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- Information Technology > Artificial Intelligence > Machine Learning > Statistical Learning (1.00)
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Towards Data-Algorithm Dependent Generalization: a Case Study on Overparameterized Linear Regression
One of the major open problems in machine learning is to characterize generalization in the overparameterized regime, where most traditional generalization bounds become inconsistent even for overparameterized linear regression. In many scenarios, this failure can be attributed to obscuring the crucial interplay between the training algorithm and the underlying data distribution. This paper demonstrate that the generalization behavior of overparameterized model should be analyzed in a both data-relevant and algorithm-relevant manner. To make a formal characterization, We introduce a notion called data-algorithm compatibility, which considers the generalization behavior of the entire data-dependent training trajectory, instead of traditional last-iterate analysis.
Qualitative Mechanism Independence
We define what it means for a joint probability distribution to be compatible with aset of independent causal mechanisms, at a qualitative level--or, more precisely with a directed hypergraph $\mathcal A$, which is the qualitative structure of a probabilistic dependency graph (PDG). When A represents a qualitative Bayesian network, QIM-compatibility with $\mathcal A$ reduces to satisfying the appropriate conditional independencies. But giving semantics to hypergraphs using QIM-compatibility lets us do much more. For one thing, we can capture functional dependencies. For another, we can capture important aspects of causality using compatibility: we can use compatibility to understand cyclic causal graphs, and to demonstrate structural compatibility, we must essentially produce a causal model. Finally, compatibility has deep connections to information theory. Applying compatibility to cyclic structures helps to clarify a longstanding conceptual issue in information theory.