Proximal Random Reshuffling Under Local Lipschitz Continuity

References

• [1] Abbe E, Bandeira AS, Bracher A, Singer A (2014) Decoding binary node labels from censored edge measurements: Phase transition and efficient recovery. IEEE Trans. Network Sci. Engrg. 1(1):10–22.Crossref, Google Scholar
• [2] Absil PA, Mahony R, Andrews B (2005) Convergence of the iterates of descent methods for analytic cost functions. SIAM J. Optim. 16(2):531–547.Crossref, Google Scholar
• [3] Attouch H, Bolte J, Svaiter BF (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):91–129.Crossref, Google Scholar
• [4] Aubin J-P, Cellina A (1984) Differential Inclusions: Set-Valued Maps and Viability Theory, Grundlehren der Mathematischen Wissenschaften (Springer, Berlin).Crossref, Google Scholar
• [5] Bauschke HH, Bolte J, Teboulle M (2017) A descent lemma beyond Lipschitz gradient continuity: First-order methods revisited and applications. Math. Oper. Res. 42(2):330–348.Link, Google Scholar
• [6] Bauschke HH, Bolte J, Chen J, Teboulle M, Wang X (2019) On linear convergence of non-Euclidean gradient methods without strong convexity and Lipschitz gradient continuity. J. Optim. Theory Appl. 182(3):1068–1087.Crossref, Google Scholar
• [7] Beck A (2017) First-Order Methods in Optimization (SIAM, Philadelphia).Crossref, Google Scholar
• [8] Benaïm M (1999) Dynamics of stochastic approximation algorithms. Azéma J, Émery M, Ledoux M, Yor M, eds. Séminaire de Probabilités XXXIII, Lecture Notes in Mathematics, vol. 1709 (Springer, Berlin), 1–68.Crossref, Google Scholar
• [9] Benaïm M, Hofbauer J, Sorin S (2005) Stochastic approximations and differential inclusions. SIAM J. Control Optim. 44(1):328–348.Crossref, Google Scholar
• [10] Benaïm M, Hofbauer J, Sorin S (2006) Stochastic approximations and differential inclusions, part II: Applications. Math. Oper. Res. 31(4):673–695.Link, Google Scholar
• [11] Bertsekas DP (2011) Incremental gradient, subgradient, and proximal methods for convex optimization: A survey. Sra S, Nowozin S, Wright SJ, eds. Optimization for Machine Learning (MIT Press, Cambridge, MA), 85–120.Crossref, Google Scholar
• [12] Bertsekas DP, Tsitsiklis JN (2000) Gradient convergence in gradient methods with errors. SIAM J. Optim. 10(3):627–642.Crossref, Google Scholar
• [13] Bianchi P, Hachem W, Schechtman S (2022) Convergence of constant step stochastic gradient descent for non-smooth non-convex functions. Set-Valued Variational Anal. 30(3):1117–1147.Crossref, Google Scholar
• [14] Bochnak J, Coste M, Roy MF (2013) Real Algebraic Geometry, vol. 36 (Springer, Berlin).Google Scholar
• [15] Bolte J, Pauwels E (2021) Conservative set valued fields, automatic differentiation, stochastic gradient methods and deep learning. Math. Programming 188(1):19–51.Crossref, Google Scholar
• [16] Bolte J, Sabach S, Teboulle M (2014) Proximal alternating linearized minimization for nonconvex and nonsmooth problems. Math. Programming 146(1–2):459–494.Crossref, Google Scholar
• [17] Bolte J, Daniilidis A, Lewis A, Shiota M (2007) Clarke subgradients of stratifiable functions. SIAM J. Optim. 18(2):556–572.Crossref, Google Scholar
• [18] Bolte J, Le T, Moulines É, Pauwels E (2025) Inexact subgradient methods for semialgebraic functions. Math. Programming, ePub ahead of print June 20, https://doi.org/10.1007/s10107-025-02245-w.Crossref, Google Scholar
• [19] Bolte J, Sabach S, Teboulle M, Vaisbourd Y (2018) First order methods beyond convexity and Lipschitz gradient continuity with applications to quadratic inverse problems. SIAM J. Optim. 28(3):2131–2151.Crossref, Google Scholar
• [20] Brézis H (1973) Opérateurs Maximaux Monotones et Semi-Groupes de Contractions Dans Les Espaces de Hilbert, North Holland Mathematics Studies, vol. 5 (North-Holland, Amsterdam).Google Scholar
