Examples of Pathological Dynamics of the Subgradient Method for Lipschitz Path-Differentiable Functions
References
- [1] (2013) Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods. Math. Programming 137(1–2):91–129.Crossref, Google Scholar
- [2] (1980) Differentiable functions. Boletim da Sociedade Brasileira de Matemática-Bull./Brazilian Math. Soc. 11(2):139–189.Crossref, Google Scholar
- [3] (2021) Conservative set valued fields, automatic differentiation, stochastic gradient methods and deep learning. Math. Program. 188:19–51, https://doi.org/10.1007/s10107-020-01501-5.Google Scholar
- [4] (2010) Characterizations of Łojasiewicz inequalities: Subgradient flows, talweg, convexity. Trans. Amer. Math. Soc. 362(6):3319–3363.Crossref, Google Scholar
- [5] (2020) Long term dynamics of the subgradient method for Lipschitz path differentiable functions. Published in JEMS, https://arxiv.org/abs/2006.00098.Google Scholar
- [6] (1998) A chain rule for essentially smooth Lipschitz functions. SIAM J. Optim. 8(2):300–308.Crossref, Google Scholar
- [7] (2001) Generalized subdifferentials: A Baire categorical approach. Trans. Amer. Math. Soc. 353(10):3875–3893.Crossref, Google Scholar
- [8] (2020) Pathological subgradient dynamics. SIAM J. Optim. 30(2):1327–1338.Crossref, Google Scholar
- [9] (2019) Stochastic subgradient method converges on tame functions. Foundations Comput. Math. 20:119–154.Crossref, Google Scholar
- [10] (2015) Measure Theory and Fine Properties of Functions (CRC Press, Boca Raton, FL).Crossref, Google Scholar
- [11] (2004) Fractal Geometry: Mathematical Foundations and Applications (John Wiley & Sons, Hoboken, NJ).Google Scholar
- [12] (2012) Differential and Riemannian manifolds. Graduate Texts in Mathematics, vol. 160 (Springer, Berlin).Google Scholar
- [13] (1991) Action minimizing invariant measures for positive definite Lagrangian systems. Math. Zeitschrift 207(2):169–207.Crossref, Google Scholar
- [14] (2012) Introduction to Piecewise Differentiable Equations (Springer Science & Business Media, Berlin).Crossref, Google Scholar
- [15] (1935) A function not constant on a connected set of critical points. Duke Math. J. 1(4):514–517.Crossref, Google Scholar

