The following lemma is standard in the literature, see e.g.

The Rashomon set is the set of models that are approximately as good as the best model in the class. That is, it is the set of all good models.

The typical training of neural networks using large stepsize gradient descent (GD) under the logistic loss often involves two distinct phases, where the empirical risk oscillates in the first phase but decreases monotonically in the second phase. We investigate this phenomenon in two-layer networks that satisfy a near-homogeneity condition. We show that the second phase begins once the empirical risk falls below a certain threshold, dependent on the stepsize. Additionally, we show that the normalized margin grows nearly monotonically in the second phase, demonstrating an implicit bias of GD in training non-homogeneous predictors. If the dataset is linearly separable and the derivative of the activation function is bounded away from zero, we show that the average empirical risk decreases, implying that the first phase must stop in finite steps. Finally, we demonstrate that by choosing a suitably large stepsize, GD that undergoes this phase transition is more efficient than GD that monotonically decreases the risk. Our analysis applies to networks of any width, beyond the well-known neural tangent kernel and mean-field regimes.

In this section, we prove Theorem 3.1, which says that it suffices to the augmented state space First, we have the following lemma. Lemma A.2. W e have D Now, we prove Theorem 3.1. By Lemma A.2, we have F Theorem 3.1 follows straightforwardly from this result. Consider the same setup as in Lemma B.1. Lemma B.1, we have F Consider the same setup as in Lemma B.1.

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In this paper we consider an alternative strategy.

Symmetries are prevalent in deep learning and can significantly influence the learning dynamics of neural networks. In this paper, we examine how exponential symmetries - a broad subclass of continuous symmetries present in the model architecture or loss function - interplay with stochastic gradient descent (SGD). We first prove that gradient noise creates a systematic motion (a "Noether flow") of the parameters θ along the degenerate direction to a unique initialization-independent fixed point θ

In this paper, we contribute to both theoretical analyses.