• [21] Calamai PH, Moré JJ (1987) Projected gradient methods for linearly constrained problems. Math. Programming 39(1):93–116.Crossref, Google Scholar
• [22] Candès EJ, Li X, Ma Y, Wright J (2011) Robust principal component analysis? J. ACM 58(3):11.Crossref, Google Scholar
• [23] Chandrasekaran V, Sanghavi S, Parrilo PA, Willsky AS (2011) Rank-sparsity incoherence for matrix decomposition. SIAM J. Optim. 21(2):572–596.Crossref, Google Scholar
• [24] Chen Y, Chi Y, Goldsmith AJ (2015) Exact and stable covariance estimation from quadratic sampling via convex programming. IEEE Trans. Inform. Theory 61(7):4034–4059.Crossref, Google Scholar
• [25] Cichocki A, Zdunek R, Amari S (2007) Hierarchical ALS algorithms for nonnegative matrix and 3D tensor factorization. Davies ME, James CJ, Abdallah SA, Plumbley MD, eds. Internat. Conf. Independent Component Anal. Signal Separation (Springer, Berlin, Heidelberg), 169–176.Google Scholar
• [26] Clarke FH (1990) Optimization and Nonsmooth Analysis (SIAM Classics in Applied Mathematics, Philadelphia).Crossref, Google Scholar
• [27] Cobzas S, Mustata C (1978) Norm preserving extension of convex Lipschitz functions. J. Approximation Theory 24(3):236–244.Crossref, Google Scholar
• [28] Cornet B (1983) Existence of slow solutions for a class of differential inclusions. J. Math. Anal. Appl. 96(1):130–147.Crossref, Google Scholar
• [29] Davis D, Drusvyatskiy D (2019) Stochastic model-based minimization of weakly convex functions. SIAM J. Optim. 29(1):207–239.Crossref, Google Scholar
• [30] Davis D, Grimmer B (2019) Proximally guided stochastic subgradient method for nonsmooth, nonconvex problems. SIAM J. Optim. 29(3):1908–1930.Crossref, Google Scholar
• [31] Davis D, Drusvyatskiy D, Charisopoulos V (2024) Stochastic algorithms with geometric step decay converge linearly on sharp functions. Math. Programming 207(1):145–190.Crossref, Google Scholar
• [32] Davis D, Drusvyatskiy D, Paquette C (2020) The nonsmooth landscape of phase retrieval. IMA J. Numer. Anal. 40(4):2652–2695.Crossref, Google Scholar
• [33] Davis D, Drusvyatskiy D, Kakade S, Lee JD (2020) Stochastic subgradient method converges on tame functions. Foundations Comput. Math. 20(1):119–154.Crossref, Google Scholar
• [34] Davis D, Drusvyatskiy D, MacPhee KJ, Paquette C (2018) Subgradient methods for sharp weakly convex functions. J. Optim. Theory Appl. 179(3):962–982.Crossref, Google Scholar
• [35] Dereich S, Kassing S (2024) Convergence of stochastic gradient descent schemes for Łojasiewicz-landscapes. J. Machine Learn. 3(3):245–281.Crossref, Google Scholar
• [36] Ding K, Xiao N, Toh K-C (2025) Adam-family methods with decoupled weight decay in deep learning. Trans. Machine Learn. Res. Preprint, submitted May 12, https://openreview.net/forum?id=xVEHiAZ7uR.Google Scholar
• [37] Dragomir RA, Taylor AB, d’Aspremont A, Bolte J (2022) Optimal complexity and certification of Bregman first-order methods. Math. Programming 194(1):41–83.Crossref, Google Scholar
• [38] Drusvyatskiy D, Paquette C (2019) Efficiency of minimizing compositions of convex functions and smooth maps. Math. Programming 178(1–2):503–558.Crossref, Google Scholar
• [39] Drusvyatskiy D, Ioffe AD, Lewis AS (2015) Curves of descent. SIAM J. Control Optim. 53(1):114–138.Crossref, Google Scholar
• [40] Duchi JC, Ruan F (2018) Stochastic methods for composite and weakly convex optimization problems. SIAM J. Optim. 28(4):3229–3259.Crossref, Google Scholar
• [41] Duchi J, Singer Y (2009) Efficient online and batch learning using forward backward splitting. J. Machine Learn. Res. 10:2899–2934.Google Scholar
