Communication-efficient sparse regression: a one-shot approach

Jason D. Lee; Yuekai Sun; Qiang Liu; Jonathan E. Taylor

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Communication-efficient sparse regression: a one-shot approach

Jason D. Lee, Yuekai Sun, Qiang Liu, Jonathan E. Taylor

arXiv:1503.04337v3Mar 14, 2015stat.MLcs.LG

TL;DR

High-dimensional distributed sparse regression must control communication while preserving centralized statistical performance. The paper debiases local lasso estimators before averaging them, and shows that the resulting one-shot estimator matches centralized lasso rates when data are not split across too many machines. The approach extends to ℓ1-regularized M-estimators, including generalized linear models.

Problem
High-dimensional distributed sparse regression needs a communication-efficient estimator that preserves centralized lasso performance.
Method
The method debiases local lasso or ℓ1-regularized M-estimators and averages the resulting estimators in one shot.
Results
The averaged estimator matches centralized lasso convergence rates, including after thresholding, under the stated machine-count conditions.
Takeaways & Limitations
Debiasing makes one-shot averaging viable for communication-efficient distributed sparse regression and extends to other ℓ1-regularized M-estimators.
Takeaways & Limitations
The averaged debiased lasso is usually dense, reducing coefficient interpretability and increasing ℓ2 and ℓ1 estimation error unless thresholded.

Abstract

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We devise a one-shot approach to distributed sparse regression in the high-dimensional setting. The key idea is to average "debiased" or "desparsified" lasso estimators. We show the approach converges at the same rate as the lasso as long as the dataset is not split across too many machines. We also extend the approach to generalized linear models.

1. Introduction.

Distributed learning addresses datasets too large for one machine, where communication is the main bottleneck. This paper averages local debiased regularized estimators to match centralized rates in high-dimensional settings.

1. Introduction.: Averaging reduces variance but not bias, so prior analyses show performance can degrade when many machines split the data.Earlier work found averaged ERM can be suboptimal in high-dimensional asymptotics.
1. Introduction.: The approach averages local debiased regularized M-estimators and converges at the same rate as the centralized regularized M-estimator.The focus is sparse linear regression, where p ≳ n makes regularization essential.

2. Background on the lasso and the debiased lasso.

The lasso is effective for sparse high-dimensional regression but remains biased because of regularization. Debiasing adds a correction based on an approximate covariance inverse, producing an estimator whose bias can be smaller than its variance under suitable conditions.

2. Background on the lasso and the debiased lasso.: The lasso is nearly as accurate as an oracle estimator under suitable design assumptions, with error governed by sparsity and dimension.
2. Background on the lasso and the debiased lasso.: Naively averaging local lasso estimators does not improve their order because averaging reduces variance but leaves bias unchanged.
2. Background on the lasso and the debiased lasso.: The debiased lasso corrects shrinkage bias using a term based on the lasso subgradient and an approximate inverse covariance matrix.When the covariance estimate is nonsingular and the inverse is used, the correction yields the ordinary least-squares estimator.
2. Background on the lasso and the debiased lasso.: Under restricted-eigenvalue and related sparsity, design, and noise conditions, the debiased estimator is consistent and its bias is higher order than its variance.The supplied bound states this comparison holds when n ≳ s^2 log p.

3. Averaging debiased lassos.

The distributed estimator averages locally debiased lasso estimates across machines. Its leading variance term matches the centralized rate, while the method requires sufficient local sample size and controlled machine count.

3. Averaging debiased lassos.: The averaged debiased lasso uses evenly partitioned local datasets and combines the machine-specific debiased estimators.
3. Averaging debiased lassos.: The leading error term matches the convergence rate of the centralized estimator when the local sample size is sufficiently large.The analysis also establishes high-probability control under independent subgaussian predictors.
3. Averaging debiased lassos.: The theorem requires conditions on the number of machines, local sample size, regularization, and debiasing parameters, with the result holding with high probability.

4. By Lemma 2.2,

Simulations support centralized-lasso-like performance for the averaged debiased estimator, while revealing degradation with too many machines and dense-coefficient effects that thresholding remedies. The thresholded estimator matches centralized lasso rates and retains one-shot communication efficiency.

4. By Lemma 2.2,: In ℓ∞ error, the averaged debiased lasso is comparable to the centralized lasso, whereas naive averaged lasso performs much worse.
4. By Lemma 2.2,: When the number of machines exceeds a threshold, a bias term growing linearly in the machine count dominates and averaged-estimator performance degrades.
4. By Lemma 2.2,: The thresholded averaged estimator matches the centralized lasso’s convergence rates in ℓ1, ℓ2, and ℓ∞ norms and can be computed by one-shot averaging.
4. By Lemma 2.2,: Thresholding reduces ℓ2 estimation error by orders of magnitude because the unthresholded averaged estimator is usually dense.After thresholding, performance is comparable to or better than the centralized lasso.

4. A distributed approach to debiasing.

The distributed estimator debiases local lasso fits before averaging, using nodewise regression to construct approximate inverse-covariance corrections. It can be formed with two communication rounds while local debiasing remains data-local.

