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Scalable Cox Regression via Grouped Risk Sets and Sharper LogSumExp Rates

Elizaveta Iashchinskaia, Egor Gladin

arXiv:2609.40120v1cs.LGmath.OC

TL;DR

Large-scale Cox regression is challenging when stochastic access is preferred to full-data passes, and mini-batch normalizer estimates generally bias gradients. The paper uses a softplus surrogate and grouped risk-set normalizers, with sharper optimization guarantees and controlled approximation to the full Cox solution. Experiments report the smallest median terminal regularized full-Cox training gap on all five datasets at matched vector-work budgets.

  • Problem

    Large-scale Cox regression relies on risk-set LogSumExp normalizers, while mini-batch estimates generally bias gradients and conventional implementations require full-data passes.

    Source: §1 · p. 1

  • Method

    The method combines a softplus surrogate with shared auxiliary normalizers for groups of neighboring Cox failures and derives uniform controls on grouping and softplus approximation error.

    Source: §1 · pp. 1–2, §4.1 · p. 5

  • Results

    At matched vector-work budgets, the method achieves the smallest median terminal regularized full-Cox training gap on all five datasets.

    Source: §6 · p. 9

  • Takeaways & Limitations

    Under the stated approximation and curvature conditions, risk-set grouping yields an eO(T^-4/5) computational rate, and under additional statistical conditions the compressed estimator is first-order equivalent to the full estimator.

    Source: §6 · p. 9

    The analysis assumes a convex linear predictor and boundedness and curvature conditions; tied-time asymptotics, time-varying covariates, and nonlinear predictors remain outside scope.

    Source: §6 · p. 9

Abstract

from arXiv · show

Motivated by the computational challenges of large-scale Cox regression, we study stochastic minimization of LogSumExp objectives over large sets. Mini-batch normalizer estimates generally yield biased gradients. We instead use a softplus surrogate that introduces one auxiliary scalar per normalizer and admits unbiased single-sample gradients. For smooth convex LogSumExp objectives, we prove an $O(T^{-1/2})$ averaged objective bound, improving the previous $T^{-1/4}$ analysis. With a strongly convex regularizer on the original variable, we also obtain a last-iterate squared-error rate of $\widetilde{O}(T^{-1})$ without strong convexity in the auxiliary variables. For Cox regression, the normalizers are defined over nested risk sets. We exploit this structure by grouping neighboring failures and sharing one auxiliary variable per group. The resulting compressed objective admits uniform score and curvature bounds that control the errors from grouping and softplus approximation. Together with the general optimization result, these bounds give a mean-square rate of $T^{-4/5}$, up to logarithmic factors, relative to the full Cox solution. The compressed estimator also matches the full estimator's asymptotic distribution. Experiments on synthetic and real survival datasets with slowly decreasing risk sets show a favorable performance relative to stochastic baselines.

1 INTRODUCTION

Large-scale Cox regression requires stochastic-access methods because full-data passes are costly or infeasible in distributed and out-of-core settings. The paper sharpens softplus optimization guarantees and groups neighboring Cox failures to compress normalizers while controlling approximation error.

1 INTRODUCTION

  • Mini-batch normalizer estimates generally bias LogSumExp gradients, while variational formulations with auxiliary variables admit unbiased single-sample gradients.The Cox model’s partial likelihood has a LogSumExp normalizer over everyone still at risk for each failure.Source: §1 · p. 1
  • Neighboring Cox failures have similar risk sets, so grouping consecutive failures lets them share one normalizer variable.A single group–event–subject draw yields an unbiased stochastic gradient; grouping perturbation is second order in relative risk-set change.Source: §1 · p. 1
  • For smooth convex LogSumExp objectives, the softplus method gives an O(T^-1/2) averaged objective bound and an O(log T/T) last-iterate squared-distance bound with strong convexity only on the original parameter.The auxiliary variables need not be strongly convex.Source: §1 · p. 2
  • Grouping and softplus error bounds yield mean squared parameter error O((log T/T)^4/5) relative to the full Cox solution, while the compressed and full estimators share the same first-order limit.The contribution also uses a sublinear number of auxiliary shifts.Source: §1 · p. 2
  • The paper’s general optimization guarantees cover smooth convex LogSumExp objectives, while its survival-analysis claims concern the convex linear Cox model.The authors state this scope separates computational approximation from statistical model error.Source: §1 · p. 2

2 RELATED WORK

Prior approaches address Cox computation through full-data solvers, approximations, or sampled objectives, while this paper targets a deterministic controlled approximation to the fixed full-data objective. Its softplus analysis improves the earlier rate and also gives a last-iterate distance guarantee.

