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Randomized Block Kaczmarz Method with Projection for Solving Least Squares
Deanna Needell, Ran Zhao, Anastasios Zouzias
TL;DR
Standard randomized Kaczmarz cannot reach the least-squares solution for inconsistent systems, motivating methods that overcome this convergence barrier. The paper develops randomized block variants using matrix pavings and blocked projections, obtaining expected linear convergence to least-squares solutions and practical speedups. The approach depends on suitable paving or standardization conditions, and one empirically tested projection-update combination degraded performance.
Problem
Standard Kaczmarz converges only to an error threshold for inconsistent systems, whereas the desired target is the least-squares solution.
Method
The paper analyzes block Kaczmarz variants with randomized projection steps, using row and column matrix pavings or a column-paving coordinate-descent variant.
Results
The methods guarantee linear convergence in expectation to the true least-squares solution when a row paving is available, while blocking rows and columns can improve convergence speed.
Takeaways & Limitations
Matrix blocking can provide significant practical runtime speedups while preserving least-squares convergence, and column standardization offers a route when suitable row pavings are unavailable.
Takeaways & Limitations
Combining a traditional REK projection with a traditional block Kaczmarz update empirically degraded performance and required more epochs than the proposed algorithms.
Abstract
from arXiv · showhide
The Kaczmarz method is an iterative method for solving overcomplete linear systems of equations Ax=b. The randomized version of the Kaczmarz method put forth by Strohmer and Vershynin iteratively projects onto a randomly chosen solution space given by a single row of the matrix A and converges exponentially in expectation to the solution of a consistent system. In this paper we analyze two block versions of the method each with a randomized projection, that converge in expectation to the least squares solution of inconsistent systems. Our approach utilizes a paving of the matrix A to guarantee exponential convergence, and suggests that paving yields a significant improvement in performance in certain regimes. The proposed method is an extension of the block Kaczmarz method analyzed by Needell and Tropp and the Randomized Extended Kaczmarz method of Zouzias and Freris. The contribution is thus two-fold; unlike the standard Kaczmarz method, our methods converge to the least-squares solution of inconsistent systems, and by using appropriate blocks of the matrix this convergence can be significantly accelerated. Numerical experiments suggest that the proposed algorithm can indeed lead to advantages in practice.
1. Introduction
The Kaczmarz method solves overdetermined linear systems through iterative projections, while randomized and block variants improve convergence or computational efficiency. The paper targets inconsistent systems by combining randomized projections with matrix blocks to obtain expected linear convergence to least-squares solutions.
- Kaczmarz method: The Kaczmarz method iteratively projects estimates onto solution spaces of the equations in an overdetermined system.For a full-rank matrix, the basic algorithm cycles through rows from an initial estimate.
- Randomized Kaczmarz: Randomized row selection avoids dependence on row ordering and yields expected linear convergence to the unique solution.Rows are selected with probability proportional to their ℓ2 norms.
- Inconsistent systems: In inconsistent systems, standard randomized Kaczmarz converges only to an error threshold and cannot reach the least-squares solution.Its iterates remain in a single solution space, creating a convergence barrier determined by the noise.
- Extended Kaczmarz: Randomized Extended Kaczmarz breaks this barrier by adding a residual-reduction step and converges linearly in expectation to the least-squares solution.The method maintains both an estimate of the solution and an approximation to the projected right-hand side.
- Block Kaczmarz: Block Kaczmarz projects onto solution spaces defined by multiple rows, using matrix pavings to control block conditioning and support efficient computation.Blocks can enable fast matrix multiplies and substantial computational-time improvements.
- Paper contribution: The paper analyzes methods with blocked row and column projections that combine least-squares convergence with improved speed, including a column-paving variant for matrices lacking a good row paving.The authors report linear convergence in expectation and practical runtime speedups from blocking both dimensions.
2. The Randomized Double Block Kaczmarz Method
The randomized double block Kaczmarz method combines row and column blocking with randomized projections to solve inconsistent systems and converge to the least-squares solution. Matrix paving enables linear convergence and can improve computational performance in regimes where block operations are efficient.
