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Confidence Sequences for Generalized Linear Models via Regret Analysis

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Pith's one-line read For any generalized linear model, a regret bound for a sequential predictor yields a valid confidence sequence for the true parameter.

desk verdict A sound and honest unification of GLM confidence-sequence construction via regret analysis, with two genuinely new bounds and one overclaim in the abstract that should be fixed. read the letter →

arxiv 2504.16555 v1 pith:4GQCVMYL submitted 2025-04-23 math.ST cs.LGstat.MLstat.TH

Eugenio Clerico , Hamish Flynn , Wojciech Kot{l}owski , Gergely Neu This is my paper · ORCID
classification math.STcs.LGstat.MLstat.TH MSC 62F2562L1262J1268Q8768Q32
keywords confidencesequencesgeneralizedlinearmodelssequentialprobabilityassignmentregretanalysisonline-to-confidence-setconversionsmartingaleconcentrationVille'sinequalitysparseregression
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 establishes that confidence-sequence construction for generalized linear models can be reduced to regret analysis in a sequential probability-assignment game. Concretely, for any GLM and any predictor that issues a distribution before seeing each label, the set of parameters whose cumulative negative log-likelihood exceeds the predictor's loss by at most the predictor's regret against a reference parameter plus $\log(1/\delta)$ is a valid confidence sequence for the true parameter. Because the covering argument is a martingale inequality, the online algorithm only needs to exist for the bound to hold, although the paper also shows how to use its predictions to center sets adaptively. The authors instantiate the reduction with exponentially weighted and normalized-maximum-likelihood forecasters, recovering earlier confidence-set constructions and producing new ones, including a width that does not grow with the sample size when the log-partition is locally strongly convex.

What carries the argument

The load-bearing object is the sequential likelihood-ratio martingale $M_n=\prod_{t=1}^n p_t(Y_t)/p(Y_t\mid X_t,\theta^\star)$, which is nonnegative and satisfies $\mathbb{E}[M_t\mid\mathcal{F}_{t-1}]=M_{t-1}$; Ville's inequality then gives $\mathbb{P}[\exists n:\log M_n\ge\log(1/\delta)]\le\delta$. The conversion identity is $$\sum_{t=1}^n(\ell_t(\$\theta$^\star)-\ell_t(\theta_n))=\mathrm{regret}_{p_n}(\theta_n)+\sum_{t=1}^n(\ell_t(\$\theta$^\star)-L_t(p_t)),$$ where the second term is exactly $\log M_n$. For algorithmic conversions the paper introduces $\eta$-shifted losses $\ell_t^{(\eta)}(\theta)=\ell_t(\eta\theta+(1-\eta)\theta^\star)$, which keep $\theta^\star$ as a fixed point and generate a second martingale, and measures the size of the resulting sets with the symmetrized convexity measure $d_\psi(z,z')=\tfrac12\psi(z)+\tfrac12\psi(z')-\psi((z+z')/2)$.

What would settle it

One settling check is simulation-based coverage of the Theorem 2.2 set for a logistic regression with known $\theta^*$ and a misspecified bound $b$: generate repeated sequences, and if $\theta^*$ falls outside the set more than $\delta$ of the time in the regime $|\langle\theta^*,X_t\rangle|\le b$, the polar-set construction used in the applications is invalid; in the opposite direction, if the set is so large that it contains essentially all parameters, the n-independent width is vacuous for that GLM.

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Extended reading notes

Core claim

The central claim is Theorem 2.2: with $\ell_t(\theta)=-\log p(Y_t\mid X_t,\theta)$, let $p_1,\dots,p_n$ be any predictions chosen before the corresponding labels are revealed, and let $\theta_n$ be any data-dependent reference parameter. Then $$\Theta_n=\left\{\$\theta$\in\mathbb{R}^d:\sum_{t=1}^n(\ell_t(\$\theta$)-\ell_t(\theta_n))\le \mathrm{regret}_{p_n}(\theta_n)+\log(1/\delta)\right\}$$ satisfies $\mathbb{P}[\exists n:\theta^\star\notin\Theta_n]\le\delta$. The proof decomposes the excess loss of the true parameter over the reference into the predictor's regret against the reference plus a sequential log-likelihood-ratio sum, which is a nonnegative martingale; Ville's inequality bounds that martingale by $\log(1/\delta)$. Thus every regret bound for sequential probability assignment with GLM losses automatically becomes a confidence sequence, and the same reduction, refined with shifted losses, produces sets whose width is governed by regret against $\theta^\star$ rather than against a reference estimator.