Scalable Cox optimization

  • Exact Cox solvers exploit nested risk sets and sparsity but require full-data passes; alternatives include MCMC, multilevel gradients, and subject subsampling.The cited MCMC guarantee uses increasing accuracy and periodic full gradients [Achab et al. (2015)]; subsampling approaches approximate or recover the full-data estimator.Source: §2 · p. 2
  • Ordinary mini-batch partial likelihood generally optimizes a batch-size-dependent population objective, whereas this method uniformly controls approximation to the fixed full-data Cox gradient.The method compresses the outer family of normalizers while retaining that deterministic approximation.Source: §2 · p. 2

LogSumExp optimization

  • The cited comparison says the prior bound optimizes to O(T^-1/4), while the paper’s expected-smoothness analysis gives O((ρT)^-1 + ρ), yielding O(T^-1/2).Source: §2 · p. 2
  • SCENT also obtains an O(T^-1/2) averaged convex-objective rate, but its general guarantee depends on trajectory-accumulated variance quantities rather than fixed problem parameters.The authors say this does not directly provide an a priori bound with explicit constants.Source: §2 · p. 2
  • With strong convexity on the original parameter, the method adds an O(log T/T) last-iterate squared-distance rate without strong convexity in auxiliary directions.For Cox, grouping reduces variational-method state from m event shifts to K group shifts and quantifies error relative to the full Cox target.Source: §2 · p. 3

3 STOCHASTIC OPTIMIZATION OF LOGSUMEXP OBJECTIVES

The section develops a softplus surrogate for a convex LogSumExp objective using one auxiliary shift per distribution, then analyzes averaged convex convergence and last-iterate convergence under strong convexity. The stated guarantees depend on bounded smooth objectives and scheduled approximation and step-size choices.

3 STOCHASTIC OPTIMIZATION OF LOGSUMEXP OBJECTIVES

  • The setup considers K distributions with positive weights summing to one and a convex objective over a compact convex parameter domain.The loss and regularizer are twice continuously differentiable with bounded values, gradients, and Hessians.Source: §3 · p. 3
  • For approximation parameter ρ, the softplus transform hρ(u) = ρ^-1 log(1 + ρe^u) defines a joint surrogate with auxiliary shifts and a profiled counterpart.When ρκ̄ < 1, the surrogate is bounded between the original objective plus an approximation term and the original objective; for ρκ̄ ≤ 1/8, optimal shifts lie in [−B−1, B].Source: §3 · p. 3
  • Each iteration samples a distribution index and a draw from that group, then applies a weighted projected update to the parameter and sampled shift.Other shift coordinates remain unchanged, and the shifts are kept within the prescribed domain.Source: §3 · p. 3
  • The general convex result establishes an O(T^-1/2) objective rate for the original LogSumExp problem.The results are presented for iterate averaging in the convex regime and last-iterate convergence in the strongly convex regime.Source: §3 · pp. 3–4
  • The last-iterate theorem assumes the surrogate objective is strongly convex and imposes conditions on the approximation parameter and step size.The regularizer’s strong convexity transfers to the surrogate, and the stated scheduled choices use ρ_T = 8C log T/(μT) and η_T = ρ_T/(4C) when the preceding conditions hold.Source: §3 · p. 4
  • The corollary gives a last-iterate bound when the original objective’s minimizer is stationary, using scheduled choices from the strongly convex theorem.The displayed bound’s full expression is not stated in the supplied passage.Source: §3 · p. 4

4 COX REGRESSION WITH GROUPED RISK SETS

The section groups neighboring Cox failure risk sets and uses one softplus shift per group instead of one shift per event. It analyzes how grouping and softplus approximation affect optimization error and statistical behavior.

