The LZ78 Source
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918450921734144 |
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| author | Sagan, Naomi Dembo, Amir Ho, Matthew Weissman, Tsachy |
| author_facet | Sagan, Naomi Dembo, Amir Ho, Matthew Weissman, Tsachy |
| contents | We study a family of processes generated according to sequential probability assignments induced by the LZ78 universal compressor. We characterize entropic and distributional properties such as their entropy and relative entropy rates, finite-state compressibility and log loss of their realizations, and the empirical distributions that they induce. Though not quite stationary, these sources are "almost stationary and ergodic;" similar to stationary and ergodic processes, they satisfy a Shannon-McMillan-Breiman-type property: the normalized log probability of their realizations converges almost surely to their entropy rate. Further, they are locally "almost i.i.d." in the sense that the finite-dimensional empirical distributions of their realizations converge almost surely to a deterministic i.i.d. law. However, unlike stationary ergodic sources, the finite-state compressibility of their realizations is almost surely strictly larger than their entropy rate by a "Jensen gap". We present simulations demonstrating the theoretical results. These sources allow to gauge the performance of sequential probability models, both classical and deep learning-based, on non-Markovian non-stationary data. As such, we apply realizations of the LZ78 source to the study of in-context learning in transformer models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_10574 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The LZ78 Source Sagan, Naomi Dembo, Amir Ho, Matthew Weissman, Tsachy Information Theory We study a family of processes generated according to sequential probability assignments induced by the LZ78 universal compressor. We characterize entropic and distributional properties such as their entropy and relative entropy rates, finite-state compressibility and log loss of their realizations, and the empirical distributions that they induce. Though not quite stationary, these sources are "almost stationary and ergodic;" similar to stationary and ergodic processes, they satisfy a Shannon-McMillan-Breiman-type property: the normalized log probability of their realizations converges almost surely to their entropy rate. Further, they are locally "almost i.i.d." in the sense that the finite-dimensional empirical distributions of their realizations converge almost surely to a deterministic i.i.d. law. However, unlike stationary ergodic sources, the finite-state compressibility of their realizations is almost surely strictly larger than their entropy rate by a "Jensen gap". We present simulations demonstrating the theoretical results. These sources allow to gauge the performance of sequential probability models, both classical and deep learning-based, on non-Markovian non-stationary data. As such, we apply realizations of the LZ78 source to the study of in-context learning in transformer models. |
| title | The LZ78 Source |
| topic | Information Theory |
| url | https://arxiv.org/abs/2503.10574 |