Abstract
Classical trust region methods were designed to solve problems in which function and gradient information are exact. This paper considers the case when there are errors (or noise) in the above computations and proposes a simple modification of the trust region method to cope with these errors. The new algorithm only requires information about the size/standard deviation of the errors in the function evaluations and incurs no additional computational expense. It is shown that, when applied to a smooth (but not necessarily convex) objective function, the iterates of the algorithm visit a neighborhood of stationarity infinitely often, assuming errors in the function and gradient evaluations are bounded. It is also shown that, after visiting the above neighborhood for the first time, the iterates cannot stray too far from it, as measured by the objective value. Numerical results illustrate how the classical trust region algorithm may fail in the presence of noise, and how the proposed algorithm ensures steady progress towards stationarity in these cases.
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References
Bellavia, S., Gurioli, G., Morini, B., Toint, P.: The impact of noise on evaluation complexity: the deterministic trust-region case. arXiv preprint arXiv:2104.02519 (2021)
Berahas, A.S., Byrd, R.H., Nocedal, J.: Derivative-free optimization of noisy functions via quasi-Newton methods. SIAM J. Optim. 29(2), 965–993 (2019)
Berahas, A.S., Cao, L., Choromanski, K., Scheinberg, K.: A theoretical and empirical comparison of gradient approximations in derivative-free optimization. Found. Comput. Math. pp. 1–54 (2021)
Berahas, A.S., Cao, L., Scheinberg, K.: Global convergence rate analysis of a generic line search algorithm with noise. SIAM J. Optim. 31(2), 1489–1518 (2021)
Berahas, A.S., Curtis, F.E., O’Neill, M.J., Robinson, D.P.: A stochastic sequential quadratic optimization algorithm for nonlinear equality constrained optimization with rank-deficient Jacobians. arXiv preprint arXiv:2106.13015 (2021)
Berahas, A.S., Curtis, F.E., Robinson, D., Zhou, B.: Sequential quadratic optimization for nonlinear equality constrained stochastic optimization. SIAM J. Optim. 31(2), 1352–1379 (2021)
Blanchet, J., Cartis, C., Menickelly, M., Scheinberg, K.: Convergence rate analysis of a stochastic trust region method via submartingales. INFORMS J. Optim. 1(2), 92–119 (2019)
Bollapragada, R., Byrd, R., Nocedal, J.: Adaptive sampling strategies for stochastic optimization. SIAM J. Optim. 28(4), 3312–3343 (2018)
Bollapragada, R., Byrd, R.H., Nocedal, J.: Exact and inexact subsampled newton methods for optimization. IMA J. Numer. Anal. (2018)
Bottou, L., Curtis, F.E., Nocedal, J.: Optimization methods for large-scale machine learning. Siam Rev. 60(2), 223–311 (2018)
Byrd, R.H., Chin, G.M., Nocedal, J., Wu, Y.: Sample size selection in optimization methods for machine learning. Math. Program. 134(1), 127–155 (2012)
Carter, R.G.: On the global convergence of trust region algorithms using inexact gradient information. SIAM J. Numer. Anal. 28(1), 251–265 (1991)
Cartis, C., Gould, N.I.M., Toint, P.: Strong evaluation complexity of an inexact trust-region algorithm with for arbitrary-order unconstrained nonconvex optimization. arXiv preprint arXiv:2001.10802 (2021)
Cartis, C., Scheinberg, K.: Global convergence rate analysis of unconstrained optimization methods based on probabilistic models. Math. Program. 169(2), 337–375 (2018)
Chen, R., Menickelly, M., Scheinberg, K.: Stochastic optimization using a trust-region method and random models. Math. Program. 169(2), 447–487 (2018)
