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Communication-Efficient Accurate Statistical Estimation

Jianqing Fan, Yongyi Guo, Kaizheng Wang

arXiv:1906.04870v2stat.MLcs.LGmath.OCstat.CO

TL;DR

Distributed statistical inference is constrained by limited bandwidth, privacy concerns, and slow convergence that fails to exploit similarity among loss functions. The paper develops CEASE distributed estimators with iterative optimization, which achieve favorable convergence in large-sample settings and linear convergence without good initialization in moderate or small samples.

  • Problem

    Distributed statistical inference faces limited bandwidth, while direct data sharing raises privacy and ownership concerns.

  • Method

    The paper develops two CEASE distributed estimators using iterative distributed optimization problems.

  • Results

    The estimators have convergence rates better than or comparable to existing methods when local sample size is sufficiently large, and converge linearly without good initialization for moderate or small n.

  • Takeaways & Limitations

    CEASE provides distributed estimators designed for large-sample regimes while retaining linear convergence guarantees beyond those regimes.

  • Takeaways & Limitations

    Existing one-shot methods are communication-efficient but can converge slowly and fail to fully exploit similarity among node-level loss functions.

Abstract

from arXiv · show

When the data are stored in a distributed manner, direct application of traditional statistical inference procedures is often prohibitive due to communication cost and privacy concerns. This paper develops and investigates two Communication-Efficient Accurate Statistical Estimators (CEASE), implemented through iterative algorithms for distributed optimization. In each iteration, node machines carry out computation in parallel and communicate with the central processor, which then broadcasts aggregated information to node machines for new updates. The algorithms adapt to the similarity among loss functions on node machines, and converge rapidly when each node machine has large enough sample size. Moreover, they do not require good initialization and enjoy linear converge guarantees under general conditions. The contraction rate of optimization errors is presented explicitly, with dependence on the local sample size unveiled. In addition, the improved statistical accuracy per iteration is derived. By regarding the proposed method as a multi-step statistical estimator, we show that statistical efficiency can be achieved in finite steps in typical statistical applications. In addition, we give the conditions under which the one-step CEASE estimator is statistically efficient. Extensive numerical experiments on both synthetic and real data validate the theoretical results and demonstrate the superior performance of our algorithms.

1 Introduction

Distributed statistical inference is constrained by data scale, communication cost, privacy concerns, and the limitations of existing one-shot and Newton-type procedures. The paper develops CEASE, multi-round estimators designed to achieve accurate inference under broader conditions.

  • Motivation: Distributed inference is motivated by datasets too large for one machine, costly communication, privacy concerns, and loss of data ownership.The setting partitions data across node machines connected to a central processor.
  • Limitations of existing methods: One-shot methods are communication-efficient but require large local samples, rely on asymptotic expansions, and can fail when practical conditions are violated.Existing methods may therefore be unreliable outside their large-local-sample regime.
  • Multi-round inference: Multi-round procedures can attain optimal statistical precision after a few communication rounds under broader settings than one-shot procedures.They alternate local computation with global aggregation, while existing approaches may not exploit similarity among local losses.
  • Limitations of existing methods: Existing Newton-type theories often require good initialization, self-concordance, or sufficiently similar local loss functions.These requirements limit methods that improve an already consistent but inefficient estimator.
  • Contribution: The paper develops two Communication-Efficient Accurate Statistical Estimators (CEASE) based on multi-round distributed algorithms.The methods extend DANE to regularized empirical risk minimization and use inexact distributed updates.
  • Results: CEASE converges linearly without good initialization for moderate or small local sample sizes, while matching or improving existing methods when local samples are sufficiently large.The paper also provides convergence guarantees for dissimilar local losses and nonsmooth regularization, with numerical verification on synthetic and real data.

2 The CEASE algorithm

The CEASE algorithms address distributed regularized risk minimization by combining adaptive gradient-enhanced local losses with proximal-point updates. They parallelize node computation, adapt to local-loss similarity, and improve stability and convergence when local sample sizes are moderate or large.

  • 2.1 Problem setup: The method targets distributed regularized empirical risk minimization, where each machine accesses only its local loss and shared penalty while the central processor coordinates computation.The data are partitioned across m machines with n = N/m samples per machine.
  • 2.2 Adaptive gradient enhancements and distributed algorithms in large-sample regimes: The gradient-enhanced loss replaces the global gradient at the current point with local curvature information while retaining the global gradient, producing a fixed point at the global empirical minimizer.This construction is formed by linearizing the global loss and replacing the remainder with a node-specific version.
  • 2.2 Adaptive gradient enhancements and distributed algorithms in large-sample regimes: CEASE improves on one-machine gradient-enhanced procedures by using all machines and averaging their local solutions to reduce estimator variance and enhance convergence.The algorithms extend the DANE-style distributed approach to regularized empirical risk minimization.
  • 2.2 Adaptive gradient enhancements and distributed algorithms in large-sample regimes: Each machine optimizes its gradient-enhanced loss in parallel, while the central processor aggregates the results and broadcasts updates.This parallel design uses the computing power of all machines rather than leaving most machines idle during optimization.
  • 2.3 The CEASE Algorithm in general regimes: Earlier gradient-enhanced algorithms can be unstable for moderate local sample sizes, whereas CEASE adapts quadratic regularization through proximal updates instead of using a fixed penalty.A fixed quadratic regularizer may converge rapidly but can target the wrong solution; CEASE adjusts regularization according to the current solution.
  • 2.3 The CEASE Algorithm in general regimes: CEASE uses proximal-point iterations as a backbone, with one-step distributed updates that simplify nested optimization loops and support a multi-step statistical interpretation.The resulting algorithms are designed to refine an initial estimator even when it is inaccurate.

