---
title: Random Greedy Fast Block Kaczmarz
url: https://www.emergentmind.com/topics/random-greedy-fast-block-kaczmarz
type: topic
---

# Random Greedy Fast Block Kaczmarz

Searching arXiv for the named method and closely related Kaczmarz variants to ground the article in current papers.
Random Greedy Fast Block Kaczmarz denotes a class of Kaczmarz-type iterative methods that combine random subsampling, greedy residual-based block selection, and low-cost block updates to solve large-scale systems, especially nonlinear systems. In the 2025 formulation for nonlinear equations, the method is presented for solving \(F(x)=0\) with \(F:\mathbb{R}^n\to\mathbb{R}^m\), and its defining feature is the replacement of Jacobian-submatrix pseudoinversion by a one-dimensional weighted residual step, yielding a pseudoinverse-free block update with linear convergence in expectation under a local tangential cone condition [2508.09596]. Related linear-system methods share the same structural motif—random sampling followed by greedy choice of high-residual rows—but differ in whether they use pseudoinverse-based projections, averaged updates, or extrapolated stepsizes [2211.16682] [2210.05128] [1902.09946].

## 1. Conceptual setting and problem class

The nonlinear RGFBK method is designed for the system
\[
F(x)=0,\qquad x\in D\subset\mathbb{R}^n,
\]
where \(F=(F_1,\dots,F_m)^T\) is continuously differentiable on a closed domain \(D\), and \(F'(x)\in\mathbb{R}^{m\times n}\) denotes the Jacobian [2508.09596]. For an index set \(I\subseteq[m]:=\{1,\dots,m\}\), the notation
\[
F_I(x)=(F_i(x))_{i\in I},\qquad F'_I(x)\in\mathbb{R}^{|I|\times n}
\]
is used to isolate sampled or greedily selected equations [2508.09596].

The method is motivated by the cost profile of classical block Kaczmarz schemes. In a block formulation, one typically enforces multiple equations simultaneously through a block correction involving the Moore–Penrose pseudoinverse of a Jacobian or coefficient submatrix. For large blocks or high-dimensional problems, that step can dominate the iteration cost. The nonlinear RGFBK method avoids this expense by using only one matrix–vector product of the form \(F'_I(x_k)^T\omega_k\), while retaining a block-selection mechanism that targets informative equations [2508.09596].

This places RGFBK at the intersection of several Kaczmarz lineages. Randomized block Kaczmarz emphasizes stochastic row or block sampling and analyzes convergence through conditioning of sampled submatrices [1902.09946]. Greedy and Motzkin-type variants prioritize equations with large residuals or violations [2011.06687]. Fast greedy block methods remove explicit pseudoinversion in the linear setting by replacing orthogonal block projections with averaged Kaczmarz-type updates [2210.05128]. Nonlinear greedy randomized sampling methods extend these ideas to Jacobian-based linearizations of \(f(x)=0\) under a local tangential cone condition [2209.06082]. The 2025 nonlinear RGFBK paper integrates these strands into a single pseudoinverse-free block method [2508.09596].

## 2. Algorithmic structure

The nonlinear RGFBK algorithm takes as inputs an initial guess \(x_0\in\mathbb{R}^n\), a control-sample size \(\alpha\in\{1,\dots,m\}\), a greedy-block size \(\beta\le\alpha\), and a relaxation parameter \(\gamma\in(0,2)\) [2508.09596]. At iteration \(k\), it first samples uniformly without replacement \(\alpha\) indices \(J_k\subseteq[m]\). It then computes the residual magnitudes \(|F_{J_k}(x_k)|\) and retains the \(\beta\) indices in \(J_k\) with largest \(|F_i(x_k)|\); this greedy subset is denoted \(\mathbb{I}_k\subseteq J_k\) [2508.09596].

The selected residual block becomes the weight vector,
\[
\omega_k=F_{\mathbb{I}_k}(x_k)\in\mathbb{R}^{\beta},
\]
and the update direction is
\[
d_k=F'_{\mathbb{I}_k}(x_k)^T\omega_k.
\]
The next iterate is computed as
\[
x_{k+1}=x_k-\gamma\cdot \frac{\omega_k^T F_{\mathbb{I}_k}(x_k)}{\|d_k\|_2}\,d_k.
\]
The iteration terminates when \(\|F(x_k)\|_2\) falls below tolerance or a maximum iteration count is reached [2508.09596].

