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Distributed Coordinate Descent Method for Learning with Big Data
Peter Richtárik, Martin Takáč
TL;DR
Hydra targets loss minimization when big data must be distributed across multiple computers. It partitions coordinates across nodes, independently samples local subsets, and applies parallel updates, while analyzing high-probability iteration bounds and dependence on data and partitioning. Experiments include a LASSO instance represented by a 3TB matrix.
Problem
Existing coordinate descent methods are not directly scalable when one computer cannot store the data efficiently, motivating methods for memory-distributed big-data optimization.
Method
Hydra partitions coordinates across cluster nodes, independently samples τ owned coordinates per node, and computes and applies local updates in parallel.
Results
For strongly convex losses, Hydra has a high-probability ϵ-accuracy iteration bound whose behavior depends on data quantities and the coordinate partition.
Takeaways & Limitations
The spectral norm σ and partition-induced norm σ′ provide data-dependent indicators of Hydra’s parallelization potential and partition quality.
Takeaways & Limitations
The advanced ring communication protocol increases storage requirements and can propagate information to all nodes only after c−1 iterations.
Abstract
from arXiv · showhide
In this paper we develop and analyze Hydra: HYbriD cooRdinAte descent method for solving loss minimization problems with big data. We initially partition the coordinates (features) and assign each partition to a different node of a cluster. At every iteration, each node picks a random subset of the coordinates from those it owns, independently from the other computers, and in parallel computes and applies updates to the selected coordinates based on a simple closed-form formula. We give bounds on the number of iterations sufficient to approximately solve the problem with high probability, and show how it depends on the data and on the partitioning. We perform numerical experiments with a LASSO instance described by a 3TB matrix.
1. Introduction
Hydra addresses big-data optimization problems that cannot fit efficiently on one computer by combining cluster-level and within-node coordinate parallelism. Its convergence and parallelization depend on the data and coordinate partition, with high-probability accuracy guarantees for strongly convex losses.
- Motivation: Existing coordinate descent methods are not directly scalable when one computer cannot store the problem data efficiently.Big-data settings therefore require memory-distributed methods across cluster nodes.
- Contribution: Hydra is presented as a distributed coordinate descent method with parallelism across cluster nodes and within individual nodes.The method is described as hybrid because it exploits both levels of parallel processing.
- Method: Each node receives a coordinate partition, stores its features locally, and independently updates τ randomly selected owned coordinates per iteration.Across c nodes, cτ coordinates are updated in each iteration.
- Main insight: Hydra’s parallelization potential depends on the data spectral norm σ and partition-induced norm σ′.Small σ can support nearly linear speedup as τ increases, while small σ′ indicates preferable partitions.
- Guarantee: For strongly convex losses, Hydra is guaranteed to produce an ϵ-accurate solution with probability at least 1 −ρ after a stated iteration bound.The supplied passage gives the bound in terms of d, β, c, τ, µ, ϵ, and ρ, though its displayed expression is truncated.
2. The problem
The paper studies minimizing a smooth convex loss plus a convex, possibly nonsmooth, separable regularizer. Its assumptions cover matrix-based losses and common regularization choices, including L1, L2, and bound constraints.
- Objective: The optimization problem minimizes a regularized loss of the form L(x) := f(x) + R(x).The objective combines a smooth loss with a regularizer.
- Assumptions: The loss f is assumed smooth and convex, while R is convex and may be nonsmooth.These are the core structural assumptions on the objective components.
- Loss model: The analysis assumes a positive definite matrix M that controls the loss geometry and can be written as M = A^T A.The matrix A is an n-by-d data matrix.
- Examples: The framework includes example losses built from n examples, d features, a single-example loss ℓ, and labels y.The supplied examples identify squared, logistic, and Huber-type losses as covered cases.
- Regularizer: The regularizer is separable, R(x) = Σ_i R_i(x_i), with each component convex and closed.Bound constraints, an L1 regularizer, and an L2 regularizer are listed as examples.
3. Distributed coordinate descent
Hydra partitions coordinates across computers and repeatedly performs independent local random sampling, parallel coordinate updates, and local application of those updates. Its updates use simple one-dimensional optimization, while communication and synchronization are explicit implementation considerations.
- Setup: Hydra partitions d coordinates into c equal-sized sets, assigns one set to each computer, and keeps the partition fixed during optimization.Each node owns s := d/c coordinates.
- Sampling: At each iteration, every computer independently samples τ owned coordinates uniformly from its partition.The sampled sets form a τ-distributed sampling rather than a uniformly random cτ-subset of all coordinates.
