Abstract
We present global convergence rates for a line-search method which is based on random first-order models and directions whose quality is ensured only with certain probability. We show that in terms of the order of the accuracy, the evaluation complexity of such a method is the same as its counterparts that use deterministic accurate models; the use of probabilistic models only increases the complexity by a constant, which depends on the probability of the models being good. We particularize and improve these results in the convex and strongly convex case. We also analyze a probabilistic cubic regularization variant that allows approximate probabilistic second-order models and show improved complexity bounds compared to probabilistic first-order methods; again, as a function of the accuracy, the probabilistic cubic regularization bounds are of the same (optimal) order as for the deterministic case.
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Notes
Note that throughout, \(f(x^k)\ne f_k\), since \(f_k=F_k(\omega _k)\) is a related measure of progress towards optimality.
Note that a recently-proposed cubic regularization variant [2] can dispense with the approximate global minimization condition altogether while maintaining the optimal complexity bound of ARC. A probabilistic variant of [2] can be constructed similarly to probabilistic ARC, and our analysis here can be extended to provide same-order complexity bounds.
References
Bandeira, A., Scheinberg, K., Vicente, L.: Convergence of trust-region methods based on probabilistic models. SIAM J. Optim. 24, 1238–1264 (2014)
Birgin, E.G., Gardenghi, J.L., Martinez, S.A.S.J.M., Toint, P.L.: Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models. Technical report naXys-05-2015, Department of Mathematics, University of Namur (2015)
Byrd, R., Nocedal, J., Oztoprak, F.: An inexact successive quadratic approximation method for convex l-1 regularized optimization. Technical report (2013)
Byrd, R.H., Chin, G.M., Nocedal, J., Wu, Y.: Sample size selection in optimization methods for machine learning. Math. Program. 134, 127–155 (2012)
Cartis, C., Gould, N., Toint, P.L.: Optimal Newton-type methods for nonconvex smooth optimization problems. Technical report, Optimization Online (2011)
Cartis, C., Gould, N., Toint, P.L.: On the oracle complexity of first-order and derivative-free algorithms for smooth nonconvex minimization. SIAM J. Optim. 22, 66–86 (2012)
Cartis, C., Gould, N.I.M., Toint, P.L.: Adaptive cubic regularisation methods for unconstrained optimization. Part I: motivation, convergence and numerical results. Math. Program. 127, 245–295 (2011)
Cartis, C., Gould, N.I.M., Toint, P.L.: Adaptive cubic regularisation methods for unconstrained optimization. Part II: worst-case function- and derivative-evaluation complexity. Math. Program. 130, 295–319 (2011)
Chen, R.: Stochastic derivative-free optimization of noisy functions. Ph.D. thesis, Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, USA (2015)
Chen, R., Menickelly, M., Scheinberg, K.: Stochastic optimization using a trust-region method and random models. Technical report, ISE Dept., Lehigh University
Devolder, O., Glineur, F., Nesterov, Y.: First-order methods of smooth convex optimization with inexact oracle. Math. Program. 146, 37–75 (2014)
Ghadimi, S., Lan, G.: Stochastic first- and zeroth-order methods for nonconvex stochastic programming. SIAM J. Optim. 23, 2341–2368 (2013)
Gratton, S., Royer, C.W., Vicente, L.N., Zhang, Z.: Direct search based on probabilistic descent. SIAM J. Optim. 25, 1515–1541 (2015)
Gratton, S., Royer, C.W., Vicente, L.N., Zhang, Z.: Complexity and global rates of trust-region methods based on probabilistic models. Technical report 17-09, Dept. Mathematics, Univ. Coimbra (2017)
Lee, J.D., Sun, Y., Saunders, M.A.: Proximal Newton-type methods for convex optimization. In: NIPS (2012)
Nesterov, Y.: Introductory Lectures on Convex Optimization. Kluwer, Dordrecht (2004)
Nesterov, Y.: Random gradient-free minimization of convex functions. Technical report 2011/1, CORE (2011)
Nesterov, Y., Polyak, B.T.: Cubic regularization of Newton method and its global performance. Math. Program. 108, 177–205 (2006)
Pasupathy, R., Glynn, P.W., Ghosh, S., Hahemi, F.: On sampling rates in stochastic recursion (under review) (2016)
Robbins, H., Monro, S.: A stochastic approximation method. Ann. Math. Stat. 22, 400–407 (1951)
Schmidt, M.W., Roux, N.L., Bach, F.: Convergence rates of inexact proximal-gradient methods for convex optimization. In: NIPS, pp. 1458–1466 (2011)
Schmidt, M.W., Roux, N.L., Bach, F.: Minimizing finite sums with the stochastic average gradient. CoRR, arXiv:1309.2388 (2013)
Shiryaev, A.: Probability, Graduate Texts on Mathematics. Springer, New York (1995)
Spall, J.: Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Trans. Autom. Control 37, 332–341 (1992)
Acknowledgements
We would like to thank Alexander Stolyar for helpful discussions on stochastic processes. We also would like to thank Zaikun Zhang, who was instrumental in helping us significantly simplify the analysis of the stochastic process in Sect. 2.
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The work of C. Cartis was partially supported by the Oxford University EPSRC Platform Grant EP/I01893X/1. The work of K. Scheinberg is partially supported by NSF Grants DMS 10-16571, DMS 13-19356, CCF-1320137, AFOSR Grant FA9550-11-1-0239, and DARPA Grant FA 9550-12-1-0406 negotiated by AFOSR.
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Cartis, C., Scheinberg, K. Global convergence rate analysis of unconstrained optimization methods based on probabilistic models. Math. Program. 169, 337–375 (2018). https://doi.org/10.1007/s10107-017-1137-4
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DOI: https://doi.org/10.1007/s10107-017-1137-4