Dynamical Systems
Lyapunov Stability Theory
GAN (Goodfellow, 2014). Two networks - generator G and discriminator D - in a minimax game. Training sometimes diverges: mode collapse, oscillation, training instability. Lyapunov theory explains when and why: if no Lyapunov function exists for the dynamics (G, D), there is no convergence guarantee. That is exactly why WGAN reformulated the game - to obtain Lyapunov-compatible dynamics. Lyapunov 1892 predicted the GAN problem one hundred twenty-two years before Goodfellow.
- **GAN and WGAN:** gradient penalty in WGAN is a Lyapunov condition for game dynamics; the W_1 norm decreases along training as a Lyapunov function
- **RL convergence:** policy gradient converges if the value function serves as a Lyapunov function; Q-learning is stable under Bellman contraction V_dot < 0
- **Robotics CLF:** Boston Dynamics uses Control Lyapunov Functions in MPC balance controllers - stability guarantees are embedded in the optimization
Предварительные знания
Lyapunov Function: Generalized Energy
2014. Goodfellow proposes the GAN - generator G versus discriminator D in a minimax game. Early experiments are impressive. Then after a few epochs: collapse. The generator produces the same image repeatedly, the discriminator stops learning, losses oscillate without converging. **Mode collapse. Training instability.** The reason: no Lyapunov function exists for the dynamics (G, D) - no convergence guarantee. Not a code bug. A theorem about the absence of guarantees.
Linearization from the previous lesson breaks down when Re(lambda) = 0 - and says nothing about large deviations. A globally applicable tool is needed. **Lyapunov's idea (1892):** instead of solving the equation, find an 'energy' function - a scalar V(x) that does not grow along trajectories. If the energy decreases, the system settles, regardless of where it started.