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Optimal algorithms for smooth and strongly convex distributed optimization in networks
Kevin Scaman, Francis Bach, Sébastien Bubeck, Yin Tat Lee, Laurent Massoulié
TL;DR
The paper asks for optimal convergence rates for smooth, strongly convex distributed optimization in centralized and decentralized networks. It establishes matching bounds and algorithms, including distributed accelerated gradient centrally and MSDA for gossip-based decentralization. The resulting rates depend on function conditioning and network structure, while the centralized method has practical robustness and synchronization limitations.
Problem
The paper addresses the lack of a coherent theoretical understanding of optimal linear convergence rates and their dependence on communication-network characteristics.
Method
It derives complexity lower bounds and matching algorithms, using distributed Nesterov acceleration centrally and a dual multi-step approach, MSDA, for decentralized gossip communication.
Results
The proposed algorithms attain the paper’s optimal convergence rates, with MSDA achieving a provable optimal linear rate for decentralized optimization.
Takeaways & Limitations
The rates expose how optimization condition numbers and communication-network quantities jointly determine distributed optimization complexity.
Takeaways & Limitations
The decentralized optimality result is stated with respect to κ_l and γ, although 1/√γ can greatly exceed the network diameter on some graphs such as star networks.
Abstract
from arXiv · showhide
In this paper, we determine the optimal convergence rates for strongly convex and smooth distributed optimization in two settings: centralized and decentralized communications over a network. For centralized (i.e. master/slave) algorithms, we show that distributing Nesterov's accelerated gradient descent is optimal and achieves a precision $\varepsilon > 0$ in time $O(\sqrt{κ_g}(1+Δτ)\ln(1/\varepsilon))$, where $κ_g$ is the condition number of the (global) function to optimize, $Δ$ is the diameter of the network, and $τ$ (resp. $1$) is the time needed to communicate values between two neighbors (resp. perform local computations). For decentralized algorithms based on gossip, we provide the first optimal algorithm, called the multi-step dual accelerated (MSDA) method, that achieves a precision $\varepsilon > 0$ in time $O(\sqrt{κ_l}(1+\fracτ{\sqrtγ})\ln(1/\varepsilon))$, where $κ_l$ is the condition number of the local functions and $γ$ is the (normalized) eigengap of the gossip matrix used for communication between nodes. We then verify the efficiency of MSDA against state-of-the-art methods for two problems: least-squares regression and classification by logistic regression.
1 Introduction
The paper studies optimal linear convergence for smooth, strongly convex distributed optimization under centralized and decentralized communication. It derives matching complexity bounds and algorithms whose rates combine optimization conditioning with network communication limits.
- The paper addresses centralized and decentralized distributed optimization for smooth, strongly convex objectives with linear convergence.
- It proposes matching upper and lower complexity bounds for this class of distributed problems.
- The lower bounds combine the square-root condition-number cost of optimization with network-dependent communication costs.
- Distributing Nesterov’s accelerated gradient is optimal centrally, while a dual algorithm attains the decentralized lower bound.
- Prior decentralized methods achieved linear convergence, but the field lacked a coherent theory of optimal rates and their dependence on network characteristics.
2 Distributed optimization setting
The setting consists of nodes holding local smooth, strongly convex functions and communicating over a connected network. Decentralized communication is modeled by a gossip matrix whose normalized eigengap captures network connectivity.
- A connected graph of n computing units supports minimizing the average of local functions, with each node accessing its own function.
- Local first-order computations take one time unit after normalization, while neighbor communication takes time τ.
- Each node maintains a local parameter θ_i, and local functions are α-strongly convex and β-smooth with local condition number κ_l.
- The global function has its own condition number κ_g, which is no larger than the local condition number κ_l.
- Decentralized communication: Gossip communication uses a matrix W whose kernel is the constant-vector space and whose nonzero entries follow the network edges.
- Decentralized communication: The normalized eigengap γ(W)=λ_n−1(W)/λ_1(W) is the principal parameter describing communication-network connectivity.
3 Optimal convergence rates
The paper establishes matching complexity lower and upper bounds for smooth, strongly convex distributed optimization in centralized and decentralized networks. Centralized accelerated gradient and decentralized MSDA attain the respective optimal rates, while the network parameter is diameter in centralized settings and gossip eigengap in decentralized settings.
