Local Maximum In Gradient Descent Algorithm
%20for%20solving%20structured%20nonconvex%20minimax%20optimization%20problems%20in%20the%20form%20of%20%24min_textbf%7Bx%7D%20max_%7Btextbf%7By%7D%20in%20Y%7D%20f(textbf%7Bx%7D%2C%20textbf%7By%7D)%24%2C%20where%20the%20objective%20function%20%24f(textbf%7Bx%7D%2C%20textbf%7By%7D)%24%20is%20nonconvex%20in%20%24textbf%7Bx%7D%24%20and%20concave%20in%20%24textbf%7By%7D%24%2C%20and%20the%20constraint%20set%20%24Y%20subseteq%20mathbb%7BR%7D%5En%24%20is%20convex%20and%20bounded.%20In%20the%20convex-concave%20setting%2C%20the%20single-timescale%20gradient%20descent%20ascent%20(GDA)%20algorithm%20is%20widely%20used%20in%20applications%20and%20has%20been%20shown%20to%20have%20strong%20convergence%20guarantees.%20In%20more%20general%20settings%2C%20however%2C%20it%20can%20fail%20to%20converge.%20Our%20contribution%20is%20to%20design%20TTGDA%20algorithms%20that%20are%20effective%20beyond%20the%20convex-concave%20setting%2C%20efficiently%20finding%20a%20stationary%20point%20of%20the%20function%20%24Phi(cdot)%20:%3D%20max_%7Btextbf%7By%7D%20in%20Y%7D%20f(cdot%2C%20textbf%7By%7D)%24.%20We%20also%20establish%20theoretical%20bounds%20on%20the%20complexity%20of%20solving%20both%20smooth%20and%20nonsmooth%20nonconvex-concave%20minimax%20optimization%20problems.%20To%20the%20best%20of%20our%20knowledge%2C%20this%20is%20the%20first%20systematic%20analysis%20of%20TTGDA%20for%20nonconvex%20minimax%20optimization%2C%20shedding%20light%20on%20its%20superior%20performance%20in%20training%20generative%20adversarial%20networks%20(GANs)%20and%20in%20other%20real-world%20application%20problems.)
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