Two aspects are structurally central. First, the method is “random greedy”: random sampling limits the search space from all \(m\) equations to a manageable subset of size \(\alpha\), and greedy filtering concentrates the update on the largest residuals within that subset [2508.09596]. Second, the update is “fast” in the specific sense used in prior fast greedy block Kaczmarz work: it avoids solving a block least-squares problem via pseudoinverse and instead performs a directionally scaled residual step [2210.05128]. In the linear fast greedy block Kaczmarz method, an analogous averaged update uses \(A^T\xi_k\) rather than \(A_{\tau_k}^\dagger(b_{\tau_k}-A_{\tau_k}x_k)\), with the same computational rationale [2210.05128].

The method therefore differs from pseudoinverse-based randomized block subsampling Kaczmarz–Motzkin schemes, which sample \(\beta\) rows, greedily retain \(\delta\) rows with largest residuals, and then compute
\[
d^k=A_{I_k}^\dagger(b_{I_k}-A_{I_k}x^k)
\]
before updating \(x^{k+1}=x^k+d^k\) [2211.16682]. RGFBK preserves the random-then-greedy logic but replaces the expensive block solve by a single Jacobian-transpose multiplication [2508.09596].

## 3. Pseudoinverse avoidance and the “fast” update

The defining computational innovation of nonlinear RGFBK is its avoidance of the classical block Kaczmarz step
\[
(F'_I(x_k))^\dagger F_I(x_k),
\]
that is, the Moore–Penrose pseudoinverse of the Jacobian submatrix applied to the selected residual block [2508.09596]. Instead, the method solves a one-dimensional weighted residual minimization and obtains an update of the form
\[
x_{k+1}=x_k-\gamma\frac{\omega_k^T F_I(x_k)}{\|F'_I(x_k)^T\omega_k\|}\,F'_I(x_k)^T\omega_k,
\]
which requires only one matrix–vector product \(F'_I(x_k)^T\omega_k\) and no pseudoinverse [2508.09596].

The computational significance is explicit in the source formulation: this reduces per-iteration cost, especially when \(I\) is large, and is better suited for high-dimensional problems [2508.09596]. The same contrast appears in linear fast greedy block Kaczmarz analysis, where pseudoinverse-based block updates can cost \(O(n|\tau_k|^2)\) or more if computed by SVD, whereas the fast averaged alternative replaces them by sparse matrix–vector operations [2210.05128].

A closely related distinction separates RGFBK from block nonlinear Kaczmarz methods in which the selected block \(\mathcal{I}_k\) is updated through
\[
x_{k+1}=x_k-(f'_{\mathcal{I}_k}(x_k))^\dagger f_{\mathcal{I}_k}(x_k),
\]
as in the BSNK1 and BSNK2 families for nonlinear systems [2209.06082]. Those methods establish linear convergence in expectation under the local tangential cone condition, but they retain the pseudoinverse bottleneck [2209.06082]. RGFBK can therefore be understood as a fast block analogue within the nonlinear Kaczmarz landscape.

This suggests a useful taxonomy. One branch comprises exact block-projection methods, which project onto the solution set of the chosen block and typically invoke a pseudoinverse [2211.16682] [2209.06082]. A second branch comprises fast or averaged methods, which replace exact block projection by a directionally scaled aggregate correction and thereby reduce arithmetic and memory overhead [2210.05128] [2508.09596]. The nonlinear RGFBK method belongs to the second branch.

## 4. Stochastic greedy conditioning and convergence theory

The convergence analysis of nonlinear RGFBK is formulated in terms of the local tangential cone condition and a lower bound on the norms of the Jacobian rows \(\nabla F_i(x)\), which are assumed to be bounded below by some \(\epsilon>0\) [2508.09596]. If \(x_*\) is a solution of \(F(x)=0\), then for the update choice \(\omega_k=F_{\mathbb{I}_k}(x_k)\) and a relaxation parameter satisfying
\[
\gamma\in(0,2(1-\eta)),
\]
where \(\eta<\tfrac12\) comes from the cone condition, the iterates satisfy
\[
E[\|x_{k+1}-x_*\|_2^2]
\le
\left[
1-\frac{2\gamma(1-\eta)-\gamma^2}{(1+\eta^2)\kappa_{\alpha,\beta}^2(F'(x_k))}
\right]
E[\|x_k-x_*\|_2^2]
\]
[2508.09596].

