Abstract
We consider optimization problems with an objective function that is the sum of two convex terms: one is smooth and given by a black-box oracle, and the other is general but with a simple, known structure. We first present an accelerated proximal gradient (APG) method for problems where the smooth part of the objective function is also strongly convex. This method incorporates an efficient line-search procedure, and achieves the optimal iteration complexity for such composite optimization problems. In case the strong convexity parameter is unknown, we also develop an adaptive scheme that can automatically estimate it on the fly, at the cost of a slightly worse iteration complexity. Then we focus on the special case of solving the \(\ell _1\)-regularized least-squares problem in the high-dimensional setting. In such a context, the smooth part of the objective (least-squares) is not strongly convex over the entire domain. Nevertheless, we can exploit its restricted strong convexity over sparse vectors using the adaptive APG method combined with a homotopy continuation scheme. We show that such a combination leads to a global geometric rate of convergence, and the overall iteration complexity has a weaker dependency on the restricted condition number than previous work.
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Agarwal, A., Negahban, S.N., Wainwright, M.J.: Fast global convergence of gradient methods for high-dimensional statistical recovery. Ann. Stat. 40(5), 2452–2482 (2012)
Beck, A., Teboulle, M.: A fast iterative shrinkage-threshold algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)
Bredies, K., Lorenz, D.A.: Linear convergence of iterative soft-thresholding. J. Fourier Anal. Appl. 14, 813–837 (2008)
Bruckstein, A.M., Donoho, D.L., Elad, M.: From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Rev. 51(1), 34–81 (2009)
Candès, E.J., Tao, T.: Decoding by linear programming. IEEE Trans. Inform. Theory 51(12), 4203–4215 (2005)
Candès, E.J., Tao, T.: Near-optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inform. Theory 52(12), 5406–5425 (2006)
Candès, E.J., Romberg, J., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theory 52(2), 489–509 (2006)
Chen, S.S., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20(1), 33–61 (1998)
Donoho, D.L.: Compressed sensing. IEEE Trans. Inform. Theory 52(4), 1289–1306 (2006)
Gonzaga, C.C., Karas, E.W.: Fine tuning Nesterov’s steepest descent algorithm for differentiable convex programming. Math. Program. Ser. A 138, 141–166 (2013)
Gu, M., Lim, L.-H., Wu, C.J.: ParNes: a rapidly convergent algorithm for accurate recovery of sparse and approximately sparse signals. Numer. Algorithms 64, 321–347 (2013)
Hale, E.T., Yin, W., Zhang, Y.: Fixed-point continuation for \(\ell _1\)-minimization: methodology and convergence. SIAM J. Optim. 19(3), 1107–1130 (2008)
Li, S., Mo, Q.: New bounds on the restricted isometry constant \(\delta _{2k}\). Appl. Comput. Harmon. Anal. 31(3), 460–468 (2011)
Luo, Z.-Q., Tseng, P.: On the linear convergence of descent methods for convex essentially smooth minimization. SIAM J. Control Optim. 30(2), 408–425 (1992)
Monteiro, R.D.C., Ortiz, C., Svaiter, B.F.: An adaptive accelerated first-order method for convex optimization. Technical report, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA (2012)
Nemirovski, A., Yudin, D.: Problem Complexity and Method Efficiency in Optimization. Wiley, New York (1983)
Nesterov, Y.: Introductory Lectures on Convex Optimization: A Basic Course. Kluwer, Boston (2004)
Nesterov, Y.: Smooth minimization of nonsmooth functions. Math. Program. 103(1), 127–152 (2005)
Nesterov, Y.: How to advance in structural convex optimization. OPTIMA: Math. Program. Soc. Newsl. 78, 2–5 (2008)
Nesterov, Y.: Gradient methods for minimizing composite objective function. Math. Program. Ser. B 140(2007/76), 125–161 (2013)
Nesterov, Y., Nemirovski, A.: Interior-Point Polynomial Algorithms in Convex Programming. SIAM Studies in Applied Mathematics. SIAM, Philadelphia (1994)
O’Donoghue, B., Candès, E.J.: Adaptive restart for accelerated gradient schemes. Found. Comput. Math. (2013). doi:10.1007/s10208-013-9150-3
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R Stat. Soc. Ser. B 58, 267–288 (1996)
Tseng, P.: On accelerated proximal gradient methods for convex–concave optimization. Manuscript (2008)
Wright, S.J., Nowad, R.D., Figueiredo, M.A.T.: Sparse reconstruction by separable approximation. IEEE Trans. Signal Process. 57(7), 2479–2493 (2009)
Wright, S.J.: Accelerated block-coordinate relaxation for regularized optimization. SIAM J. Optim. 22(1), 159–186 (2012)
Wen, Z., Yin, W., Goldfarb, D., Zhang, Y.: A fast algorithm for sparse reconstruction based on shrinkage, subspace optimization and continuation. SIAM J. Sci. Comput. 32(4), 1832–1857 (2010)
Xiao, L., Zhang, T.: A proximal-gradient homotopy method for the sparse least-squares problem. SIAM J. Optim. 23, 1062–1091 (2013)
Zhang, C.-H., Huang, J.: The sparsity and bias of the lasso selection in high-dimensional linear regression. Ann. Stat. 36, 1567–1594 (2008)
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We thank Professor Tong Zhang for helpful discussions on the APG homotopy method.
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Lin, Q., Xiao, L. An adaptive accelerated proximal gradient method and its homotopy continuation for sparse optimization. Comput Optim Appl 60, 633–674 (2015). https://doi.org/10.1007/s10589-014-9694-4
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DOI: https://doi.org/10.1007/s10589-014-9694-4