4. A distributed approach to debiasing.: The estimator requires two communication rounds because debiasing uses only each machine’s locally stored data after shared residual averages are formed.Machines send local residual-related quantities to a central server, receive their averages, and then debias assigned coefficient blocks locally.
4. A distributed approach to debiasing.: Each machine forms local lasso estimates and nodewise-regression corrections, then participates in averaging the debiased estimates.The correction matrix is constructed as an approximate component-wise inverse of the empirical covariance matrix.
4. A distributed approach to debiasing.: The averaged debiased estimator matches the centralized lasso’s convergence rate under assumptions on the design, sparsity, and related quantities.The theorem’s bound includes a variance term and a remainder involving smax log p / n.

3. By the triangle inequality,

The analysis decomposes the debiased estimator’s error into stochastic and residual terms. A thresholded version achieves centralized-lasso rates, with the variance term dominant when the machine count is sufficiently limited.

3. By the triangle inequality,: Hard-thresholding the averaged debiased estimator yields convergence to β∗ at the same rates as the centralized lasso.The threshold is chosen on the order of the estimator’s sup-norm error.
3. By the triangle inequality,: For m ≲ n / (smax^2 log p), the variance term dominates the thresholded estimator’s error.This condition identifies the regime in which splitting the data across machines preserves the intended rate behavior.
3. By the triangle inequality,: Amortizing debiasing across machines provides computational gains over forming the fully debiased estimator centrally.The distributed estimator retains the advantages of the averaged debiased estimator while distributing the correction cost.

5. Averaging debiased ℓ1 regularized M-estimators.

The debiasing-and-averaging construction extends from linear regression to ℓ1-regularized M-estimators. Under the stated assumptions, both the averaged estimator and its thresholded version attain centralized-M-estimator convergence rates.

5. Averaging debiased ℓ1 regularized M-estimators.: The distributed debiasing approach extends to ℓ1-regularized M-estimators with convex loss functions.Each machine computes a local regularized M-estimator and constructs the correction using a weighted design matrix.
5. Averaging debiased ℓ1 regularized M-estimators.: The averaged estimator achieves the convergence rate of the centralized ℓ1-regularized M-estimator under assumptions (B1)–(B7).Assumption (B5) is stated for exposition but is implied by the other assumptions.
5. Averaging debiased ℓ1 regularized M-estimators.: Hard-thresholding the averaged M-estimator at an appropriate threshold gives convergence to β∗ at centralized-M-estimator rates.The threshold is selected on the order of the corresponding sup-norm error.
5. Averaging debiased ℓ1 regularized M-estimators.: The paper concludes that debiased averaging extends beyond sparse linear regression to other ℓ1-regularized M-estimators.This extension is presented as part of the paper’s communication-efficient distributed-learning approach.

6. Summary and discussion.

The approach targets communication-efficient sparse regression, where communication is the main distributed-learning bottleneck. Its communication cost is near-optimal in the stated high-dimensional regime.

6. Summary and discussion.: The paper relates its communication result to lower bounds for sparse linear-regression risk established in concurrent and prior work.The cited lower-bound results concern the communication required to achieve a target risk.
6. Summary and discussion.: The approach communicates Õ(mp) bits to achieve risk s log p / N, and is communication-optimal when p ≲ n.The optimality claim is stated for the regime where the ambient dimension is no larger than the per-machine sample size.

APPENDIX A: PROOFS OF LEMMAS

The appendix proves concentration and consistency ingredients for the debiased estimator using subgaussian and subexponential bounds, union bounds, and substitutions into the linear model.

The proofs bound empirical covariance and related quantities by treating products of transformed predictors as independent subexponential variables and applying concentration inequalities.The argument then uses union bounds over coordinates or matrix entries to obtain uniform control.
The proofs substitute the linear model into the debiasing expression, separate terms, and bound them using lasso error control and concentration results.The derivation takes norms and combines the resulting bounds to establish the stated conclusions.
For subgaussian predictors, the appendix derives tail bounds for coordinatewise and matrix quantities, with exponents involving n and predictor-tail parameters.These bounds are obtained through propositions on subgaussian and subexponential variables followed by union bounds.

APPENDIX B: A SHARPER CONSISTENCY RESULT

This appendix sharpens consistency for the averaged debiased estimator by relaxing the norm requirement and hard-thresholding its coefficients. Under the stated conditions, the resulting machine-splitting range can recover centralized lasso rates.

Hard-thresholding ˜β at a threshold based on an estimated sparsity level yields the sharper consistency result under Theorem B.1.The estimator is thresholded at t = |˜β|(ˆs0) for some ˆs0 ∼ s0.
The sharper result relies on controlling the debiased estimator in an (∞, l) norm that interpolates between ℓ∞ and rescaled ℓ2 norms.The bias bound for the debiased lasso is supplied through the cited Javanmard and Montanari result and accompanying lemmas.
When m ≲ N/(s0 log p), the variance term dominates and the convergence rates simplify to those of the centralized lasso estimator.This regime is identified after hard-thresholding the averaged estimator under Theorem B.1.
By forgoing ℓ∞-norm consistency, the averaged estimator can use the sharper sample-complexity condition m ≲ s0 log p.The appendix states that m = 1 recovers the centralized lasso estimator’s sample complexity.
The thresholding theorem requires an estimate of s0, which can be obtained from the empirical sparsity of a local lasso estimator.The empirical sparsity is defined as the size of the lasso estimator’s equicorrelation set.