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Straggler Mitigation in Distributed Optimization Through Data Encoding

Can Karakus, Yifan Sun, Suhas Diggavi, Wotao Yin

arXiv:1711.04969v2stat.MLcs.DCcs.ITcs.LG

TL;DR

Straggler delays hinder synchronous distributed optimization, motivating a method that embeds redundancy in data while leaving computation unaware of the encoding. The paper develops encoding schemes for gradient descent and L-BFGS and reports deterministic linear convergence to an approximate original solution using subsets of nodes. Experiments show advantages over uncoded and replicated data strategies.

  • Problem

    Straggler nodes, network delays, and failures can significantly hamper distributed optimization, while prior coding approaches encode computational steps rather than the data.

  • Method

    The paper encodes X and y with an overcomplete matrix S, then applies gradient descent or L-BFGS directly to the encoded problem without waiting for all workers.

  • Results

    Gradient descent and L-BFGS deterministically achieve sample path linear convergence to an approximate original solution using only a fraction of nodes, with performance advantages over uncoded and replication strategies.

  • Takeaways & Limitations

    Redundancy and the number of responding nodes can control the approximation while mitigating straggler delays in synchronous distributed optimization.

  • Takeaways & Limitations

    The theoretical focus is limited to quadratic objectives and gradient descent and L-BFGS, although broader generalizations are discussed.

Abstract

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Slow running or straggler tasks can significantly reduce computation speed in distributed computation. Recently, coding-theory-inspired approaches have been applied to mitigate the effect of straggling, through embedding redundancy in certain linear computational steps of the optimization algorithm, thus completing the computation without waiting for the stragglers. In this paper, we propose an alternate approach where we embed the redundancy directly in the data itself, and allow the computation to proceed completely oblivious to encoding. We propose several encoding schemes, and demonstrate that popular batch algorithms, such as gradient descent and L-BFGS, applied in a coding-oblivious manner, deterministically achieve sample path linear convergence to an approximate solution of the original problem, using an arbitrarily varying subset of the nodes at each iteration. Moreover, this approximation can be controlled by the amount of redundancy and the number of nodes used in each iteration. We provide experimental results demonstrating the advantage of the approach over uncoded and data replication strategies.

1 Introduction

The paper addresses straggler-induced delays in synchronous distributed optimization by encoding data rather than computation. Coding-oblivious gradient descent and L-BFGS can converge linearly to an approximate original solution while using only responding nodes.

  • Approach: Encoding X and y adds redundancy directly to the stored data, allowing workers to compute without exposing or explicitly handling the encoding.The encoded data are eX = SX and ey = Sy, and node computations proceed as if they held uncoded data.
  • Guarantees: The approximation can be tightened by increasing the redundancy and the number of node updates used in each iteration.The method targets sufficiently accurate solutions rather than exact optima.
  • Guarantees: Gradient descent and L-BFGS achieve deterministic sample path linear convergence to a neighborhood of the original solution using a strict subset of nodes per iteration.The guarantee holds under a spectral condition on the encoding matrix and does not require waiting for stragglers.
  • Contributions: The paper presents ETF, fast-transform, and random coding matrices, and evaluates the approach against uncoded and replication strategies.Experiments cover ridge regression on synthetic AWS-cluster data and matrix factorization on the Movielens 1-M task.
  • Related work: Unlike computation-level coding, the proposed method embeds redundancy in the dataset and avoids encoding and decoding operations at every computational step.Related coding approaches perform certain linear computations in coded form and explicitly encode or decode during each step.

2 Encoded Optimization Framework

The encoded framework stores transformed data across workers and lets the server update from whichever k nodes respond first. It is designed to approximate the original objective despite changing responder subsets and straggler delays.

  • Worker computation: Workers operate coding-obliviously by computing gradient terms from their stored encoded partitions as though they contained raw data.The server broadcasts wt, and each worker computes the gradient contribution associated with its own encoded partition.
  • Straggler handling: At each iteration, the server uses gradient updates from the first k responding workers instead of waiting for all m nodes.The responding set At can vary by iteration, while slower workers may be interrupted or their updates dropped.
  • Server update: The server forms a descent direction from gradient and parameter history, then selects a constant, decaying, or exact-line-search step size.For line search, workers provide the encoded matrix-vector products needed to determine the step size.
  • Convergence target: The design goal is universal convergence of the encoded objective toward a neighborhood of the original objective for arbitrary sequences of responding sets.With k < m, convergence is not automatically guaranteed for traditional batch methods such as L-BFGS, motivating the encoding design.

