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arxiv: 2412.07999 · v3 · submitted 2024-12-11 · 🧮 math.ST · math.PR· stat.ML· stat.TH

Fast Mixing of Data Augmentation Algorithms: Bayesian Probit, Logit, and Lasso Regression

Holden Lee , Kexin Zhang This is my paper

Pith reviewed 2026-05-23 07:47 UTC · model grok-4.3

classification 🧮 math.ST math.PRstat.MLstat.TH
keywords mixing timedata augmentationGibbs samplerBayesian probitBayesian logitBayesian lassoconductance methodnon-asymptotic bounds
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The pith

Probit and logit data augmentation samplers mix in O(n log(log η/ε)) steps with high probability over data.

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

The paper develops a modified conductance method to bound the mixing times of data augmentation Gibbs samplers for Bayesian regression models. It proves the first polynomial upper bounds on the number of steps needed for ProbitDA and LogitDA to reach ε-accuracy in total variation, KL, or chi-squared distance. The bounds depend explicitly on the design matrix and prior, and simplify to O(n log(log η/ε)) when data are sub-Gaussian or log-concave and scaled properly. For LassoDA the bound is polynomial in d and n but higher order. These results apply to large-scale settings with imbalanced responses.

Core claim

Using a modified conductance-based method, the first non-asymptotic polynomial upper bounds are proved for the mixing times of ProbitDA, LogitDA, and LassoDA, with explicit dependence on design matrix for the first two, leading to O(n log(log η/ε)) under data assumptions.

What carries the argument

A modified conductance-based method for analyzing the mixing time of two-block Gibbs samplers in data augmentation algorithms.

If this is right

  • ProbitDA and LogitDA achieve ε-mixing in O(n log(log η/ε)) steps with η-warm start.
  • LassoDA requires O(d²(d log d + n log n)² log(η/ε)) steps for TV distance ε.
  • The bounds hold with high probability for data from sub-Gaussian or log-concave distributions.
  • These bounds are applicable even with highly imbalanced response data.
  • The results provide comparisons to Langevin Monte Carlo methods.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The conductance method could be adapted to analyze other two-block samplers beyond DA.
  • Fast mixing suggests these algorithms are practical for large n in Bayesian inference.
  • Extensions might include other priors or models where DA is used.
  • Initialization strategies could be further optimized based on these bounds.

Load-bearing premise

The data must be independently generated from a sub-Gaussian or log-concave distribution and properly scaled.

What would settle it

Finding a dataset generated from a sub-Gaussian distribution where the mixing time of ProbitDA exceeds the stated O(n log(log η/ε)) bound would falsify the result.

Figures

Figures reproduced from arXiv: 2412.07999 by Holden Lee, Kexin Zhang.

Figure 1
Figure 1. Figure 1: Illustration of the transition kernels of ProbitDA, LogitDA, and LassoDA. Here, the arrow represents conditional dependency [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Simulation results for ProbitDA with imbalance factor Υ = 0.6 [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Simulation results for ProbitDA with imbalance factor Υ = 1 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Simulation results for LogitDA with imbalance factor Υ = 0.6. demonstrated in Theorem 3.6, is tight in n, but the d’s dependency can be potentially im￾proved. In [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Simulation results for LogitDA with imbalance factor Υ = 1 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Simulation results for LassoDA. We report the autocorrelation time for both the v￾coordinate and the first coordinate of β. We plot the autocorrelation time for the three scenarios in [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Illustration of the kernel of T -transformed LassoDA = TV  φ (m) , φ(m) π  + TV  φ (m−1)PTφ→ρ , φ(m−1) π PTφ→ρ  ≤(i) 2 TV  φ (m−1), φ(m−1) π  = 2 TV  νTφ P m−1 Tφ , πTφ  , where (i) is due to data processing equality. Overall, we have TV (νP m, π) = TV (νTP m T , πT ) ≤ 2 TV  νTφ P m−1 Tφ , πTφ  (31) . Equation (31) gives us a way to control the mixing time of the LassoDA by that of φ-marginal of… view at source ↗
read the original abstract

