Min-Max Optimization Requires Exponentially Many Queries
Martino Bernasconi, Matteo Castiglioni, Andrea Celli, Alexandros Hollender
TLDR
Min-max optimization for nonconvex-nonconcave functions demands exponentially many queries to find an approximate stationary point.
Key contributions
- Studies query complexity for min-max optimization of nonconvex-nonconcave functions.
- Demonstrates that finding an $\varepsilon$-approximate stationary point is hard.
- Proves query complexity is exponential in $1/\varepsilon$ or problem dimension $d$.
Why it matters
This research reveals a fundamental computational bottleneck in min-max optimization for nonconvex-nonconcave settings. It implies that current algorithms may face inherent scalability challenges, guiding future research towards more efficient approaches or problem reformulations.
Original Abstract
We study the query complexity of min-max optimization of a nonconvex-nonconcave function $f$ over $[0,1]^d \times [0,1]^d$. We show that, given oracle access to $f$ and to its gradient $\nabla f$, any algorithm that finds an $\varepsilon$-approximate stationary point must make a number of queries that is exponential in $1/\varepsilon$ or $d$.
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