Parallel Blockwise Computation Scheme

• Parallel blockwise computation schemes are techniques that partition complex optimization problems into independent or weakly coupled blocks for efficient distributed processing.
• The approach employs blockwise local approximations and quadratic regularization to ensure strong convergence and scalability on modern multicore and distributed architectures.
• Empirical evidence in high-dimensional settings, such as Lasso regression, shows enhanced performance and resource utilization compared to traditional sequential methods.

A parallel blockwise computation scheme is a computational strategy that partitions large-scale optimization, inference, or algebraic problems into independent or weakly coupled blocks, enabling distributed or parallel computation of updates or subproblems specific to each block. Such schemes are increasingly central to modern optimization, deep learning, large-scale data analysis, and scientific computing, as they permit significant improvements in scalability, resource utilization, and overall solution efficiency.

1. Mathematical Formulation and Problem Structure

Parallel blockwise schemes typically target composite objective functions of the form

$$\min_{x \in X = X_1 \times \cdots \times X_N} \quad V(x) = F(x) + G(x),$$

where $F$ is a potentially nonconvex, smooth function (e.g., a loss or data-fidelity term) with partial coupling across blocks, and $G$ is a block-separable and possibly nonsmooth convex function (e.g., blockwise regularization or constraint indicator) (Facchinei et al., 2013). The variable $x$ is partitioned into $N$ distinct blocks $x_i \in X_i$, and the strategy proceeds by solving blockwise subproblems—each involving only $x_i$—in parallel, either exactly or approximately.

Key approaches include:

• Blockwise local approximations $P_i(z; w)$ to $F$ at the current iterate, with convexity, gradient matching, and Lipschitz continuity properties.
• Quadratic regularization for strong convexity of the surrogate subproblems.
• Parallelism through selection and update of a subset of blocks at each iteration (from full Jacobi to Southwell/coordinate descent).

2. Update Rules and Parallelization Mechanisms

The central update mechanism involves, for each block $i$, minimizing a strongly convex surrogate

$$\widetilde{h}_i(x_i; x^k) = P_i(x_i; x^k) + \frac{\tau_i}{2} (x_i - x_i^k)^T Q_i(x^k)(x_i - x_i^k) + g_i(x_i),$$

yielding an in-block update

$$\hat{x}_i(x^k, \tau_i) = \arg\min_{x_i \in X_i} \widetilde{h}_i(x_i; x^k).$$

Updates across blocks are executed in parallel according to a selected index set $S^k$; the new iterate is assembled via

$$x^{k+1} = x^k + \gamma^k (\hat{z}^k - x^k), \quad \text{with} \quad \hat{z}_i^k = \begin{cases} z_i^k & i \in S^k \ x_i^k & i \notin S^k \end{cases}$$

and where $z_i^k$ may be computed to within prescribed inexactness.

Flexibility is achieved by:

• Varying the selection strategy (from all blocks, yielding a full Jacobi step, to a single block as in Gauss-Seidel/Southwell).
• Adapting the approximation $P_i$ for linear/quadratic/second-order information or block convex structure.
• Allowing inexact solves and arbitrary (possibly diminishing) step sizes $\gamma^k$.

3. Theoretical Convergence and Complexity Properties

The theoretical guarantees are established under broad assumptions:

• Convexity of each $X_i$ and separability of $G$.
• Lipschitz continuity of $\nabla F$ and coercivity of $V$.
• Properties (P1–P3) for $P_i$ and positive definiteness of $Q_i$.

The main convergence theorem (Theorem 1 in (Facchinei et al., 2013)) shows:

• For step sizes $\gamma^k \to 0$ with $\sum_k \gamma^k = \infty$ and $\sum_k (\gamma^k)^2 < \infty$, and if approximation errors decrease appropriately,
• Every limit point of $\{x^k\}$ is stationary, even for nonconvex $F$ and with arbitrary block update selection.
• A strong descent property is established at each iteration: $$V(x^{k+1}) \leq V(x^k) - \gamma^k \beta \| \hat{x}(x^k) - x^k \|^2 + \text{(small error terms)}$$ for some $\beta > 0$, ensuring steady decrease of the objective until convergence.

This generalized framework improves upon prior block-parallel schemes that required strong contraction assumptions or limited update rules.

4. Algorithmic Flexibility and Realization

The decomposition framework subsumes many familiar parallel and blockwise algorithms:

• Jacobi-type (all blocks updated in parallel each iteration), crucial for taking advantage of many-core or distributed architectures.
• Gauss-Seidel/Southwell-type (one or a subset of blocks chosen greedily via error bounds or heuristics).
• Proximal block coordinate descent as a special case.
• Second-order (blockwise Newton) variants via richer choices of $P_i$.

Trade-offs between approaches include:

• Full parallelism yields better scalability on hardware but may incur increased per-iteration cost or communication.
• Selective updates (e.g., Southwell rules based on blockwise error magnitudes) can yield faster convergence with fewer updates but may impede parallelism if not balanced carefully.

The error bound mechanism via $E_i(x^k)$ ensures that blocks with sufficiently large suboptimality are prioritized.

5. Empirical Performance and Applications

Empirical evaluation focuses on high-dimensional regularized regression, specifically:

• Lasso problem setting: $F(x) = \|Ax - b\|^2$, $G(x) = c\|x\|_1$, $X = \mathbb{R}^n$.
• Direct blockwise soft-thresholding solution for each subproblem via the closed-form proximal operator.

Comparative results show:

• FPA (Flexible Parallel Algorithm) outperforms parallel FISTA, sparse coordinate-update Grock, sequential Gauss-Seidel coordinate descent, and ADMM, particularly in large, high-sparsity settings.
• Sequential methods scale poorly with problem size; FISTA is fast for approximate solutions but less competitive at high accuracy.
• FPA demonstrates robust, high-parallelism scaling and superior performance as the number of updated blocks increases.

6. Practical Implementation and Deployment Considerations

Implementation notes include:

• Each block subproblem is often strongly convex and efficiently solvable in parallel.
• The method is well-suited for distributed-memory and multicore systems, as blockwise independence minimizes the need for synchronization.
• Inexact subproblem solves are supported, provided that the accuracy tolerance decreases with step size.
• The flexibility to match the block update granularity to hardware—full, partial, or single block—makes the method easily adaptable to a range of practical deployment environments.

The method’s robust convergence under mild conditions (and even for nonconvex $F$) makes it particularly attractive for real-world big-data and machine learning workloads characterized by partial separability and structural regularization.

7. Summary and Broader Impact

The parallel blockwise computation scheme outlined in (Facchinei et al., 2013) provides:

• A mathematically principled, highly flexible framework for blockwise parallel optimization, unifying Jacobi, Gauss-Seidel/Southwell, and proximal block coordinate approaches.
• Generalized convergence guarantees under minimal assumptions, including inexact block solves and arbitrary update selection.
• Strong empirical performance on large-scale penalized regression problems, outperforming established solvers.
• Direct applicability and scalability on modern parallel architectures, offering tangible benefits in convergence speed and resource efficiency.

This scheme forms the foundation for numerous scalable optimization algorithms central to contemporary large-scale data analysis, variable selection, and structured convex or nonconvex learning.