4.1 COX OBJECTIVE AND RISK-SET GROUPING

  • Replacing each group’s average event normalizer with Φ_k gives a grouped objective whose gradient error is quadratic in within-group risk-set change.Groups are consecutive failures, extended according to the user-chosen δ threshold; the construction uses one shift per group in the softplus objective.Source: §4.1 · p. 5

4.2 GRADIENT AND CURVATURE PRESERVATION

  • Theorem 4.1 uniformly controls gradient and Hessian differences under stated dispersion and grouping conditions, with softplus profiling contributing an O(ρκ) error.The grouping gradient error is second order in within-group risk-set change; when the full objective has curvature ν, the minimizer comparison is also controlled.Source: §4.2 · p. 6

4.3 END-TO-END COMPUTATIONAL RATE

  • The auxiliary state is O((T/log T)^1/5) until singleton groups cause finite-data saturation; the stated stochastic error and squared grouping bias are balanced in the end-to-end rate analysis.At saturation, grouping bias is zero and the bound crosses toward the ordinary eO(m/T) finite-problem rate.Source: §4.3 · p. 6

4.4 FIRST-ORDER STATISTICAL EFFICIENCY

  • For varying sample size, maximal grouping uses O_p(N^1/4ℓ_N)=o_p(N) shifts under the stated risk-set and ℓ_N conditions, while the grouped-softplus estimator can share the full estimator’s first-order limit under numerical-accuracy conditions.The statistical transfer result assumes distinct failures; finite-data compression also accommodates tied failures under the Breslow convention.Source: §4.1 · p. 4, §4.4 · p. 7

5 EXPERIMENTS

The experiments compare five methods under matched vector-work budgets and evaluate optimization, approximation errors, and test prediction. The proposed method has the lowest median terminal training gap on all five datasets, while test-loss winners vary by dataset.

5.1 BENCHMARK PROTOCOL

  • The benchmark uses stratified 60/20/20 splits across SUPPORT2, NWTCO, and three synthetic datasets, with Breslow handling of ties.Configurations are selected by validation loss and final runs use ten optimizer seeds under equal per-fit vector-work budgets.Source: §5.1 · pp. 7–8

5.2 OPTIMIZATION AND PREDICTION

  • The method achieves the lowest median terminal regularized full-Cox training gap on all five datasets at matched vector-work budgets.BigSurv is the closest baseline, with median terminal gaps 1.22–26.15 times ours.Source: §5.2 · p. 8, Figure 1 · p. 8
  • At δ = 0.05, grouping reduces auxiliary shifts by factors of 22.8–337.8 relative to one shift per training event.Table 1 reports datasets, training event counts, group counts, and per-fit vector budgets.Source: Table 1 · p. 7, §5.2 · p. 8
  • Grouping-error slopes are 1.37–1.87, while softplus error is nearly linear in ρ; discrete partition changes mean the O(δ^2) bound does not require a finite-grid slope of two.Figure 2 isolates grouping with exact group normalizers and softplus profiling at δ = 0.05.Source: §5.2 · p. 9, Figure 2 · p. 9
  • The method has the lowest mean test Cox loss on NWTCO, Correlated 100k, and Independent 100k; Minibatch Cox leads on SUPPORT2 and BigSurv on Correlated 1M.The comparisons use validation-selected checkpoints and unpenalized predictive Cox loss.Source: §5.1 · p. 8, §5.2 · p. 9

6 LIMITATIONS AND CONCLUSION

The conclusion presents risk-set grouping as a stochastic full-Cox approach with controlled approximation error and compact auxiliary state, but its theory and statistical transfer have stated scope conditions. The appendices provide proofs and experimental details, and the authors report AI assistance in manuscript development.

6 LIMITATIONS AND CONCLUSION

  • Risk-set grouping yields an eO(T^-4/5) computational rate under stated approximation and curvature conditions, and first-order equivalence to the full Cox estimator under additional statistical conditions and parameter schedules.The auxiliary state is sublinear under the stated risk-set size condition; experiments do not establish an end-to-end runtime advantage over optimized cumulative-sum solvers.Source: §6 · p. 9
  • The analysis assumes a convex linear predictor and boundedness and curvature conditions; adverse risk-weight heterogeneity can make Lq and κ large.Tied-time asymptotics, time-varying covariates, and nonlinear predictors remain outside scope; the main statistical transfer result is stated for distinct failures.Source: §6 · p. 9

REPRODUCIBILITY STATEMENT

  • The appendix provides complete proofs and details of the experimental setup, implementation, tuning, and evaluation procedures.The paper also states that the experimental source code is publicly available.Source: REPRODUCIBILITY STATEMENT · p. 10

