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Communication Complexity of Distributed Convex Learning and Optimization
Yossi Arjevani, Ohad Shamir
TL;DR
The paper asks how communication-efficient distributed methods can solve convex learning and optimization when local objectives differ in geometry and similarity. It develops lower bounds under a structural algorithm assumption and, for one-round protocols, without that assumption, finding optimality in some settings while leaving others open. The results show that unrelated objectives can require many rounds even with unbounded local computation, whereas related objectives admit faster bounds in restricted cases.
Problem
Distributed convex learning must limit communication because machines hold separate, large, high-dimensional data or local objectives, but the fundamental round complexity under different assumptions is not fully characterized.
Method
The paper proves communication-round lower bounds under a mild structural assumption and separately studies assumption-free algorithms restricted to one communication round.
Results
Unrelated objectives require polynomially many rounds in 1/ε or 1/λ, smooth-function bounds are matched by straightforward accelerated gradient descent, and related quadratic strongly convex objectives are matched up to constants and logarithmic factors.
Takeaways & Limitations
Similarity between local objectives can materially reduce communication difficulty, while without similarity even unlimited local computation cannot avoid many rounds in the covered model.
Takeaways & Limitations
The paper leaves assumption-free lower bounds for multiple communication rounds, runtime-efficient algorithms, and other distributed scenarios for future work.
Abstract
from arXiv · showhide
We study the fundamental limits to communication-efficient distributed methods for convex learning and optimization, under different assumptions on the information available to individual machines, and the types of functions considered. We identify cases where existing algorithms are already worst-case optimal, as well as cases where room for further improvement is still possible. Among other things, our results indicate that without similarity between the local objective functions (due to statistical data similarity or otherwise) many communication rounds may be required, even if the machines have unbounded computational power.
1 Introduction
The paper studies fundamental communication limits for distributed convex learning and optimization, contrasting unrelated and related local objectives. It proves lower bounds showing when existing methods are worst-case optimal and when optimal algorithms remain unknown.
- Problem and motivation: Distributed convex learning minimizes an aggregate objective whose local convex functions are distributed across machines, making communication the central efficiency constraint.Machines typically exchange dimension-sized vectors or gradients rather than entire high-dimensional datasets.
- Problem and motivation: The paper reverses the usual algorithm-design perspective by studying fundamental performance limitations and identifying settings where existing algorithms are optimal or improvable.
- Unrelated objectives: Under a mild structural algorithm assumption, unrelated local functions require Ω(1/ε) communication rounds for smooth convex optimization, matching distributed accelerated gradient descent.The lower bound persists even with unbounded local computational power.
- Unrelated objectives: For λ-strongly convex and general convex functions in the unrelated setting, the lower bounds are Ω(1/λε) and Ω(1/ε), respectively.The paper suggests acceleration combined with Moreau proximal smoothing may match these bounds, but leaves a full derivation for future work.
- Related objectives: For δ-related smooth λ-strongly convex functions, the lower bound is Ω(δ/λ log(1/ε)); it is matched for quadratics, while general optimal algorithms remain open.
- One-round communication: With one communication round and fewer than Ω(d^2) communicated bits, distributed algorithms may perform no better than returning one local minimizer.Thus no communication-efficient one-round method has non-trivial worst-case performance in the stated broad regime.
2 Notation and Framework
The framework models iterative distributed optimization with dimension-scaled broadcasts, separates function geometry from inter-machine similarity, and parameterizes relatedness through δ. Its lower bounds cover the δ-related setting but do not directly apply to randomly partitioned data.
- Communication model: Each round consists of unrestricted local computation followed by broadcasts to all machines, after which a designated machine outputs the solution.
- Communication model: Messages are typically about Õ(d) bits, sufficient for a d-dimensional real-valued vector, whereas unrestricted communication would allow sending each local function in one round.
- Problem classes: The framework asks how many communication rounds are needed to solve the aggregate problem to accuracy ε under convexity, strong convexity, smoothness, and relatedness assumptions.
- Problem classes: Quadratic local functions have the form Fi(w) := w⊤Aiw + biw + ci, and δ-relatedness defines how similar the local functions are across machines.
- Problem classes: For randomly split bounded-norm linear-regression data, δ can be O(1/√n); δ = Ω(1) represents unrelated objectives, while δ = 0 gives common minimizers and zero communication.
