domain assumption Posterior contraction machinery of [GGVDV00] (existence of tests, Bernstein-type concentration)
    Used in the proofs of Lemmas 2.4 and 2.5 to establish high-probability contraction of the (α-)posterior; cited and invoked in Section B.4 and B.5.
  • domain assumption Poisson moment-matching bound [WY16, Lemma 3]
    Used in Lemma 1.4 and Lemma 2.2 to approximate any prior on [0,A] by an L-point mixture with small total-variation error; cited in Section A.2 and B.2.
  • domain assumption Hellinger-regret inequality [JPW25, Lemma 4]
    The key bridge from posterior contraction (Hellinger distance) to squared-error regret; used in Lemma 2.1 and the proof of Theorem 1.3.
  • standard math Minimax theorem (Ky Fan) and Prokhorov's theorem for compactness of the prior space
    Used in Lemma 1.2 to interchange inf and sup and establish existence of least-favorable PoPs; stated in Section A.1.
  • domain assumption The trained transformer attains the global ERM minimizer (or approximates the Bayes estimator E_Π[θ^n|X^n])
    All regret bounds (Theorems 1.1, 1.3, 1.5) are conditional on this. The paper states it as a black-box assumption in Section 1 and provides only partial numerical verification in Section 4.2 for finite-mixture PoPs.
  • domain assumption Length-generalization structural model: transformer output is T_i(X^n) = f(X_i, μ_n) for a fixed map f, independent of n
    Stated in Section 1.2 and used in Theorem 1.5; motivated by self-attention structure and [FdHP25, Section 2.2]. Numerically supported in Section 4.2.
  • domain assumption Assumption 1 (Universal PoP): Π places mass at least e^{-B_n} in a χ²-neighbourhood of every prior
    Defines the sufficient condition for the regret bounds; not an unsupported axiom since Lemma 1.4 shows Algorithm 1's PoP satisfies it, but it is an assumption about the training prior that the theorems rely on.
invented entities (1)
  • Universal prior-on-prior (PoP) independent evidence
    purpose: A training distribution over priors such that ERM pretraining attains near-optimal regret uniformly over all test priors
    Not a physical entity but a new mathematical construction. The paper provides an explicit falsifiable handle—the regret bound of Theorem 1.1—which is confirmed in Section 4.1 for a concrete instance. The construction (Algorithm 1) is explicit and does not depend on the target result.

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Cite this review

Pith. "Pith review of Universal priors: solving empirical Bayes via Bayesian inference and pretraining." pith.science (2026). https://pith.science/paper/XOCHYLDH

@misc{pith2026260215136,
  author       = {Pith},
  title        = {Pith review of: Universal priors: solving empirical Bayes via Bayesian inference and pretraining},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XOCHYLDH}},
  note         = {Machine review of arXiv:2602.15136}
}
abstract

We theoretically justify the recent empirical finding of [Teh et al., 2025] that a transformer pretrained on synthetically generated data achieves strong performance on empirical Bayes (EB) problems. We take an indirect approach to this question: rather than analyzing the model architecture or training dynamics, we ask why a pretrained Bayes estimator, trained under a prespecified training distribution, can adapt to arbitrary test distributions. Focusing on Poisson EB problems, we identify the existence of universal priors such that training under these priors yields a near-optimal regret bound of $\widetilde{O}(\frac{1}{n})$ uniformly over all test distributions. Our analysis leverages the classical phenomenon of posterior contraction in Bayesian statistics, showing that the pretrained transformer adapts to unknown test distributions precisely through posterior contraction. This perspective also explains the phenomenon of length generalization, in which the test sequence length exceeds the training length, as the model performs Bayesian inference using a generalized posterior.

Figures

Figures reproduced from arXiv: 2602.15136 by the authors.

Figure 1
Figure 1. Regrets of different estimators with different test sequence lengths and test priors. For all [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. The regret of the hierarchical Bayes estimator, as well as its mean squared distance to [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Plots of the mean squared distance between transformer output and the hierarchical Bayes [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Mean squared distance between transformer output and the hierarchical Bayes estimator [PITH_FULL_IMAGE:figures/full_fig_p034_4.png]
Figure 5
Figure 5. Figure 5: Mean squared distance between transformer output and the hierarchical Bayes estimator [PITH_FULL_IMAGE:figures/full_fig_p035_5.png]

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Forward citations

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Reference graph

Works this paper leans on

18 extracted references · 13 linked inside Pith · cited by 2 Pith papers

  1. [1]

    The Bayesian geometry of transformer attention.arXiv preprint arXiv:2512.22471,

    [ADM25] Naman Aggarwal, Siddhartha R Dalal, and Vishal Misra. The Bayesian geometry of transformer attention.arXiv preprint arXiv:2512.22471,

    arXiv

  2. [4]

    Trans- formers Simulate MLE for Sequence Generation in Bayesian Networks.arXiv preprint arXiv:2501.02547,

    [CHW+25] Yuan Cao, Yihan He, Dennis Wu, Hong-Yu Chen, Jianqing Fan, and Han Liu. Trans- formers Simulate MLE for Sequence Generation in Bayesian Networks.arXiv preprint arXiv:2501.02547,

    arXiv

  3. [7]

    Learning minimax estimators via online learning.arXiv preprint arXiv:2006.11430,

    [GSP+20] Kartik Gupta, Arun Sai Suggala, Adarsh Prasad, Praneeth Netrapalli, and Pradeep Ravikumar. Learning minimax estimators via online learning.arXiv preprint arXiv:2006.11430,

    arXiv 2006

  4. [8]

