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AIDE: Fast and Communication Efficient Distributed Optimization
Sashank J. Reddi, Jakub Konečný, Peter Richtárik, Barnabás Póczós, Alex Smola
TL;DR
The paper addresses communication-efficient distributed optimization for large-scale empirical risk minimization. It develops INEXACTDANE for approximate local solves and AIDE as an accelerated first-order method, with AIDE nearly matching communication lower bounds and reported empirical superiority in relevant machine-learning settings.
Problem
Distributed empirical risk minimization must manage computational and communication constraints, while DANE does not match communication-complexity lower bounds.
Method
The paper introduces INEXACTDANE, which permits approximate local DANE solves, and AIDE, an accelerated first-order variant.
Results
AIDE nearly matches communication lower bounds and uses only first-order oracle access, while INEXACTDANE improves robustness to bad data partitioning.
Takeaways & Limitations
AIDE combines DANE’s practical local-computation advantages with near-optimal communication complexity for large-scale distributed optimization.
Takeaways & Limitations
The SVRG connection and corresponding guarantees apply only under specific settings, such as sufficiently similar quadratic local functions.
Abstract
from arXiv · showhide
In this paper, we present two new communication-efficient methods for distributed minimization of an average of functions. The first algorithm is an inexact variant of the DANE algorithm that allows any local algorithm to return an approximate solution to a local subproblem. We show that such a strategy does not affect the theoretical guarantees of DANE significantly. In fact, our approach can be viewed as a robustification strategy since the method is substantially better behaved than DANE on data partition arising in practice. It is well known that DANE algorithm does not match the communication complexity lower bounds. To bridge this gap, we propose an accelerated variant of the first method, called AIDE, that not only matches the communication lower bounds but can also be implemented using a purely first-order oracle. Our empirical results show that AIDE is superior to other communication efficient algorithms in settings that naturally arise in machine learning applications.
1 Introduction
The paper studies communication-efficient distributed optimization for large-scale empirical risk minimization, where communication can dominate computation. It introduces INEXACTDANE and AIDE to combine local computation with reduced communication while retaining first-order implementability.
- Large-scale distributed learning makes computational and communication constraints central because datasets can reach hundreds of terabytes.
- Distributed empirical risk minimization partitions data across K machines, each accessing only its local functions, including potentially nonconvex losses relevant to deep learning.
- Gradient descent remains communication-intensive because machines communicate at every iteration, leaving computation under-utilized during communication delays.
- DANE and DISCO reduce communication through local computation but require strong oracle access, whereas COCOA+ permits approximate local solves.
- INEXACTDANE approximately solves DANE subproblems, improving robustness to bad data partitioning while retaining partial convergence guarantees and connections to distributed SVRG.
- AIDE builds on INEXACTDANE to match communication lower bounds up to logarithmic factors using only a first-order oracle.
2 Algorithm: INEXACTDANE
INEXACTDANE modifies DANE by allowing local subproblems to be solved approximately rather than exactly. This preserves parallel implementation while permitting flexible local algorithms and explicit communication-round analysis.
- INEXACTDANE replaces DANE’s exact local subproblem solves with approximate minimization of each local objective.
- The inexactness parameter γ controls local accuracy, with smaller γ corresponding to higher-accuracy solutions.
- Each INEXACTDANE iteration is embarrassingly parallel because machines independently solve local objectives before communicating.
- Communication complexity counts rounds needed to reach an ϵ-accurate solution when each round transmits information linear in dimension d.
3 Analysis of INEXACTDANE: Quadratic Case
For quadratic objectives, the analysis establishes contraction of INEXACTDANE under similarity and spectral assumptions, and translates this into communication-round guarantees. The required local accuracy need not be perfect.
- Theorem 1 gives a contraction bound for INEXACTDANE iterates under positive-definite quadratic structure, with suitable parameters ensuring ρ<1.
- For δ-related quadratic objectives, smaller δ supports stronger communication guarantees, motivating the analysis of similar local functions.
- Under the quadratic assumptions of Theorem 2, INEXACTDANE iterates contract geometrically: ∥w_t−ŵ∥≤˜ρ∥w_{t−1}−ŵ∥.
- The local subproblems should be solved only to an accuracy depending on δ and λ, because greater accuracy does not improve the overall complexity bound.
- When local Hessian averages come from n i.i.d. quadratic samples, δ=O(L/√n) and the communication complexity becomes ˜O((L/λ)^2/n log(1/ϵ)).
4 Analysis of INEXACTDANE: General Case
The general-case analysis extends INEXACTDANE to strongly convex, weakly convex, and nonconvex objectives. Constant local approximation accuracy has only a weak effect on the stated convergence rates.