- Scope boundary: The paper’s δ-related lower bounds do not apply to random partitioning, where better performance is possible in certain strong-convexity and smoothness regimes.The random-partition result relies on statistical independence preserved by limiting how often each data point is accessed.
3 Lower Bounds Using a Structural Assumption
Under a mild structural assumption covering many black-box distributed optimization methods, the paper proves communication-round lower bounds across smoothness, convexity, and local-function similarity regimes. Unrelated functions can require many rounds even with unbounded local computation, while related-function bounds are partly matched and partly open.
- Structural assumption: The structural assumption permits linear operations on points, gradients, Hessians, inverses, and local optimization, without restricting between-round computation.It is substantially weaker than standard span-of-gradients assumptions and includes exact local optimization, gradient-based methods, and Hessian preconditioning.
- Unrelated local functions: For smooth convex functions with unrelated local objectives, the communication-round lower bound is Ω(1/ε), matching accelerated gradient descent up to constants and logarithmic factors.Thus, high-accuracy optimization may inherently require many rounds despite unbounded computational power at the machines.
- Related local functions: For δ-related smooth λ-strongly convex functions, the lower bound is Ω(δ/λ log(1/ε)); DISCO matches its δ and λ dependence for quadratics.Optimal algorithms remain open for general smooth strongly convex functions and for non-smooth or non-strongly convex functions.
- Lower-bound construction: The hard quadratic construction restricts progress after T rounds to the first T + 1 coordinates, so additional coordinates require further communication.This coordinate-growth mechanism produces the optimization-error lower bound.
- Non-smooth local functions: For non-smooth unrelated objectives, the lower bounds are Ω(1/λε) for λ-strongly convex functions and Ω(1/ε) for general convex functions.Acceleration combined with Moreau proximal smoothing is suggested as a way to match these bounds, although the full derivation is left for future work.
- Implications: Both strong convexity and smoothness are necessary for communication rounds to scale logarithmically with accuracy ε within the structural-assumption framework.The conclusion continues to hold when machines may perform arbitrarily many permitted local operations between rounds.
4 One Round of Communication
Without the structural assumption, the paper studies one-round distributed optimization, a realistic non-interactive setting in which machines broadcast one message before a designated machine outputs a solution. It shows that limited one-round communication can prevent non-trivial optimization accuracy, even for related strongly convex quadratic functions.
- Setting: The unrestricted analysis is limited to one communication round, covering non-interactive algorithms such as one-shot averaging.Each machine broadcasts a single message, after which a designated machine produces the output.
- Lower bound: With at most d^2/128 communicated bits, there exist δ-related, λ-strongly convex, and 9λ-smooth quadratic functions that are hard to optimize in one round.The result applies to possibly randomized algorithms and assumes δ ≥ 3λ > 0.
- Lower-bound consequence: Unless the communication budget is quadratic in dimension, one-round algorithms may achieve no better accuracy than returning the optimum of a single local function.This establishes a lower bound on attainable error relative to a zero-communication baseline.
- Hardness mechanism: The hard instance makes the global optimum depend on a particular column of a matrix known to one machine but not the designated output machine.Communicating enough information about that column is difficult when the message is much smaller than the matrix, whose size is d^2.
5 Summary and Open Questions
The paper establishes communication-round lower bounds for distributed convex learning across several settings, including unrelated and related local functions and one-round algorithms. It concludes that many-round optimality is settled in some cases, while assumption-free multi-round bounds, computational efficiency, and other scenarios remain open.
- Summary and Open Questions: For unrelated local functions, many communication rounds may be necessary regardless of local computational power, with accelerated gradient descent worst-case optimal for smooth functions.When functions are related, the algorithm of [26] is optimal for quadratic strongly convex functions, while optimal algorithms for more general functions remain open.
- Summary and Open Questions: Assumption-free one-round lower bounds show that even strongly convex quadratic problems can have only trivial performance.This complements the lower bounds requiring a mild structural assumption on the algorithm.
- Summary and Open Questions: Assumption-free lower bounds for algorithms using multiple communication rounds remain an open question.The paper also identifies runtime-efficient algorithms and lower bounds for other distributed learning and optimization scenarios as open directions.
- Summary and Open Questions: The paper leaves computational complexity of local computations as an important open issue alongside communication complexity.The authors specifically ask what performance is attainable with simple, runtime-efficient algorithms.
