We present an approach to automate the process of discovering optimization methods, with a focus on deep learning architectures. We train a Recurrent Neural Network controller to generate a string in a domain specific language that describes a mathematical update equation based on a list of primitive functions, such as the gradient, running average of the gradient, etc. The controller is trained with Reinforcement Learning to maximize the performance of a model after a few epochs. On CIFAR-10, our method discovers several update rules that are better than many commonly used optimizers, such as Adam, RMSProp, or SGD with and without Momentum on a ConvNet model. We introduce two new optimizers, named PowerSign and AddSign, which we show transfer well and improve training on a variety of different tasks and architectures, including ImageNet classification and Google's neural machine translation system.
Many of the above update rules are designed by borrowing ideas from convex analysis, even though optimization problems in neural networks are non-convex. Recent empirical results with non-monotonic learning rate heuristics @cite_4 suggest that there are still many unknowns in training neural networks and that many ideas in non-convex optimization can be used to improve it.
@cite_4
2167967601 2963534251 2963076198 2785523195
Convexity has recently received a lot of attention in the machine learning community, and the lack of convexity has been seen as a major disadvantage of many learning algorithms, such as multi-layer artificial neural networks. We show that training multi-layer neural networks in which the number of hidden units is learned can be viewed as a convex optimization problem. This problem involves an infinite number of variables, but can be solved by incrementally inserting a hidden unit at a time, each time finding a linear classifier that minimizes a weighted sum of errors. We consider neural networks with a single hidden layer and non-decreasing positively homogeneous activation functions like the rectified linear units. By letting the number of hidden units grow unbounded and using classical non-Euclidean regularization tools on the output weights, they lead to a convex optimization problem and we provide a detailed theoretical analysis of their generalization performance, with a study of both the approximation and the estimation errors. We show in particular that they are adaptive to unknown underlying linear structures, such as the dependence on the projection of the input variables onto a low-dimensional subspace. Moreover, when using sparsity-inducing norms on the input weights, we show that high-dimensional non-linear variable selection may be achieved, without any strong assumption regarding the data and with a total number of variables potentially exponential in the number of observations. However, solving this convex optimization problem in infinite dimensions is only possible if the nonconvex subproblem of addition of a new unit can be solved efficiently. We provide a simple geometric interpretation for our choice of activation functions and describe simple conditions for convex relaxations of the finite-dimensional non-convex subproblem to achieve the same generalization error bounds, even when constant-factor approximations cannot be found. We were not able to find strong enough convex relaxations to obtain provably polynomialtime algorithms and leave open the existence or non-existence of such tractable algorithms with non-exponential sample complexities. With the increasing interest in deeper understanding of the loss surface of many non-convex deep models, this paper presents a unifying framework to study the local global optima equivalence of the optimization problems arising from training of such non-convex models. Using the "local openness" property of the underlying training models, we provide simple sufficient conditions under which any local optimum of the resulting optimization problem is globally optimal. We first completely characterize the local openness of matrix multiplication mapping in its range. Then we use our characterization to: 1) show that every local optimum of two layer linear networks is globally optimal. Unlike many existing results in the literature, our result requires no assumption on the target data matrix Y, and input data matrix X. 2) develop almost complete characterization of the local global optima equivalence of multi-layer linear neural networks. We provide various counterexamples to show the necessity of each of our assumptions. 3) show global local optima equivalence of non-linear deep models having certain pyramidal structure. Unlike some existing works, our result requires no assumption on the differentiability of the activation functions and can go beyond "full-rank" cases. Several recently proposed stochastic optimization methods that have been successfully used in training deep networks such as RMSProp, Adam, Adadelta, Nadam, etc are based on using gradient updates scaled by square roots of exponential moving averages of squared past gradients. It has been empirically observed that sometimes these algorithms fail to converge to an optimal solution (or a critical point in nonconvex settings). We show that one cause for such failures is the exponential moving average used in the algorithms. We provide an explicit example of a simple convex optimization setting where Adam does not converge to the optimal solution, and describe the precise problems with the previous analysis of Adam algorithm. Our analysis suggests that the convergence issues may be fixed by endowing such algorithms with "long-term memory" of past gradients, and propose new variants of the Adam algorithm which not only fix the convergence issues but often also lead to improved empirical performance.