We study which machine learning algorithms have tight generalization bounds with respect to a given collection of population distributions. Our results build on and extend the recent work of Gastpar et al. (2024). First, we present conditions that preclude the existence of tight generalization bounds. Specifically, we show that algorithms that have certain inductive biases that cause them to be unstable do not admit tight generalization bounds. Next, we show that algorithms that are sufficiently loss-stable do have tight generalization bounds. We conclude with a simple characterization that relates the existence of tight generalization bounds to the conditional variance of the algorithm's loss.

We propose a multi-step inertial Forward-Backward splitting algorithm for minimizing the sum of two non-necessarily convex functions, one of which is proper lower semi-continuous while the other is differentiable with a Lipschitz continuous gradient. We first prove global convergence of the algorithm with the help of the Kurdyka-Łojasiewicz property. Then, when the non-smooth part is also partly smooth relative to a smooth submanifold, we establish finite identification of the latter and provide sharp local linear convergence analysis. The proposed method is illustrated on several problems arising from statistics and machine learning.

Let M denote M = 2 1 1 2 1 . . . . . . . . . 1 2 1 1 2 R

The conditions on the representation that imply global convergence are different between these two algorithms.

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These shifts are defined via parametric changes in the causal mechanisms of observed variables, where constraints on parameters yield a "robustness set" of plausible distributions and acorresponding worst-case loss overthe set.