This paper studies the properties of solutions to multi-task shallow ReLU neural network learning problems, wherein the network is trained to fit a dataset with minimal sum of squared weights. Remarkably, the solutions learned for each individual task resemble those obtained by solving a kernel regression problem, revealing a novel connection between neural networks and kernel methods. It is known that single-task neural network learning problems are equivalent to a minimum norm interpolation problem in a non-Hilbertian Banach space, and that the solutions of such problems are generally non-unique. In contrast, we prove that the solutions to univariate-input, multi-task neural network interpolation problems are almost always unique, and coincide with the solution to a minimum-norm interpolation problem in a Sobolev (Reproducing Kernel) Hilbert Space.

We explore online learning in episodic Markov decision processes on non-stationary environments (changing losses and probability transitions). Our focus is on the Concave Utility Reinforcement Learning problem (CURL), an extension of classical RL for handling convex performance criteria in state-action distributions induced by agent policies. While various machine learning problems can be written as CURL, its non-linearity invalidates traditional Bellman equations.

In this section, we discuss the hyperbolic operations used in HNN formulations and set up the meta-learning problem. This particular setup is also known as the N-ways K-shot learning problem. This section provides the theoretical proofs of the theorems presented in our main paper. Note that points in the local tangent space follow Euclidean algebra. The columns present the number of tasks in each batch (# Tasks), HNN update learning rate (), meta update learning rate (), and size of hidden dimensions (d).

Penalizing the nuclear norm of a function's Jacobian encourages it to locally behave like a low-rank linear map. Such functions vary locally along only a handful of directions, making the Jacobian nuclear norm a natural regularizer for machine learning problems. However, this regularizer is intractable for high-dimensional problems, as it requires computing a large Jacobian matrix and taking its SVD. We show how to efficiently penalize the Jacobian nuclear norm using techniques tailormade for deep learning. We prove that for functions parametrized as compositions f = g h, one may equivalently penalize the average squared Frobenius norms of Jg and Jh. We then propose a denoising-style approximation that avoids Jacobian computations altogether. Our method is simple, efficient, and accurate, enabling Jacobian nuclear norm regularization to scale to high-dimensional deep learning problems. We complement our theory with an empirical study of our regularizer's performance and investigate applications to denoising and representation learning.

Equivariant deep learning architectures exploit symmetries in learning problems to improve the sample efficiency of neural-network-based models and their ability to generalise. However, when modelling real-world data, learning problems are often not exactly equivariant, but only approximately. For example, when estimating the global temperature field from weather station observations, local topographical features like mountains break translation equivariance. In these scenarios, it is desirable to construct architectures that can flexibly depart from exact equivariance in a data-driven way. Current approaches to achieving this cannot usually be applied out-of-the-box to any architecture and symmetry group. In this paper, we develop a general approach to achieving this using existing equivariant architectures. Our approach is agnostic to both the choice of symmetry group and model architecture, making it widely applicable. We consider the use of approximately equivariant architectures in neural processes (NPs), a popular family of meta-learning models. We demonstrate the effectiveness of our approach on a number of synthetic and realworld regression experiments, showing that approximately equivariant NP models can outperform both their non-equivariant and strictly equivariant counterparts.

In this paper we consider the problem of how a reinforcement learning agent that is tasked with solving a sequence of reinforcement learning problems (a sequence of Markov decision processes) can use knowledge acquired early in its lifetime to improve its ability to solve new problems. We argue that previous experience with similar problems can provide an agent with information about how it should explore when facing a new but related problem. We show that the search for an optimal exploration strategy can be formulated as a reinforcement learning problem itself and demonstrate that such strategy can leverage patterns found in the structure of related problems. We conclude with experiments that show the benefits of optimizing an exploration strategy using our proposed framework.

Despite the success of contrastive learning (CL) in vision and language, its theoretical foundations and mechanisms for building representations remain poorly understood. In this work, we build connections between noise contrastive estimation losses widely used in CL and distribution alignment with entropic optimal transport (OT). This connection allows us to develop a family of different losses and multistep iterative variants for existing CL methods. Intuitively, by using more information from the distribution of latents, our approach allows a more distribution-aware manipulation of the relationships within augmented sample sets. We provide theoretical insights and experimental evidence demonstrating the benefits of our approach for generalized contrastive alignment. Through this framework, it is possible to leverage tools in OT to build unbalanced losses to handle noisy views and customize the representation space by changing the constraints on alignment. By reframing contrastive learning as an alignment problem and leveraging existing optimization tools for OT, our work provides new insights and connections between different self-supervised learning models in addition to new tools that can be more easily adapted to incorporate domain knowledge into learning.

This paper considers the problem of efficient exploration of unseen environments, a key challenge in AI. We propose a'learning to explore' framework where we learn a policy from a distribution of environments. At test time, presented with an unseen environment from the same distribution, the policy aims to generalize the exploration strategy to visit the maximum number of unique states in a limited number of steps. We particularly focus on environments with graph-structured state-spaces that are encountered in many important real-world applications like software testing and map building. We formulate this task as a reinforcement learning problem where the'exploration' agent is rewarded for transitioning to previously unseen environment states and employ a graph-structured memory to encode the agent's past trajectory. Experimental results demonstrate that our approach is extremely effective for exploration of spatial maps; and when applied on the challenging problems of coverage-guided software-testing of domain-specific programs and real-world mobile applications, it outperforms methods that have been hand-engineered by human experts.

Although optimization is the longstanding algorithmic backbone of machine learning, new models still require the time-consuming implementation of new solvers. As a result, there are thousands of implementations of optimization algorithms for machine learning problems. A natural question is, if it is always necessary to implement a new solver, or if there is one algorithm that is sufficient for most models. Common belief suggests that such a one-algorithm-fits-all approach cannot work, because this algorithm cannot exploit model specific structure and thus cannot be efficient and robust on a wide variety of problems. Here, we challenge this common belief. We have designed and implemented the optimization framework GENO (GENeric Optimization) that combines a modeling language with a generic solver. GENO generates a solver from the declarative specification of an optimization problem class. The framework is flexible enough to encompass most of the classical machine learning problems. We show on a wide variety of classical but also some recently suggested problems that the automatically generated solvers are (1) as efficient as well-engineered specialized solvers, (2) more efficient by a decent margin than recent state-of-the-art solvers, and (3) orders of magnitude more efficient than classical modeling language plus solver approaches.

We propose a generalization of constrained Markov decision processes (CMDPs) that we call the semi-infinitely constrained Markov decision process (SICMDP). Particularly, we consider a continuum of constraints instead of a finite number of constraints as in the case of ordinary CMDPs. We also devise a reinforcement learning algorithm for SICMDPs that we call SI-CRL. We first transform the reinforcement learning problem into a linear semi-infinitely programming (LSIP) problem and then use the dual exchange method in the LSIP literature to solve it. To the best of our knowledge, we are the first to apply tools from semi-infinitely programming (SIP) to solve constrained reinforcement learning problems.