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Conjecture 6.11: Hybrid Lyapunov with Memory Consolidation

REFORMULATED A hybrid Lyapunov function satisfies monotone decrease between consolidation events and bounded jumps at consolidation times.

Statement

A hybrid Lyapunov function \( V_H \) for systems with episodic memory satisfies: (i) monotone decrease between consolidation events, \( V_H(\theta_{t+1}) \leq \rho \, V_H(\theta_t) \) for \( t \notin \mathcal{T}_c \); and (ii) bounded jumps at consolidation times \( t_c \in \mathcal{T}_c \), with \( V_H(\theta_{t_c}^+) \leq V_H(\theta_{t_c}^-) + \Delta_c \) where \( \Delta_c \leq (1 - \rho^{\tau_{\min}}) V_H(\theta_{t_c}^-) \) and \( \tau_{\min} \) is the minimum inter-consolidation interval. The net effect is convergence provided \( \rho^{\tau_{\min}} + \Delta_c / V_H < 1 \), i.e., the contraction between jumps exceeds the jump magnitude.

Status: Reformulated

Supported. Preliminary experiments confirm that consolidation discontinuities are bounded and the overall trajectory of \( V_H \) is decreasing when the inter-consolidation interval is sufficiently long relative to the jump magnitude.

Evidence Summary

The experiment exp_memory_dynamics.sx tracks the Lyapunov function during learning with memory consolidation:

  • Between consolidation events: \( V \) decreases smoothly at rate \( \dot{V} \leq -\gamma V \), consistent with Theorem 3
  • At consolidation events: \( V \) jumps by \( \Delta V > 0 \) (increases), then resumes smooth decrease
  • The jumps have magnitude proportional to the amount of information consolidated
  • Despite the jumps, \( V \) trends downward overall — the smooth decrease between jumps more than compensates

The pattern resembles a sawtooth wave with a downward envelope. Standard Lyapunov theory (which assumes \( \dot{V} \leq 0 \) everywhere) cannot handle the upward jumps. A hybrid Lyapunov analysis would need to show that:

\[ V(t_{k+1}^-) < V(t_k^-) \]

where \( t_k^- \) denotes the moment just before the \( k \)-th consolidation event. This "net decrease" condition is observed experimentally but not yet proven.

Relevant Experiments

  • exp_memory_dynamics.sx — Lyapunov trajectory with consolidation jumps
  • exp_lyapunov.sx — standard Lyapunov analysis (no consolidation)

What This Means

This conjecture identifies a gap in the current theory. The Unified Adaptation Theorem proves convergence for systems with smooth dynamics, but real adaptive systems often have discrete events (memory consolidation, cache clearing, model checkpointing) that temporarily increase the Lyapunov function. A hybrid analysis would extend the theorem to cover these practical cases. The mathematical tools exist in the hybrid dynamical systems literature (Goebel, Sanfelice, Teel), but applying them to the specific structure of composed adaptive systems requires new work.