3 Deterministic analysis

The deterministic analysis establishes contraction guarantees for CEASE under local-loss homogeneity and strong convexity, with regularization providing convergence when similarity is insufficient. For smooth objectives, averaging can substantially accelerate contraction at low computational cost.

  • Large local sample sizes improve homogeneity, making contraction factors smaller and enabling faster communication-efficient convergence.The analysis links larger n to smaller δ and contraction factors that can approach zero.
  • The deterministic results cover smooth losses with potentially nonsmooth convex penalties, including the ℓ1 norm.This extends the analyzed setting beyond methods restricted to smooth problems.
  • CEASE algorithms converge linearly when local losses are sufficiently homogeneous, with contraction factors determined by structural parameters and regularization.The homogeneity parameter δ measures local-loss similarity, and suitable α ensures contraction even when unregularized methods may fail.
  • Regularization α ≍δ^2/ρ provides a default trade-off that preserves fast contraction in homogeneous settings while improving robustness in general settings.The recommended choice combines the general convergence guarantee with near-unregularized rates when δ/ρ is small.
  • For smooth loss-plus-penalty objectives, CEASE with averaging is roughly as effective as two optimization iterations while using only one optimization round.The averaging step can reduce error as much as an optimization step and requires much less time.

4 Statistical analysis

The statistical analysis treats CEASE as a multi-step estimator and studies how iterations and initialization affect optimization and statistical error. Under generalized linear-model conditions, finite iterations can make optimization error negligible relative to statistical error and achieve centralized statistical efficiency.

  • With smooth regularization, averaging improves the contraction rate to approximately κ^2p/n.Without averaging, the contraction factor is approximately κp/n up to logarithmic terms.
  • Each CEASE iteration reduces optimization error by an amount controlled by local sample size, eventually making it negligible relative to statistical error.Under typical scaling of total sample size, the distributed multi-step estimator performs as well as the centralized global minimizer.
  • Both algorithms achieve linear convergence under suitable regularization and local sample size, with rates explicitly depending on n and the condition number κ.The theory permits inaccurate initialization such as θ0 = 0 and gives explicit contraction behavior even when p and κ diverge.
  • A one-step CEASE estimator is statistically efficient when n^3 ≫ N(κ^2p log N)(p + κ^2 log p), while t-step estimators require weaker local-sample-size conditions.The t-step condition is n^(t+2) ≫ N(κ^2p log N)^t(p + κ^2 log p).
  • Algorithms 3 and 4 reach statistical efficiency in O(log∥θ0−bθ∥2 + log(N/p) log(1/η)) iterations.Here η is the contraction-scale quantity determined by κ, p, log N, and n.

5 Numerical experiments

Synthetic and real-data experiments examine CEASE across local-sample-size, initialization, penalty, and algorithm-comparison settings. They report rapid convergence across scenarios, stronger performance with averaging, and failures of unregularized alternatives in difficult regimes.

  • With proper regularization, CEASE is the only tested approach that converges rapidly across all synthetic scenarios.ADMM and accelerated gradient descent are reliable but slow, whereas unregularized CEASE and GIANT can fail with small n or poor initialization.
  • Averaged CEASE outperforms the non-averaged version; at (n, m) = (1000, 10), averaged CEASE with α = 0 converges while the other version does not.The experiments attribute the better performance to the averaging step.
  • CEASE algorithms converge rapidly, usually within several steps, across the tested scenarios.The experiments report this pattern for synthetic settings including varying local sample sizes and initialization quality.
  • The experiments indicate that CEASE benefits from statistical similarity among local losses and from averaging with smooth loss functions.They also report convergence guarantees under small local sample sizes and uninformative initialization.
  • Fashion-MNIST experiments on distributed logistic regression support the paper’s theoretical findings.The real-data evaluation uses classification error on a held-out test set and compares CEASE with ADMM, GIANT, and AGD.

6 Discussions

The discussion highlights CEASE’s theoretical guarantees and empirical performance while identifying extensions to broader distributed and optimization settings.

  • Two CEASE distributed estimators combine theoretical guarantees with superior performance on real data.
  • The paper assumes exact computation, leaving finer analysis of inexact updates for future work.
  • Future directions include decentralized and asynchronous settings, distributed confidence regions and hypothesis tests, and non-convex problems such as mixture models and deep learning.
  • On Fashion-MNIST, Figure 2 compares CEASE variants and shows that GIANT and CEASE-single(0) do not converge to the optimal solution.
  • CEASE’s gradient-enhanced loss-function idea is proposed as a basis for further research.

