Convergence Rates of Gradient Methods for Convex Optimization in the Space of Measures

Chizat, Lénaïc

Open Journal of Mathematical Optimization, Tome 3 (2022), article no. 8, 19 p.

We study the convergence rate of Bregman gradient methods for convex optimization in the space of measures on a $d$-dimensional manifold. Under basic regularity assumptions, we show that the suboptimality gap at iteration $k$ is in $O(log\left(k\right)k^{-1})$ for multiplicative updates, while it is in $O\left(k^{-q/(d+q)}\right)$ for additive updates for some $q\in \{1,2,4\}$ determined by the structure of the objective function. Our flexible proof strategy, based on approximation arguments, allows us to painlessly cover all Bregman Proximal Gradient Methods (PGM) and their acceleration (APGM) under various geometries such as the hyperbolic entropy and $L^p$ divergences. We also prove the tightness of our analysis with matching lower bounds and confirm the theoretical results with numerical experiments on low dimensional problems. Note that all these optimization methods must additionally pay the computational cost of discretization, which can be exponential in $d$.

Mots clés : Gradient Descent, Convergence rate, Space of measures, Convex Optimization, Banach space

[1] Amid, Ehsan; Warmuth, Manfred K. Winnowing with gradient descent, Conference on Learning Theory, PMLR (2020), pp. 163-182

[2] Auslender, Alfred; Teboulle, Marc Interior gradient and proximal methods for convex and conic optimization, SIAM J. Optim., Volume 16 (2006) no. 3, pp. 697-725 | DOI | MR | Zbl

[3] Azulay, Shahar; Moroshko, Edward; Nacson, Mor Shpigel; Woodworth, Blake; Srebro, Nathan; Globerson, Amir; Soudry, Daniel On the Implicit Bias of Initialization Shape: Beyond Infinitesimal Mirror Descent (2021) (https://arxiv.org/abs/2102.09769)

[4] Bach, Francis Breaking the curse of dimensionality with convex neural networks, J. Mach. Learn. Theory, Volume 18 (2017) no. 1, pp. 629-681

[5] Bach, Francis Learning Theory from First Principles Draft, 2021

[6] Bauschke, Heinz H.; Borwein, Jonathan M.; Combettes, Patrick L. Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces, Commun. Contemp. Math., Volume 3 (2001) no. 04, pp. 615-647 | DOI | MR | Zbl

[7] Bauschke, Heinz H.; Borwein, Jonathan M.; Combettes, Patrick L. Bregman monotone optimization algorithms, SIAM J. Control Optim., Volume 42 (2003) no. 2, pp. 596-636 | DOI | MR | Zbl

[8] Beck, Amir; Teboulle, Marc A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., Volume 2 (2009) no. 1, pp. 183-202 | DOI | MR | Zbl

[9] Bengio, Yoshua; Roux, Nicolas; Vincent, Pascal; Delalleau, Olivier; Marcotte, Patrice Convex Neural Networks, Advances in Neural Information Processing Systems (Weiss, Y.; Schölkopf, B.; Platt, J., eds.), Volume 18, MIT Press (2006)

[10] Boyd, Nicholas; Schiebinger, Geoffrey; Recht, Benjamin The alternating descent conditional gradient method for sparse inverse problems, SIAM J. Optim., Volume 27 (2017) no. 2, pp. 616-639 | DOI | MR | Zbl

[11] Bredies, Kristian; Lorenz, Dirk A. Linear convergence of iterative soft-thresholding, J. Fourier Anal. Appl., Volume 14 (2008) no. 5-6, pp. 813-837 | DOI | MR | Zbl

[12] Bredies, Kristian; Pikkarainen, Hanna Katriina Inverse problems in spaces of measures, ESAIM, Control Optim. Calc. Var., Volume 19 (2013) no. 1, pp. 190-218 | DOI | Numdam | MR | Zbl

[13] Bubeck, Sébastien Convex Optimization: Algorithms and Complexity, Found. Trends Mach. Learn., Volume 8 (2015) no. 3-4, pp. 231-357 | DOI | Zbl

[14] Candès, Emmanuel J.; Fernandez-Granda, Carlos Towards a mathematical theory of super-resolution, Commun. Pure Appl. Math., Volume 67 (2014) no. 6, pp. 906-956 | DOI | MR | Zbl

[15] Chambolle, Antonin; Tovey, Robert "FISTA" in Banach spaces with adaptive discretisations (2021) (https://arxiv.org/abs/2101.09175)

[16] Chizat, Lénaïc Sparse optimization on measures with over-parameterized gradient descent, Math. Program. (2021), pp. 1-46

[17] Chizat, Lénaïc; Bach, Francis Implicit bias of gradient descent for wide two-layer neural networks trained with the logistic loss, Conference on Learning Theory, PMLR (2020), pp. 1305-1338

[18] Condat, Laurent Fast projection onto the simplex and the ${\ell }_1$ ball, Math. Program., Volume 158 (2016) no. 1, pp. 575-585 | DOI | MR | Zbl

[19] d’Aspremont, Alexandre; Scieur, Damien; Taylor, Adrien Acceleration methods (2021) (https://arxiv.org/abs/2101.09545)

[20] De Castro, Yohann; Gamboa, Fabrice Exact reconstruction using Beurling minimal extrapolation, J. Math. Anal. Appl., Volume 395 (2012) no. 1, pp. 336-354 | DOI | MR | Zbl