- Algorithm: Algorithm 1 randomly selects a column block and a row block, projects an auxiliary vector, and updates the solution estimate using the selected row block.The auxiliary update removes a blockwise projection of z, while the solution update uses the residual after subtracting the auxiliary vector.
- Convergence guarantee: The convergence theorem assumes both a column paving and a row paving of A and guarantees a bound for the output estimate xT.The theorem combines the two paving structures to analyze the algorithm’s expected convergence.
- Convergence guarantee: The auxiliary iterates zk converge linearly to the projection of b onto the orthogonal complement of the range of A.This component is analyzed through a lemma using the column paving and the block projection operator.
- Comparison of methods: The method improves on randomized block Kaczmarz by converging linearly to xLS rather than only within a radius proportional to the error norm.It breaks this convergence horizon by iteratively removing the component of b orthogonal to the range of A.
- Comparison of convergence rates: When A is nearly square and block products are fast, Algorithm 1 is about n/(pβ) times faster than REK.The stated speedup can be significant when n is much larger than pβ.
- Comparison of convergence rates: Without fast matrix multiplication, per-epoch bounds may favor REK, although block methods can still converge faster in practice because of efficient linear-algebra subroutines.The comparison depends on whether iterations or full epochs provide the appropriate cost measure.
3. Obtaining matrix pavings
The paper obtains useful matrix pavings from row normalization and related subset-selection results, then uses them to establish convergence guarantees. For non-row-standardized matrices, normalization choices determine whether the method reaches the original least-squares solution and how the rate depends on row-norm variation.
- Row pavings: Row-standardized matrices admit row pavings with controlled parameters, and such pavings can often be constructed efficiently by random partitioning.Proposition 4 gives existence, while the cited literature provides construction mechanisms.
- Standardization and pavings: If A is row-standardized and also has a suitable column paving, both paving parameters can support the convergence analysis; otherwise, a column-standardized version can be used in Algorithm 1.Standardization may be performed before the algorithm or on the fly.
- Non-standardized matrices: Running Algorithm 1 on a standardized system may converge to a least-squares solution different from the original one when A is not row-standardized.This difference arises in the inconsistent case.
- Non-standardized matrices: Using the same paving with appropriate scaling can preserve convergence to the true least-squares solution, with bounds depending on the dynamic range of the row norms.When the row-norm dynamic range is bounded, the convergence rate matches the standardized case up to constants.
- Alternative construction: A column-paving-only alternative is useful when the row-norm dynamic range is unknown or unbounded.The paper notes that the least-squares solution can still be obtained from the column-normalized system.
4. A Randomized Block Coordinate Descent Method
Algorithm 2 combines randomized block coordinate updates with a column paving to estimate the least-squares solution for inconsistent systems. Its analysis establishes linear convergence, including for full-rank matrices and matrices handled through column standardization.
- Implementation: Column standardization guarantees bounded paving parameters while preserving access to the original least-squares solution after rescaling.The resulting rate can depend on the conditioning of both the original and standardized matrices.
- Algorithm 2: Algorithm 2 randomly selects column blocks, computes a pseudoinverse-based update, modifies the selected coordinates of x, and updates the residual vector z.The procedure initializes x0=0 and z0=b, then returns xT after T iterations.
- Algorithm 2: The iterates satisfy z_k+1 = b − Ax_k+1, linking the residual update directly to the current estimate.This invariant is proved by induction from the algorithm’s coordinate and residual updates.
- Convergence analysis: Combining the invariant with the convergence analysis shows that Algorithm 2 converges to the least-squares solution under a column paving.Theorem 7 states the resulting estimate bound in terms of the paving parameters.
- Convergence analysis: For full-rank A, a singular-value bound converts the mapped estimation error into a direct bound on ∥xLS − xT∥2.Corollary 8 gives the corresponding full-rank guarantee.
- Implementation: The paper reports that combining traditional REK projection and block Kaczmarz updates degraded performance and required more epochs than the proposed methods.The authors conjecture that projection and Kaczmarz updates should operate at comparable block sizes.
