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Randomized Block Adaptive Linear System Solvers
Vivak Patel, Mohammad Jahangoshahi, Daniel Adrian Maldonado
TL;DR
Randomized solvers can be theoretically effective yet inefficient in practice, prompting the question of whether highly adapted methods retain convergence. The paper gives sufficient conditions for randomized block adaptive solvers, with worst-case exponential convergence results for two solver archetypes. It also provides numerical evidence and applies the theory across existing and novel solvers.
Problem
Randomized solvers can have a substantial disconnect between theoretical convergence rates and practical efficiency, while highly adapted solvers raise questions about whether and how fast they converge.
Source: §1 · pp. 1–2Method
The paper develops sufficient conditions for randomized block adaptive solvers, including Markovianity, N, π-exploration, and either finite projection spaces or uniform nontriviality.
Source: §1 · p. 2, §6 · p. 20Results
Under the stated conditions, the two RBAS archetypes converge exponentially fast to a solution, with numerical evidence illustrating key aspects of the theory.
Source: §6 · p. 20Takeaways & Limitations
The theory provides a basis for designing tailored randomized adaptive solvers whose exponential convergence can be established under the stated conditions.
Source: §1 · p. 2, §6 · p. 20The cited corollaries have scope boundaries: the row-action proof strategy does not extend directly to infinite projection sets, and Corollary 3.18 does not cover Example 3.14 when nk < rank(A).
Source: §3.1 · p. 7, §3.2 · p. 10
Abstract
from arXiv · showhide
Randomized linear solvers randomly compress and solve a linear system with compelling theoretical convergence rates and computational complexities. However, such solvers suffer a substantial disconnect between their theoretical rates and actual efficiency in practice. Fortunately, these solvers are quite flexible and can be adapted to specific problems and computing environments to ensure high efficiency in practice, even at the cost of lower effectiveness (i.e., having a slower theoretical rate of convergence). While highly efficient adapted solvers can be readily designed by application experts, will such solvers still converge and at what rate? To answer this, we distill three general criteria for randomized adaptive solvers, which, as we show, will guarantee a worst-case exponential rate of convergence of the solver applied to consistent and inconsistent linear systems irrespective of whether such systems are over-determined, under-determined or rank-deficient. As a result, we enable application experts to design randomized adaptive solvers that achieve efficiency and can be verified for effectiveness using our theory. We demonstrate our theory on twenty-six solvers, nine of which are novel or novel block extensions of existing methods to the best of our knowledge.
1. Introduction.
Randomized solvers can have strong theoretical convergence yet perform poorly in practice, motivating adaptation to specific systems and computing environments. The paper gives sufficient conditions for adapted block solvers to retain worst-case exponential convergence and applies the theory to 26 solvers.
1. Introduction.
- Randomized solvers can be theoretically fast yet practically inefficient, including when access patterns violate data locality.The paper’s motivating example shows this disconnect for an oracle solver and block Kaczmarz.Source: §1 · p. 1
- Adaptation can improve practical efficiency for a specific system or computing environment, but raises the question of whether convergence persists and at what rate.The paper frames this trade-off as a practitioner’s concern when designing highly tailored methods.Source: §1 · p. 2
- Earlier exponential-convergence results for adaptive solvers were limited, and existing theory did not cover high-efficiency block operations.The cited prior work imposed finite-projection and exactness conditions or addressed vector-operation methods.Source: §1 · p. 2
- Adapted block solvers satisfying the stated sufficient conditions have worst-case exponential convergence for row-action methods on consistent systems and column-action methods on least-squares problems.Both settings include overdetermined, underdetermined, and rank-deficient systems.Source: §1 · p. 2
- The authors apply their results to 26 solvers, including nine they describe as novel methods or novel block extensions, to support solver design for specific problems and environments.Novelty is qualified as being to the authors’ best knowledge.Source: §1 · p. 2
2. Counter Example.
The example illustrates that theoretical operation counts can diverge from actual solver efficiency. The oracle solver is slower in practice than block Kaczmarz because its access pattern sacrifices data locality.
2. Counter Example.
- The example uses a consistent system with 107 equations and 100 unknowns, split into 0.5 Gigabyte chunks for a computer with 4 Gigabytes of memory.The system is represented in double precision.Source: §2 · p. 3
- The oracle solver replaces 100 equations with identity rows and then applies Kaczmarz to those rows, requiring about 100 arithmetic operations in the example.The solver uses knowledge of the indices of the replaced equations.Source: §2 · p. 3
- Despite its theoretical advantage, the oracle solver is slower in practice than random block Kaczmarz because it must read new chunks to access embedded equations, violating data locality.Block Kaczmarz uses the information in a single chunk, which the example says suffices for a high-quality solution in one iteration.Source: §2 · p. 3
- The example motivates adapting solvers for efficiency while establishing the need for conditions that preserve worst-case exponential convergence.The authors warn that efficient adaptation can otherwise result in arbitrarily slow convergence.Source: §2 · p. 3
3. Randomized Block Adaptive Solvers.
This section presents randomized block adaptive solvers as row- or column-action methods for linear systems that may be consistent or inconsistent, over- or under-determined, or rank-deficient. It gives column-action guarantees for consistent and inconsistent systems and notes a limitation of Corollary 3.18 for Example 3.14 when n_k < rank(A).
3.1. Row-action RBASs.
- Row-action updates may choose blocks adaptively and retain finite prior state; the stated convergence framework requires Markovian, N,π-exploratory behavior.For example, random-permutation block Kaczmarz eventually updates a nonsolution iterate because it encounters a block with nonzero residual within a cycle.Source: §3.1 · pp. 4, 6
- For consistent systems with finite row-space choices, Corollary 3.8 gives finite-expected-time termination or probability-one worst-case convergence with controlled expected stopping times.In the convergence branch, E[τj] ≤ j[(rank(A) − 1)(N/π) + 1]; otherwise, xτ = PHx0.Source: §3.1 · pp. 6–7
- For infinite row-space choices, uniformly nontrivial mean behavior replaces a finite-set bound and supports an analogous convergence result without controlling higher moments.The infinite-set case can defeat the finite-set proof because the supremum contraction factor may equal one; the result needs only expected behavior, despite dependence among the factors.Source: §3.1 · pp. 7–8
3.2. Column-action RBASs.
- Column-action RBASs do not require consistency, and under the finite-set, Markovian, and N,π-exploratory conditions, Corollary 3.18 allows finite-expected-time attainment of Axτ − b = r*.Here r* = −Pker(A⊺)b.Source: §3.2 · pp. 8, 10
- Corollary 3.18 does not cover Example 3.14 when n_k < rank(A).Source: §3.2 · p. 10
- Under Definitions 3.15, 3.17, and 3.19, Corollary 3.20 gives either finite-expected-time attainment of r* or stopping times with E[τ_j] ≤ j[(rank(A)−1)(N/π)+1].The stopping-time alternative also states the existence of γ̄ ∈ (0, 1) and a sequence γ_j ∈ (0, 1).Source: §3.2 · pp. 10–11