Randomized Sketches of Convex Programs with Sharp Guarantees (1404.7203v1)

Abstract: Random projection (RP) is a classical technique for reducing storage and computational costs. We analyze RP-based approximations of convex programs, in which the original optimization problem is approximated by the solution of a lower-dimensional problem. Such dimensionality reduction is essential in computation-limited settings, since the complexity of general convex programming can be quite high (e.g., cubic for quadratic programs, and substantially higher for semidefinite programs). In addition to computational savings, random projection is also useful for reducing memory usage, and has useful properties for privacy-sensitive optimization. We prove that the approximation ratio of this procedure can be bounded in terms of the geometry of constraint set. For a broad class of random projections, including those based on various sub-Gaussian distributions as well as randomized Hadamard and Fourier transforms, the data matrix defining the cost function can be projected down to the statistical dimension of the tangent cone of the constraints at the original solution, which is often substantially smaller than the original dimension. We illustrate consequences of our theory for various cases, including unconstrained and $\ell_1$-constrained least squares, support vector machines, low-rank matrix estimation, and discuss implications on privacy-sensitive optimization and some connections with de-noising and compressed sensing.

• The paper introduces a random projection approach that reduces high-dimensional convex programs to lower dimensions while preserving sharp accuracy bounds.
• It evaluates optimization methods across least squares, Lasso, compressed sensing, and SVMs, demonstrating significant reductions in computational demand.
• The study employs geometric and statistical analyses to establish rigorous guarantees for randomized sketches, paving the way for efficient and privacy-aware optimizations.

Summary of "Randomized Sketches of Convex Programs with Sharp Guarantees"

This paper presents an in-depth analysis of random projection (RP) techniques for approximating convex programs, aiming to offer solutions with reduced computational and storage demand. RP methods are of particular significance in computationally constrained scenarios where direct optimization is prohibitive due to the large dimensionality and complexity of convex sets. The authors rigorously explore how RP-based approaches can deliver approximate solutions by projecting operations onto lower dimensions while maintaining defined accuracy bounds.

Key Contributions

The paper methodically evaluates the effectiveness of RP schemes across several types of optimization problems. These include quadratic, semidefinite, and second-order cone programs, as well as practical cases like least squares, $\ell_1$-constrained minimizations, low-rank matrix estimation, support vector machines (SVMs), and compressed sensing. The authors focus on bounding the approximation ratio in terms of geometric properties of constraint sets, offering insights into the statistical dimension of tangent cones as pivotal in determining the projection efficiency.

Numerical Results and Implications

• Unconstrained Least Squares: Dimensionality can be effectively reduced to the rank of the data matrix, significantly cutting the computational cost compared to solving high-dimensional problems directly.
• $\ell_1$-Constrained Least Squares (Lasso): The authors propose a method for solving Lasso with sketching dimensions scaled to the sparsity of solutions, offering an improvement compared to previous standards by lowering the dimensionality required for accurate approximations.
• Compressed Sensing: The paper demonstrates how RP techniques extend to noiseless and noisy cases, emphasizing how randomness in projection not only aids computation but also serves privacy by minimizing information retention.
• Support Vector Machines: An RP approach reduces the sample size needed to effectively approximate solutions, highlighting its potential to streamline real-world binary classification tasks in machine learning.

Theoretical Advances and Future Directions

From a theoretical standpoint, the paper consolidates RP methodologies by leveraging geometric analysis, such as Banach space theory and empirical processes, to sharpen bounds on sketch sizes and probabilistic guarantees. The results hint at potential for further investigation into optimization problems where RP could allow private data usage with minimal leakage, aligning with data privacy interests in sensitive domains.

Moreover, the authors offer techniques for sharpening existing RP bounds for randomized orthogonal systems, such as for subspaces, $\ell_1$-cones, and nuclear norm cones.

Conclusion

The paper of randomized projections as expounded in this paper advances the practical and theoretical understanding of convex program approximation methods. The rigorous derivation of accuracy bounds in relation to dimensional reductions provides a promising outlook for scaling optimization solutions in large-scale data environments. Future research may explore exploiting RP in privacy-concerned contexts and explore extending the framework to a broader array of convex optimization problems.

In sum, while the results deliver concrete guidelines applicable to immediate computational challenges, the theoretical groundings set a course for ongoing enhancements in efficient and private convex optimization techniques.