Load-bearing premise

The binding assumption is that the user knows an interval on which the log-partition is strongly convex and can guarantee the true parameter's linear predictions stay inside it; when that curvature constant is tiny the width explodes, and a wrong interval invalidates the comparator.

Editorial extensions

If this is right

  • With adaptively chosen covariates and an $M$-smooth log-partition, the analytic conversion gives confidence width $\|\hat\theta_n\|^2/(2\gamma^2)+\tfrac12\log\det(\gamma^2M\Lambda_n+I_d)+\log(1/\delta)$, recovering linear-model ellipsoid bounds and removing the uniform lower-curvature assumptions in earlier GLM confidence sets.
  • With obliviously chosen covariates and a locally $m$-strongly convex log-partition, the confidence set has width $d\log(1+2M/m)+2\log(1/\delta)$, independent of the sample size $n$ and invariant to invertible linear reparametrization of the covariates.
  • For $s$-sparse parameters, the algorithmic conversion with a sparsity-inducing prior yields width $(4s/m)\log(2ed\sqrt{1+MB^2L_\infty^2n/2}/s)+(4/m)\log(2\sqrt e/\delta)$, which removes the factor $\max_tY_t^2$ that appears in earlier deterministic-forecaster confidence sets.
  • Any future regret bound for sequential probability assignment with GLM log losses can be plugged into Theorem 2.2 to yield a valid confidence sequence, so the statistical task reduces to an algorithmic task.
  • The framework recovers several published confidence-set constructions as special cases of one reduction, including linear, logistic, and general exponential-family settings.

Reading between the lines

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

  • One extension the paper only sketches is that the same martingale argument survives when the exponential-family moment condition is relaxed to an inequality, so the reduction should transfer to sub-exponential families and generalized additive models; checking that transfer would be a natural next test.
  • The n-independent width of Theorem 4.3 is most useful when the strong-convexity interval is short; for logistic regression the curvature constant $m$ is typically exponentially small in the dimension, so a practical development would be to replace global strong convexity on $[-b,b]$ with curvature measured only on the realized prediction interval of $\theta^*$.
  • The analytic/algorithmic distinction suggests a design rule: in analytic mode the user should choose the predictor solely for its regret bound, since it never has to be run, whereas in algorithmic mode the predictor's output defines the pseudo-labels and therefore must be computable; the sparse-GLM result shows when paying that computation removes the label-magnitude factor.
  • The rank-aware regret bound indicates that confidence widths could adapt to the effective dimension of the covariates, shrinking on sequences where the design matrix is low rank; this is implicit in the determinant bound but not developed as a separate adaptive procedure.
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Editorial analysis

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

Referee Report

0 major / 5 minor

Summary. This paper develops a framework for constructing confidence sets and confidence sequences for GLM parameters by reducing the statistical problem to regret analysis in a sequential probability assignment game. The central reduction (Theorem 2.2) decomposes the excess loss of the true parameter against a data-dependent reference estimator into the online learner's regret plus a sequential log-likelihood-ratio martingale, which is controlled by Ville's inequality. The authors distinguish analytic conversions, where only the existence of a low-regret forecaster is needed, from algorithmic conversions, where the forecaster's predictions are used actively to define the confidence set. The framework is instantiated with EWA and NML forecasters. The resulting applications include confidence sets for adaptive covariates (Theorem 4.1), an n-independent width for transductive strongly convex GLMs (Theorem 4.3), and sparse-model confidence sets obtained from algorithmic conversions with pseudo-labels (Theorems 4.4-4.7). The manuscript also positions the work relative to Abbasi-Yadkori, Pal and Szepesvari (2012), Jun et al. (2017), Kirschner et al. (2025), and related online-to-confidence-set literature.