Conn, A.R., Gould, N.I.M., Toint, P.L.: Numerical experiments with the LANCELOT package (Release A) for large-scale nonlinear optimization. Math. Program. Ser. A 73(1), 73–110 (1996)
Curtis, F.E., Scheinberg, K.: Adaptive stochastic optimization: a framework for analyzing stochastic optimization algorithms. IEEE Signal Process. Mag. 37(5), 32–42 (2020)
Curtis, F.E., Scheinberg, K., Shi, R.: A stochastic trust region algorithm based on careful step normalization. INFORMS J. Optim. 1(3), 200–220 (2019)
Friedlander, M.P., Schmidt, M.: Hybrid deterministic–stochastic methods for data fitting. SIAM J. Sci. Comput. 34(3), A1380–A1405 (2012)
Jin, B., Scheinberg, K., Xie, M.: High probability complexity bounds for line search based on stochastic oracles. Adv. Neural Inf. Process. Syst. 34, 9193–9203 (2021)
Cao, L., Berahas, A.S., Scheinberg, K.: First-and second-order high probability complexity bounds for trust-region methods with noisy oracles. arXiv preprint arXiv:2205.03667 (2022)
Micikevicius, P., Narang, S., Alben, J., Diamos, G., Elsen, E., Garcia, D., Ginsburg, B., Houston, M., Kuchaiev, O., Venkatesh, G., et al.: Mixed precision training. arXiv preprint arXiv:1710.03740 (2017)
Moré, J.J., Wild, S.M.: Estimating computational noise. SIAM J. Sci. Comput. 33(3), 1292–1314 (2011)
Nedić, A., Bertsekas, D.: Convergence rate of incremental subgradient algorithms. In: Stochastic Optimization: Algorithms and Applications, pp. 223–264. Springer (2001)
Nesterov, Y., Spokoiny, V.: Random gradient-free minimization of convex functions. Found. Comput. Math. 17(2), 527–566 (2017)
Ng, L.W., Willcox, K.E.: Multifidelity approaches for optimization under uncertainty. Int. J. Numer. Methods Eng. 100(10), 746–772 (2014)
Nocedal, J., Wright, S.: Numerical Optimization, 2nd edn. Springer, New York (1999)
Öztoprak, F., Byrd, R., Nocedal, J.: Constrained optimization in the presence of noise. arXiv preprint arXiv:2110.04355 (2021)
Paquette, C., Scheinberg, K.: A stochastic line search method with convergence rate analysis. arXiv preprint arXiv:1807.07994 (2018)
Pasupathy, R., Glynn, P., Ghosh, S., Hashemi, F.S.: On sampling rates in simulation-based recursions. SIAM J. Optim. 28(1), 45–73 (2018)
Peherstorfer, B., Willcox, K., Gunzburger, M.: Survey of multifidelity methods in uncertainty propagation, inference, and optimization. Siam Rev. 60(3), 550–591 (2018)
Polyak, B.T.: Introduction to Optimization. Optimization Software, vol. 1, p. 32. Inc., Publications Division, New York (1987)
Schittkowski, K.: More Test Examples for Nonlinear Programming Codes. Lecture Notes in Economics and Mathematical Systems, vol. 282 (1987)
Shi, H.J.M., Xie, Y., Xuan, M.Q., Nocedal, J.: Adaptive finite-difference interval estimation for noisy derivative-free optimization. arXiv preprint arXiv:2110.06380 (2021)
Xie, Y., Byrd, R.H., Nocedal, J.: Analysis of the BFGS method with errors. SIAM J. Optim. 30(1), 182–209 (2020)
Acknowledgements
We thank Richard Byrd and the referees for many valuable suggestions. We are also grateful to Figen Öztoprak, Andreas Wächter and Melody Xuan for proofreading the paper and providing useful feedback.
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Sun was supported by NSF Grant DMS-1620022. Nocedal was supported by AFOSR Grant FA95502110084 and ONR Grant N00014-21-1-2675.
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Sun, S., Nocedal, J. A trust region method for noisy unconstrained optimization. Math. Program. 202, 445–472 (2023). https://doi.org/10.1007/s10107-023-01941-9
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DOI: https://doi.org/10.1007/s10107-023-01941-9