- Lower-bound framework: The analysis proves oracle-complexity lower bounds for both centralized and gossip-based decentralized distributed optimization.The procedures are modeled as black-box algorithms with local memory, computation, communication, and output constraints.
- Centralized algorithms: Distributing Nesterov’s accelerated gradient descent attains the centralized optimal convergence rate.The master aggregates local gradients and broadcasts updated parameters, requiring communication across the network.
- Centralized algorithms: Ω(√κg ln(1/ε)) gradient steps and Ω(∆√κg ln(1/ε)) communication steps are necessary in centralized networks.The communication term reflects the network diameter ∆, while κg is the global condition number.
- Algorithmic limitations: The centralized method is limited by master-node failures, the need for a precomputed spanning tree, and synchronization with the slowest machine.These constraints reduce suitability for failures, time-varying graphs, and heterogeneous computation latencies.
- Decentralized algorithms: Decentralized lower bounds depend on the local condition number κl and the gossip-matrix eigengap γ.The analysis requires at least Ω(√κl ln(1/ε)) gradient steps and corresponding communication complexity governed by γ.
- Decentralized algorithms: MSDA, a dual algorithm, meets the decentralized lower bound with respect to κl and γ.The paper cautions that 1/√γ need not be the right network quantity on general graphs, despite its connection to diameter on several simple networks.
4 Optimal decentralized algorithms
The decentralized framework uses a dual formulation with accelerated gradient methods and gossip-based communication. MSDA accelerates gossip through Chebyshev-polynomial multi-step updates, attaining an optimal convergence rate with respect to the local condition number and eigengap.
- Dual formulation: The decentralized approach reformulates the constrained optimization problem through a convex unconstrained dual objective.The constraint θ1 = · · · = θn is represented using a gossip matrix, and Fenchel conjugates define the dual formulation.
- Single-step dual acceleration: SSDA applies Nesterov’s accelerated gradient descent to the dual problem and converges to the common optimizer.Its gradient step can be interpreted as gossiping gradients of local conjugate functions.
- Single-step dual acceleration: Theorem 3 guarantees that SSDA reaches any precision ε > 0 with a convergence time governed by the local condition number and gossip eigengap.The proof bounds the dual objective’s condition number using κ_l and γ, then invokes accelerated-gradient convergence.
- Multi-step gossip: Replacing W with P_K(W) enables more gossip steps than gradient steps when communication is cheap relative to local computation.The polynomial P_K has degree at most K and is chosen so that repeated gossip propagates gradients beyond local neighborhoods.
- Multi-step gossip: K = ⌊1/√γ⌋ yields MSDA, whose convergence rate is optimal with respect to γ and κ_l.The construction uses Chebyshev polynomials and verifies that the resulting polynomial transformation remains a gossip matrix.
- Extensions and discussion: The methods can be extended to composite functions, average strong convexity, warm starts, and asynchronous optimization variants.Warm starts address the practical difficulty of computing conjugate gradients; asynchronous extensions using SVRG or SAGA are left for future work.
5 Experiments
The experiments compare SSDA and MSDA with four linearly convergent distributed algorithms on generated least-squares and logistic-classification problems over grid and random networks. DAGD is strongest centrally for least-squares, MSDA is strongest decentrally, and multiple communication rounds improve MSDA relative to SSDA when communication is faster than computation.
- Experimental setup: The experiments compare SSDA and MSDA with D-ADMM, EXTRA, DIGing, and DAGD on generated least-squares and logistic-classification problems.The datasets are distributed across 100 network nodes, and the comparisons use grid and Erdős–Rényi networks.
- Least-squares regression: All compared algorithms converge linearly, but their convergence rates differ by several orders of magnitude across the tested networks.The least-squares experiments use a 10×10 grid and an Erdős–Rényi random graph of average degree 6.
- Least-squares regression: DAGD has the best convergence rate centrally, while MSDA has the best convergence rate among decentralized methods for least-squares regression.This comparison is reported for both the grid and Erdős–Rényi network experiments.
- Least-squares regression: When communication is faster than computation (τ ≫1), several communication rounds per gradient iteration improve efficiency, and MSDA substantially outperforms SSDA.The paper reports this effect for the least-squares experiments.
