Parent: optimization
Where the objective itself is learned, adversarial, nested, or over distributions.
Bilevel & meta-optimization
- An optimization inside an optimization: hyperparameter optimization, learning-to-optimize (learn the update rule), meta-learning (MAML);
- "LLMs as optimizers" (OPRO) — the model proposes/refines solutions; in-context learning ≈ implicit gradient descent. → our LLM-as-inner-optimizer (stages-gates-as-hard-optimization).
Minimax / games / adversarial
- Saddle-point problems
min_x max_y f(x,y): GANs, adversarial training, robust optimization; - Solved by no-regret dynamics / gradient-descent-ascent / monotone operators; convergence is subtle (cycling, not descent).
Optimal transport & gradient flows on measures
- Optimal transport / Wasserstein distance; optimizing over probability distributions;
- Wasserstein gradient flows = PDEs on measure space (Fokker–Planck) — the math under diffusion models.
Structuring hard optimization (the continuation family)
- Homotopy / continuation, graduated non-convexity, curriculum learning, annealing — solve a path of easy→hard problems, each warm-starting the next.
- → this is stages-and-gates: stages-gates-as-hard-optimization.
The unifying abstractions
- Monotone operators / fixed-point theory — GD, proximal, ADMM, primal-dual all as fixed-point iterations of (firmly) nonexpansive operators;
- Everything is: iterate an operator to its fixed point; converges iff it's a contraction / the flow is stable → dynamics-to-a-fixed-point.
The theme
The clean "minimize a fixed convex f" dissolves: the objective is learned (bilevel), adversarial (minimax), over distributions (OT), or too hard to descend directly (continuation). This frontier is exactly where the agent/LLM optimization thread sits.