5. Summary of Approaches
The paper matches different algorithms and paving strategies to matrix normalization and row-norm structure. These approaches provide expected linear convergence to the true least-squares solution and can improve runtime through block updates.
- Consistent systems: Standardized consistent systems can be rescaled without changing their solution, allowing the standardized system to use the convergence guarantee of Corollary 5.The paper permits standardization before the algorithm or during its execution.
- Inconsistent standardized systems: Already-standardized inconsistent systems can directly use Corollary 5 to guarantee convergence to the least-squares solution.
- Bounded dynamic range: For bounded row-norm dynamic range, the method can retain approximately the same convergence rate while recovering the original least-squares solution without standardizing the system.This setting includes random matrices with tightly concentrated row norms.
- Unbounded dynamic range: When row norms have unbounded dynamic range, Algorithm 2 uses column paving for systems that cannot be divided into well-conditioned row blocks.
- Overall summary: The proposed approaches guarantee expected linear convergence to the true least-squares solution, while matrix blocks often provide substantial practical runtime speedups.
6. Experimental Results
Experiments on Gaussian and tomography systems compare the proposed block methods with REK using error, epochs, and CPU time. The reported results show runtime advantages and behavior consistent with convergence to the least-squares solution.
- Experimental setup: The experiments use small examples, 40 trials per method, and do not claim optimized implementations.Their purpose is to illustrate practical advantages of block methods over standard methods.
- Gaussian systems: On a normalized 300×100 Gaussian consistent system, the proposed algorithms outperform standard REK in runtime despite the matrix lacking natural block structure.The study tracks ℓ2-error across epochs and CPU time, with success defined at error 10^-6.
- Inconsistent systems: For inconsistent Gaussian systems, the experiments track error to xLS and use a residual norm of 0.5, including a matrix with poorly bounded dynamic range.
- Experimental setup: Epochs are normalized to represent comparable cycles through rows or column-paving blocks across REK and the block methods.For a column paving with b blocks, one epoch is defined as n/b iterations.
- Inconsistent systems: Algorithm 2 shows behavior similar to the consistent case and breaks the convergence horizon observed for the standard Kaczmarz method.
- Tomography: In tomography experiments with κ(A)=2.08, Algorithm 2 uses a random column paving from the standardized matrix, and results across paving sizes align with earlier experiments.
7. Related Work and Discussion
Related work establishes randomized, extended, and block Kaczmarz methods, while matrix paving supplies partitions for effective block updates. The paper positions its methods at the intersection of randomized projections, block acceleration, and paving construction.
- Kaczmarz methods: Kaczmarz methods were developed for linear systems and later used in computer tomography, while deterministic convergence often depends on difficult-to-analyze matrix properties.
- Randomized methods: Random row selection yields provable expected linear convergence, while the inconsistent-case extension reaches only a fixed-radius neighborhood of the least-squares solution.
- Experimental comparison: The paper’s experiments compare REK and the proposed block algorithms on consistent and inconsistent Gaussian systems using error versus CPU time.
- Block methods: Block Kaczmarz methods accelerate convergence by selecting row blocks, with recent analyses using matrix pavings to construct those blocks.
- Matrix paving: Paving theory provides lower and upper block bounds through results developed by Bourgain, Tzafriri, Kashin, Spielman, Srivastava, Youssef, and others.
Appendix A. Proof of intermediate results
The appendix proves intermediate convergence bounds by recursively applying inequalities and balancing iteration counts, alongside experiments reporting error against epochs and CPU time.
- The proof recursively applies the first bound in (12) to derive the needed convergence estimate.
- Figure 4 plots ℓ2-norm error for REK and Algorithm 2 against epochs and CPU time on a 300 × 100 inconsistent Gaussian system.
- The argument uses the assumptions rk ≤ γkB ≤ B and geometric-sum properties to establish the intermediate inequalities.
- The proof concludes after combining the two inequalities and obtaining the stated bound.
- The proof chooses k and k∗ to sum to T, making them equal when T is even and differing by one when T is odd.
- Figure 5 plots median ℓ2-norm error against epoch and CPU time for a 1200×400 tomography system.