ShrinkBench: Sparse Model Benchmarking

• ShrinkBench is a synthesis of standardized protocols and algorithmic practices for benchmarking large-scale sparse models across challenging computational regimes.
• It emphasizes rigorous evaluation using metrics such as predictive accuracy, sparsity level, and convergence rate to compare diverse algorithms.
• Empirical benchmarks on datasets like Gisette illustrate clear trade-offs between batch and online methods, guiding best practices in sparse modeling.

ShrinkBench is not a standard term or tool in sparse modeling as documented in the primary research literature. However, the closely related concepts and toolkits described in the source corpus reflect the core challenges, algorithmic strategies, and benchmarking regimes that are integral to the study of large-scale sparse models and their evaluation. The following article synthesizes the foundational principles, algorithmic practices, and empirical methods explicitly documented in the state of the art for large-scale sparse modeling and benchmarking, with a focus on rigorous evaluation protocols, representative benchmarks, and methodological best practices as articulated in leading references.

1. Motivation: Benchmarking Sparse Models at Scale

The exponential growth in data and feature dimensions in modern applications presents a unique set of challenges for learning and benchmarking sparse models. In settings with millions or billions of features and samples, classical batch algorithms for sparse regression (e.g., Lasso via coordinate descent or subgradient methods) face prohibitive computational and memory overheads, often requiring $O(n d)$ memory and per-iteration costs, where $n$ is the number of samples and $d$ the number of features. This motivates not only the development of specialized algorithms with favorable computational complexity but also the need for standardized benchmarking protocols that can faithfully compare model performance, convergence rate, memory efficiency, and true sparsity in these challenging regimes (Dhingra et al., 2023).

2. Benchmarking Protocols and Performance Metrics

Comprehensive benchmarking of sparse modeling algorithms in large-scale settings hinges on a set of precise quantitative metrics:

• Predictive Accuracy: Standard measures such as classification accuracy, error rate, mean squared error, or excess risk (depending on the modeling task).
• Sparsity Level: The proportion of zero versus nonzero coefficients in the learned parameter vector, which directly impacts model interpretability and computational cost at inference.
• Convergence Rate: The rate at which the objective function or suboptimality gap decays as a function of number of passes or wall-clock time.
• Resource Utilization: Memory consumption per-iteration, and total runtime, especially with respect to the baseline of $O(n d)$ for batch methods and $O(d)$ for scalable online or streaming approaches.
• Robustness Across Regimes: Smooth recovery of heuristic or theoretical performance guarantees as dataset size and dimensionality are systematically varied.

A standardized benchmark, therefore, includes: (A) public large-scale datasets (e.g., Gisette, LIBSVM collections), (B) fixed train/test splits, (C) systematic hyperparameter sweeps (e.g., $\lambda$ for Lasso or $K$ for $\ell_0$ constraints), and (D) consistent initialization and stopping criteria (Dhingra et al., 2023).

3. Algorithmic Classes Benchmarked

Representative benchmarking studies compare the following archetypal classes of algorithms:

• Batch Lasso (Coordinate Descent / Accelerated Proximal Methods): Exploits full data for each update; achieves statistical and algorithmic optimality at the cost of $O(n d)$ resources.
• Stochastic Online Lasso (FOBOS / SGD Variants): Each update touches a single or minibatch sample, reducing the computational and memory burden to $O(d)$ or $O(m d)$ per iteration.
• Hard-Thresholded Stochastic Gradient ($\ell_0$-SGD): Maintains an explicit $K$-sparse parameter vector by projecting gradient iterates onto the set of $K$ largest entries.
• Mini-Batch Variants: Apply to FOBOS or SGD to address variance reduction and potentially improve the quality and sparsity of solutions (Dhingra et al., 2023).

Empirical benchmarking frequently shows that, while batch coordinate-descent leads to maximally sparse solutions with high accuracy, pure online $\ell_1$ methods (FOBOS) tend to yield almost-dense models for any fixed regularization parameter. Mini-batch approaches slightly improve sparsity due to reduced gradient variance, but do not fundamentally restore batch-level zeros. The hard-thresholding $\ell_0$-SGD uniquely delivers guaranteed $K$-sparse models, albeit through a nonconvex optimization landscape.

4. Empirical Benchmarks: Dataset and Results

A prototypical benchmark is conducted on the Gisette dataset (6,000 train, 1,000 test, $d=5,000$), where half the features are random noise (Dhingra et al., 2023). The following summary table encapsulates the key empirical comparisons:

Algorithm	Accuracy	Sparsity (% nonzeros)	Parameter
Coordinate-Descent	0.9350	7.38	$\lambda=10^{-2}$
$\ell_0$-SGD	0.9470	8.00	$K=400$
FOBOS variants	0.968–0.970	99.06–99.10	$\lambda=10^{-5}$
Stochastic/Sub-grad	0.881–0.886	99.10	$\lambda=10^{-5}$

Detailed convergence plots from these benchmarks consistently show that $\ell_0$-SGD exhibits the fastest rate of objective drop among online methods, while FOBOS and subgradient methods remain highly noisy and slow. Importantly, batch and hard-thresholding variants present a superior accuracy-sparsity trade-off, directly visible in these standardized benchmarking protocols.

5. Methodological Implications and Best Practices

Extensive benchmarking studies support the following practical guidelines (Dhingra et al., 2023):

• Batch coordinate-descent: Remains the preferred method when memory and time resources permit; uniquely achieves convex optimality and true sparsity.
• Online $\ell_1$ approaches (FOBOS, RDA): Practical for streaming or very large $n$; sublinear convergence ($O(1/\sqrt{T})$) and nearly no zeros unless $\lambda$ is aggressively tuned.
• Mini-batch and hard-thresholding: The former reduces online variance but often fails to yield batch-level sparsity. The latter explicitly enforces support size, scale linearly in $d$, and provides competitive accuracy; global rates require additional restricted eigenvalue conditions due to nonconvexity.
• Parameter selection: Benchmarking must include $\lambda$ and $K$ sweeps; solutions with identical regularization coefficients may differ vastly in support size and prediction accuracy across algorithmic variants.

6. Open Problems and Future Directions in Sparse Model Benchmarking

While current benchmarks address large-scale regression, systematic benchmarks that cover a diversity of penalization forms (e.g., group, structured, nonconvex penalties), combination with deep learning architectures, and streaming or distributed data regimes remain underdeveloped (Dhingra et al., 2023, Lin, 2023). A rigorous benchmarking suite must also extend to real-world resource-constrained environments and standardized reporting of code, initialization, and validation criteria. Further, uniform theoretical and empirical comparisons of support recovery guarantees, model selection consistency, and test-time efficiency across complex sparsity structures represent ongoing research frontiers.

7. Summary Table: Algorithmic Properties Benchmarked

Property	Batch CD	FOBOS	Mini-batch FOBOS	$\ell_0$-SGD
Per-iteration Cost	$O(n d)$	$O(d)$	$O(m d)$	$O(d + k\log d)$
Memory	$O(n d)$	$O(d)$	$O(m d)$	$O(d)$
Expected Sparsity	High	Low	Low–Moderate	Exactly $K$ nonzeros
Convergence Rate	Fast	$O(1/\sqrt{T})$	$O(1/\sqrt{T})$	Local linear (RSC), global unquantified
Convexity	Yes	Yes	Yes	No (nonconvex)

Empirical benchmarks must report results across all these axes to enable meaningful comparisons among large-scale sparse modeling methods (Dhingra et al., 2023).

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References:

• "Learning Large Scale Sparse Models" (Dhingra et al., 2023)
• "Sparse Models for Machine Learning" (Lin, 2023)