In this section, we compare the symmetric solutions found in erf [2] and ReLU networks [5] to our one-neuron solution (n =1). The main difference is that both earlier studies constrain the search space to the symmetric subspace whereas we first prove that the non-trivial critical points are contained in this subspace in Theorem 5.1 for a broad class of activation functions, including erf and ReLU. Solving the low-dimensional loss, we recover the same solution for ReLU and erf as in [2, 5] for unit-orthonormal teachers.

Any continuous function f can be approximated arbitrarily well by a neural network with sufficiently many neurons k. We consider the case when f itself is a neural network with one hidden layer and k neurons. Approximating f with a neural network with n < k neurons can thus be seen as fitting an under-parameterized "student" network with nneurons to a "teacher" network with k neurons. As the student has fewer neurons than the teacher, it is unclear, whether each of the n student neurons should copy one of the teacher neurons or rather average a group of teacher neurons. For shallow neural networks with erf activation function and for the standard Gaussian input distribution, we prove that "copy-average" configurations are critical points if the teacher's incoming vectors are orthonormal and its outgoing weights are unitary. Moreover, the optimum among such configurations is reached when n 1student neurons each copy one teacher neuron and the n-th student neuron averages the remaining k n+1 teacher neurons. For the student network with n = 1 neuron, we provide additionally a closed-form solution of the non-trivial critical point(s) for commonly used activation functions through solving an equivalent constrained optimization problem. Empirically, we find for the erf activation function that gradient flow converges either to the optimal copy-average critical point or to another point where each student neuron approximately copies a different teacher neuron. Finally, we find similar results for the ReLU activation function, suggesting that the optimal solution of underparameterized networks has a universal structure.

We consider a contextual bandit problem with S contexts and K actions. In each round t = 1,2,... the learner observes a random context and chooses an action based on its past experience. The learner then observes a random reward whose mean is a function of the context and the action for the round. Under the assumption that the contexts can be lumped into r min{S,K}groups such that the mean reward for the various actions is the same for any two contexts that are in the same group, we give an algorithm that outputs an ε-optimal policy after using at most eO(r(S+K)/ε2) samples with high probability and provide a matching Ω(r(S + K)/ε2) lower bound. In the regret minimization setting, we give an algorithm whose cumulative regret up to time T is bounded by eO( p r3(S+K)T). To the best of our knowledge, we are the first to show the near-optimal sample complexity in the PAC setting and eO( p poly(r)(S+K)T)minimax regret in the online setting for this problem. We also show our algorithms can be applied to more general low-rank bandits and get improved regret bounds in some scenarios.

The extraordinary ability of generative models to generate photographic images has intensified concerns about the spread of disinformation, thereby leading to the demand for detectors capable of distinguishing between AI-generated fake images and real images. However, the lack of large datasets containing images from the most advanced image generators poses an obstacle to the development of such detectors. In this paper, we introduce the GenImage dataset, which has the following advantages: 1) Plenty of Images, including over one million pairs of AI-generated fake images and collected real images.

We follow the guidelines of the datasheets for datasets [1] to explain the composition, collection, recommended use case, and other details of the CityRefer dataset. For what purpose was the dataset created? We created this CityRefer dataset to facilitate research toward city-scale 3D visual grounding. Who created the dataset (e.g., which team, research group) and on behalf of which entity (e.g., company, institution, organization)? Who funded the creation of the dataset? What do the instances that comprise the dataset represent?

Some of the most successful knowledge graph embedding (KGE) models for link prediction - CP, RESCAL, TUCKER, COMPLEX - can be interpreted as energy-based models. Under this perspective they are not amenable for exact maximum-likelihood estimation (MLE), sampling and struggle to integrate logical constraints. This work re-interprets the score functions of these KGEs as circuits - constrained computational graphs allowing efficient marginalisation. Then, we design two recipes to obtain efficient generative circuit models by either restricting their activations to be non-negative or squaring their outputs. Our interpretation comes with little or no loss of performance for link prediction, while the circuits framework unlocks exact learning by MLE, efficient sampling of new triples, and guarantee that logical constraints are satisfied by design.