In this paper, we establish an upper bound on the return gap between the oracle expert policy and an optimal decision tree policy. This enables us to recast the DT extraction problem into a novel non-euclidean clustering problem over the local observation and action values space of each agent, with action values as cluster labels and the upper bound on the return gap as clustering loss.

In this work, we provide a comparative analysis of the learning dynamics for the Euclidean and cosine-based asymmetric losses in the eigenspace of the closed-form predictor DirectPred.

Further, we clarify that the term 'self-supervised' has different meanings in DA and in pretraining [S23, S24, S25, S26]. Problem definitions for both papers are different.

High dimensional representation drift is commonly quantified using Euclidean or cosine distances, which presuppose fixed coordinates when comparing representations across time, training or preprocessing stages. While effective in many settings, these measures entangle intrinsic changes in the data with variations induced by arbitrary parametrizations. We introduce a projective geometric view of representation drift grounded in the Fubini Study metric, which identifies representations that differ only by gauge transformations such as global rescalings or sign flips. Applying this framework to empirical high dimensional datasets, we explicitly construct representation trajectories and track their evolution through cumulative geometric drift. Comparing Euclidean, cosine and Fubini Study distances along these trajectories reveals that conventional metrics systematically overestimate change whenever representations carry genuine projective ambiguity. By contrast, the Fubini Study metric isolates intrinsic evolution by remaining invariant under gauge-induced fluctuations. We further show that the difference between cosine and Fubini Study drift defines a computable, monotone quantity that directly captures representation churn attributable to gauge freedom. This separation provides a diagnostic for distinguishing meaningful structural evolution from parametrization artifacts, without introducing model-specific assumptions. Overall, we establish a geometric criterion for assessing representation stability in high-dimensional systems and clarify the limits of angular distances. Embedding representation dynamics in projective space connects data analysis with established geometric programs and yields observables that are directly testable in empirical workflows.

In this work, we report what happens when two large language models respond to each other for many turns without any outside input in a multi-agent setup. The setup begins with a short seed sentence. After that, each model reads the other's output and generates a response. This continues for a fixed number of steps. We used Mistral Nemo Base 2407 and Llama 2 13B hf. We observed that most conversations start coherently but later fall into repetition. In many runs, a short phrase appears and repeats across turns. Once repetition begins, both models tend to produce similar output rather than introducing a new direction in the conversation. This leads to a loop where the same or similar text is produced repeatedly. We describe this behavior as a form of convergence. It occurs even though the models are large, trained separately, and not given any prompt instructions. To study this behavior, we apply lexical and embedding-based metrics to measure how far the conversation drifts from the initial seed and how similar the outputs of the two models becomes as the conversation progresses.