Significance. Subject to the stated assumptions, the central reduction is correct and the paper is a genuine conceptual contribution: it provides a single template that recovers several known confidence-set constructions and yields new results, including the n-independent transductive width and label-independent sparse confidence sets. I checked the proofs of Propositions 2.1, 2.3, 3.1, and 3.2 and Theorems 2.2, 2.4, 4.1, 4.3, 4.5, 4.6, and 4.7; the martingale decompositions are sound and the Gaussian integral computations are standard. The authors deserve credit for being explicit about the severe limitations of strong-convexity assumptions, in particular the exponentially small m for logistic regression, and about the need for the polar set S_{n,b}. The main issues are local: a technically incorrect regularized-loss comparison around Eq. (18), an overbroad statement in the abstract, and several notational slips. These do not affect the validity of the central reduction and are readily fixable.

minor comments (5)
  1. [Section 4.1.1, Eq. (18)] The passage around Eq. (18) is not a correct rewrite of Theorem 4.1 for general lambda. Theorem 4.1 is stated with lambda = 1 and with a width for the unregularized loss; after writing \tilde{\ell}_t = \ell_t + \rho, the exact regularized width is \tilde{\beta}_n = \rho(\theta_\star) + \gamma^{\rho}_{n,1} + \log(1/\delta) at lambda = 1, or, for general lambda, \tilde{\beta}_n = (\rho(\hat{\theta}_n) + \gamma^{\rho}_{n,\lambda})/\lambda + \rho(\theta_\star) - \rho(\hat{\theta}_n) + \log(1/\delta). The printed expression with 1/(2\lambda) prefactors and lambda inside the determinant omits the term (1/\lambda - 1)\rho(\hat{\theta}_n) and is only valid when lambda = 1 or \hat{\theta}_n = 0. Please derive this step explicitly or restrict the display to lambda = 1.
  2. [Abstract] The abstract's claim that the methodology 'recovers all state-of-the-art confidence set constructions within a single framework' overstates the scope. Several constructions in the cited literature rely on additional assumptions such as compact parameter sets or known strong-convexity constants, and the Discussion itself highlights severe limitations of the strong-convexity results. I suggest softening this to 'several' or to 'the constructions covered by our assumptions'.
  3. [Section 4.1.2 and Theorem 4.3] The theorem's validity depends on the user knowing b and m and on the true parameter lying in the polar set S_{n,b}. These conditions are stated in the text before the theorem, but the 'first to have width independent of n' claim in Section 5 and the abstract do not carry the caveat. Since a misspecified b invalidates the strong-convexity inequality (21) and the first-order optimality step, this caveat deserves more prominence in the theorem statement and the discussion.
  4. [Throughout] Notation and typos: Eq. (20) writes the reference estimator as \hat{\theta}_{b,n} while the theorem and proof use \hat{\theta}_{n,b}; the proof of Theorem 4.3 first writes \rho(\theta) = \gamma^{-2}\|\theta-\theta_\star\|^2_{\Lambda_n} where the subsequent argument uses 1/(2\gamma^2); Proposition 2.3's proof contains '\ell_n' and 'q_n' where '\ell_t' and 'q_t' are meant; in Section 2.1, 'we we will play special attention' should be 'we will pay special attention'; Eq. (12) has a missing parenthesis in the denominator; and in Section 3.2, 'Our of the many possibilities' should be 'Out of the many possibilities'.
  5. [Section 2.3, last paragraph] The sentence 'We provide concrete examples (as well as a tighter confidence set of a similar shape) in Section 2.3' appears to be a cross-reference error; the concrete examples of convex sets built from d_\psi are actually given in Section 4.2.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the online-to-confidence-set reduction is self-contained; the strong-convexity caveat in Theorem 4.3 is a disclosed limitation, not a circular step.

