Exponential divergence
Rapid, multiplicative growth in the separation between two states, outputs, or parameter vectors such that small perturbations amplify exponentially over iterations or time, producing instability or compounding errors in AI systems.
A rapid multiplicative increase in the difference between two trajectories, model outputs, or parameter states where initially small perturbations grow roughly as e^{λt} (λ>0), causing trajectories to separate exponentially and undermining stability in learning or inference.
Exponential divergence is a formal way to describe instability in dynamical systems applied to AI: when the largest local expansion rate (e.g., the maximal Lyapunov exponent or the spectral radius of the Jacobian) is positive, errors, perturbations, or model mismatches are amplified exponentially across time steps or iterative updates. In ML (Machine Learning) contexts this manifests as exploding gradients in recurrent or deep networks, compounding prediction errors in autoregressive models and imitation learning, policy drift in reinforcement learning when updates step outside a trust region, and numerical blow-up in iterative solvers. The concept provides a unifying lens linking chaos theory, stability analysis, and optimization dynamics: mathematically one studies linearized updates (Jacobian spectrum), growth rates of divergences (Lyapunov exponents), and information-theoretic divergence measures (KL/JSD) as they propagate through model dynamics. Understanding exponential divergence guides mitigation strategies—gradient clipping, spectral/weight normalization, stable architectural choices (LSTM/GRU, orthogonal initialization), trust-region methods (TRPO/PPO), and data/algorithmic approaches (scheduled sampling, DAGGER)—and is central for designing robust training procedures, trustworthy deployment, and interpretable long-horizon behavior.
First formalizations trace to stability theory and Lyapunov’s work in the late 19th/early 20th century, the phenomenon gained broad scientific prominence with chaos theory in the 1960s (Lorenz), and the phrase and practical attention in AI/ML grew during the 1990s–2010s as recurrent network training (vanishing/exploding gradients) and large-scale deep models highlighted instability.
Key contributors include Aleksandr Lyapunov and Henri Poincaré (stability and dynamical-systems foundations) and Edward Lorenz (chaos and sensitive dependence); in AI/ML contexts, Hochreiter & Schmidhuber (analysis and LSTM to address gradient problems), Bengio and colleagues (on long-term dependency issues), Pascanu, Mikolov & Bengio (formal analysis of exploding/vanishing gradients), Ross, Gordon & Bagnell (compounding error and DAGGER for imitation learning), and later practitioners who developed normalization and trust-region techniques (e.g., spectral normalization, gradient clipping, TRPO/PPO).