We propose using a modified conductance-based method to study the mixing time of an important class of two-block Gibbs samplers, the data augmentation (DA) algorithm. %, which is of prominent interest in both theoretical and empirical research. Using this method, we prove the first non-asymptotic polynomial upper bounds on mixing times of three important DA algorithms: DA algorithms for Bayesian Probit regression (Albert and Chib, 1993, ProbitDA) and Bayesian Logit regression (Polson, Scott, and Windle, 2013, LogitDA), and Bayesian Lasso Regression (Park and Casella, 2008, Rajaratnam et al., 2015, LassoDA). Concretely, for ProbitDA and LogitDA, we demonstrate a tight bound that explicitly depends on the design matrix and prior covariance matrix. Under the assumption that data are independently generated from either a sub-Gaussian or log-concave distribution and properly scaled, the bound implies that with $\eta$-warm start, parameter dimension $d$, and sample size $n$, with high probability over data, the two algorithms require $\mathcal{O}\left(n\log \left(\frac{\log \eta}{\epsilon}\right)\right)$ steps to obtain samples with at most $\epsilon$ error in TV, KL, or $\chi^2$ distance. Meanwhile, we show that under minimal data assumptions, LassoDA requires $\mathcal{O}\left(d^2(d\log d +n \log n)^2 \log \left(\frac{\eta}{\epsilon}\right)\right)$ steps to achieve $\epsilon$-accuracy in TV distance. The results are generally applicable to settings with large $n$ and large $d$, including settings with highly imbalanced response data in Probit and Logit regression. We compare them with the best known guarantees of Langevin Monte Carlo and Metropolis Adjusted Langevin Algorithm. We evaluate our theoretical results using numerical examples, and discuss the mixing times of the three algorithms under feasible initialization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a modified conductance-based method to derive non-asymptotic upper bounds on the mixing times of data augmentation (DA) Gibbs samplers for Bayesian probit regression (ProbitDA), logit regression (LogitDA), and lasso regression (LassoDA). It claims the first polynomial bounds: for ProbitDA and LogitDA, under independent sub-Gaussian or log-concave data with proper scaling, an O(n log(log η / ε)) mixing time (with high probability over data) from an η-warm start to ε-accuracy in total variation, KL, or χ² distance; for LassoDA, an O(d²(d log d + n log n)² log(η/ε)) bound in TV distance. The work includes comparisons to Langevin Monte Carlo and MALA, plus numerical validation, and applies to large-n, large-d regimes including imbalanced responses.

Significance. If the derivations hold, the results supply the first explicit, non-asymptotic polynomial mixing-time guarantees for these standard DA algorithms, which are widely used in Bayesian computation. The explicit dependence on the design matrix and prior covariance, the high-probability simplification under standard data assumptions, and the direct comparison with gradient-based samplers are useful for both theory and practice. The numerical examples provide concrete support for the claimed rates.

minor comments (3)
  1. The abstract states that the general bound 'explicitly depends on the design matrix and prior covariance matrix' before simplifying under the sub-Gaussian/log-concave assumption; the main text should state this general bound (with the precise matrix dependence) as a numbered theorem early in the results section so readers can see the reduction.
  2. The phrase 'properly scaled' in the data assumption for the O(n log(...)) bound is used without an explicit definition or reference to a scaling condition on the design matrix; a short clarifying sentence or display equation would remove ambiguity.
  3. The comparison with LMC and MALA is mentioned but the precise regimes (e.g., step-size choices or dimension dependence) under which the DA bounds are competitive are not summarized in a table or remark; adding such a comparison would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of our manuscript. Their summary correctly identifies the main contributions: the first explicit non-asymptotic polynomial mixing-time bounds for ProbitDA, LogitDA, and LassoDA via a modified conductance argument, together with high-probability simplifications under standard data assumptions and comparisons to gradient-based methods. We are pleased that the referee finds the explicit dependence on the design matrix, prior, and the applicability to large-n/large-d and imbalanced regimes useful.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives non-asymptotic mixing time bounds for the three DA algorithms by applying a modified conductance method to the two-block Gibbs samplers. The resulting bounds are stated to depend explicitly on the design matrix and prior covariance; they simplify to the claimed O(n log(log η / ε)) form only after imposing the external sub-Gaussian or log-concave data assumptions and proper scaling. No equation reduces by construction to a fitted parameter, self-referential definition, or load-bearing self-citation chain; the conductance argument supplies independent analytic content that is not presupposed by the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard Markov chain conductance theory and the data distribution assumptions stated in the abstract; no free parameters or invented entities are apparent from the abstract.

axioms (2)
  • standard math Standard conductance bounds for Markov chains apply after the proposed modification
    Invoked to obtain the mixing time upper bounds for the two-block Gibbs samplers
  • domain assumption Data are i.i.d. from sub-Gaussian or log-concave distributions and properly scaled
    Required for the high-probability O(n log(log η / ε)) bound to hold

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