3 Algorithms and Convergence Analysis

The paper analyzes coding-oblivious gradient descent and L-BFGS on encoded data, showing deterministic linear convergence to a neighborhood of the original solution under conditions on the encoding matrix and responding-node subsets.

  • Encoding condition: The encoding condition requires every relevant submatrix S_A to have nearly uniform eigenvalues, analogous to a restricted isometry condition.The condition is needed for worst-case guarantees, while practical performance may remain good when most eigenvalues lie in a small interval.
  • Gradient descent: Gradient descent uses a constant step size, and its convergence proof treats every sequence of fastest-worker subsets with fixed cardinality k.The theorem specifies a step size based on the largest eigenvalue M and the encoding distortion parameter ε.
  • Gradient descent: The analysis shows each instantaneous encoded problem has a solution in a bounded objective set, attracting iterates toward a neighborhood of the original optimum.The contraction requires κγ1 < 1, ensured by the encoding property.
  • Gradient descent: Gradient descent using only k < m workers deterministically converges linearly to a neighborhood of the true solution, with the approximation controlled by redundancy and k.When k = m and S is properly designed, the original optimum is reached.
  • L-BFGS: L-BFGS adapts its Hessian estimate using gradient components shared by consecutive responding-node subsets and requires their overlap to be sufficiently large.Under additional encoding conditions, the inverse Hessian estimates remain uniformly bounded and convergence is deterministic for all allowed update sequences.
  • L-BFGS: For L-BFGS, coding enables deterministic linear convergence, unlike the stochastic and multi-batch variants, while still converging toward a neighborhood of the true solution.The result assumes the encoding conditions for both the responding-node subsets and their consecutive-iteration overlaps.
  • Generalizations: Although the analysis focuses on quadratic objectives and gradient descent or L-BFGS, the framework is described as extendable to convex regularization, constrained problems, and algorithms such as FISTA.The paper presents these as potential generalizations rather than established convergence results in this section.

4 Code Design

The paper considers tight frames, fast transforms, and random matrices as data-encoding designs, emphasizing how their spectral properties affect approximation and implementation. Tight frames preserve the original optimum under full participation, while numerical and theoretical evidence characterizes redundancy requirements and approximate recovery under partial participation.

  • Encoding classes: The paper evaluates three coding-matrix classes: tight frames, fast transforms such as FFT and FWHT, and i.i.d. random matrices.Fast transforms offer computational efficiency, while random matrices have analytically characterizable eigenvalue behavior.
  • Tight frames: Tight frames satisfy S^T S = βI, preserving the original optimization optimum when all nodes participate and the objective is strongly convex.The encoded and original optimality conditions are equivalent for tight frames; the equivalence also extends to convex constraints and nonsmooth convex terms.
  • Tight frames: Equiangular tight frames minimize element correlations, making each submatrix S^T A S close to orthogonal and potentially reducing the redundancy required by the spectral condition.The paper specifically evaluates Paley and Hadamard ETFs and discusses Steiner ETFs for efficient implementation.
  • Experimental comparison: Figure 4 compares uncoded, replicated, and FWHT-coded evolution and runtime under k = 12 and m = 32, with runtime primarily reflecting network delay profiles.The visual reports matched-iteration runtimes rather than relative convergence rates across methods.
  • Random matrices: For finite β, random matrices generally do not recover the original optimum exactly, even when all nodes participate, although they become suitable candidates as redundancy and problem dimension grow.Their singular-value behavior supports suitability asymptotically, but the finite-redundancy limitation remains.
  • Spectral behavior: Numerical spectra suggest ETFs may satisfy the spectral condition with smaller ϵ than random matrices, while bulk eigenvalues can support close approximation even with modest redundancy such as β = 2.The latter observation follows from a practical, rather than worst-case, view of gradient energy across the eigenspectrum.

5 Numerical Results

Experiments evaluate coded distributed optimization on synthetic ridge regression and MovieLens matrix factorization, comparing coding, replication, and uncoded schemes under straggler conditions. The results show that coded schemes improve robustness when the server uses only a small subset of nodes, while encoding overhead is included in runtime measurements.