- Strongly convex case: For L-smooth, λ-strongly convex objectives, INEXACTDANE achieves geometric convergence in function suboptimality under parameters yielding 0<˜ρ<1.
- Strongly convex case: With γ=1/8, η=1, and µ=6L−λ, the strongly convex method reaches ϵ-suboptimality in O((L/λ) log(1/ϵ)) iterations.
- Weakly convex case: For weakly convex functions, a quadratic perturbation yields an O(ϵ)-optimal solution after ˜O(L log(1/ϵ)/ϵ) iterations.
- Across the analyzed cases, the convergence rate depends only weakly on the inexactness parameter γ.
- Nonconvex case: For nonconvex functions, constant γ=1/8 can yield an O(1/t) convergence rate to a stationary point when µ is sufficiently large.
5 Accelerated Distributed Optimization
AIDE accelerates INEXACTDANE using catalyst to close communication-complexity gaps while retaining first-order implementability. The analysis establishes lower-bound-matching rates in quadratic, strongly convex, and weakly convex settings.
- Accelerated Distributed Optimization: AIDE applies catalyst acceleration to INEXACTDANE and matches communication lower bounds up to logarithmic factors in specific settings.The method is designed to retain practical distributed-optimization properties while improving communication complexity.
- Quadratic Case: For quadratic objectives, suitable τ yields convergence rates matching the communication lower bounds.
- Strongly Convex Case: In the general strongly convex case, AIDE reaches ϵ-accuracy with communication complexity matching lower bounds for unrelated strongly convex functions.The result uses smoothness and strong-convexity assumptions, with constant-accuracy local subproblem solves.
- Weakly Convex Case: For weakly convex objectives, AIDE applies to a regularized strongly convex problem and obtains an ϵ-accuracy guarantee.The perturbed objective is made ϵ-strongly convex before applying AIDE.
6 Connection to a Practical Distributed Version of SVRG
The paper connects INEXACTDANE with a practical distributed SVRG variant when SVRG is used as the local solver. This connection explains the observed procedure but leaves its direct analysis open beyond a special parameter setting.
- Connection to Distributed SVRG: Using SVRG as INEXACTDANE’s local solver yields a distributed SVRG procedure with parallel inner loops over each machine’s local data.
- Connection to Distributed SVRG: With µ = 0 and η = 1, one SVRG outer iteration on the INEXACTDANE subproblem produces an identical iterate sequence.
- Limitations: Direct convergence analysis remains open, and the paper’s guarantees cover this special procedure only under specific quadratic δ-related conditions.
7 Experiments
Experiments compare INEXACTDANE, AIDE, and COCOA+ on distributed binary classification under varied losses, local computation, and node counts. AIDE performs best on the reported rcv1 setting, while additional local passes eventually yield little improvement.
- Experimental Setup: The study uses normalized rcv1, covtype, real-sim, and url data with random partitions that mimic i.i.d. distributions.
- Main Comparison: AIDE outperforms INEXACTDANE and COCOA+ on rcv1 with smoothed hinge loss across the tested regularization settings.The comparison uses eight nodes and λ = 1/(cN) for c ∈ {1, 10, 100}.
- Main Comparison: The benefits of AIDE are particularly pronounced when the condition number satisfies κ = L/λ > N.
- Local Computation: Running SVRG beyond four passes through local data provides little improvement in overall convergence.The experiment varies local passes per INEXACTDANE iteration over {1/6, 1/3, 1, 2, 4}.
8 Discussion
The discussion positions AIDE as communication-efficient, first-order, and parallelizable, while summarizing partial results for distributed SVRG. It emphasizes lower-bound matching in selected settings and practical communication–computation tradeoffs.
- Communication Complexity: AIDE matches communication lower bounds in quadratic δ-related and unrelated strongly convex settings, outperforming the O(κ log(1/ϵ)) rates of DANE, INEXACTDANE, and COCOA+ in the latter case.For unrelated strongly convex functions, AIDE and DISCO achieve ˜O(√κ log(1/ϵ)).
- Nature of Oracle Access: AIDE and INEXACTDANE use simple first-order oracles, unlike DANE’s exact-subproblem oracle and DISCO’s second-order oracle.
- Parallelism and Implementation: AIDE, INEXACTDANE, DANE, and COCOA+ are embarrassingly parallel because each iteration solves local objectives independently across nodes.DISCO instead has an asymmetric master-node workload.
- Distributed SVRG: The analysis provides partial convergence guarantees for a distributed SVRG variant observed to perform well in practice.
