# Conditional Gradient Sliding for Convex Optimization

@article{Lan2016ConditionalGS, title={Conditional Gradient Sliding for Convex Optimization}, author={Guanghui Lan and Yi Zhou}, journal={SIAM J. Optim.}, year={2016}, volume={26}, pages={1379-1409} }

In this paper, we present a new conditional gradient type method for convex optimization by calling a linear optimization ($LO$) oracle to minimize a series of linear functions over the feasible set. Different from the classic conditional gradient method, the conditional gradient sliding (CGS) algorithm developed herein can skip the computation of gradients from time to time and, as a result, can achieve the optimal complexity bounds in terms of not only the number of calls to the $LO$ oracle…

## 141 Citations

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## References

SHOWING 1-10 OF 42 REFERENCES

### Gradient sliding for composite optimization

- Computer Science, MathematicsMath. Program.
- 2016

If the smooth component in the composite function is strongly convex, the developed gradient sliding algorithms can significantly reduce the number of graduate and subgradient evaluations for the smooth and nonsmooth component to O(1/ϵ), respectively.

### The Complexity of Large-scale Convex Programming under a Linear Optimization Oracle

- Computer Science
- 2013

This paper formally establishes the theoretical optimality or nearly optimality, in the large-scale case, for the CG method and its variants to solve different classes of CP problems, including smooth, nonsmooth and certain saddle-point problems.

### Iterated Hard Shrinkage for Minimization Problems with Sparsity Constraints

- MathematicsSIAM J. Sci. Comput.
- 2008

It is shown that the hard shrinkage algorithm is a special case of the generalized conditional gradient method with quadratic discrepancy term and strong convergence properties of the iterates with convergence rates $\mathcal{O}(n^{-1/2})$ and $\lambda^n)$ for $p=1$ and $1 < p \leq 2$, respectively.

### Sparse Convex Optimization Methods for Machine Learning

- Computer Science
- 2011

A convergence proof guaranteeing e-small error is given after O( 1e ) iterations, and the sparsity of approximate solutions for any `1-regularized convex optimization problem (and for optimization over the simplex), expressed as a function of the approximation quality.

### A Linearly Convergent Conditional Gradient Algorithm with Applications to Online and Stochastic Optimization

- Computer Science
- 2013

A novel conditional gradient algorithm for smooth and strongly convex optimization over polyhedral sets that performs only a single linear optimization step over the domain on each iteration and enjoys a linear convergence rate, which gives an exponential improvement in convergence rate over previous results.

### An optimal method for stochastic composite optimization

- Computer Science, MathematicsMath. Program.
- 2012

The accelerated stochastic approximation (AC-SA) algorithm based on Nesterov’s optimal method for smooth CP is introduced, and it is shown that the AC-SA algorithm can achieve the aforementioned lower bound on the rate of convergence for SCO.

### Revisiting Frank-Wolfe: Projection-Free Sparse Convex Optimization

- Computer ScienceICML
- 2013

A new general framework for convex optimization over matrix factorizations, where every Frank-Wolfe iteration will consist of a low-rank update, is presented, and the broad application areas of this approach are discussed.

### Convergence Rates for Conditional Gradient Sequences Generated by Implicit Step Length Rules

- Mathematics
- 1980

Conditional gradient algorithms with implicit line minimization and Goldstein–Armijo step length rules are considered for the problem $\min _\Omega F$ with $\Omega $ a bounded convex subset of a real…

### Mini-batch stochastic approximation methods for nonconvex stochastic composite optimization

- Computer ScienceMath. Program.
- 2016

A randomized stochastic projected gradient (RSPG) algorithm, in which proper mini-batch of samples are taken at each iteration depending on the total budget of Stochastic samples allowed, is proposed, which shows nearly optimal complexity of the algorithm for convex stoChastic programming.

### Dual subgradient algorithms for large-scale nonsmooth learning problems

- Computer ScienceMath. Program.
- 2014

This work proposes a novel approach to solving nonsmooth optimization problems arising in learning applications where Fenchel-type representation of the objective function is available and requires the problem domain to admit a Linear Optimization oracle—the ability to efficiently maximize a linear form on the domain of the primal problem.