A.1 Proof of Thm. 1
The proof constructs finite-dimensional restrictions of local functions whose feasible-point sets expand by at most one coordinate per communication round. This coordinate-growth limitation yields lower bounds on the rounds needed to reach a target accuracy.
- A.1 Proof of Thm. 1: The proof restricts infinite-dimensional local functions and their average to the first d coordinates, producing finite-dimensional instances in R^d.The restrictions set all coordinates beyond d to zero and use leading principal submatrices of the associated matrices.
- A.1 Proof of Thm. 1: After each communication round, feasible-point sets can expand from E_T,d to at most E_T+1,d under Assumption 1.Lemma 1 establishes this invariant, and Corollary 1 applies it recursively after T ≤ d − 1 rounds.
- A.1 Proof of Thm. 1: Machines holding the two restricted function types become stuck in alternating coordinate subspaces before communication, preventing further local progress.The proof shows F1 machines remain in E_2k+1,d while F2 machines reach E_2k+2,d before the next round.
- A.1 Proof of Thm. 1: The average function’s minimizer is represented by a geometric sequence, enabling a lower bound from the coordinates that remain inaccessible after T rounds.For T ≤ d − 2, all coordinates after the first T + 1 vanish, which is used to lower-bound suboptimality.
- A.1 Proof of Thm. 1: Choosing d sufficiently large and requiring the lower bound to fall below ǫ yields the theorem’s communication-round lower bound.The construction can alternatively use exact local minimizers, avoiding dependence of d on problem parameters except the number of rounds, but that analysis is more involved.
A.2 Proof of Thm. 2
The proof for non-smooth functions uses paired absolute-value constructions whose subgradients preserve a coordinate-support invariant. Applying this invariant recursively bounds the suboptimality achievable after a given number of communication rounds.
- A.2 Proof of Thm. 2: The constructed local functions are convex, λ-strongly convex, and (1+λ)-Lipschitz over the unit Euclidean ball.The λ = 0 case gives 1-Lipschitz functions, while λ > 0 supplies strong convexity through the quadratic term.
- A.2 Proof of Thm. 2: At most one new non-zero coordinate can be gained at each iteration, regardless of how subgradients are selected.Lemma 2 formalizes the same feasible-set expansion invariant for the non-differentiable setting.
- A.2 Proof of Thm. 2: After T communication rounds, the feasible-point sets are confined to E_T+1,d for all machines.The proof tracks F1 machines in E_2p+1,d and F2 machines in E_2p+2,d before communication, then applies the update recursively.
- A.2 Proof of Thm. 2: The lower-bound construction sets k = T + 2 so that the output has w[T + 2] = 0 after T rounds.The missing coordinate creates a gap from the optimum, which is converted into an objective-value lower bound using the triangle inequality.
- A.2 Proof of Thm. 2: The resulting suboptimality lower bound implies that achieving an ǫ-suboptimal solution requires at least the stated number of communication rounds for sufficiently small ǫ.The proof separately establishes the strongly convex case and extracts the non-strongly convex result.
A.3 Proof of Thm. 3
The proof constructs two related quadratic local functions and reduces one-round optimization to recovering information about a hidden matrix column. An information-theoretic argument shows that limited communication prevents accurate recovery and therefore forces nontrivial optimization error.
- A.3 Proof of Thm. 3: The constructed functions F1 and F2 are λ-strongly convex, 9λ-smooth, and δ-related.The proof verifies these properties through Hessian spectra, invertibility, and bounds on the difference between quadratic terms.
- A.3 Proof of Thm. 3: The optimum depends on the j-th column of M, which machines holding F1 must communicate to the designated output machine holding F2.The output machine knows j through F2 but not M, creating the information-transfer bottleneck.
- A.3 Proof of Thm. 3: A one-round algorithm communicating at most d^2/128 bits has a valid instance on which its returned predictor incurs the theorem’s lower-bounded error.Lemma 5 applies to possibly randomized algorithms and takes expectation over the algorithm’s randomness.
- A.3 Proof of Thm. 3: Yao’s minimax principle converts the distributional lower bound into a hard deterministic choice of M and j for randomized algorithms.The random-matrix spectral-norm concentration argument ensures the selected instance satisfies the required norm constraint.
- A.3 Proof of Thm. 3: The information bound follows because the relevant upper-right quadrant of M contains Θ(d^2) random variables, while the machines communicate fewer than d^2 bits.The proof bounds how much the broadcast transcript reveals about entries of the hidden column using mutual information and Pinsker’s inequality.