Appendices

The appendices provide supplementary variants, regression results, logistic-regression iterates, regularized logistic-regression experiments, proofs, and technical lemmas.

  • The appendices introduce a variant of Algorithm 4 and present deferred distributed linear-regression results.
  • They also include distributed logistic-regression iterates and numerical results for distributed ℓ1-regularized logistic regression.
  • The remaining appendices present proofs of the main results and technical lemmas used in those proofs.

A A variant of Algorithm 4

Algorithm 5 stabilizes distributed updates through smaller step-sizes and proximal regularization, improving convergence when local sample sizes are small.

  • Choosing αt ∈(0, 1) stabilizes iterates, while Algorithm 2 is the special case αt = 1.
  • When (n, m) = (250, 40), Algorithm 2 fails to converge, motivating step-size stabilization in Algorithm 5.
  • Algorithm 5 with αt = 1/4 performs similarly to CEASE, αt = 1/2 converges faster, and αt = 1/8 converges more slowly.
  • The analysis uses constant step-sizes, while decaying schemes such as αt ≍t−β remain an open direction.
  • The method assumes sufficiently large local samples for reliable optimization; when n < p, local loss functions may lack unique minima.
  • The quadratic proximity term in CEASE makes each objective strongly convex and ensures unique local solutions by shrinking toward the previous iterate.

B Distributed linear regression

For distributed linear regression, CEASE achieves linear convergence under broad local-sample regimes, with regularization adapting to p/n and avoiding condition-number dependence in the contraction factor.

  • Algorithm 4 averages local estimators, a step described as reducing variance and accelerating convergence.
  • In the large-sample regime, CEASE attains contraction factor O(p/n) without condition-number dependence, while general regimes retain linear convergence.
  • Algorithm 4 guarantees linear convergence regardless of the relationship between n and p when proper regularization is used.
  • Regularization is unnecessary when n/p is large, although choosing α ≍p/n does not substantially hurt performance.
  • With α ≍p/n, the contraction factor is below one even when local sample sizes are not large relative to dimension.
  • The condition number affects iteration complexity only logarithmically and does not enter Theorem B.1’s contraction factor.

D Numerical results on distributed ℓ1-regularized logistic regression

The distributed ℓ1-regularized logistic-regression experiments show that properly regularized CEASE methods generally perform well, while unregularized CEASE can fail with small local samples and uninformative initialization. In this nonsmooth setting, averaging offers no clear advantage over the single-machine-update version, whereas ADMM slows near the minimizer.

  • The experiment fixes N = 5000 and p = 1001 and compares large, moderate, and small local-sample regimes with good or zero initialization.
  • Properly regularized CEASE algorithms work well in the distributed ℓ1-regularized logistic-regression experiments.
  • Without regularization, CEASE fails to converge when local sample size is small and initialization is uninformative.
  • Averaging does not appear advantageous over the single version for this nonsmooth problem.
  • ADMM converges quickly near the minimizer but then proceeds slowly.

E.4 Proof of Theorem 3.2

The proof of Theorem 3.2 combines contraction and averaging lemmas to establish the behavior of the distributed updates under local-loss similarity and regularity assumptions.

  • Lemma E.2 bounds each local update's distance from the global minimizer by δ/ρ0 times the current distance.
  • Lemma E.3 supplies an averaging result under Assumptions 3.1, 3.2, and 3.3.
  • The proof defines local and global regularized losses and represents the local update through a gradient-corrected minimization problem.
  • The argument uses the zero average of the local gradient discrepancies and Hessian bounds to combine the two lemmas.

E.5 Proof of Theorem 4.1

The proof of Theorem 4.1 derives stochastic contraction guarantees by controlling local Hessian differences and regularized covariance deviations under sample-size and initialization conditions.

  • The stochastic proof controls local Hessian differences in generalized linear models, yielding a contraction rate.
  • Under the stated initialization and contraction conditions, both Algorithms 3 and 4 satisfy the theorem's contraction result.
  • The analysis assumes the initial iterate lies within B(bθ, R/2) and requires [bδ/(ρ0 + α)]2 < ρ/(ρ + 2α).
  • The proof uses Neumann expansions to control inverse-matrix approximations and obtain the contraction inequalities.
  • High-probability concentration arguments bound normalized covariance deviations when local sample size is sufficiently large relative to dimension or effective trace.

F Technical lemmas

The technical lemmas establish foundational properties of strong convexity, proximal mappings, matrix expansions, and concentration for sub-Gaussian means and covariances.

  • Strong convexity is characterized through equivalent function, subgradient, and quadratic-interpolation inequalities.
  • Strong convexity converts bounded subgradient differences into bounds on the distance between points.
  • The proximal mapping contracts toward a minimizer by α/(α + ρ) within a strongly convex neighborhood.
  • A Neumann expansion bounds the error from approximating an inverse matrix by I + M.
  • For sub-Gaussian vectors, sample means and covariance matrices concentrate with exponentially small failure probabilities under sample-size conditions.
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