{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/138938"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/138938","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"On the Complexity of Nonconvex-Strongly-Concave Smooth Minimax Optimization Using First-Order Methods","abstract":"The problem of minimax optimization arises in a wide range of applications. When the objective function is convex-concave, almost the full picture is known. However, the general nonconvex-concave setting is less understood. In this work, we study the complexity of nonconvex-strongly-concave minimax optimization using first-order methods. First, we provide a first-order oracle complexity lower bound for finding stationary points of nonconvex-strongly-concave smooth min-max optimization problems. We establish a lower bound of Ω ( √ 𝜅𝜖⁻²) for deterministic oracles, where 𝜖 defines the level of approximate stationarity and 𝜅 is the condition number, which matches the existing upper bound achieved in (Lin et al., 2020b) up to logarithmic factors. For stochastic oracles, we provide a lower bound of Ω (︀√ 𝜅𝜖⁻² + 𝜅 ¹/³ 𝜖 ⁻⁴)︀ . Second, we study the specific first-order algorithm, gradient descent-ascent (GDA). We show that for quadratic or nearly quadratic nonconvex-strongly-concave functions under our assumptions, two-time-scale GDA with appropriate stepsizes achieves a linear convergence rate. Then we also extend our result to stochastic gradient descent-ascent (SGDA).","abstract_html":"The problem of minimax optimization arises in a wide range of applications. When the objective function is convex-concave, almost the full picture is known. However, the general nonconvex-concave setting is less understood. In this work, we study the complexity of nonconvex-strongly-concave minimax optimization using first-order methods. First, we provide a first-order oracle complexity lower bound for finding stationary points of nonconvex-strongly-concave smooth min-max optimization problems. We establish a lower bound of Ω ( √ 𝜅𝜖⁻²) for deterministic oracles, where 𝜖 defines the level of approximate stationarity and 𝜅 is the condition number, which matches the existing upper bound achieved in (Lin et al., 2020b) up to logarithmic factors. For stochastic oracles, we provide a lower bound of Ω (︀√ 𝜅𝜖⁻² + 𝜅 ¹/³ 𝜖 ⁻⁴)︀ . Second, we study the specific first-order algorithm, gradient descent-ascent (GDA). We show that for quadratic or nearly quadratic nonconvex-strongly-concave functions under our assumptions, two-time-scale GDA with appropriate stepsizes achieves a linear convergence rate. Then we also extend our result to stochastic gradient descent-ascent (SGDA).","abstract_has_math":false,"creators":["Li, Haochuan"],"institution":"Massachusetts Institute of Technology","degree_name":"Master","degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science","school":null,"contributors":[],"advisors":["Jadbabaie, Ali","Rakhlin, Alexander"],"committee_chairs":[],"committee_members":[],"year":2021,"date_issued":"2021-06","date_published":"2021-06","updated_at":"2026-07-22T22:21:34Z","subjects":[],"languages":[],"rights":["In Copyright - Educational Use Permitted","Copyright MIT"],"rights_urls":["http://rightsstatements.org/page/InC-EDU/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1721.1/138938","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Jadbabaie, Ali","Rakhlin, Alexander"]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. 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When the objective function is convex-concave, almost the full picture is known. However, the general nonconvex-concave setting is less understood. In this work, we study the complexity of nonconvex-strongly-concave minimax optimization using first-order methods. First, we provide a first-order oracle complexity lower bound for finding stationary points of nonconvex-strongly-concave smooth min-max optimization problems. We establish a lower bound of Ω ( √ 𝜅𝜖⁻²) for deterministic oracles, where 𝜖 defines the level of approximate stationarity and 𝜅 is the condition number, which matches the existing upper bound achieved in (Lin et al., 2020b) up to logarithmic factors. For stochastic oracles, we provide a lower bound of Ω (︀√ 𝜅𝜖⁻² + 𝜅 ¹/³ 𝜖 ⁻⁴)︀ . Second, we study the specific first-order algorithm, gradient descent-ascent (GDA). We show that for quadratic or nearly quadratic nonconvex-strongly-concave functions under our assumptions, two-time-scale GDA with appropriate stepsizes achieves a linear convergence rate. Then we also extend our result to stochastic gradient descent-ascent (SGDA)."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["S.M."]},{"key":"dc:title","label":"Title","values":["On the Complexity of Nonconvex-Strongly-Concave Smooth Minimax Optimization Using First-Order Methods"]}]}],"canonical_facts":{"dc:contributor.advisor":["Jadbabaie, Ali","Rakhlin, Alexander"],"dc:contributor.department":["Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science"],"dc:creator":["Li, Haochuan"],"dc:date.accessioned":["2022-01-14T14:39:36Z"],"dc:date.available":["2022-01-14T14:39:36Z"],"dc:date.issued":["2021-06"],"dc:description.abstract":["The problem of minimax optimization arises in a wide range of applications. When the objective function is convex-concave, almost the full picture is known. However, the general nonconvex-concave setting is less understood. In this work, we study the complexity of nonconvex-strongly-concave minimax optimization using first-order methods. First, we provide a first-order oracle complexity lower bound for finding stationary points of nonconvex-strongly-concave smooth min-max optimization problems. We establish a lower bound of Ω ( √ 𝜅𝜖⁻²) for deterministic oracles, where 𝜖 defines the level of approximate stationarity and 𝜅 is the condition number, which matches the existing upper bound achieved in (Lin et al., 2020b) up to logarithmic factors. For stochastic oracles, we provide a lower bound of Ω (︀√ 𝜅𝜖⁻² + 𝜅 ¹/³ 𝜖 ⁻⁴)︀ . Second, we study the specific first-order algorithm, gradient descent-ascent (GDA). We show that for quadratic or nearly quadratic nonconvex-strongly-concave functions under our assumptions, two-time-scale GDA with appropriate stepsizes achieves a linear convergence rate. Then we also extend our result to stochastic gradient descent-ascent (SGDA)."],"dc:description.degree":["S.M."],"dc:identifier.uri":["https://hdl.handle.net/1721.1/138938"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["In Copyright - Educational Use Permitted","Copyright MIT"],"dc:rights.uri":["http://rightsstatements.org/page/InC-EDU/1.0/"],"dc:title":["On the Complexity of Nonconvex-Strongly-Concave Smooth Minimax Optimization Using First-Order Methods"],"dc:type":["Thesis"],"thesis:degree_name":["Master","Master of Science in Electrical Engineering and Computer Science"]},"updated_at":"2026-07-22T22:21:34Z"}