    Besting Good– Turing: Optimality of Non-Parametric Maximum Likelihood for Distribution Esti- mation.arXiv preprint arXiv:2509.07355,

    [HNWSW25] Yanjun Han, Jonathan Niles-Weed, Yandi Shen, and Yihong Wu. Besting Good– Turing: Optimality of Non-Parametric Maximum Likelihood for Distribution Esti- mation.arXiv preprint arXiv:2509.07355,

    arXiv

  5. [9]

    Quantitative bounds for length generalization in transformers.arXiv preprint arXiv:2510.27015,

    [INL25] Zachary Izzo, Eshaan Nichani, and Jason D Lee. Quantitative bounds for length generalization in transformers.arXiv preprint arXiv:2510.27015,

    arXiv

  6. [12]

    In- context parametric inference: Point or distribution estimators?arXiv preprint arXiv:2502.11617,

    [MBML25] Sarthak Mittal, Yoshua Bengio, Nikolay Malkin, and Guillaume Lajoie. In- context parametric inference: Point or distribution estimators?arXiv preprint arXiv:2502.11617,

    arXiv

  7. [15]

    [Rob51] Herbert Robbins. Asymptotically subminimax solutions of compound statistical deci- sion problems.Proceedings of the second Berkeley symposium on mathematical statis- tics and probability, abs/1904.10040:131–149,

    arXiv 1904

  8. [17]

    Solving empirical Bayes via trans- formers.arXiv preprint arXiv:2502.09844,

    [TJP25] Anzo Teh, Mark Jabbour, and Yury Polyanskiy. Solving empirical Bayes via trans- formers.arXiv preprint arXiv:2502.09844,

    arXiv

Show all 18 references
  1. [18]

    Transformers can achieve length generalization but not robustly.arXiv preprint arXiv:2402.09371,

    39 [ZAC+24] Yongchao Zhou, Uri Alon, Xinyun Chen, Xuezhi Wang, Rishabh Agarwal, and Denny Zhou. Transformers can achieve length generalization but not robustly.arXiv preprint arXiv:2402.09371,

    arXiv

  2. [1953]

    Epsilon-Minimax Solutions of Statistical Decision Problems

    [FBO+25] Andrés Aradillas Fernández, José Blanchet, José Luis Montiel Olea, Chen Qiu, Jörg Stoye, and Lezhi Tan. Epsilon-Minimax Solutions of Statistical Decision Problems. arXiv preprint arXiv:2509.08107,

  3. [1956]

    Empirical Bayes estimation: When doesg-modeling beatf-modeling in theory (and in practice)?arXiv preprint arXiv:2211.12692,

    [SW22] Yandi Shen and Yihong Wu. Empirical Bayes estimation: When doesg-modeling beatf-modeling in theory (and in practice)?arXiv preprint arXiv:2211.12692,

    arXiv

  4. [1983]

    Amortized in-context bayesian posterior estimation.arXiv preprint arXiv:2502.06601,

    [MBL+25] Sarthak Mittal, Niels Leif Bracher, Guillaume Lajoie, Priyank Jaini, and Mar- cus Brubaker. Amortized in-context bayesian posterior estimation.arXiv preprint arXiv:2502.06601,

    arXiv

  5. [2000]

    Stein’s unbi- ased risk estimate and Hyvärinen’s score matching.arXiv preprint arXiv:2502.20123,

    [GIKL25] Sulagna Ghosh, Nikolaos Ignatiadis, Frederic Koehler, and Amber Lee. Stein’s unbi- ased risk estimate and Hyvärinen’s score matching.arXiv preprint arXiv:2502.20123,

  6. [2009]

    Function estimation in the empirical Bayes setting.arXiv preprint arXiv:2601.18689,

    [KPT26] Benjamin Kang, Yury Polyanskiy, and Anzo Teh. Function estimation in the empirical Bayes setting.arXiv preprint arXiv:2601.18689,

  7. [2019]

    A nonparametric regression alterna- tive to empirical bayes approaches to simultaneous estimation.arXiv preprint arXiv:2205.00336,

    [BZ22] Alton Barbehenn and Sihai Dave Zhao. A nonparametric regression alterna- tive to empirical bayes approaches to simultaneous estimation.arXiv preprint arXiv:2205.00336,

    arXiv

  8. [2022]

    Tabdpt: Scaling tabular foundation models.arXiv preprint arXiv:2410.18164,

    [MTH+24] Junwei Ma, Valentin Thomas, Rasa Hosseinzadeh, Hamidreza Kamkari, Alex Labach, Jesse C Cresswell, Keyvan Golestan, Guangwei Yu, Maksims Volkovs, and An- thony L Caterini. Tabdpt: Scaling tabular foundation models.arXiv preprint arXiv:2410.18164,

  9. [2024]

    Sharp regret bounds for empirical Bayes and com- pound decision problems.arXiv preprint arXiv:2109.03943,

    [PW21] Yury Polyanskiy and Yihong Wu. Sharp regret bounds for empirical Bayes and com- pound decision problems.arXiv preprint arXiv:2109.03943,

    arXiv

  10. [2025]

    On provable length and compositional generaliza- tion.arXiv preprint arXiv:2402.04875,

    [AM24] Kartik Ahuja and Amin Mansouri. On provable length and compositional generaliza- tion.arXiv preprint arXiv:2402.04875,

    arXiv

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