AI USE STATEMENT

  • Generative AI assisted with language editing, literature searches, and proof development, and the authors say AI-assisted mathematical content was independently verified.The authors take full responsibility for the manuscript.Source: AI USE STATEMENT · p. 10

A GENERIC GROUPED LOGSUMEXP OPTIMIZATION

A.1 SOFTPLUS IDENTITIES AND PROFILE REGULARITY

  • For each group, the softplus profile is bounded relative to its LogSumExp term using a normalized second-moment condition, with corresponding bounds on the optimal shift.The proof applies the softplus approximation and optimal-shift bounds of Gladin et al. (2025).Source: §A.1 · p. 12

A.2 WEIGHTED STOCHASTIC GRADIENT METHOD

The appendix develops smoothness and curvature tools for stochastic optimization of the joint softplus surrogate and applies them to grouped Cox objectives, bounding grouping and softplus approximation errors and comparing the profiled surrogate with the exact Cox objective. The supplied excerpt on the weighted stochastic gradient method adds only that nothing remains to prove when h = 0.

A.2 WEIGHTED STOCHASTIC GRADIENT METHOD

  • The generic stochastic-gradient analysis uses convex sample functions that are C/ρ-smooth in the weighted norm, with its stated lemma requiring ρ\barκ ≤ 1/8.The appendix derives the sample smoothness and variance bounds used in the stochastic-gradient analysis.Source: §A.2 · pp. 12–13

A.3 CURVATURE OF THE JOINT SURROGATE

  • Strong convexity of the profiled objective supplies curvature for fast convergence even when the auxiliary shift variables are not strongly convex.The Cox application can obtain this profile curvature from the partial likelihood, allowing λ = 0.Source: §A.3 · pp. 14–15

A.4 GENERIC CONVERGENCE BOUNDS

  • The generic convergence analysis transfers surrogate bounds to the profiled objective, with the approximation error controlled when ρT\barκ ≤ 1/8.For the strongly convex case, the stated schedule yields the result after dropping shift coordinates; the squared-error guarantee uses stationarity.Source: §A.4 · pp. 15–16

B COX REGRESSION

  • The Cox section specializes the generic recursion using a uniform comparison theorem and retains constants in its resulting bound.The section states that it proves Theorem 4.1 and derives Corollary 4.2.Source: B COX REGRESSION · p. 16

B.1 NOTATION AND DATA CONSTANTS

  • The Cox setup uses bounded covariates and linear predictors, with constants κ and Lq independent of the grouping.For local statistical comparisons, the suprema over the parameter set may instead be taken over a fixed neighborhood of the target.Source: §B.1 · p. 16

B.2 GROUPING BOUNDS

  • Grouping neighboring failures controls differences in log normalizers, gradients, and Hessians by bounds proportional to rδ.The stated bounds are 2Lq rδ for log normalizers, 2X Lq rδ for gradients, and 5X^2 Lq rδ for Hessians.Source: §B.2 · p. 17

B.3 SOFTPLUS GRADIENT AND CURVATURE

  • The group-moment envelope is at most 9κ/8, enabling a uniform softplus-gradient and curvature comparison when the stated condition on ρ holds.The proof gives softplus gradient error below 4Xρκ and Hessian loss below 20X^2ρκ before adding grouping errors.Source: §B.2 · p. 18, §B.3 · p. 19

B.4 CURVATURE TRANSFER AND DETERMINISTIC MINIMIZER COMPARISON

  • The Cox comparison establishes curvature transfer and a deterministic minimizer-distance bound for the profiled surrogate under the stated assumptions.The distance bound does not require the additional curvature budget; when the ridge coefficient is zero, the curvature assumption concerns the exact partial likelihood rather than a penalty.Source: §B.4 · p. 20
  • When h = 0, the proof requires no further argument.Source: §B.4 · p. 20

B.5 END-TO-END COMPUTATIONAL RATE

The end-to-end Cox guarantee combines stochastic optimization with grouping and softplus approximation under stated curvature and risk-set conditions. Its rate depends on iteration horizon, while finite datasets eventually leave the grouping-limited regime.

