PRAXIS: Learning Dynamics of Self-Improving Models with Symbolic Archives

Self-improving learning systems adapt data selection, optimization, and auxiliary symbolic components, inducing nonstationary objectives outside standard learning assumptions. We introduce PRAXIS, a co-evolutionary framework that models generators, learners, and symbolic archives as interacting dynamical processes. We prove that KL-constrained generator updates bound finite-horizon objective drift, and that archive updates exponentially suppress persistently non-competitive programs under sub-Gaussian noise. These results characterize selection and suppression as structural consequences of the update dynamics. Experiments across visual robustness, relational graph reasoning, and algorithmic graph reasoning show generator drift, learner tracking, and archive concentration consistent with the theory.