The quantity controlling the rate is the stochastic greedy condition number. Let
\[
\mathbb{I}^{\alpha,\beta}
=
\{\text{any }\beta\text{-subset chosen greedily from some }\alpha\text{-subset of }[m]\}.
\]
For a matrix \(A\in\mathbb{R}^{m\times n}\), define
\[
\kappa_{\alpha,\beta}(A)
=
\max_{I\in\mathbb{I}^{\alpha,\beta}}\kappa(A_I)
=
\max_{I\in\mathbb{I}^{\alpha,\beta}}
\frac{\sigma_{\max}(A_I)}{\sigma_{\min}(A_I)}.
\]
In the nonlinear method one substitutes \(A=F'(x_k)\) [2508.09596].

The rate factor is therefore
\[
\rho_k
=
1-
\frac{2\gamma(1-\eta)-\gamma^2}
{(1+\eta^2)\kappa_{\alpha,\beta}^2(F'(x_k))}.
\]
The optimal choice \(\gamma=1-\eta\) yields
\[
\rho_k^*
=
1-
\frac{1-2\eta+\eta^2}
{(1+\eta^2)\kappa_{\alpha,\beta}^2(F'(x_k))}
\]
[2508.09596].

The proof strategy begins from a descent lemma showing
\[
\|x_{k+1}-x_*\|_2^2
\le
\|x_k-x_*\|_2^2
-
[2\gamma(1-\eta)-\gamma^2]
\cdot
\frac{\|F_I(x_k)\|_2^2}{\sigma_{\max}^2(F'_I(x_k))}
\]
for \(\omega_k=F_I(x_k)\); conditional expectation over the random greedy choice \(I=\mathbb{I}_k\), the cone condition, and singular-value bounds then introduce \(\kappa_{\alpha,\beta}(F'(x_k))\) [2508.09596].

This mode of analysis parallels earlier randomized block Kaczmarz theory in the linear case, where convergence rates depend on geometric properties of sampled submatrices and on block conditioning [1902.09946]. It also echoes the expectation-based convergence results of nonlinear greedy randomized sampling Kaczmarz methods, which obtain linear convergence factors involving cone-condition constants and Jacobian-dependent quantities for both single-sample and block methods [2209.06082]. The distinctive feature of RGFBK is that conditioning is encoded through a stochastic greedy condition number tailored to random-then-greedy block formation [2508.09596].

## 5. Parameterization, practical tuning, and large-scale use

The core algorithmic parameters are the control-sample size \(\alpha\), the greedy-block size \(\beta\), and the relaxation parameter \(\gamma\) [2508.09596]. Their practical roles are distinct. The sample size \(\alpha\) governs how much random exploration occurs at each iteration; the block size \(\beta\) controls how many high-residual equations are aggregated into the update; and \(\gamma\) scales the correction [2508.09596].

For the nonlinear method, the stated convergence requirement is \(\gamma\in(0,2(1-\eta))\) if \(\eta\) is known, and the practical guidance is that \(\eta\in[0,0.5)\) often leads to good performance with \(\gamma\approx1\) [2508.09596]. If an estimate of \(\eta\) is unavailable, a safe default is \(\gamma\in[0.8,1.2]\), and one may perform a short line-search on \(\gamma\) [2508.09596]. The source also recommends selecting \(\alpha\) moderately large, for example \(50\%\) to \(75\%\) of \(m\), and \(\beta\approx30\%\) to \(60\%\) of \(\alpha\), in order to balance block update power against cost [2508.09596].

Memory-aware implementation is emphasized. Only the Jacobian rows for indices in \(\mathbb{I}_k\) need to be precomputed, stored, or computed efficiently, which helps keep memory under control [2508.09596]. Monitoring \(\|F(x_k)\|_2\) and adapting \(\gamma\) downward if stagnation or oscillation appears is also recommended [2508.09596].

These guidelines are consistent with the broader Kaczmarz literature. Randomized block subsampling methods report a U-shaped total-time curve when varying sample and greedy block sizes: too small a block yields many iterations, while too large a block raises per-iteration cost [2211.16682]. Fast greedy block Kaczmarz for linear systems similarly notes that smaller selection thresholds produce larger blocks and alter the balance between progress and cost [2210.05128]. This suggests that the empirical tuning rules in RGFBK reflect a general block-Kaczmarz trade-off rather than an isolated peculiarity of the nonlinear method.

## 6. Numerical behavior and relation to adjacent methods

The nonlinear RGFBK paper reports experiments on two large-scale test problems: the Chandrasekhar \(H\)-equation, with dense Jacobians and discretized dimension \(m\) up to \(10^4\), and the Broyden tridiagonal function, with sparse Jacobian and \(m\) up to \(10^3\) [2508.09596]. The stopping criterion is
\[
\|F(x)\|_2 \le 10^{-6}+10^{-8}\|F(x_0)\|_2
\]
or \(10^5\) iterations, and the reported parameter choice for RGFBK is \(\alpha=\lfloor0.75m\rfloor\), \(\beta=\lfloor0.5\alpha\rfloor\), \(\gamma=1.2\) [2508.09596].