• [42] Fattahi S, Sojoudi S (2020) Exact guarantees on the absence of spurious local minima for non-negative rank-1 robust principal component analysis. J. Machine Learn. Res. 21(59):1–51.Google Scholar
• [43] Frankel P, Garrigos G, Peypouquet J (2015) Splitting methods with variable metric for Kurdyka–Łojasiewicz functions and general convergence rates. J. Optim. Theory Appl. 165(3):874–900.Crossref, Google Scholar
• [44] Gillis N (2020) Nonnegative Matrix Factorization (SIAM, Philadelphia).Crossref, Google Scholar
• [45] Goffin JL (1977) On convergence rates of subgradient optimization methods. Math. Programming 13(1):329–347.Crossref, Google Scholar
• [46] Goldstein A (1977) Optimization of Lipschitz continuous functions. Math. Programming 13(1):14–22.Crossref, Google Scholar
• [47] Grimmer B (2018) Radial subgradient method. SIAM J. Optim. 28(1):459–469.Crossref, Google Scholar
• [48] Grimmer B (2019) Convergence rates for deterministic and stochastic subgradient methods without Lipschitz continuity. SIAM J. Optim. 29(2):1350–1365.Crossref, Google Scholar
• [49] Gürbüzbalaban M, Ozdaglar A, Parrilo PA (2019) Convergence rate of incremental gradient and incremental Newton methods. SIAM J. Optim. 29(4):2542–2565.Crossref, Google Scholar
• [50] Gürbüzbalaban M, Ozdaglar A, Parrilo PA (2021) Why random reshuffling beats stochastic gradient descent. Math. Programming 186(1):49–84.Crossref, Google Scholar
• [51] Gwanyama PW (2004) The HM-GM-AM-QM inequalities. College Math. J. 35(1):47–50.Crossref, Google Scholar
• [52] Hà H-V, Phạm T-S (2016) Genericity in Polynomial Optimization, vol. 3 (World Scientific, London).Google Scholar
• [53] Hajinezhad D, Chang TH, Wang X, Shi Q, Hong M (2016) Nonnegative matrix factorization using ADMM: Algorithm and convergence analysis. Ching PC, Ho DKC, eds. Proc. 2016 IEEE Internat. Conf. Acoustics Speech Signal Processing (ICASSP) (IEEE, Piscataway, NJ), 4742–4746.Google Scholar
• [54] Haochen J, Sra S (2019) Random shuffling beats SGD after finite epochs. Chaudhuri K, Salakhutdinov R, eds. Proc. 36th Internat. Conf. Machine Learn., vol. 97 (PMLR, Cambridge, MA), 2624–2633.Google Scholar
• [55] Jia X, Kanzow C, Mehlitz P (2023) Convergence analysis of the proximal gradient method in the presence of the Kurdyka–Łojasiewicz property without global Lipschitz assumptions. SIAM J. Optim. 33(4):3038–3056.Crossref, Google Scholar
• [56] Josz C (2023) Global convergence of the gradient method for functions definable in o-minimal structures. Math. Programming 202(1):355–383.Crossref, Google Scholar
• [57] Josz C, Lai L (2023) Lyapunov stability of the subgradient method with constant step size. Math. Programming 202(1):387–396.Crossref, Google Scholar
• [58] Josz C, Lai L (2024) Global stability of first-order methods for coercive tame functions. Math. Programming 207(1–2):551–576.Crossref, Google Scholar
• [59] Josz C, Li X (2023) Certifying the absence of spurious local minima at infinity. SIAM J. Optim. 33(3):1416–1439.Crossref, Google Scholar
• [60] Josz C, Lai L, Li X (2023) Convergence of the momentum method for semialgebraic functions with locally Lipschitz gradients. SIAM J. Optim. 33(4):3012–3037.Crossref, Google Scholar
• [61] Kanzow C, Mehlitz P (2022) Convergence properties of monotone and nonmonotone proximal gradient methods revisited. J. Optim. Theory Appl. 195(2):624–646.Crossref, Google Scholar
• [62] Khaled A, Richtárik P (2023) Better theory for SGD in the nonconvex world. Trans. Machine Learn. Res. Preprint, submitted March 1, https://openreview.net/forum?id=AU4qHN2VkS.Google Scholar
• [63] Kim H, Park H (2008) Nonnegative matrix factorization based on alternating nonnegativity constrained least squares and active set method. SIAM J. Matrix Anal. Appl. 30(2):713–730.Crossref, Google Scholar