- Algorithm: Algorithm 1 runs the computers in parallel, selecting random coordinate sets before computing and applying their updates.The algorithm parameters include x0, the coordinate partition, β, and τ.
- Updates: Coordinate updates are computed using the corresponding partial derivative and a simple one-dimensional optimization problem.For some regularizers, the update subproblem has a closed-form solution.
- Stepsize: The stepsize β is central to convergence and speed, and its safe choice is linked to σ and σ′ or their computable upper bounds ω and ω′.The iteration count is proportional to β for fixed c and τ.
- Implementation: Derivative computation requires communication because the iterate and loss data are distributed, whereas applying local coordinate updates is communication-free.The analyzed method is synchronous, although the paper notes that asynchronous implementations may be faster in practice.
4. Convergence rate analysis
The convergence analysis defines data- and partition-dependent quantities for selecting Hydra’s stepsize, then proves exponential convergence under strong convexity and discusses bounds when those quantities are difficult to compute.
- Stepsize and data-dependent quantities: σ′ and σ govern the stepsize β, convergence rate, and potential speedup from parallelization and distribution.The paper also introduces easily computable upper bounds ω′ and ω because σ′ and σ may be difficult to compute.
- Stepsize and data-dependent quantities: ω′ measures how many partition blocks contain a nonzero in each data row, while ω measures the number of row nonzeros.These quantities provide interpretable sparsity-based upper bounds associated with the partition and the full data matrix.
- Stepsize and data-dependent quantities: Replacing σ′ with its upper bound ω′ eliminates the need to compute σ′ but can double β and the required number of iterations.The iteration consequence follows because the iteration count is proportional to β for fixed c and τ.
- Separable approximation: The analysis derives an expected quadratic-form identity for Hydra’s distributed sampling and uses it to construct a separable quadratic upper bound.The proof transforms the sampled quadratic form using Q and bounds its component quadratics using σ and the block structure BQ.
- Fast rates: Expected Separable Overapproximation inequalities support convergence analysis, but Hydra extends prior work to a different loss-function class and distributed sampling protocol.The cited prior analyses did not provide an efficient distributed protocol for the setting considered here.
- Fast rates: Under strong convexity with µf + µR > 0 and β ≥ β∗, Hydra decreases the objective at an exponential rate in ϵ.The theorem gives a high-probability iteration guarantee for reaching an approximate solution from an initial point x0.
5. Discussion
Hydra’s convergence depends on stepsize, strong convexity, parallelism, and data partitioning. The discussion also compares its distributed setting and guarantees with related coordinate-descent methods.
- Convergence rate: Strong convexity can mitigate the effect of a large β, leaving a rate term that yields linear speedup in c and τ.For large µR, (β + µR)/(µf + µR) approaches 1, and the bound is dominated by d/(cτ).
- Stepsize selection: Hydra uses β = β∗ as a safe small stepsize, while smaller practical values may produce larger steps and faster convergence.The safe choice avoids divergence risks associated with overly large stepsizes; replacing difficult quantities with upper bounds can increase β.
- Price of distribution: Using more nodes can incur a stepsize penalty determined by σ′, although favorable partitions can keep σ′ near 1.In general σ′ ≥ cσ/d and can be as large as c; choosing τ = s removes the effect associated with β∗².
- Stepsize selection: At most doubling β to avoid estimating σ′ translates into at most doubling the number of iterations.This provides a simpler alternative when σ′ is difficult to compute.
- Parallelism: Updating all coordinates at each node can require fewer iterations than updating one coordinate, and using c > 1 loses very little in iteration count.The comparison defines γτ = β∗cτ and reports γ1 ≥ γs; Figure 1 summarizes the small iteration-count difference between distributed and single-node settings.
- Price of distribution: Distribution slightly weakens performance in the illustrated comparison, but big-data instances may require multiple nodes because one computer cannot store the data efficiently.The reported comparison ignores communication cost.
- Comparison with other methods: Hydra is presented as a distributed coordinate-descent method, while related methods include Shotgun, PCDM, SPCDM, and mini-batch SDCA.The comparisons emphasize Hydra’s distributed setting, specialized loss class, stepsize analysis, and strong-convexity guarantees.
6. Distributed computation of the gradient
Hydra distributes feature data across nodes and computes selected-coordinate derivatives locally, while advanced protocols overlap computation with communication and reduce communication bottlenecks. ASL lowers per-iteration communication but propagates information more slowly and requires additional storage.
- Distributed storage: Hydra partitions coordinates and the data matrix columnwise across c computers, storing each feature partition locally.Each node owns its partition throughout the iterative process.
