Understanding the representational power of Deep Neural Networks (DNNs) and how their structural properties (e.g., depth, width, type of activation unit) affect the functions they can compute, has been an important yet challenging question in deep learning and approximation theory. In a seminal paper, Telgarsky highlighted the benefits of depth by presenting a family of functions (based on simple triangular waves) for which DNNs achieve zero classification error, whereas shallow networks with fewer than exponentially many nodes incur constant error. Even though Telgarsky's work reveals the limitations of shallow neural networks, it does not inform us on why these functions are difficult to represent and in fact he states it as a tantalizing open question to characterize those functions that cannot be well-approximated by smaller depths. In this work, we point to a new connection between DNNs expressivity and Sharkovsky's Theorem from dynamical systems, that enables us to characterize the depth-width trade-offs of ReLU networks for representing functions based on the presence of generalized notion of fixed points, called periodic points (a fixed point is a point of period 1). Motivated by our observation that the triangle waves used in Telgarsky's work contain points of period 3 - a period that is special in that it implies chaotic behavior based on the celebrated result by Li-Yorke - we proceed to give general lower bounds for the width needed to represent periodic functions as a function of the depth. Technically, the crux of our approach is based on an eigenvalue analysis of the dynamical system associated with such functions.

We propose to study the generalization error of a learned predictor $\hat h$ in terms of that of a surrogate (potentially randomized) classifier that is coupled to $\hat h$ and designed to trade empirical risk for control of generalization error. In the case where $\hat h$ interpolates the data, it is interesting to consider theoretical surrogate classifiers that are partially derandomized or rerandomized, e.g., fit to the training data but with modified label noise. We show that replacing $\hat h$ by its conditional distribution with respect to an arbitrary $\sigma$-field is a viable method to derandomize. We give an example, inspired by the work of Nagarajan and Kolter (2019), where the learned classifier $\hat h$ interpolates the training data with high probability, has small risk, and, yet, does not belong to a nonrandom class with a tight uniform bound on two-sided generalization error. At the same time, we bound the risk of $\hat h$ in terms of a surrogate that is constructed by conditioning and shown to belong to a nonrandom class with uniformly small generalization error.

Inverse reinforcement learning has proved its ability to explain state-action trajectories of expert agents by recovering their underlying reward functions in increasingly challenging environments. Recent advances in adversarial learning have allowed extending inverse RL to applications with non-stationary environment dynamics unknown to the agents, arbitrary structures of reward functions and improved handling of the ambiguities inherent to the ill-posed nature of inverse RL. This is particularly relevant in real time applications on stochastic environments involving risk, like volatile financial markets. Moreover, recent work on simulation of complex environments enable learning algorithms to engage with real market data through simulations of its latent space representations, avoiding a costly exploration of the original environment. In this paper, we explore whether adversarial inverse RL algorithms can be adapted and trained within such latent space simulations from real market data, while maintaining their ability to recover agent rewards robust to variations in the underlying dynamics, and transfer them to new regimes of the original environment.

Conditional GANs [29] (cGANs) are a type of GAN that use conditional information such as class labels to guide the training of the discriminator and the generator. Most frameworks of cGANs either augment a GAN by injecting (embedded) class information into the architecture of the real/fake discriminator [31], or add an auxiliary loss that is class based [36]. We place the class conditional structure at the forefront of the generative model by proposing a loss that ensures generator updates are always class specific. Rather than training a function that measures the information theoretic distance between the generative distribution and one target distribution, we generalize the successful hinge-loss [28] that has become an essential ingredient of many GANs [38, 7] to the multi-class setting and use it to train a single generator classifier pair [38]. While the canonical hinge loss made generator updates according to a class agnostic margin a real/fake discriminator learned [28], our multi-class hinge-loss GAN updates the generator according to many classification margins. With this modification, we are able to accelerate training and achieve state of the art Inception Scores on CIFAR10, CIFAR100, and STL10.

Structured CNN designed using the prior information of problems potentially improves efficiency over conventional CNNs in various tasks in solving PDEs and inverse problems in signal processing. This paper introduces BNet2, a simplified Butterfly-Net and inline with the conventional CNN. Moreover, a Fourier transform initialization is proposed for both BNet2 and CNN with guaranteed approximation power to represent the Fourier transform operator. Experimentally, BNet2 and the Fourier transform initialization strategy are tested on various tasks, including approximating Fourier transform operator, end-to-end solvers of linear and nonlinear PDEs in 1D, and denoising and deblurring of 1D signals. On all tasks, under the same initialization, BNet2 achieves similar accuracy as CNN but has fewer parameters. Fourier transform initialized BNet2 and CNN consistently improve the training and testing accuracy over the randomly initialized CNN.

We present a deep learning system for testing graphics units by detecting novel visual corruptions in videos. Unlike previous work in which manual tagging was required to collect labeled training data, our weak supervision method is fully automatic and needs no human labelling. This is achieved by reproducing driver bugs that increase the probability of generating corruptions, and by making use of ideas and methods from the Multiple Instance Learning (MIL) setting. In our experiments, we significantly outperform unsupervised methods such as GAN-based models and discover novel corruptions undetected by baselines, while adhering to strict requirements on accuracy and efficiency of our real-time system.

We design a new provably efficient algorithm for episodic reinforcement learning with generalized linear function approximation. We analyze the algorithm under a new expressivity assumption that we call "optimistic closure," which is strictly weaker than assumptions from prior analyses for the linear setting. With optimistic closure, we prove that our algorithm enjoys a regret bound of $\tilde{O}(\sqrt{d^3 T})$ where $d$ is the dimensionality of the state-action features and $T$ is the number of episodes. This is the first statistically and computationally efficient algorithm for reinforcement learning with generalized linear functions.

Recent progress in recommender system research has shown the importance of including temporal representations to improve interpretability and performance. Here, we incorporate temporal representations in continuous time via recurrent point process for a dynamical model of reviews. Our goal is to characterize how changes in perception, user interest and seasonal effects affect review text.

We consider the initial situation where a dataset has been over-partitioned into $k$ clusters and seek a domain independent way to merge those initial clusters. We identify the total variation distance (TVD) as suitable for this goal. By exploiting the relation of the TVD to the Bayes accuracy we show how neural networks can be used to estimate TVDs between all pairs of clusters in parallel. Crucially, the needed memory space is decreased by reducing the required number of output neurons from $k^2$ to $k$. On realistically obtained over-clusterings of ImageNet subsets it is demonstrated that our TVD estimates lead to better merge decisions than those obtained by relying on state-of-the-art unsupervised representations. Further the generality of the approach is verified by evaluating it on a a point cloud dataset.

We have derived theoretical maximum likelihood estimators of trends for standard dynamics and implemented them. We have reframed the problem of trend detection into a classification problem amenable to machine learning methods. We have shown that RNN are in a way a generalization of simple moving average techniques and motivated this by theory. In a simple case, we have shown that this generalization transforms the trend estimation problem into simply locating the state vector into convex polytopes cells. Finally, we have showed empirically that GRU or LSTM cells are on average the best building block to use compared to a broad range of estimators in order to detect trends in time series. Putting the emphasis on learning stylized data and then transferring to real data rather than building complex structures fitted to data is also an important takeaway of this paper. Ongoing preliminary research seems to validate our approach for financial applications. This might pave the way to building efficient market estimators protected against over-fitting.