The Role of Randomness in Stability
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909488569647104 |
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| author | Hopkins, Max Moran, Shay |
| author_facet | Hopkins, Max Moran, Shay |
| contents | Stability is a central property in learning and statistics promising the output of an algorithm $A$ does not change substantially when applied to similar datasets $S$ and $S'$. It is an elementary fact that any sufficiently stable algorithm (e.g.\ one returning the same result with high probability, satisfying privacy guarantees, etc.) must be randomized. This raises a natural question: can we quantify how much randomness is needed for algorithmic stability?
We study the randomness complexity of two influential notions of stability in learning: replicability, which promises $A$ usually outputs the same result when run over samples from the same distribution (and shared random coins), and differential privacy, which promises the output distribution of $A$ remains similar under neighboring datasets. The randomness complexity of these notions was studied recently in (Dixon et al. ICML 2024) and (Cannone et al. ITCS 2024) for basic $d$-dimensional tasks (e.g. estimating the bias of $d$ coins), but little is known about the measures more generally or in complex settings like classification.
Toward this end, we prove a `weak-to-strong' boosting theorem for stability: the randomness complexity of a task $M$ (either under replicability or DP) is tightly controlled by the best replication probability of any deterministic algorithm solving the task, a weak measure called `global stability' that is universally capped at $\frac{1}{2}$ (Chase et al. FOCS 2023). Using this, we characterize the randomness complexity of PAC Learning: a class has bounded randomness complexity iff it has finite Littlestone dimension, and moreover scales at worst logarithmically in the excess error of the learner. This resolves a question of (Chase et al. STOC 2024) who asked for such a characterization in the equivalent language of (error-dependent) `list-replicability'. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_08007 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Role of Randomness in Stability Hopkins, Max Moran, Shay Machine Learning 68Q32 Stability is a central property in learning and statistics promising the output of an algorithm $A$ does not change substantially when applied to similar datasets $S$ and $S'$. It is an elementary fact that any sufficiently stable algorithm (e.g.\ one returning the same result with high probability, satisfying privacy guarantees, etc.) must be randomized. This raises a natural question: can we quantify how much randomness is needed for algorithmic stability? We study the randomness complexity of two influential notions of stability in learning: replicability, which promises $A$ usually outputs the same result when run over samples from the same distribution (and shared random coins), and differential privacy, which promises the output distribution of $A$ remains similar under neighboring datasets. The randomness complexity of these notions was studied recently in (Dixon et al. ICML 2024) and (Cannone et al. ITCS 2024) for basic $d$-dimensional tasks (e.g. estimating the bias of $d$ coins), but little is known about the measures more generally or in complex settings like classification. Toward this end, we prove a `weak-to-strong' boosting theorem for stability: the randomness complexity of a task $M$ (either under replicability or DP) is tightly controlled by the best replication probability of any deterministic algorithm solving the task, a weak measure called `global stability' that is universally capped at $\frac{1}{2}$ (Chase et al. FOCS 2023). Using this, we characterize the randomness complexity of PAC Learning: a class has bounded randomness complexity iff it has finite Littlestone dimension, and moreover scales at worst logarithmically in the excess error of the learner. This resolves a question of (Chase et al. STOC 2024) who asked for such a characterization in the equivalent language of (error-dependent) `list-replicability'. |
| title | The Role of Randomness in Stability |
| topic | Machine Learning 68Q32 |
| url | https://arxiv.org/abs/2502.08007 |