- Gradient computation: Local data suffices to compute each node’s update contribution, after which a reduce-all operation aggregates updates for all nodes.Nodes use the aggregated vector to compute required partial derivatives for their owned coordinates.
- Advanced protocols: The basic protocol alternates computation and serial MPI communication, whereas Fully Parallel dedicates one thread to communication while others compute.Figure 2 contrasts the parallel-serial and fully parallel approaches.
- Advanced protocols: ASL replaces reduce-all with asynchronous ring communication, sending one message to a neighboring node and receiving one from another.The ring orders nodes so each node receives from l− and sends to l+.
- Advanced protocols: ASL needs less communication per iteration but takes c−1 iterations to propagate information to all nodes and increases storage requirements.At iteration k, nodes store update vectors from iterations k−c through k.
7. Experiments
Experiments evaluate Hydra on controlled LASSO problems, including a 3TB sparse matrix, using multiple communication protocols. ASL-FP provides the largest gains over RA-PS, with speedups depending on the computation-to-communication balance.
- Experimental setup: Experiments use a C++ Hydra implementation with Boost.MPI and OpenMP on a 128-node Cray XE6 cluster for LASSO problems.Each node has two 16-core AMD Opteron processors and 32 GB of RAM.
- Experimental setup: The benchmark uses 128 nodes, 512 MPI processes, 4,096 cores, and a 3TB double-precision matrix with n = 10^9 rows and d = 5 × 10^8 columns.Table 4 compares average iteration times across three communication protocols and three values of τ.
- Protocol comparison: ASL-FP yields the largest gains over the benchmark RA-PS protocol across the tested settings.The basic RA-PS protocol remains the theoretically analyzed benchmark.
- Protocol comparison: 1.62 times speedup occurs for τ = 10, while ASL-FP is 3.11 times faster than RA-PS for τ = 10^2.For τ = 10, communication overhead limits speedup; for τ = 10^2, computation and communication durations are comparable.
- Protocol comparison: For τ = 10^3, the gain is moderate because computation takes much longer than communication, making the auxiliary-vector strategy less significant.Larger τ also requires larger β and may increase the number of iterations in the worst case.
- Large-scale LASSO: On the 3TB problem, ASL-FP significantly outperforms RA-FP and pushes L(x^k) − L* down by 25 orders of magnitude in under 30 minutes.Figure 4 compares loss evolution over time for the two protocols.
8. Extensions
The paper states that Hydra’s results extend beyond coordinate updates to block-coordinate settings and partially separable losses.
- Extensions: Hydra’s results can be extended to blocks of coordinates and to partially separable losses.The stated extensions connect Hydra with block-coordinate methods and partially separable-loss settings.
A. Proof Lemma 1
The proof establishes block-wise smoothness properties for the quadratic model and relates the relevant constants through bounds involving the coordinate partition. These relationships support interpretable bounds for distributed coordinate-descent analysis.
- Constant bounds: The proof establishes 1 ≤ σ′ ≤ ω′ ≤ c for the partition-based quantities.The bounds are derived using the block structure and quadratic-form relations.
- Block Lipschitz property: For each coordinate block P_k, the quadratic model’s gradient is block Lipschitz with respect to the block norm and constant 1.The passage states that this property is satisfied with equality.
- Special case: When c = d and each partition contains one coordinate, the partition-based constants reduce to the coordinate-wise constants σ′ = σ and ω′ = ω.In this special case, BQ = DQ and the relevant quadratic forms coincide.
B. Proof of Lemma 2
The proof bounds β∗_2 using the relationship between σ′ and c = d, then reduces the required inequality to a clearly nonnegative expression.
- β∗_2 is increasing in σ′, while Lemma 1 gives σ′ ≤ c = d.
- The argument therefore reduces to proving a simpler inequality.
- After simplification, the inequality becomes (s − τ) + (τ − 2)σ + σ/d (s + τ) ≥ 0.
C. Proof of Lemma 3
The proof handles s = 1 directly and analyzes s > 1 through joint-selection probabilities for coordinates within the same or different partitions.
- When s = 1, τ = 1 and the sampled set contains all coordinates with probability one.
- For s > 1, the proof invokes Lemma 3 from Richtárik and Takáč (2012a).
- The joint-selection probability p_ij is defined as Prob(i ∈ Ŝ & j ∈ Ŝ).
- For coordinates in the same partition, p_ij equals s(s−1) in the stated case; for different partitions, it equals τ^2/s^2.
- The proof substitutes these probabilities into equation (22) and transforms the result into the desired form.