This generalization ability cannot be explained by the capacity of the explicitly specified model class (namely, the functions representable in the chosen architecture). Instead, it seems that the optimization algorithm biases us toward a "simple" model, minimizing

Learning with deep neural networks has enjoyed huge empirical success in recent years across a wide variety of tasks.

Jastrz ebski et al. [6] suggested that the ratio between the learning rate and the batch

We study the set of continuous functions that admit no spurious local optima (i.e.

Jastrz ebski et al. [6] suggested that the ratio between the learning rate and the batch

In recent years, transformer-based models have revolutionized deep learning, particularly in sequence modeling.

We study finite sample expressivity, i.e., memorization power of ReLU networks.

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We prove this by contradiction. The original problem in Eq. (2) is now equivalently reduced following problem because r Namely, it is the solution of a ridgeless linear regression problem. In the second case, one needs to discern the saddle points from the global minima. The objective (9) can be upper-bounded by Eq. (9) γ We see that the above inequality is equivalent to Eq. (9) γ B.5.1 Proposition 4 Proposition 4. Any global minimum of Eq. (9) is of the form U = b By the induction assumption, the global minimum of this problem takes the form of Eq. B.6 Lemma 3 Lemma 3. At any global minimum of Eq. (9), let b ( D 1) We first apply Lemma 3 to determine the condition for the nontrivial solution to exist.

SGD until 100% accuracy is achieved, in four different settings: 1. Random initialization + Training with true labels.