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January  2020, 7(1): 79-96. doi: 10.3934/jdg.2020005

A regularization interpretation of the proximal point method for weakly convex functions

Tim Hoheisel , , Maxime Laborde and  Adam Oberman

Department of Mathematics and Statistics, McGill University, Montreal, Canada

* Corresponding author: tim.hoheisel@mcgill.ca

Received  January 2019 Published  February 2020

Fund Project: The third author is supported by AFOSR FA9550-18-1-0167.

Empirical evidence and theoretical results suggest that the proximal point method can be computed approximately and still converge faster than the corresponding gradient descent method, in both the stochastic and exact gradient case. In this article we provide a perspective on this result by interpreting the method as gradient descent on a regularized function. This perspective applies in the case of weakly convex functions where proofs of the faster rates are not available. Using this analysis we find the optimal value of the regularization parameter in terms of the weak convexity.

Keywords: Proximal-point method, weak convexity, Moreau envelope, forward Euler method, $ \theta $-method.
Mathematics Subject Classification: Primary: 26B25, 65K10; Secondary: 90C25.
Citation: Tim Hoheisel, Maxime Laborde, Adam Oberman. A regularization interpretation of the proximal point method for weakly convex functions. Journal of Dynamics & Games, 2020, 7 (1) : 79-96. doi: 10.3934/jdg.2020005
References:
[1]

Z. Allen-Zhu, Katyusha: The first direct acceleration of stochastic gradient methods, Journal of Machine Learning Research (JMLR), 18 (2017), 51 pp.  Google Scholar

[2]

L. T. H. An and P. D Tao, Convex analysis approach to d.c. programming: Theory, algorithms and applications, Acta Mathematica Vietnamica, 22 (1997), 289-355.   Google Scholar

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H. Attouch and D. Azé, Approximation and regularization of arbitrary functions in Hilbert spaces by the Lasry-Lions method, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 289-319.  doi: 10.1016/S0294-1449(16)30214-1.  Google Scholar

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H. Attouch and J. Bolte, On the convergence of the proximal algorithm for nonsmooth functions involving analytic features, Math. Program., 116 (2009), 5-16.  doi: 10.1007/s10107-007-0133-5.  Google Scholar

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J.-B. Baillon and G. Haddad, Quelques propriétés des opérateurs angle-bornés et $n$-cycliquement monotones, Israel J. Math., 26 (1977), 137-150.  doi: 10.1007/BF03007664.  Google Scholar

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H. H. Bauschke and P. L. Combettes, The Baillon-Haddad theorem revisited, J. Convex Anal., 17 (2010), 781-787.   Google Scholar

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H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, 2011. doi: 10.1007/978-1-4419-9467-7.  Google Scholar

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A. Beck, First-order Methods in Optimization, MOS-SIAM Series on Optimization, 25. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, Mathematical Optimization Society, Philadelphia, PA, 2017. doi: 10.1137/1.9781611974997.ch1.  Google Scholar

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L. Bottou, F. E. Curtis and J. Nocedal, Optimization methods for large-scale machine learning, SIAM Rev., 60 (2018), 223-311.  doi: 10.1137/16M1080173.  Google Scholar

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S. Bubeck, Convex optimization: Algorithms and complexity, Found. Trends Mach. Learn., 8 (2015), 231-357.  doi: 10.1561/2200000050.  Google Scholar

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Y. Carmon, J. C. Duchi, O. Hinder and A. Sidford, Accelerated methods for nonconvex optimization, SIAM J. Optim., 28 (2018), 1751-1772.  doi: 10.1137/17M1114296.  Google Scholar

[12]

P. Chaudhari, A. Oberman, S. Osher, S. Soatto and G. Carlier, Deep Relaxation: Partial differential equations for optimizing deep neural networks, Res. Math. Sci., 5 (2018), 30 pp. doi: 10.1007/s40687-018-0148-y.  Google Scholar

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L. C. Evans, Partial Differential Equations, Second edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 978-0-8218-4974-3.  Google Scholar

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N. Flammarion and F. Bach, From averaging to acceleration, there is only a step-size, Conference on Learning Theory, (2015), 658–695. Google Scholar

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A. Kaplan and R. Tichatschke, Proximal point methods and nonconvex optimization, J. Global Optim., 13 (1998), 389-406.  doi: 10.1023/A:1008321423879.  Google Scholar

[16]

J.-M. Lasri and P.-L. Lions, A remark on regularization in Hilbert spaces, Israel J. Math., 55 (1986), 257-266.  doi: 10.1007/BF02765025.  Google Scholar

[17]

L. Lessard, B. Recht and A. Packard, Analysis and design of optimization algorithms via integral quadratic constraints, SIAM J. Optim., 26 (2016), 57-95.  doi: 10.1137/15M1009597.  Google Scholar

[18]

