Simple and Sharp Generalization Bounds via Lifting

We develop an information-theoretic framework for bounding the supremum of stochastic processes, offering a simpler and sharper alternative to classical chaining and slicing arguments for generalization bounds. The key idea is a lifting argument that produces information-theoretic analogues of empirical process bounds, such as Dudley’s entropy integral. Lifting introduces permutation symmetry, yielding sharp bounds when the classical Dudley integral is loose. This gives a simple proof of the majorizing measure theorem via the sharpness of Dudley’s entropy integral for stationary processes, a result known well before the proof of the majorizing measure theorem. Furthermore, the information-theoretic formulation provides soft versions of classical localized complexity bounds in generalization theory, but is simpler and does not require the slicing argument. We apply this approach to empirical risk minimization over Sobolev ellipsoids, obtaining sharp convergence rates in settings where previous methods are suboptimal.

💡 Research Summary

The paper introduces a novel information‑theoretic “lifting” technique that replaces the traditional chaining and slicing (peeling) arguments used to bound the supremum of stochastic processes in learning theory. The key idea is to replicate the stochastic process and impose permutation symmetry on the replicas, thereby converting a possibly non‑stationary process into a stationary one. This symmetry enables the authors to derive a rate‑distortion integral that is a two‑sided (upper and lower) bound on the expected supremum, a result that was previously unavailable for general Gaussian processes.

Formally, for a centered sub‑Gaussian process ((X_t){t\in T}) and a probability measure (\mu) on the index set, the authors define the set of couplings (\Pi\sigma(\mu)) with mean‑squared error (\sigma^2). They then prove the central equivalence

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