"equiconnected", which is a stronger result that we will define and introduce later in the paper.

We present an application of the connectedness of these superlevel sets to the derivation of minimax theorems for robust reinforcement learning.

"equiconnected", which is a stronger result that we will define and introduce later in the paper.

We discuss several aspects of the loss landscape of regularized neural networks: the structure of stationary points, connectivity of optimal solutions, path with nonincreasing loss to arbitrary global optimum, and the nonuniqueness of optimal solutions, by casting the problem into an equivalent convex problem and considering its dual. Starting from two-layer neural networks with scalar output, we first characterize the solution set of the convex problem using its dual and further characterize all stationary points. With the characterization, we show that the topology of the global optima goes through a phase transition as the width of the network changes, and construct counterexamples where the problem may have a continuum of optimal solutions. Finally, we show that the solution set characterization and connectivity results can be extended to different architectures, including two-layer vector-valued neural networks and parallel three-layer neural networks.

The aim of this paper is to improve the understanding of the optimization landscape for policy optimization problems in reinforcement learning. Specifically, we show that the superlevel set of the objective function with respect to the policy parameter is always a connected set both in the tabular setting and under policies represented by a class of neural networks. In addition, we show that the optimization objective as a function of the policy parameter and reward satisfies a stronger "equiconnectedness" property. To our best knowledge, these are novel and previously unknown discoveries. We present an application of the connectedness of these superlevel sets to the derivation of minimax theorems for robust reinforcement learning. We show that any minimax optimization program which is convex on one side and is equiconnected on the other side observes the minimax equality (i.e. has a Nash equilibrium). We find that this exact structure is exhibited by an interesting robust reinforcement learning problem under an adversarial reward attack, and the validity of its minimax equality immediately follows. This is the first time such a result is established in the literature.

We are interested in two-layer ReLU neural networks from an optimization perspective. We prove that the path-connected sublevel set, i.e., valleys, of a neural network which is Clarke stationary with respect to the training loss with weight decay regularization contains a specific, simpler and more structured neural network, which we call its minimal representation. We provide an explicit construction of a continuous path between the neural network and its minimal counterpart. Importantly, we show that characterizing the optimality properties of a neural network can be reduced to characterizing those of its minimal representation. Thanks to the specific structure of minimal neural networks, we show that we can embed them into a convex optimization landscape. Leveraging convexity, we are able to (i) characterize the minimal size of the hidden layer so that the neural network optimization landscape has no spurious valleys and (ii) provide a polynomial-time algorithm for checking if a neural network is a global minimum of the training loss. Overall, we provide a rich framework for studying the landscape of the neural network training loss through our embedding to a convex optimization landscape.

The signature transform is a 'universal nonlinearity' on the space of continuous vector-valued paths, and has received attention for use in machine learning on time series. However, real-world temporal data is typically observed at discrete points in time, and must first be transformed into a continuous path before signature techniques can be applied. We make this step explicit by characterising it as an imputation problem, and empirically assess the impact of various imputation strategies when applying signature-based neural nets to irregular time series data. For one of these strategies, Gaussian process (GP) adapters, we propose an extension~(GP-PoM) that makes uncertainty information directly available to the subsequent classifier while at the same time preventing costly Monte-Carlo (MC) sampling. In our experiments, we find that the choice of imputation drastically affects shallow signature models, whereas deeper architectures are more robust. Next, we observe that uncertainty-aware predictions (based on GP-PoM or indicator imputations) are beneficial for predictive performance, even compared to the uncertainty-aware training of conventional GP adapters. In conclusion, we have demonstrated that the path construction is indeed crucial for signature models and that our proposed strategy leads to competitive performance in general, while improving robustness of signature models in particular.

We study sublevel sets of the loss function in training deep neural networks. For linearly independent data, we prove that every sublevel set of the loss is connected and unbounded. We then apply this result to prove similar properties on the loss surface of deep over-parameterized neural nets with piecewise linear activation functions.

A number of modern learning tasks involve estimation from heterogeneous information sources. This includes classification with labeled and unlabeled data as well as other problems with analogous structure such as competitive (game theoretic) problems. The associated estimation problems can be typically reduced to solving a set of fixed point equations (consistency conditions). We introduce a general method for combining a preferred information source with another in this setting by evolving continuous paths of fixed points at intermediate allocations. We explicitly identify critical points along the unique paths to either increase the stability of estimation or to ensure a significant departure from the initial source. The homotopy continuation approach is guaranteed to terminate at the second source, and involves no combinatorial effort. We illustrate the power of these ideas both in classification tasks with labeled and unlabeled data, as well as in the context of a competitive (min-max) formulation of DNA sequence motif discovery.