H. Lin, J. Mairal and Z. Harchaoui, A universal catalyst for first-order optimization, Advances in Neural Information Processing Systems 28, Curran Associates, Inc., (2015), 3384–3392. Google Scholar

[19]

B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler scheme and applications, Commun. Pure Appl. Anal., 9 (2010), 685-702.  doi: 10.3934/cpaa.2010.9.685.  Google Scholar

[20]

Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, Applied Optimization, 87. Kluwer Academic Publishers, Boston, MA, 2004. doi: 10.1007/978-1-4419-8853-9.  Google Scholar

[21]

A. Nitanda, Stochastic proximal gradient descent with acceleration techniques, Advances in Neural Information Processing Systems 27, Curran Associates, Inc., (2014), 1574–1582. Google Scholar

[22]

C. Paquette, H. Lin, D. Drusvyatskiy, J. Mairal and Z. Harchaoui, Catalyst for gradient-based nonconvex optimization, Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics, Proceedings of Machine Learning Research, 84 (2018), 613-622.   Google Scholar

[23]

N. Parikh and S. Boyd, Proximal algorithms, Found. Trends Optim., 1 (2014), 127-239.  doi: 10.1561/2400000003.  Google Scholar

[24]

B. T. Polyak, Some methods of speeding up the convergence of iteration methods, Ž. Vyčisl. Mat i Fiz., 4 (1964), 791–803.  Google Scholar

[25]

R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Grundlehren der Mathematischen Wissenschaften, 317. Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[26]

L. Rosasco, S. Villa and B. C. Vũ, Convergence of stochastic proximal gradient algorithm, preprint, arXiv: 1403.5074. Google Scholar

[27]

M. Schmidt, N. Le Roux and F. Bach, Convergence rates of inexact proximal-gradient methods for convex optimization, Advances in Neural Information Processing Systems 24, Curran Associates, Inc., (2011), 1458–1466. Google Scholar

[28]

D. Scieur, V. Roulet, F. Bach and A. D'Aspremont, Integration methods and optimization algorithms, Proceedings of the 31st International Conference on Neural Information Processing Systems, Curran Associates Inc., (2017), 1109–1118. Google Scholar

[29]

A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge Monographs on Applied and Computational Mathematics, 2. Cambridge University Press, Cambridge, 1996.  Google Scholar

[30]

W. J. Su, S. Boyd and E. J. Candes, A differential equation for modeling Nesterov's accelerated gradient method: Theory and insights, J. Mach. Learn. Res., 17 (2016), 43 pp.  Google Scholar

[31]

P. H. Yin, M. Pham, A. Oberman and S. Osher, Stochastic backward Euler: An implicit gradient descent algorithm for $k$-means clustering, J. Sci. Comput., 77 (2018), 1133-1146.  doi: 10.1007/s10915-018-0744-4.  Google Scholar

show all references

References:
[1]

Z. Allen-Zhu, Katyusha: The first direct acceleration of stochastic gradient methods, Journal of Machine Learning Research (JMLR), 18 (2017), 51 pp.  Google Scholar

[2]

L. T. H. An and P. D Tao, Convex analysis approach to d.c. programming: Theory, algorithms and applications, Acta Mathematica Vietnamica, 22 (1997), 289-355.   Google Scholar

[3]

H. Attouch and D. Azé, Approximation and regularization of arbitrary functions in Hilbert spaces by the Lasry-Lions method, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 289-319.  doi: 10.1016/S0294-1449(16)30214-1.  Google Scholar

[4]

H. Attouch and J. Bolte, On the convergence of the proximal algorithm for nonsmooth functions involving analytic features, Math. Program., 116 (2009), 5-16.  doi: 10.1007/s10107-007-0133-5.  Google Scholar

[5]

J.-B. Baillon and G. Haddad, Quelques propriétés des opérateurs angle-bornés et $n$-cycliquement monotones, Israel J. Math., 26 (1977), 137-150.  doi: 10.1007/BF03007664.  Google Scholar

[6]

H. H. Bauschke and P. L. Combettes, The Baillon-Haddad theorem revisited, J. Convex Anal., 17 (2010), 781-787.   Google Scholar

[7]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, 2011. doi: 10.1007/978-1-4419-9467-7.  Google Scholar

[8]

A. Beck, First-order Methods in Optimization, MOS-SIAM Series on Optimization, 25. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, Mathematical Optimization Society, Philadelphia, PA, 2017. doi: 10.1137/1.9781611974997.ch1.  Google Scholar

[9]

L. Bottou, F. E. Curtis and J. Nocedal, Optimization methods for large-scale machine learning, SIAM Rev., 60 (2018), 223-311.  doi: 10.1137/16M1080173.  Google Scholar

[10]

S. Bubeck, Convex optimization: Algorithms and complexity, Found. Trends Mach. Learn., 8 (2015), 231-357.  doi: 10.1561/2200000050.  Google Scholar

[11]