• [64] Kim J, He Y, Park H (2014) Algorithms for nonnegative matrix and tensor factorizations: A unified view based on block coordinate descent framework. J. Global Optim. 58(2):285–319.Crossref, Google Scholar
• [65] Kimura T, Takahashi N (2015) Global convergence of a modified HALS algorithm for nonnegative matrix factorization. Gini F, Richard C, eds. Pro. 2015 IEEE 6th Internat. Workshop Comput. Adv. Multi-Sensor Adaptive Processing (CAMSAP) (IEEE, Piscataway, NJ), 21–24.Google Scholar
• [66] Kurdyka K (1998) On gradients of functions definable in o-minimal structures. Annales de L’Institut Fourier 48(3):769–783.Crossref, Google Scholar
• [67] Kushner H (1977) Convergence of recursive adaptive and identification procedures via weak convergence theory. IEEE Trans. Automatic Control 22(6):921–930.Crossref, Google Scholar
• [68] Kushner HJ (1977) General convergence results for stochastic approximations via weak convergence theory. J. Math. Anal. Appl. 61(2):490–503.Crossref, Google Scholar
• [69] LeCun Y, Bengio Y, Hinton G (2015) Deep learning. Nature 521(7553):436–444.Crossref, Google Scholar
• [70] Lee DD, Seung HS (1999) Learning the parts of objects by non-negative matrix factorization. Nature 401(6755):788–791.Google Scholar
• [71] Lee D, Seung HS (2000) Algorithms for non-negative matrix factorization. Leen TK, Dietterich TG, Tresp V, eds. Adv. Neural Inform. Processing Systems, vol. 13 (MIT Press, Cambridge, MA), 556–562.Google Scholar
• [72] Lewis AS, Tian T (2025) Identifiability, the KL property in metric spaces, and subgradient curves. Foundations Computational Math. 25(3):905–942.Crossref, Google Scholar
• [73] Li X (2025) Bounded subgradient trajectories in semialgebraic optimization. PhD thesis, Columbia University, New York.Google Scholar
• [74] Li X, Milzarek A, Qiu J (2023) Convergence of random reshuffling under the Kurdyka–Łojasiewicz inequality. SIAM J. Optim. 33(2):1092–1120.Crossref, Google Scholar
• [75] Li X, Zhao L, Zhu D, So AMC (2025) Revisiting subgradient method: Complexity and convergence beyond Lipschitz continuity. Vietnam J. Math. 53(4):735–755.Crossref, Google Scholar
• [76] Li X, Zhu Z, Man-Cho So A, Vidal R (2020) Nonconvex robust low-rank matrix recovery. SIAM J. Optim. 30(1):660–686.Crossref, Google Scholar
• [77] Lin CJ (2007) Projected gradient methods for nonnegative matrix factorization. Neural Comput. 19(10):2756–2779.Crossref, Google Scholar
• [78] Ljung L (1977) Analysis of recursive stochastic algorithms. IEEE Trans. Automatic Control 22(4):551–575.Crossref, Google Scholar
• [79] Lu Z, Mei S (2023) Accelerated first-order methods for convex optimization with locally Lipschitz continuous gradient. SIAM J. Optim. 33(3):2275–2310.Crossref, Google Scholar
• [80] Luo ZQ (1991) On the convergence of the lms algorithm with adaptive learning rate for linear feedforward networks. Neural Comput. 3(2):226–245.Crossref, Google Scholar
• [81] Ma J, Fattahi S (2023) Global convergence of sub-gradient method for robust matrix recovery: Small initialization, noisy measurements, and over-parameterization. J. Machine Learn. Res. 24(96):1–84.Google Scholar
• [82] Majewski S, Miasojedow B, Moulines E (2018) Analysis of nonsmooth stochastic approximation: The differential inclusion approach. Preprint, submitted May 4, https://arxiv.org/abs/1805.01916.Google Scholar
• [83] Mangasarian O, Solodov M (1994) Serial and parallel backpropagation convergence via nonmonotone perturbed minimization. Optim. Methods Software 4(2):103–116.Crossref, Google Scholar
• [84] Marcellin S, Thibault L (2006) Evolution problems associated with primal lower nice functions. J. Convex Anal. 13(2):385–421.Google Scholar
