In this paper we study the problem of learning Rectified Linear Units (ReLUs) which are functions of the form $\vct{x}\mapsto \max(0,\langle \vct{w},\vct{x}\rangle)$ with $\vct{w}\in\R^d$ denoting the weight vector. We study this problem in the high-dimensional regime where the number of observations are fewer than the dimension of the weight vector. We assume that the weight vector belongs to some closed set (convex or nonconvex) which captures known side-information about its structure. We focus on the realizable model where the inputs are chosen i.i.d.~from a Gaussian distribution and the labels are generated according to a planted weight vector. We show that projected gradient descent, when initialized at $\vct{0}$, converges at a linear rate to the planted model with a number of samples that is optimal up to numerical constants. Our results on the dynamics of convergence of these very shallow neural nets may provide some insights towards understanding the dynamics of deeper architectures.

By reparameterizing the weights in this way we improve the conditioning of the optimization problem and we speed up convergence of stochastic gradient descent. Our reparameterization is inspired by batch normalization but does not introduce any dependencies between the examples in a minibatch. This means that our method can also be applied successfully to recurrent models such as LSTMs and to noise-sensitive applications such as deep reinforcement learning or generative models, for which batch normalization is less well suited. Although our method is much simpler, it still provides much of the speed-up of full batch normalization. In addition, the computational overhead of our method is lower, permitting more optimization steps to be taken in the same amount of time. We demonstrate the usefulness of our method on applications in supervised image recognition, generative modelling, and deep reinforcement learning.

Suppose an n x d design matrix in a linear regression problem is given, but the response for each point is hidden unless explicitly requested. The goal is to sample only a small number k << n of the responses, and then produce a weight vector whose sum of squares loss over points is at most 1+epsilon times the minimum. When k is very small (e.g., k=d), jointly sampling diverse subsets of points is crucial. One such method called volume sampling has a unique and desirable property that the weight vector it produces is an unbiased estimate of the optimum. It is therefore natural to ask if this method offers the optimal unbiased estimate in terms of the number of responses k needed to achieve a 1+epsilon loss approximation. Surprisingly we show that volume sampling can have poor behavior when we require a very accurate approximation -- indeed worse than some i.i.d.

We show that it provides significant benefits to RL agents on a suite of gridworld, CARL and MetaWorld tasks.

Each sublayer adds a skip-connection (ADD) and batch normalization (BN). The decoder sequentially chooses a node according to a probability distribution produced by the node embeddings to construct a solution. The scaled symmetric sampling method is shown in Algorithm 2. The scaled factor The uniform division of the weight space is illustrated as follows. Thus, its approximate Pareto optimal solutions are commonly pursued. V ehicles must serve all the customers and finally return to the depot.

Recently, neural heuristics based on deep reinforcement learning have exhibited promise in solving multi-objective combinatorial optimization problems (MO-COPs).

The cost-accuracy trade-off has been a longstanding and critical issue for deep neural networks.

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Neural Information Processing Systems http://nips.cc/