Y. Carmon, J. C. Duchi, O. Hinder and A. Sidford, Accelerated methods for nonconvex optimization, SIAM J. Optim., 28 (2018), 1751-1772.  doi: 10.1137/17M1114296.  Google Scholar

[12]

P. Chaudhari, A. Oberman, S. Osher, S. Soatto and G. Carlier, Deep Relaxation: Partial differential equations for optimizing deep neural networks, Res. Math. Sci., 5 (2018), 30 pp. doi: 10.1007/s40687-018-0148-y.  Google Scholar

[13]

L. C. Evans, Partial Differential Equations, Second edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 978-0-8218-4974-3.  Google Scholar

[14]

N. Flammarion and F. Bach, From averaging to acceleration, there is only a step-size, Conference on Learning Theory, (2015), 658–695. Google Scholar

[15]

A. Kaplan and R. Tichatschke, Proximal point methods and nonconvex optimization, J. Global Optim., 13 (1998), 389-406.  doi: 10.1023/A:1008321423879.  Google Scholar

[16]

J.-M. Lasri and P.-L. Lions, A remark on regularization in Hilbert spaces, Israel J. Math., 55 (1986), 257-266.  doi: 10.1007/BF02765025.  Google Scholar

[17]

L. Lessard, B. Recht and A. Packard, Analysis and design of optimization algorithms via integral quadratic constraints, SIAM J. Optim., 26 (2016), 57-95.  doi: 10.1137/15M1009597.  Google Scholar

[18]

H. Lin, J. Mairal and Z. Harchaoui, A universal catalyst for first-order optimization, Advances in Neural Information Processing Systems 28, Curran Associates, Inc., (2015), 3384–3392. Google Scholar

[19]

B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler scheme and applications, Commun. Pure Appl. Anal., 9 (2010), 685-702.  doi: 10.3934/cpaa.2010.9.685.  Google Scholar

[20]

Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, Applied Optimization, 87. Kluwer Academic Publishers, Boston, MA, 2004. doi: 10.1007/978-1-4419-8853-9.  Google Scholar

[21]

A. Nitanda, Stochastic proximal gradient descent with acceleration techniques, Advances in Neural Information Processing Systems 27, Curran Associates, Inc., (2014), 1574–1582. Google Scholar

[22]

C. Paquette, H. Lin, D. Drusvyatskiy, J. Mairal and Z. Harchaoui, Catalyst for gradient-based nonconvex optimization, Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics, Proceedings of Machine Learning Research, 84 (2018), 613-622.   Google Scholar

[23]

N. Parikh and S. Boyd, Proximal algorithms, Found. Trends Optim., 1 (2014), 127-239.  doi: 10.1561/2400000003.  Google Scholar

[24]

B. T. Polyak, Some methods of speeding up the convergence of iteration methods, Ž. Vyčisl. Mat i Fiz., 4 (1964), 791–803.  Google Scholar

[25]

R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Grundlehren der Mathematischen Wissenschaften, 317. Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[26]

L. Rosasco, S. Villa and B. C. Vũ, Convergence of stochastic proximal gradient algorithm, preprint, arXiv: 1403.5074. Google Scholar

[27]

M. Schmidt, N. Le Roux and F. Bach, Convergence rates of inexact proximal-gradient methods for convex optimization, Advances in Neural Information Processing Systems 24, Curran Associates, Inc., (2011), 1458–1466. Google Scholar

[28]

D. Scieur, V. Roulet, F. Bach and A. D'Aspremont, Integration methods and optimization algorithms, Proceedings of the 31st International Conference on Neural Information Processing Systems, Curran Associates Inc., (2017), 1109–1118. Google Scholar

[29]

A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge Monographs on Applied and Computational Mathematics, 2. Cambridge University Press, Cambridge, 1996.  Google Scholar

[30]

W. J. Su, S. Boyd and E. J. Candes, A differential equation for modeling Nesterov's accelerated gradient method: Theory and insights, J. Mach. Learn. Res., 17 (2016), 43 pp.  Google Scholar

[31]

P. H. Yin, M. Pham, A. Oberman and S. Osher, Stochastic backward Euler: An implicit gradient descent algorithm for $k$-means clustering, J. Sci. Comput., 77 (2018), 1133-1146.  doi: 10.1007/s10915-018-0744-4.  Google Scholar

Figure 1.  Illustration of Example 1
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Figure 2.  Top: Iterations of gradient descent and the proximal point method. The proximal point method gets closer to the global mininum with the same number of total gradient evaluations. Bottom: function values at each iteration of gradient descent and at each outer proximal point iteration for the Rosenbrock function (a fair comparison is used by counting total gradient evaluations, using 10 or 20 for the approximate proximal point). After 2000 gradient evaluations the function value for gradient descent is still order 1, while the proximal point method is order $ 10^{-2} $, moreover the function values appear to decrease at a first order rate
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