• [85] Maréchal P (2005) On a functional operation generating convex functions, part 1: Duality. J. Optim. Theory Appl. 126(1):175–189.Crossref, Google Scholar
• [86] Mishchenko K, Khaled A, Richtárik P (2020) Random reshuffling: Simple analysis with vast improvements. Larochelle H, Ranzato M, Hadsell R, Balcan MF, Lin HT, eds. Adv. Neural Inform. Processing Systems, vol. 33 (Curran Associates, Inc., Red Hook, NY), 17309–17320.Google Scholar
• [87] Mishchenko K, Khaled A, Richtarik P (2022) Proximal and federated random reshuffling. Chaudhuri K, Jegelka S, Song L, Szepesvari C, Niu G, Sabato S, eds. Proc. 39th Internat. Conf. Machine Learn., vol. 162 (PMLR, Cambridge, MA), 15718–15749.Google Scholar
• [88] Moreau JJ (1965) Proximité et dualité dans un espace hilbertien. Bull. de la Société Mathématique de France 93:273–299.Crossref, Google Scholar
• [89] Mukkamala MC, Ochs P (2019) Beyond alternating updates for matrix factorization with inertial Bregman proximal gradient algorithms. Wallach HM, Larochelle H, Beygelzimer A, dAlche Buc F, Fox EB, Garnett R, eds. Adv. Neural Inform. Processing Systems, vol. 32 (Curran Associates, Inc., Red Hook, NY), 4266–4276.Google Scholar
• [90] Nesterov Y (2013) Gradient methods for minimizing composite functions. Math. Programming 140(1):125–161.Crossref, Google Scholar
• [91] Nguyen LM, Tran-Dinh Q, Phan DT, Nguyen PH, van Dijk M (2021) A unified convergence analysis for shuffling-type gradient methods. J. Machine Learn. Res. 22(207):1–44.Google Scholar
• [92] Nurminskii EA (1973) The quasigradient method for the solving of the nonlinear programming problems. Cybernetics 9(1):145–150.Crossref, Google Scholar
• [93] Ochs P, Chen Y, Brox T, Pock T (2014) iPiano: Inertial proximal algorithm for nonconvex optimization. SIAM J. Imaging Sci. 7(2):1388–1419.Crossref, Google Scholar
• [94] Paatero P, Tapper U (1994) Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values. Environmetrics 5(2):111–126.Crossref, Google Scholar
• [95] Paszke A, Gross S, Massa F, Lerer A, Bradbury J, Chanan G, Killeen T, et al. (2019) Pytorch: An imperative style, high-performance deep learning library. Wallach HM, Larochelle H, Beygelzimer A, dAlche Buc F, Fox EB, Garnett R, eds. Adv. Neural Inform. Processing Systems, vol. 32 (Curran Associates, Inc., Red Hook, NY), 8026–8037.Google Scholar
• [96] Pauwels E (2021) Incremental without replacement sampling in nonconvex optimization. J. Optim. Theory Appl. 190(1):274–299.Crossref, Google Scholar
• [97] Poliquin R (1991) Integration of subdifferentials of nonconvex functions. Nonlinear Anal. Theory Methods Appl. 17(4):385–398.Crossref, Google Scholar
• [98] Polyak BT (1969) Minimization of unsmooth functionals. USSR Comput. Math. Math. Phys. 9(3):14–29.Crossref, Google Scholar
• [99] Qiu J, Li X, Milzarek A (2025) A new random reshuffling method for nonsmooth nonconvex finite-sum optimization. J. Machine Learn. Res. 26(191):1–46.Google Scholar
• [100] Rakotomamonjy A (2013) Direct optimization of the dictionary learning problem. IEEE Trans. Signal Processing 61(22):5495–5506.Crossref, Google Scholar
• [101] Renegar J (2016) “Efficient” subgradient methods for general convex optimization. SIAM J. Optim. 26(4):2649–2676.Crossref, Google Scholar
• [102] Robbins H, Monro S (1951) A stochastic approximation method. Ann. Math. Statist. 22(3):400–407.Crossref, Google Scholar
• [103] Rockafellar R (1970) Convex Analysis, Princeton Mathematical Series (Princeton University Press, Princeton, NJ).Crossref, Google Scholar
• [104] Rockafellar RT, Wets RJB (2009) Variational Analysis, vol. 317 (Springer Science & Business Media, Berlin).Google Scholar