[21] Denoyelle, Quentin; Duval, Vincent; Peyré, Gabriel; Soubies, Emmanuel The sliding Frank–Wolfe algorithm and its application to super-resolution microscopy, Inverse Probl., Volume 36 (2019) no. 1, 014001, 42 pages | DOI | MR | Zbl

[22] Dieuleveut, Aymeric Stochastic approximation in Hilbert spaces, Ph. D. Thesis, PSL Research University (2017)

[23] Domingo-Enrich, Carles; Jelassi, Samy; Mensch, Arthur; Rotskoff, Grant; Bruna, Joan A mean-field analysis of two-player zero-sum games, Advances in Neural Information Processing Systems (Larochelle, H.; Ranzato, M.; Hadsell, R.; Balcan, M. F.; Lin, H., eds.), Volume 33, Curran Associates, Inc. (2020), pp. 20215-20226

[24] Garrigos, Guillaume; Rosasco, Lorenzo; Villa, Silvia Thresholding gradient methods in Hilbert spaces: support identification and linear convergence, ESAIM, Control Optim. Calc. Var., Volume 26 (2020), 28, 20 pages | DOI | MR | Zbl

[25] Ghai, Udaya; Hazan, Elad; Singer, Yoram Exponentiated gradient meets gradient descent, Algorithmic Learning Theory, PMLR (2020), pp. 386-407

[26] Gray, Alfred; Vanhecke, Lieven Riemannian geometry as determined by the volumes of small geodesic balls, Acta Math., Volume 142 (1979) no. 1, pp. 157-198 | DOI | MR | Zbl

[27] Gray, Alfred; Willmore, Tom J. Mean-value theorems for Riemannian manifolds, Proc. R. Soc. Edinb., Sect. A, Math., Volume 92 (1982) no. 3-4, pp. 343-364 | DOI | MR | Zbl

[28] Jacobs, Matt; Léger, Flavien; Li, Wuchen; Osher, Stanley Solving large-scale optimization problems with a convergence rate independent of grid size, SIAM J. Numer. Anal., Volume 57 (2019) no. 3, pp. 1100-1123 | DOI | MR | Zbl

[29] Kan, Chao; Song, Wen The Moreau envelope function and proximal mapping in the sense of the Bregman distance, Nonlinear Anal., Theory Methods Appl., Volume 75 (2012) no. 3, pp. 1385-1399 | MR | Zbl

[30] Kivinen, Jyrki; Warmuth, Manfred K. Exponentiated gradient versus gradient descent for linear predictors, Inf. Comput., Volume 132 (1997) no. 1, pp. 1-63 | DOI | MR | Zbl

[31] Lasserre, Jean B. Global optimization with polynomials and the problem of moments, SIAM J. Optim., Volume 11 (2001) no. 3, pp. 796-817 | DOI | MR | Zbl

[32] Liang, Jingwei; Fadili, Jalal M.; Peyré, Gabriel Local linear convergence of forward–backward under partial smoothness, Proceedings of the 27th International Conference on Neural Information Processing Systems-Volume 2 (2014), pp. 1970-1978

[33] Mei, Song; Montanari, Andrea; Nguyen, Phan-Minh A mean field view of the landscape of two-layer neural networks, Proc. Natl. Acad. Sci. USA, Volume 115 (2018) no. 33, p. E7665-E7671 | MR | Zbl

[34] Nemirovsky, Arkadij Semenovič; Yudin, David Borisovich Problem complexity and method efficiency in optimization, Wiley-Interscience series in discrete mathematics, John Wiley & Sons, 1983

[35] Nesterov, Yurii On an approach to the construction of optimal methods of minimization of smooth convex functions, Ekonom. i. Mat. Metody, Volume 24 (1988) no. 3, pp. 509-517

[36] Nesterov, Yurii Introductory lectures on convex optimization: A basic course, 87, Springer, 2003

[37] Nitanda, Atsushi; Wu, Denny; Suzuki, Taiji Particle Dual Averaging: Optimization of Mean Field Neural Networks with Global Convergence Rate Analysis (2020) (https://arxiv.org/abs/2012.15477)

[38] Poon, Clarice; Keriven, Nicolas; Peyré, Gabriel The geometry of off-the-grid compressed sensing (2018) (https://arxiv.org/abs/1802.08464)

[39] Rockafellar, Ralph Integrals which are convex functionals. II, Pac. J. Math., Volume 39 (1971) no. 2, pp. 439-469 | DOI | MR | Zbl

[40] Tseng, Paul Approximation accuracy, gradient methods, and error bound for structured convex optimization, Math. Program., Volume 125 (2010) no. 2, pp. 263-295 | DOI | MR | Zbl

[41] Vaskevicius, Tomas; Kanade, Varun; Rebeschini, Patrick Implicit Regularization for Optimal Sparse Recovery, Advances in Neural Information Processing Systems, Volume 32, Curran Associates, Inc. (2019)

[42] Wojtowytsch, Stephan; E, Weinan Can shallow neural networks beat the curse of dimensionality? A mean field training perspective, IEEE Trans. Artif. Intell., Volume 1 (2020) no. 2, pp. 121-129 | DOI

[43] Yao, Yuan; Rosasco, Lorenzo; Caponnetto, Andrea On early stopping in gradient descent learning, Computing, Volume 26 (2007) no. 2, pp. 289-315 | MR | Zbl

[44] Yu, Yao-Liang The strong convexity of Von Neumann’s entropy, 2013 (Unpublished note)