[Paper Review] Smooth Optimization with Approximate Gradient

Alexandre d’Aspremont|ArXiv.org|Dec 14, 2005

Sparse and Compressive Sensing Techniques9 references4 citations

TL;DR

This paper demonstrates that Nesterov's smooth first-order optimization algorithm maintains its optimal $O(1/ heta)$ complexity even when gradients are computed approximately, with bounded error. By replacing full matrix exponentials with only a few leading eigenvalues in semidefinite programs, the method achieves significant computational savings while preserving convergence guarantees, enabling efficient solution of large-scale sparse and structured problems.

ABSTRACT

We show that the optimal complexity of Nesterov's smooth first-order optimization algorithm is preserved when the gradient is only computed up to a small, uniformly bounded error. In applications of this method to semidefinite programs, this means in some instances computing only a few leading eigenvalues of the current iterate instead of a full matrix exponential, which significantly reduces the method's computational cost. This also allows sparse problems to be solved efficiently using sparse maximum eigenvalue packages.

Motivation & Objective

• To establish that Nesterov's optimal complexity bound is preserved when gradients are computed with bounded error.
• To reduce computational cost in semidefinite programming by approximating gradients using only a few leading eigenvalues instead of full matrix exponentials.
• To enable efficient solution of sparse and structured problems by leveraging approximate gradient information.
• To analyze the impact of problem structure on the number of eigenvalues required per iteration.

Proposed method

• Theoretical analysis of Nesterov's smooth optimization algorithm under approximate gradient oracles satisfying $| abla f(x) - \tilde{\nabla}f(x), y-z\rangle| \leq \delta$ for all $x,y,z \in Q$.
• Adaptation of the smoothing technique from [15] to non-smooth problems, using a smooth approximation of the objective with Lipschitz continuous gradient.
• Use of prox-regularization with a strongly convex prox-function $d(x)$ to ensure convergence under inexact gradient information.
• Derivation of a convergence bound in terms of the approximation error $\delta$ and the number of iterations, showing that the $O(1/\epsilon)$ complexity is preserved.
• Application of the method to maximum eigenvalue minimization problems by approximating the gradient via leading eigenvalues of the Hessian matrix.
• Empirical validation on random and biological data sets to measure computational savings and eigenvalue requirements across different problem structures.

Experimental results

Research questions

• RQ1 Can the optimal $O(1/\epsilon)$ complexity of Nesterov's smooth first-order method be preserved when gradients are computed approximately with bounded error?
• RQ2 What is the required accuracy of the gradient approximation to maintain convergence and optimal complexity?
• RQ3 How does the number of leading eigenvalues needed in the gradient computation evolve during optimization for different problem structures?
• RQ4 To what extent can computational savings be achieved in semidefinite programs by replacing full matrix exponentials with partial eigenvalue computations?
• RQ5 How does problem structure—such as spectrum distribution or sparsity—affect the effective number of eigenvalues required per iteration?

Key findings

• The optimal $O(1/\epsilon)$ complexity bound of Nesterov's method is preserved even when gradients are approximated with uniformly bounded error.
• Computational cost is significantly reduced in semidefinite programs by computing only a few leading eigenvalues instead of full matrix exponentials.
• The number of eigenvalues required per iteration varies substantially with problem structure, with Gaussian-distributed matrices requiring more eigenvalues than structured or uniform spectra.
• Empirical results show that computational savings are substantial and highly dependent on spectral properties, with some instances achieving significant speedups.
• The coalescence of largest eigenvalues near optimality does not consistently predict increased eigenvalue requirements, suggesting complex interplay between spectrum and convergence behavior.
• Problem structure, including signal-to-noise ratio and sparsity, influences effective complexity, though no simple predictive model exists for the number of eigenvalues needed at each iteration.