• [105] Rosasco L, Villa S, Vũ BC (2020) Convergence of stochastic proximal gradient algorithm. Appl. Math. Optim. 82(3):891–917.Crossref, Google Scholar
• [106] Salim A (2018) Random monotone operators and application to stochastic optimization. PhD thesis, Université Paris-Saclay (ComUE), Gif-sur-Yvette, France.Google Scholar
• [107] Scardapane S, Comminiello D, Hussain A, Uncini A (2017) Group sparse regularization for deep neural networks. Neurocomputing 241:81–89.Crossref, Google Scholar
• [108] Schechtman S (2023) Stochastic proximal subgradient descent oscillates in the vicinity of its accumulation set. Optim. Lett. 17(1):177–190.Crossref, Google Scholar
• [109] Schulz K, Schwartz B (1979) Finite extensions of convex functions. Mathematische Operationsforschung und Statistik Series Optimization 10(4):501–509.Crossref, Google Scholar
• [110] Solodov MV (2025) Convergence properties of proximal (sub) gradient methods without convexity or smoothness of any of the functions. SIAM J. Optim. 35(1):28–41.Crossref, Google Scholar
• [111] Tadić VB (2015) Convergence and convergence rate of stochastic gradient search in the case of multiple and non-isolated extrema. Stochastic Processes Appl. 125(5):1715–1755.Crossref, Google Scholar
• [112] Takahashi N, Hibi R (2014) Global convergence of modified multiplicative updates for nonnegative matrix factorization. Comput. Optim. Appl. 57(2):417–440.Crossref, Google Scholar
• [113] Teboulle M, Vaisbourd Y (2020) Novel proximal gradient methods for nonnegative matrix factorization with sparsity constraints. SIAM J. Imaging Sci. 13(1):381–421.Crossref, Google Scholar
• [114] Tian L, So AMC (2021) On the hardness of computing near-approximate stationary points of Clarke regular nonsmooth nonconvex problems and certain dc programs. ICML Workshop Beyond First-Order Methods ML Systems, https://sites.google.com/view/optml-icml2021/accepted-papers.Google Scholar
• [115] Tian L, So AMC (2024) No dimension-free deterministic algorithm computes approximate stationarities of Lipschitzians. Math. Programming 208(1–2):51–74.Crossref, Google Scholar
• [116] Van den Dries L (1998) Tame Topology and o-Minimal Structures, vol. 248 (Cambridge University Press, Cambridge, UK).Crossref, Google Scholar
• [117] van den Dries L, Miller C (1996) Geometric categories and o-minimal structures. Duke Math. J. 84(2):497–540.Crossref, Google Scholar
• [118] Vial JP (1983) Strong and weak convexity of sets and functions. Math. Oper. Res. 8(2):231–259.Link, Google Scholar
• [119] Wang Y-X, Zhang Y-J (2013) Nonnegative matrix factorization: A comprehensive review. IEEE Trans. Knowledge Data Engrg. 25(6):1336–1353.Crossref, Google Scholar
• [120] Xiao N, Hu X, Toh KC (2023) Stochastic subgradient methods with guaranteed global stability in nonsmooth nonconvex optimization. Preprint, submitted July 19, https://arxiv.org/abs/2307.10053.Google Scholar
• [121] Yu H, Li X (2023) High probability guarantees for random reshuffling. Preprint, submitted November 20, https://arxiv.org/abs/2311.11841.Google Scholar
• [122] Yuan M, Lin Y (2006) Model selection and estimation in regression with grouped variables. J. Roy. Statist. Soc. Ser. B Statist. Methodology 68(1):49–67.Crossref, Google Scholar
• [123] Zhang J, Hong M (2024) First-order algorithms without Lipschitz gradient: A sequential local optimization approach. INFORMS J. Optim. 6(2):118–136.Link, Google Scholar
• [124] Zhang J, Lin H, Jegelka S, Sra S, Jadbabaie A (2020) Complexity of finding stationary points of nonconvex nonsmooth functions. Daume H III, Singh A, eds. Proc. 37th Internat. Conf. Machine Learn., vol. 119 (PMLR, Cambridge, MA), 11173–11182.Google Scholar