Intrinsically motivated graph exploration using network theories of human curiosity

Intrinsically motivated exploration has proven useful for reinforcement learning, even without additional extrinsic rewards. When the environment is naturally represented as a graph, how to guide exploration best remains an open question. In this work, we propose a novel approach for exploring graph-structured data motivated by two theories of human curiosity: the information gap theory and the compression progress theory. The theories view curiosity as an intrinsic motivation to optimize for topological features of subgraphs induced by the visited nodes in the environment. We use these proposed features as rewards for graph neural-network-based reinforcement learning. On multiple classes of synthetically generated graphs, we find that trained agents generalize to larger environments and to longer exploratory walks than are seen during training. Our method computes more efficiently than the greedy evaluation of the relevant topological properties. The proposed intrinsic motivations bear particular relevance for recommender systems. We demonstrate that curiosity-based recommendations are more predictive of human behavior than PageRank centrality for several real-world graph datasets, including MovieLens, Amazon Books, and Wikispeedia.

Multi-task Bias-Variance Trade-off Through Functional Constraints

Multi-task learning aims to acquire a set of functions, either regressors or classifiers, that perform well for diverse tasks. At its core, the idea behind multi-task learning is to exploit the intrinsic similarity across data sources to aid in the learning process for each individual domain. In this paper we draw intuition from the two extreme learning scenarios -- a single function for all tasks, and a task-specific function that ignores the other tasks dependencies -- to propose a bias-variance trade-off. To control the relationship between the variance (given by the number of i.i.d. samples), and the bias (coming from data from other task), we introduce a constrained learning formulation that enforces domain specific solutions to be close to a central function. This problem is solved in the dual domain, for which we propose a stochastic primal-dual algorithm. Experimental results for a multi-domain classification problem with real data show that the proposed procedure outperforms both the task specific, as well as the single classifiers.

Training Graph Neural Networks on Growing Stochastic Graphs

Graph Neural Networks (GNNs) rely on graph convolutions to exploit meaningful patterns in networked data. Based on matrix multiplications, convolutions incur in high computational costs leading to scalability limitations in practice. To overcome these limitations, proposed methods rely on training GNNs in smaller number of nodes, and then transferring the GNN to larger graphs. Even though these methods are able to bound the difference between the output of the GNN with different number of nodes, they do not provide guarantees against the optimal GNN on the very large graph. In this paper, we propose to learn GNNs on very large graphs by leveraging the limit object of a sequence of growing graphs, the graphon. We propose to grow the size of the graph as we train, and we show that our proposed methodology -- learning by transference -- converges to a neighborhood of a first order stationary point on the graphon data. A numerical experiment validates our proposed approach.

Learning Globally Smooth Functions on Manifolds

Smoothness and low dimensional structures play central roles in improving generalization and stability in learning and statistics. The combination of these properties has led to many advances in semi-supervised learning, generative modeling, and control of dynamical systems. However, learning smooth functions is generally challenging, except in simple cases such as learning linear or kernel models. Typical methods are either too conservative, relying on crude upper bounds such as spectral normalization, too lax, penalizing smoothness on average, or too computationally intensive, requiring the solution of large-scale semi-definite programs. These issues are only exacerbated when trying to simultaneously exploit low dimensionality using, e.g., manifolds. This work proposes to overcome these obstacles by combining techniques from semi-infinite constrained learning and manifold regularization. To do so, it shows that, under typical conditions, the problem of learning a Lipschitz continuous function on a manifold is equivalent to a dynamically weighted manifold regularization problem. This observation leads to a practical algorithm based on a weighted Laplacian penalty whose weights are adapted using stochastic gradient techniques. We prove that, under mild conditions, this method estimates the Lipschitz constant of the solution, learning a globally smooth solution as a byproduct. Numerical examples illustrate the advantages of using this method to impose global smoothness on manifolds as opposed to imposing smoothness on average.

Federated Representation Learning via Maximal Coding Rate Reduction

We propose a federated methodology to learn low-dimensional representations from a dataset that is distributed among several clients. In particular, we move away from the commonly-used cross-entropy loss in federated learning, and seek to learn shared low-dimensional representations of the data in a decentralized manner via the principle of maximal coding rate reduction (MCR2). Our proposed method, which we refer to as FLOW, utilizes MCR2 as the objective of choice, hence resulting in representations that are both between-class discriminative and within-class compressible. We theoretically show that our distributed algorithm achieves a first-order stationary point. Moreover, we demonstrate, via numerical experiments, the utility of the learned low-dimensional representations.

Training Stable Graph Neural Networks Through Constrained Learning

Graph Neural Networks (GNN) rely on graph convolutions to learn features from network data. GNNs are stable to different types of perturbations of the underlying graph, a property that they inherit from graph filters. In this paper we leverage the stability property of GNNs as a typing point in order to seek for representations that are stable within a distribution. We propose a novel constrained learning approach by imposing a constraint on the stability condition of the GNN within a perturbation of choice. We showcase our framework in real world data, corroborating that we are able to obtain more stable representations while not compromising the overall accuracy of the predictor.

Increase and Conquer: Training Graph Neural Networks on Growing Graphs

Graph neural networks (GNNs) use graph convolutions to exploit network invariances and learn meaningful features from network data. However, on large-scale graphs convolutions incur in high computational cost, leading to scalability limitations. Leveraging the graphon -- the limit object of a graph -- in this paper we consider the problem of learning a graphon neural network (WNN) -- the limit object of a GNN -- by training GNNs on graphs sampled Bernoulli from the graphon. Under smoothness conditions, we show that: (i) the expected distance between the learning steps on the GNN and on the WNN decreases asymptotically with the size of the graph, and (ii) when training on a sequence of growing graphs, gradient descent follows the learning direction of the WNN. Inspired by these results, we propose a novel algorithm to learn GNNs on large-scale graphs that, starting from a moderate number of nodes, successively increases the size of the graph during training. This algorithm is benchmarked on both a recommendation system and a decentralized control problem where it is shown to retain comparable performance, to its large-scale counterpart, at a reduced computational cost.

Multi-task Supervised Learning via Cross-learning

In this paper we consider a problem known as multi-task learning, consisting of fitting a set of classifier or regression functions intended for solving different tasks. In our novel formulation, we couple the parameters of these functions, so that they learn in their task specific domains while staying close to each other. This facilitates cross-fertilization in which data collected across different domains help improving the learning performance at each other task. First, we present a simplified case in which the goal is to estimate the means of two Gaussian variables, for the purpose of gaining some insights on the advantage of the proposed cross-learning strategy. Then we provide a stochastic projected gradient algorithm to perform cross-learning over a generic loss function. If the number of parameters is large, then the projection step becomes computationally expensive. To avoid this situation, we derive a primal-dual algorithm that exploits the structure of the dual problem, achieving a formulation whose complexity only depends on the number of tasks. Preliminary numerical experiments for image classification by neural networks trained on a dataset divided in different domains corroborate that the cross-learned function outperforms both the task-specific and the consensus approaches.