# Global convergence rate analysis of unconstrained optimization methods based on probabilistic models

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## 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.

## Keywords

Line-search methods Cubic regularization methods Random models Global convergence analysis## Mathematics Subject Classification

90C30 90C56 49M37## Notes

### 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.

## References

- 1.Bandeira, A., Scheinberg, K., Vicente, L.: Convergence of trust-region methods based on probabilistic models. SIAM J. Optim.
**24**, 1238–1264 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 2.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)Google Scholar
- 3.Byrd, R., Nocedal, J., Oztoprak, F.: An inexact successive quadratic approximation method for convex l-1 regularized optimization. Technical report (2013)Google Scholar
- 4.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)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Cartis, C., Gould, N., Toint, P.L.: Optimal Newton-type methods for nonconvex smooth optimization problems. Technical report, Optimization Online (2011)Google Scholar
- 6.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)MathSciNetCrossRefzbMATHGoogle Scholar - 7.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)MathSciNetCrossRefzbMATHGoogle Scholar - 8.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)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Chen, R.: Stochastic derivative-free optimization of noisy functions. Ph.D. thesis, Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, USA (2015)Google Scholar
- 10.Chen, R., Menickelly, M., Scheinberg, K.: Stochastic optimization using a trust-region method and random models. Technical report, ISE Dept., Lehigh UniversityGoogle Scholar
- 11.Devolder, O., Glineur, F., Nesterov, Y.: First-order methods of smooth convex optimization with inexact oracle. Math. Program.
**146**, 37–75 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Ghadimi, S., Lan, G.: Stochastic first- and zeroth-order methods for nonconvex stochastic programming. SIAM J. Optim.
**23**, 2341–2368 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Gratton, S., Royer, C.W., Vicente, L.N., Zhang, Z.: Direct search based on probabilistic descent. SIAM J. Optim.
**25**, 1515–1541 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 14.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)Google Scholar
- 15.Lee, J.D., Sun, Y., Saunders, M.A.: Proximal Newton-type methods for convex optimization. In: NIPS (2012)Google Scholar
- 16.Nesterov, Y.: Introductory Lectures on Convex Optimization. Kluwer, Dordrecht (2004)CrossRefzbMATHGoogle Scholar
- 17.Nesterov, Y.: Random gradient-free minimization of convex functions. Technical report 2011/1, CORE (2011)Google Scholar
- 18.Nesterov, Y., Polyak, B.T.: Cubic regularization of Newton method and its global performance. Math. Program.
**108**, 177–205 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 19.Pasupathy, R., Glynn, P.W., Ghosh, S., Hahemi, F.: On sampling rates in stochastic recursion (under review) (2016)Google Scholar
- 20.Robbins, H., Monro, S.: A stochastic approximation method. Ann. Math. Stat.
**22**, 400–407 (1951)MathSciNetCrossRefzbMATHGoogle Scholar - 21.Schmidt, M.W., Roux, N.L., Bach, F.: Convergence rates of inexact proximal-gradient methods for convex optimization. In: NIPS, pp. 1458–1466 (2011)Google Scholar
- 22.Schmidt, M.W., Roux, N.L., Bach, F.: Minimizing finite sums with the stochastic average gradient. CoRR, arXiv:1309.2388 (2013)
- 23.Shiryaev, A.: Probability, Graduate Texts on Mathematics. Springer, New York (1995)Google Scholar
- 24.Spall, J.: Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Trans. Autom. Control
**37**, 332–341 (1992)MathSciNetCrossRefzbMATHGoogle Scholar