Deep neural network learning can be formulated as a non-convex optimization problem. Existing optimization algorithms, e.g., Adam, can learn the models fast, but may get stuck in local optima easily. In this paper, we introduce a novel optimization algorithm, namely GADAM (Genetic-Evolutionary Adam). GADAM learns deep neural network models based on a number of unit models generations by generations: it trains the unit models with Adam, and evolves them to the new generations with genetic algorithm. We will show that GADAM can effectively jump out of the local optima in the learning process to obtain better solutions, and prove that GADAM can also achieve a very fast convergence. Extensive experiments have been done on various benchmark datasets, and the learning results will demonstrate the effectiveness and efficiency of the GADAM algorithm.
SGD and SGDM scale the gradient uniformly in all directions (i.e., all the variables), which makes the tuning of learning rate @math very tedious and laborious. To resolve such a problem, several adaptive optimization methods have been proposed, including Adam @cite_14 , RMSprop @cite_24 and Adagrad @cite_6 , which uses distinct learning rates for different variables. Formally, the variable updating equations adopted in Adagrad @cite_6 and RMSprop @cite_24 can be represented as follows respectively: In Adagrad, vector @math increases monotonically as the update iteration continues, and its scaling factor will keep decreasing when updating vector @math . RMSprop addresses the problem by employing an average scale instead of a cumulative scale, and Adam @cite_14 resolves such a problem with a bias correction, which adopts an exponential moving average for the step in lieu of the gradient.
@cite_24 @cite_14 @cite_6
2964121744 2146502635
Adam and RMSProp are two of the most influential adaptive stochastic algorithms for training deep neural networks, which have been pointed out to be divergent even in the convex setting via a few simple counterexamples. Many attempts, such as decreasing an adaptive learning rate, adopting a big batch size, incorporating a temporal decorrelation technique, seeking an analogous surrogate, etc., have been tried to promote Adam RMSProp-type algorithms to converge. In contrast with existing approaches, we introduce an alternative easy-to-check sufficient condition, which merely depends on the parameters of the base learning rate and combinations of historical second-order moments, to guarantee the global convergence of generic Adam RMSProp for solving large-scale non-convex stochastic optimization. Moreover, we show that the convergences of several variants of Adam, such as AdamNC, AdaEMA, etc., can be directly implied via the proposed sufficient condition in the non-convex setting. In addition, we illustrate that Adam is essentially a specifically weighted AdaGrad with exponential moving average momentum, which provides a novel perspective for understanding Adam and RMSProp. This observation coupled with this sufficient condition gives much deeper interpretations on their divergences. At last, we validate the sufficient condition by applying Adam and RMSProp to tackle a certain counterexample and train deep neural networks. Numerical results are exactly in accord with our theoretical analysis. Adaptive optimization methods, which perform local optimization with a metric constructed from the history of iterates, are becoming increasingly popular for training deep neural networks. Examples include AdaGrad, RMSProp, and Adam. We show that for simple overparameterized problems, adaptive methods often find drastically different solutions than gradient descent (GD) or stochastic gradient descent (SGD). We construct an illustrative binary classification problem where the data is linearly separable, GD and SGD achieve zero test error, and AdaGrad, Adam, and RMSProp attain test errors arbitrarily close to half. We additionally study the empirical generalization capability of adaptive methods on several state-of-the-art deep learning models. We observe that the solutions found by adaptive methods generalize worse (often significantly worse) than SGD, even when these solutions have better training performance. These results suggest that practitioners should reconsider the use of adaptive methods to train neural networks. Adaptive gradient methods such as AdaGrad and its variants update the stepsize in stochastic gradient descent on the fly according to the gradients received along the way; such methods have gained widespread use in large-scale optimization for their ability to converge robustly, without the need to fine-tune the stepsize schedule. Yet, the theoretical guarantees to date for AdaGrad are for online and convex optimization. We bridge this gap by providing theoretical guarantees for the convergence of AdaGrad for smooth, nonconvex functions. We show that the norm version of AdaGrad (AdaGrad-Norm) converges to a stationary point at the @math rate in the stochastic setting, and at the optimal @math rate in the batch (non-stochastic) setting -- in this sense, our convergence guarantees are 'sharp'. In particular, the convergence of AdaGrad-Norm is robust to the choice of all hyper-parameters of the algorithm, in contrast to stochastic gradient descent whose convergence depends crucially on tuning the step-size to the (generally unknown) Lipschitz smoothness constant and level of stochastic noise on the gradient. Extensive numerical experiments are provided to corroborate our theory; moreover, the experiments suggest that the robustness of AdaGrad-Norm extends to state-of-the-art models in deep learning, without sacrificing generalization. Several recently proposed stochastic optimization methods that have been successfully used in training deep networks such as RMSProp, Adam, Adadelta, Nadam are based on using gradient updates scaled by square roots of exponential moving averages of squared past gradients. In many applications, e.g. learning with large output spaces, it has been empirically observed that 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 can 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.