  • Ridge regression: More than 40% runtime reduction was achieved by reducing η from 1 to 0.375, while uncoded L-BFGS failed to converge and Hadamard-coded L-BFGS remained stable.The comparison used 20 aggregated trials and included uncoded and β = 2 replication baselines.
  • MovieLens matrix factorization: The MovieLens task used an 80/20 train/test split to evaluate matrix factorization on sparse movie-rating data.The model estimates user and movie vectors together with user, movie, and global biases.
  • MovieLens matrix factorization: The selected MovieLens configuration achieved a test RMSE of 0.861, close to the current best test RMSE reported for matrix factorization on the dataset.The configuration used μ = 3, p = 15, and λ = 10.
  • Experimental setup: MovieLens runtime measurements included the initial data-encoding overhead and simulated network latency on a single 32-core machine.An artificial delay Δ ∼ exp(10 ms) was imposed after worker tasks, and encoding matrices were precomputed for several problem sizes.
  • MovieLens matrix factorization: Coded schemes were most robust for small k in distributed L-BFGS across the evaluated epochs.Figure 5 reports test RMSE for m = 8 and m = 24 nodes with k = m/8 and k = m/2; “Perfect” denotes k = m.

A Lemmas

The lemmas establish spectral, convexity, and recursive bounds needed to analyze optimization on changing encoded data subsets. Together, they support contraction toward a neighborhood of the original solution.

  • Changing subsets: The instantaneous minimizer is defined for the effective problem associated with the active node subset at iteration t.The lemmas distinguish this solution from the minimizer of the original objective and analyze their relationship.
  • Approximation bound: A condition-number bound yields an objective comparison of f(ew∗) ≤ κ^2 f(w∗).The bound is obtained after choosing c = λ_max + λ_min and completing the preceding inequality argument.
  • Convergence recursion: A recursive inequality bounds the current error by a contraction term plus a persistent approximation term.The proof re-arranges the inequality and sums weighted inequalities over previous iterations.
  • Spectral properties: The analysis uses minimum and maximum eigenvalue bounds to control encoded objectives and their condition numbers.The proof develops positive-definiteness and eigenvalue bounds for matrices associated with changing node subsets.
  • Convexity: The effective encoded objective is shown to be strongly convex under the stated spectral conditions.Lemma 4 states that ef_A(w) is λ_min μ-strongly convex.

B Proofs of Theorem 1 and Theorem 2

This proof section defines the changing active-node objective and the encoded loss used to analyze each iteration. It also records the eigenvalue notation and normalization convention used throughout the proofs.

  • Iteration-specific notation: The proof analyzes a particular iteration t while allowing the active set A to vary with t.The notation treats the effective problem and its minimizer as iteration-specific.
  • Spectral notation: λ_min(·) and λ_max(·) denote the minimum and maximum eigenvalues of a matrix.These quantities are used in the spectral conditions underlying the theorems.
  • Encoded objective: The encoded objective is defined as ef_A(w) := ∥eX_Aw − ey_A∥^2, with normalization constants omitted.The same section introduces the effective instantaneous problem associated with the selected subset.

B.1 Proof of Theorem 1

The proof uses convexity and matrix spectral bounds to derive the recursive inequality underlying the convergence result. The displayed steps then invoke earlier lemmas to complete the bound.

  • Proof of Theorem 1: Convexity is combined with the choices d_t = −e g_t and α_t = α to obtain the key inequality.The subsequent proof steps use spectral upper bounds and strong convexity to rearrange it.

B.2 Proof of Theorem 2

The proof derives the claimed convergence bound by combining the step-size expression with convexity, matrix assumptions, and prior lemmas.

  • Convexity and the closed-form step size establish the starting inequality for the convergence proof.
  • The proof applies the encoding-matrix assumption and the bound X⊤X ⪯ MI to control the relevant quadratic term.
  • Rearrangement followed by Lemmas 3 and 2 yields the desired result.

C Full results of Movielens 1-M experiment

The Movielens 1-M experiments report train/test RMSE and runtime for distributed L-BFGS under two cluster sizes, including uncoded baselines.

  • The experiment uses a random 80/20 train/test split for the Movielens 1-M recommendation task.
  • 0.756 / 0.861 train/test RMSE and 9.58 hours are reported for uncoded full-batch L-BFGS on 8 nodes.
  • 0.757 / 0.862 train/test RMSE and 14.11 hours are reported for uncoded full-batch L-BFGS on 24 nodes.

D Efficient encoding using Steiner ETF

The section describes Steiner ETF construction and distributed implementation, including efficient block processing and memory-conscious computation.

  • Steiner ETF constructs the encoding matrix from a Hadamard matrix and incidence vectors of two-element subsets.
  • The construction yields redundancy factor β = 2v/(v−1), with larger redundancy levels possible in general.
  • Multiplication by each encoding block can use identified nonzero columns and a Fast Hadamard Transform, with block processing parallelized across nodes and cores.
  • Shuffling encoded rows improves observed Steiner ETF performance, but exchanging rows between nodes incurs significant communication cost.
  • For sparse datasets, computing encoded gradients through matrix-vector multiplications can limit memory growth to a constant factor on the order of redundancy.
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