B.5 END-TO-END COMPUTATIONAL RATE

  • The rate analysis uses a profiled-surrogate curvature bound; its curvature budget yields at least ν/2 curvature, while intrinsic risk-weight factors may remain large.The stated curvature lower bound gives a T-independent restricted-secant constant, but Lq and κ may be large under heterogeneous risk weights.Source: §B.5 · pp. 20–21
  • The algorithm samples a group, then an event uniformly within it, then a subject uniformly from that event’s risk set.The grouping tolerance is chosen for a declared horizon, and the resulting groups remain fixed during the run.Source: §B.5 · p. 20
  • Under the stated approximation and curvature conditions, the computational rate is eO(T^-4/5).Corollary B.9 assumes every event risk set contains at least a fixed fraction c of the subjects and constants uniform in T.Source: §B.5 · p. 22
  • For a fixed finite dataset, singleton groups eventually eliminate grouping bias, and the bound crosses toward the ordinary eO(m/T) finite-problem rate.The transition occurs as δT decreases until each event has its own group.Source: §B.5 · p. 22
  • Local uniqueness can translate objective convergence into parameter convergence, but does not ensure the uniform profile curvature required along the optimization path.The theorem therefore imposes condition (57) on B.Source: §B.5 · p. 22

C STATISTICAL CONSEQUENCE

The statistical consequence transfers the full Cox estimator’s assumed limit to the compressed estimator under stated model, risk-set, and approximation conditions. The accompanying experimental protocol uses fixed dataset splits, selected configurations, and implementation-specific accounting, with test results described as exploratory rather than independent confirmation.

C STATISTICAL CONSEQUENCE

  • If inf0≤t≤tmax Pr{Y(t)=1}>0, every risk set through this fixed finite horizon contains a fixed positive fraction of the sample with probability tending to one; listed conditions suffice for the full estimator’s classical limit.Source: C STATISTICAL CONSEQUENCE · p. 23
  • Under the statistical conditions, the compressed estimator inherits the full Cox estimator’s assumed first-order limit by Slutsky’s theorem.The argument requires approximation errors to vanish and the segment between estimators to lie in the stated convex neighborhood.Source: C STATISTICAL CONSEQUENCE · p. 23
  • The computational schedule and statistical schedule address different limits: T controls numerical approximation, while N governs statistical approximation.An unconditional mean-square conclusion additionally requires integrable bounds, such as deterministic computational assumptions and a uniformly integrable prefactor.Source: C STATISTICAL CONSEQUENCE · p. 23

D.1 DATA PREPARATION

  • The real-data preparation retains 8,873 of 9,105 SUPPORT2 subjects and uses time to death with 14 covariates; NWTCO uses relapse outcomes and specified histology, stage, study, and age predictors.The datasets use event-stratified splits and training-fitted preprocessing; ties use Breslow handling.Source: §D.1 · pp. 23–24

D.2 NATIVE OBJECTIVES AND OPTIMIZER CONVENTIONS

  • The proposed method samples groups and uses unbiased stochastic gradients, while the compared methods use differing sampled risk sets and native objectives.All methods are evaluated with the same regularized full-Cox objective; the proposed method uses fixed ρ=10^-4 and δ=0.05, with batch size and initial step tuned.Source: §D.2 · p. 24

D.3 TUNING MENUS AND SELECTION

  • Each method’s selected configuration minimizes the mean across two tuning seeds of its minimum saved validation Cox loss within budget.Twelve candidate configurations are considered per method, with selected settings then trained using ten optimizer seeds.Source: §D.3 · p. 25

D.4 VECTOR-WORK ACCOUNTING AND TIMING

  • Vector-work units count specified passes and operations rather than hardware FLOPs, while elapsed time includes setup and optimizer execution but excludes shared preparation and diagnostics.Runtime comparisons characterize these implementations and were recorded in separate execution sessions.Source: §D.4 · pp. 25–26

D.5 METRICS, CHECKPOINTS, AND AGGREGATION

This section defines how full-Cox training accuracy, predictive metrics, and target attainment are measured and aggregated. Test results are descriptive, not an independent confirmation, because the test cohorts were evaluated during earlier exploratory work.