The comparison set includes MR-BSNK, MD-BSNK, RB-CNK, and RBWNK [2508.09596]. According to the reported summary, RGFBK required roughly half the iterations and CPU time of the best competing block methods on the \(H\)-equation for \(m\) up to \(10^4\). On the Broyden problem, competing methods either diverged or slowed dramatically as \(m\) grew, while RGFBK remained robust and grew slowly in CPU. A parameter study further identified an optimal trade-off around \(\alpha\approx0.6m\), \(\beta\approx0.4\alpha\) [2508.09596].

The source interprets these outcomes through the stochastic greedy condition number and the relaxation parameter: when the Jacobian matrix exhibits a favorable stochastic greedy condition number and an appropriate relaxation parameter is selected, convergence is significantly accelerated [2508.09596]. A plausible implication is that the method derives practical gains not only from cheaper iterations but also from more effective submatrix selection.

The performance claims fit a broader empirical pattern across related methods. In the linear setting, fast greedy block Kaczmarz often uses fewer CPU seconds and fewer iterations than earlier greedy or deterministic block Kaczmarz schemes [2210.05128]. Randomized block subsampling Kaczmarz–Motzkin methods outperform greedy block methods without random subsampling when \((\beta,\delta)\) is tuned well [2211.16682]. Nonlinear greedy randomized sampling methods report that block methods outperform single-sample variants and that greedy selection rules give faster convergence rates than random ones in applications including the Brown almost linear function and generalized linear model [2209.06082]. The nonlinear RGFBK results therefore extend an established empirical trend into the pseudoinverse-free nonlinear block regime [2508.09596].

## 7. Historical lineage, distinctions, and common misconceptions

The phrase “Random Greedy Fast Block Kaczmarz” can refer to more than one nearby construction, and distinguishing these is essential. The 2025 nonlinear method is specifically a solver for large-scale nonlinear systems \(F(x)=0\) that uses random sampling, greedy block selection, and a pseudoinverse-free update based on \(F'_I(x_k)^T\omega_k\) [2508.09596]. Earlier fast greedy block Kaczmarz work addressed consistent linear systems \(Ax=b\), with a fully greedy thresholding rule and an averaged update \(x_{k+1}=x_k+\alpha_k A^T\xi_k\), but without the initial random subsampling stage that defines the nonlinear RGFBK sampling mechanism [2210.05128].

A second source of confusion is the relation to randomized block subsampling Kaczmarz–Motzkin methods. Those methods also sample a subset of rows, greedily retain the largest residuals, and update on a block; however, the update is a classical block projection using \(A_{I_k}^\dagger\) [2211.16682]. The 2025 nonlinear RGFBK method explicitly avoids that pseudoinverse [2508.09596]. Referring to both methods as “random greedy block Kaczmarz” captures a family resemblance, but obscures the computational distinction between exact block projection and fast averaged correction.

A third distinction concerns the notion of “greedy.” In some Kaczmarz papers, greediness means choosing the maximum residual, as in Motzkin-type rules [2011.06687] [2209.06082]. In others, it refers to selecting all rows above a threshold derived from the maximum scaled residual [2210.05128]. In RGFBK, the greedy rule is top-\(\beta\) selection among a random control set \(J_k\) according to largest \(|F_i(x_k)|\) [2508.09596]. These are related but not identical selection logics.

Finally, the method should not be conflated with generic randomized block Kaczmarz frameworks that rely primarily on well-conditioned row pavings and extrapolated stepsizes. Faster randomized block Kaczmarz theory shows that well-conditioned blocks and extrapolated stepsizes can yield linear convergence in expectation with rates governed by stochastic block geometry [1902.09946]. RGFBK shares the dependence on submatrix conditioning, but its central object is the stochastic greedy condition number induced by random-then-greedy selection, and its analysis is tied to the nonlinear cone-condition setting [2508.09596].

Taken together, these distinctions situate Random Greedy Fast Block Kaczmarz as a specific member of a broader Kaczmarz family: a pseudoinverse-free, random-then-greedy block method whose nonlinear 2025 formulation combines computational economy with expectation-linear convergence guarantees and strong large-scale empirical behavior [2508.09596].

Source: https://www.emergentmind.com/topics/random-greedy-fast-block-kaczmarz