## Abstract

In this paper we develop a randomized block-coordinate descent method for minimizing the sum of a smooth and a simple nonsmooth block-separable convex function and prove that it obtains an \(\varepsilon \)-accurate solution with probability at least \(1-\rho \) in at most \(O((n/\varepsilon ) \log (1/\rho ))\) iterations, where \(n\) is the number of blocks. This extends recent results of Nesterov (SIAM J Optim 22(2): 341–362, 2012), which cover the smooth case, to composite minimization, while at the same time improving the complexity by the factor of 4 and removing \(\varepsilon \) from the logarithmic term. More importantly, in contrast with the aforementioned work in which the author achieves the results by applying the method to a regularized version of the objective function with an unknown scaling factor, we show that this is not necessary, thus achieving first true iteration complexity bounds. For strongly convex functions the method converges linearly. In the smooth case we also allow for arbitrary probability vectors and non-Euclidean norms. Finally, we demonstrate numerically that the algorithm is able to solve huge-scale \(\ell _1\)-regularized least squares problems with a billion variables.

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## Notes

A function \(F: \mathbb{R }^N\rightarrow \mathbb{R }\) is

*isotone*if \(x\ge y\) implies \(F(x)\ge F(y)\).Note that in [12] Nesterov considered the composite setting and developed standard and accelerated gradient methods with iteration complexity guarantees for minimizing composite objective functions. These can be viewed as block coordinate descent methods with a

*single*block.This will not be the case for certain types of matrices, such as those arising from wavelet bases or FFT.

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## Acknowledgments

We thank anonymous referees and Hui Zhang (National University of Defense Technology, China) for useful comments that helped to improve the manuscript.

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An extended abstract of a preliminary version of this paper appeared in [15]. The work of the first author was supported in part by EPSRC grant EP/I017127/1 “Mathematics for vast digital resources”. The second author was supported in part by the Centre for Numerical algorithms and Intelligent Software (funded by EPSRC grant EP/G036136/1 and the Scottish Funding Council)

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Richtárik, P., Takáč, M. Iteration complexity of randomized block-coordinate descent methods for minimizing a composite function.
*Math. Program.* **144**, 1–38 (2014). https://doi.org/10.1007/s10107-012-0614-z

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DOI: https://doi.org/10.1007/s10107-012-0614-z

### Keywords

- Block coordinate descent
- Huge-scale optimization
- Composite minimization
- Iteration complexity
- Convex optimization
- LASSO
- Sparse regression
- Gradient descent
- Coordinate relaxation
- Gauss–Seidel method