D.5 METRICS, CHECKPOINTS, AND AGGREGATION

  • The training gap is L(β) − L(βref), measured against a full-risk-set reference whose regularized gradient norm is below 10^-10.The reference minimizes the event-normalized Breslow objective with λ = 10^-3; its computation is separate from the compared trajectories.Source: §D.5 · p. 26
  • Validation Cox loss selects the saved checkpoint; test Cox loss is unpenalized, and Harrell C-index is reporting-only.Checkpoints include the first completed update crossing each shared work target, plus initial and terminal states.Source: §D.5 · p. 26
  • A target counts only when the gap stays below its threshold at every later saved checkpoint through the cap; costs use observed checkpoints without interpolation.Thresholds are 10^-3, 10^-5, 10^-7, and 10^-9; nonattainment is reported separately from median cost among attainers.Source: §D.5 · p. 26
  • Figures aggregate ten optimizer-seed traces using medians and percentile bands, while variability reflects optimizer randomness on one split, not independent cohorts.Aggregation uses linear interpolation on common observed support, without extrapolation or best-so-far smoothing.Source: §D.5 · p. 26
  • Because the test cohorts were used during earlier exploratory work, predictive results are descriptive rather than a new independent confirmation.The 50 validation-selected models per dataset were fixed before their test evaluation.Source: §D.5 · p. 26

E.1 OPTIMIZATION ACCURACY AND TARGET ATTAINMENT

The method has the lowest median terminal full-Cox training gap on all five datasets at matched vector-work caps. Target attainment and predictive-metric leaders vary by threshold, dataset, and metric.

E.1 OPTIMIZATION ACCURACY AND TARGET ATTAINMENT

  • Our method has the lowest median terminal full-Cox training gap on all five datasets at matched vector-work caps.BigSurv is the closest baseline, with median gaps 3.74, 2.38, 16.14, 26.15, and 1.22 times ours in table order.Source: §E.1 · p. 26, Table 4 · p. 27
  • At 10^-5, only our method attains the target within the final caps: seven of ten runs on Correlated 100k and all ten on Correlated 1M.At 10^-3, our method and BigSurv attain the target in all ten runs on every dataset; no method attains 10^-7 or 10^-9.Source: §E.1 · p. 26, Table 5 · p. 27

E.2 PREDICTIVE PERFORMANCE

  • Mean test Cox-loss leaders are our method on NWTCO, Correlated 100k, and Independent 100k; Minibatch Cox on SUPPORT2; and BigSurv on Correlated 1M.Metrics are validation-selected and reported as mean ± sample SD over ten optimizer seeds on one fixed split.Source: Table 6 · p. 28, §E.2 · p. 28
  • Mean test C-index leaders are our method on NWTCO and Correlated 100k, BigSurv on SUPPORT2 and Independent 100k, and Cox-CC on Correlated 1M.Higher C-index is better.Source: Table 6 · p. 28, §E.2 · p. 28

F APPROXIMATION DIAGNOSTICS

The approximation diagnostics compare grouped and softplus objectives at fixed coefficient vectors, reporting fitted error trends and checking stated compression conditions. Complete trajectory figures show training and validation metrics against work and time.

F APPROXIMATION DIAGNOSTICS

  • At fixed coefficient vectors, diagnostics compare exact-normalizer grouping with softplus objectives whose shifts are profiled over RK.The coefficient-gradient differences are evaluated over grids of reference-scaled vectors, grouping values δ, and softplus values ρ.Source: F APPROXIMATION DIAGNOSTICS · p. 29
  • 1.37–1.87 (grouping) and 0.997–1.000 (softplus) are the fitted log–log error slopes; softplus error is nearly linear in ρ.These fits use every grid point at βref.Source: F APPROXIMATION DIAGNOSTICS · p. 29, Table 8 · p. 29
  • Grouping’s finite-grid slopes reflect changing partitions; the O(δ^2) bound requires stated compression conditions and these diagnostics do not estimate optimizer convergence.The applicable pointwise conditions held in 80/100 grouping and 96/120 softplus comparisons, but were not certified uniformly over the radius-10 ball.Source: F APPROXIMATION DIAGNOSTICS · p. 29

G COMPLETE TRAJECTORY FIGURES

  • Trajectory figures plot full-Cox training gap, validation Cox loss, and validation C-index against vector work and setup-plus-optimizer time; compare matched panels.The C-index is higher-is-better, while validation Cox loss is lower-is-better.Source: Figure 3 · p. 30, Figure 4 · p. 31, Figure 5 · p. 32
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