- Conclusion: AIDE combines DANE’s practical advantages with optimal communication complexity and supports communication–computation tradeoffs for large-scale optimization.
A.4 Proof of Theorem 3
The proof bounds the error propagation caused by inexact local subproblem solutions and derives convergence for INEXACTDANE under smoothness and convexity conditions.
- A.4 Proof of Theorem 3: The virtual iterate and Bregman-divergence identities organize the proof of the local-to-global convergence bound.The virtual iterate is introduced before bounding the resulting terms using optimality and Lipschitz-gradient properties.
- A.4 Proof of Theorem 3: The proof combines the inexactness condition with smoothness, convexity, and optimality relations to bound successive iterate errors.The argument proceeds through bounds on auxiliary gradient and iterate-difference terms before establishing the required result.
- A.4 Proof of Theorem 3: The proof extends the argument to weakly convex and nonconvex settings using regularization and telescoping inequalities.The supplied steps invoke smoothness and strong convexity of the regularized subproblem, followed by telescoping to obtain the desired result.
B.1 Proof of Theorem 5
This proof analyzes AIDE by applying INEXACTDANE to regularized subproblems and bounding the number of inner iterations needed for an accurate solution.
- B.1 Proof of Theorem 5: AIDE applies INEXACTDANE to regularized functions whose smoothness and strong convexity increase with the acceleration parameter.The proof uses the regularized function f^t and its curvature properties to invoke the INEXACTDANE guarantee.
- B.1 Proof of Theorem 5: The contraction factor is controlled by choosing the inexactness parameter γ ≤ 1/8, yielding ρ̃ = 2/3 in the analyzed case.The proof explicitly attributes the value ρ̃ = 2/3 to the inexactness choice and the relevant curvature condition.
- B.1 Proof of Theorem 5: The resulting iteration bound follows by combining the INEXACTDANE contraction with Proposition 3.2 and the selected acceleration parameters.The proof states that the total number of INEXACTDANE iterations required for ε-accuracy is obtained from this contraction analysis.
- B.1 Proof of Theorem 5: The corollary assumes stochastic positive-semidefinite Hessian averages with eigenvalues bounded by L and holds with probability at least 1 − α.These are the stated stochastic conditions for the corollary underlying the analyzed result.
C Auxiliary Results
The auxiliary material supplies matrix and smoothness lemmas, connects INEXACTDANE with distributed SVRG, and reports extended experiments on datasets, node counts, and local computation.
- C Auxiliary Results: The matrix lemmas bound deviations involving local Hessians through δ, λ, and μ, supporting the convergence analysis.The bounds are derived under spectral assumptions on the aggregate and local matrices.
- C Auxiliary Results: A smooth strongly convex function satisfies the auxiliary inequality used to control distances and function errors in the main proofs.The lemma assumes L-smoothness and λ-strong convexity and states the resulting bound for all pairs of points.
- C Auxiliary Results: Running SVRG as the local solver in INEXACTDANE yields a distributed SVRG procedure in the specified parameter setting.The paper notes that the same algorithm can also be obtained by reformulating the local subproblem and applying local SGD.
- C Auxiliary Results: Across classification experiments, AIDE performs much better than INEXACTDANE and COCOA+, especially when κ = L/μ exceeds N.The reported pattern appears across multiple datasets and losses, with a larger performance gap at larger condition numbers.
- C Auxiliary Results: Four passes through local data usually provide little additional improvement, although covtype shows a notable exception.The local-computation experiment varies passes from 1/6 to 4 and reports the exception explicitly.
- C Auxiliary Results: Increasing node count degrades performance with fixed local work, while matching local SVRG iterations across nodes generally yields similar behavior.The reported scaling study also notes slower convergence for rcv1 and realsim at 64 nodes under the matched-work setting.
E.4 Inconvenient data partitioning
INEXACTDANE remains robust under inconvenient data partitions where DANE diverges, including partitions by output label. This robustness matters for fixed data layouts, although AIDE may require more tuning in practice.
- Output-label partitioning: DANE diverges under output-label partitioning, whereas INEXACTDANE’s performance drops only slightly.The comparison uses random versus output-label partitions on rcv1, covtype, and realsim.
- Practical relevance: INEXACTDANE’s robustness is practically relevant because large-scale data are often partitioned as given rather than randomly reshuffled.
- Practical limitations: AIDE may not always outperform alternatives in practice because its catalyst acceleration parameter τ lacks a simple selection rule.INEXACTDANE also requires tuning a local-solver hyperparameter, while COCOA+ can use a data-independent aggregation parameter.