- Logistic classification: In Figure 4(a), D-ADMM requires 383 iterations to reach the error obtained by SSDA after 10 iterations.The authors present this as a substantial improvement over the compared state-of-the-art method.
6 Conclusion
The paper derives optimal convergence rates for strongly convex and smooth distributed optimization with centralized and decentralized network communications. It introduces MSDA for decentralized optimization, proves its optimal linear rate, and reports high efficiency against state-of-the-art methods.
- Conclusion: The paper derives optimal convergence rates for strongly convex and smooth distributed optimization with centralized and decentralized communications.The two settings are treated as networked optimization regimes.
- Conclusion: MSDA is introduced for decentralized optimization with a provable optimal linear convergence rate.The paper also reports high efficiency relative to distributed ADMM and EXTRA.
- Conclusion: The approach is described as simple and flexible, with proposed extensions to time-varying networks and non-strongly-convex functions.The paper also identifies extending lower bounds to delays, variable computational speeds, or machine failures as future work.
A.1 Complexity lower bounds
The lower-bound analysis combines optimization and communication constraints in distributed networks. It uses graph distance and structured local functions to show that computation and communication steps jointly limit information propagation.
- Lower-bound construction: The proof uses a Nesterov-based construction for strongly convex and smooth functions to establish complexity lower bounds.The construction assigns structured functions to nodes in a graph.
- Lower-bound construction: Local gradients can increase only selected coordinate dimensions depending on whether nodes lie in A or its distant complement A_d.The block structure also applies to gradients of the dual functions.
- Communication barrier: Reaching a later coordinate requires a local computation in A followed by d communication steps before a node in A_d can access the needed information.The argument yields at least k local computation steps and (k−1)d communication steps for the stated progress condition.
- Centralized lower bound: Taking d as the graph diameter and A as one endpoint produces the desired centralized lower-bound result.The proof applies the distance-based lemma to a pair of nodes separated by the diameter.
- Decentralized lower bound: The construction relates global and local condition numbers through κ_g ≥ κ_l/16.This relation is obtained from the chosen graph and set A in the lower-bound construction.
- Decentralized lower bound: For γ ≤ 1/3, the constructed network yields a distance-related factor at least 1/(5√γ).The proof obtains this bound from the selected weighted graph family.
A.2 Convergence rates of SSDA and MSDA
The convergence analysis decomposes each dual-algorithm step into local gradient computation and gossip communication. It then bounds the time to reach precision ε using the relevant condition number and gossip-matrix spectrum.
- Per-step cost: Each algorithm step consists of computing gradients and communicating them across neighborhoods, so one step takes time 1 + τ.The unit computation time and communication time are combined in the per-step cost.
- Dual-space analysis: The iterates remain in the image of W ⊗ I_d, the subspace of matrices x satisfying x^⊤1 = 0.This restriction is used in analyzing the dual function’s condition number.
- Gossip acceleration: Nesterov acceleration requires √γ communication steps to achieve any given precision in the gossip setting.The communication complexity is stated in terms of the gossip spectral parameter γ.
- Spectral normalization: Scaling the gossip matrix produces a spectrum in [1−c^-1/2, 1+c^-1/2], enabling the convergence analysis of the multi-step method.The proof applies an accelerated polynomial result after this spectral normalization.
B Composite problems for machine learning
The section develops a dual accelerated proximal-gradient approach for composite distributed optimization and analyzes its convergence under smoothness and operator-norm assumptions.
- Dual formulation: The composite formulation introduces a Lagrange multiplier so the Fenchel conjugate of each g_i appears in the dual problem.This transformation starts from the primal problem and uses an arbitrary parameter ρ > 0.
- Optimization method: Accelerated proximal gradient is used to maximize the resulting dual problem.The method is specified through proximal-gradient updates.
- Convergence analysis: The convergence analysis assumes each g_i is μ-smooth and each B_i has largest singular value at most M.These assumptions support computing the condition number of the associated quadratic function.
- Convergence result: The resulting rate is essentially the same as in the non-composite case with Nesterov and Chebyshev accelerations.The condition-number discussion identifies γ as a natural upper bound on κ_l/γ in the stated analysis.