Meta-Learning and Universality: Deep Representations and Gradient Descent can Approximate any Learning Algorithm

arXiv.org Artificial Intelligence

Learning to learn is a powerful paradigm for enabling models to learn from data more effectively and efficiently. A popular approach to meta-learning is to train a recurrent model to read in a training dataset as input and output the parameters of a learned model, or output predictions for new test inputs. Alternatively, a more recent approach to meta-learning aims to acquire deep representations that can be effectively fine-tuned, via standard gradient descent, to new tasks. In this paper, we consider the meta-learning problem from the perspective of universality, formalizing the notion of learning algorithm approximation and comparing the expressive power of the aforementioned recurrent models to the more recent approaches that embed gradient descent into the meta-learner. In particular, we seek to answer the following question: does deep representation combined with standard gradient descent have sufficient capacity to approximate any learning algorithm? We find that this is indeed true, and further find, in our experiments, that gradient-based meta-learning consistently leads to learning strategies that generalize more widely compared to those represented by recurrent models.


The Expressive Power of Neural Networks: A View from the Width

Neural Information Processing Systems

The expressive power of neural networks is important for understanding deep learning. Most existing works consider this problem from the view of the depth of a network. In this paper, we study how width affects the expressiveness of neural networks. Classical results state that depth-bounded (e.g. depth-2) networks with suitable activation functions are universal approximators. We show a universal approximation theorem for width-bounded ReLU networks: width-(n + 4) ReLU networks, where n is the input dimension, are universal approximators. Moreover, except for a measure zero set, all functions cannot be approximated by width-n ReLU networks, which exhibits a phase transition. Several recent works demonstrate the benefits of depth by proving the depth-efficiency of neural networks. That is, there are classes of deep networks which cannot be realized by any shallow network whose size is no more than an exponential bound. Here we pose the dual question on the width-efficiency of ReLU networks: Are there wide networks that cannot be realized by narrow networks whose size is not substantially larger? We show that there exist classes of wide networks which cannot be realized by any narrow network whose depth is no more than a polynomial bound. On the other hand, we demonstrate by extensive experiments that narrow networks whose size exceed the polynomial bound by a constant factor can approximate wide and shallow network with high accuracy. Our results provide more comprehensive evidence that depth may be more effective than width for the expressiveness of ReLU networks.


Universal Approximation of Markov Kernels by Shallow Stochastic Feedforward Networks

arXiv.org Machine Learning

We establish upper bounds for the minimal number of hidden units for which a binary stochastic feedforward network with sigmoid activation probabilities and a single hidden layer is a universal approximator of Markov kernels. We show that each possible probabilistic assignment of the states of $n$ output units, given the states of $k\geq1$ input units, can be approximated arbitrarily well by a network with $2^{k-1}(2^{n-1}-1)$ hidden units.


Value Functions for RL-Based Behavior Transfer: A Comparative Study

AAAI Conferences

Temporal difference (TD) learning methods (Sutton & Barto 1998) have become popular reinforcement learning techniques in recent years. TD methods, relying on function approximators to generalize learning to novel situations, have had some experimental successes and have been shown to exhibit some desirable properties in theory, but have often been found slow in practice. This paper presents methods for further generalizing across tasks, thereby speeding up learning, via a novel form of behavior transfer. We compare learning on a complex task with three function approximators, a CMAC, a neural network, and an RBF, and demonstrate that behavior transfer works well with all three. Using behavior transfer, agents are able to learn one task and then markedly reduce the time it takes to learn a more complex task. Our algorithms are fully implemented and tested in the RoboCup-soccer keepaway domain.


Feasibility of random basis function approximators for modeling and control

arXiv.org Artificial Intelligence

We discuss the role of random basis function approximators in modeling and control. We analyze the published work on random basis function approximators and demonstrate that their favorable error rate of convergence O(1/n) is guaranteed only with very substantial computational resources. We also discuss implications of our analysis for applications of neural networks in modeling and control.