full rationale

The central derivation is self-contained and non-circular. Theorem 2.2's proof is an equality: the excess log-likelihood of the true parameter relative to the reference theta_n equals regret_pn(theta_n) plus the sequential log-likelihood-ratio martingale M_n(theta_star); Proposition 2.1, proved in-text via Ville's inequality, gives P(sup_n M_n(theta_star) >= log(1/delta)) <= delta, so the claim follows from a martingale bound and an algorithmic regret bound, with no fitted parameter and no target-dependent input. The regret bounds in Section 3 are deterministic, hold for every comparator, and are instantiated with data-dependent references only after being proved pointwise, so the confidence width is not renamed as a fitted prediction. In Theorem 4.3, the EWA prior is centered at theta_star, but this is a legitimate existence device: the algorithm is never run, the resulting confidence set depends only on b, delta, kappa, and the observed data, and the proof uses theta_star only to derive a regret bound for a hypothetical forecaster. The strong-convexity and polar-set assumptions are stated explicitly, and their severity (e.g., exponentially small m for logistic regression) is acknowledged in Sections 4 and 5, so they are disclosed limitations rather than hidden inputs. Self-citations to prior work by the authors are contextual and none is load-bearing for the main reduction.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central results depend on standard GLM assumptions (exponential family form, smoothness of psi) and on user-chosen tuning parameters (gamma, lambda) that are set analytically rather than fitted to data. No new entities are postulated. The framework uses classic martingale concentration (Ville's inequality) and existing regret bounds as building blocks; the novel content is the reduction scheme and its instantiations.

free parameters (4)
  • gamma (regularization scale) = sqrt(2/m) in Theorem 4.3; user-chosen in Theorem 4.1
    Controls the strength of the Gaussian prior in the EWA regret bound; the confidence width depends on it, and the optimal choice requires knowing m.
  • lambda (EWA learning rate) = 1 in Theorems 4.1 and 4.3; 1/2 in Theorems 4.5-4.7
    Learning rate of the exponentially weighted average forecaster; set to a constant for each application, not fitted to data.
  • b (bound on |<theta_star,X_t>|) = assumed known
    Required for strong convexity on [-b,b] and for the polar set S_{n,b}; if unknown, the n-independent bound cannot be evaluated.
  • s, B, L_infinity (sparsity, norm bound, coordinate covariate bound) = assumed known in Theorem 4.7
    Inputs to the sparse prior; the confidence width depends on them.
assumptions (5)
  • domain assumption Labels follow a GLM: p(y|X,theta_star)=exp(<theta_star,X>y - psi(<theta_star,X>)) h(y) with psi convex.
    Defines the entire setting (Section 1).
  • domain assumption psi is M-smooth (psi'' <= M) for adaptive-covariate results.
    Used in Lemma 4.2 to bound the Bregman information gain (Section 4.1.1).
  • domain assumption psi is m-strongly convex on [-b,b] and |<theta_star,X_t>| <= b for the n-independent and algorithmic results.
    Critical for Theorem 4.3 and Theorems 4.4-4.7; the authors note m can be exponentially small for logistic regression.
  • standard math The regularized MLE and Gaussian integrals exist (for example, Lambda_n full rank, integrable prior).
    Needed for the EWA prior and Bregman information gain computations (Sections 3.1 and 4.1.2).
  • standard math Ville's inequality for nonnegative supermartingales (Proposition 2.1).
    The backbone of all conversions; classic result referenced in Section 2.1.

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Pith. "Pith review of Confidence Sequences for Generalized Linear Models via Regret Analysis." pith.science (2026). https://pith.science/paper/4GQCVMYL

@misc{pith2026250416555,
  author       = {Pith},
  title        = {Pith review of: Confidence Sequences for Generalized Linear Models via Regret Analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4GQCVMYL}},
  note         = {Machine review of arXiv:2504.16555}
}
read the original abstract

We develop a methodology for constructing confidence sets for parameters of statistical models via a reduction to sequential prediction. Our key observation is that for any generalized linear model (GLM), one can construct an associated game of sequential probability assignment such that achieving low regret in the game implies a high-probability upper bound on the excess likelihood of the true parameter of the GLM. This allows us to develop a scheme that we call online-to-confidence-set conversions, which effectively reduces the problem of proving the desired statistical claim to an algorithmic question. We study two varieties of this conversion scheme: 1) analytical conversions that only require proving the existence of algorithms with low regret and provide confidence sets centered at the maximum-likelihood estimator 2) algorithmic conversions that actively leverage the output of the online algorithm to construct confidence sets (and may be centered at other, adaptively constructed point estimators). The resulting methodology recovers all state-of-the-art confidence set constructions within a single framework, and also provides several new types of confidence sets that were previously unknown in the literature.

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Reviewed August 16, 2026 · model on record in the stance chip above.