Computation & AI - Computational Complexity of Neural Networks
Problem Statement: Develop a comprehensive theoretical framework characterizing the computational complexity, approximation capabilities, and optimization landscape of neural networks, explaining why deep learning works, when it fails, and providing principled design guidelines.
Why This Exemplifies the Field: This addresses the fundamental gap between the empirical success of deep learning and our theoretical understanding, representing the core challenge of making AI a mathematically rigorous science rather than an empirical art.
Evaluation Criteria:
Rigorous mathematical characterization of expressivity vs. network architecture
Proof of computational complexity bounds for training and inference
Theoretical explanation for empirical phenomena like double descent
Precise characterization of optimization landscape and training dynamics
Mathematically justified architecture design principles validated on benchmark tasks
Theory that accurately predicts generalization performance on novel architectures
Feasibility Assessment: Very challenging, likely requiring 10-20 years. Current understanding is largely empirical with fragmented theoretical insights. Requires synthesis across statistical learning theory, differential geometry, and dynamical systems. May need entirely new mathematical tools for analyzing high-dimensional non-convex optimization.
Impact on the Field: Would transform deep learning from an empirical discipline to one with solid theoretical foundations. Would enable systematic architecture design rather than trial-and-error approaches. Would clarify fundamental limits of neural computation and guide research toward promising new paradigms when appropriate.
