Iteration and Stochastic First-order Oracle Complexities of Stochastic Gradient Descent using Constant and Decaying Learning Rates

The performance of stochastic gradient descent (SGD), which is the simplest first-order optimizer for training deep neural networks, depends on not only the learning rate but also the batch size. They both affect the number of iterations and the stochastic first-order oracle (SFO) complexity needed for training. In particular, the previous numerical results indicated that, for SGD using a constant learning rate, the number of iterations needed for training decreases when the batch size increases, and the SFO complexity needed for training is minimized at a critical batch size and that it increases once the batch size exceeds that size. Here, we study the relationship between batch size and the iteration and SFO complexities needed for nonconvex optimization in deep learning with SGD using constant or decaying learning rates and show that SGD using the critical batch size minimizes the SFO complexity. We also provide numerical comparisons of SGD with the existing first-order optimizers and show the usefulness of SGD using a critical batch size. Moreover, we show that measured critical batch sizes are close to the sizes estimated from our theoretical results.

Role of Momentum in Smoothing Objective Function in Implicit Graduated Optimization

While stochastic gradient descent (SGD) with momentum has fast convergence and excellent generalizability, a theoretical explanation for this is lacking. In this paper, we show that SGD with momentum smooths the objective function, the degree of which is determined by the learning rate, the batch size, the momentum factor, the variance of the stochastic gradient, and the upper bound of the gradient norm. This theoretical finding reveals why momentum improves generalizability and provides new insights into the role of the hyperparameters, including momentum factor. We also present an implicit graduated optimization algorithm that exploits the smoothing properties of SGD with momentum and provide experimental results supporting our assertion that SGD with momentum smooths the objective function.

Using Stochastic Gradient Descent to Smooth Nonconvex Functions: Analysis of Implicit Graduated Optimization with Optimal Noise Scheduling

The graduated optimization approach is a heuristic method for finding globally optimal solutions for nonconvex functions and has been theoretically analyzed in several studies. This paper defines a new family of nonconvex functions for graduated optimization, discusses their sufficient conditions, and provides a convergence analysis of the graduated optimization algorithm for them. It shows that stochastic gradient descent (SGD) with mini-batch stochastic gradients has the effect of smoothing the function, the degree of which is determined by the learning rate and batch size. This finding provides theoretical insights on why large batch sizes fall into sharp local minima, why decaying learning rates and increasing batch sizes are superior to fixed learning rates and batch sizes, and what the optimal learning rate scheduling is. To the best of our knowledge, this is the first paper to provide a theoretical explanation for these aspects. Moreover, a new graduated optimization framework that uses a decaying learning rate and increasing batch size is analyzed and experimental results of image classification that support our theoretical findings are reported.

Relationship between Batch Size and Number of Steps Needed for Nonconvex Optimization of Stochastic Gradient Descent using Armijo Line Search

Stochastic gradient descent (SGD) is the simplest deep learning optimizer with which to train deep neural networks. While SGD can use various learning rates, such as constant or diminishing rates, the previous numerical results showed that SGD performs better than other deep learning optimizers using when it uses learning rates given by line search methods. In this paper, we perform a convergence analysis on SGD with a learning rate given by an Armijo line search for nonconvex optimization. The analysis indicates that the upper bound of the expectation of the squared norm of the full gradient becomes small when the number of steps and the batch size are large. Next, we show that, for SGD with the Armijo-line-search learning rate, the number of steps needed for nonconvex optimization is a monotone decreasing convex function of the batch size; that is, the number of steps needed for nonconvex optimization decreases as the batch size increases. Furthermore, we show that the stochastic first-order oracle (SFO) complexity, which is the stochastic gradient computation cost, is a convex function of the batch size; that is, there exists a critical batch size that minimizes the SFO complexity. Finally, we provide numerical results that support our theoretical results. The numerical results indicate that the number of steps needed for training deep neural networks decreases as the batch size increases and that there exist the critical batch sizes that can be estimated from the theoretical results.

Critical Bach Size Minimizes Stochastic First-Order Oracle Complexity of Deep Learning Optimizer using Hyperparameters Close to One

Practical results have shown that deep learning optimizers using small constant learning rates, hyperparameters close to one, and large batch sizes can find the model parameters of deep neural networks that minimize the loss functions. We first show theoretical evidence that the momentum method (Momentum) and adaptive moment estimation (Adam) perform well in the sense that the upper bound of the theoretical performance measure is small with a small constant learning rate, hyperparameters close to one, and a large batch size. Next, we show that there exists a batch size called the critical batch size minimizing the stochastic first-order oracle (SFO) complexity, which is the stochastic gradient computation cost, and that SFO complexity increases once the batch size exceeds the critical batch size. Finally, we provide numerical results that support our theoretical results. That is, the numerical results indicate that Adam using a small constant learning rate, hyperparameters close to one, and the critical batch size minimizing SFO complexity has faster convergence than Momentum and stochastic gradient descent (SGD).

Theoretical analysis of Adam using hyperparameters close to one without Lipschitz smoothness

Convergence and convergence rate analyses of adaptive methods, such as Adaptive Moment Estimation (Adam) and its variants, have been widely studied for nonconvex optimization. The analyses are based on assumptions that the expected or empirical average loss function is Lipschitz smooth (i.e., its gradient is Lipschitz continuous) and the learning rates depend on the Lipschitz constant of the Lipschitz continuous gradient. Meanwhile, numerical evaluations of Adam and its variants have clarified that using small constant learning rates without depending on the Lipschitz constant and hyperparameters ($\beta_1$ and $\beta_2$) close to one is advantageous for training deep neural networks. Since computing the Lipschitz constant is NP-hard, the Lipschitz smoothness condition would be unrealistic. This paper provides theoretical analyses of Adam without assuming the Lipschitz smoothness condition in order to bridge the gap between theory and practice. The main contribution is to show theoretical evidence that Adam using small learning rates and hyperparameters close to one performs well, whereas the previous theoretical results were all for hyperparameters close to zero. Our analysis also leads to the finding that Adam performs well with large batch sizes. Moreover, we show that Adam performs well when it uses diminishing learning rates and hyperparameters close to one.

Conjugate Gradient Method for Generative Adversarial Networks

While the generative model has many advantages, it is not feasible to calculate the Jensen-Shannon divergence of the density function of the data and the density function of the model of deep neural networks; for this reason, various alternative approaches have been developed. Generative adversarial networks (GANs) can be used to formulate this problem as a discriminative problem with two models, a generator and a discriminator whose learning can be formulated in the context of game theory and the local Nash equilibrium. Since this optimization is more difficult than minimization of a single objective function, we propose to apply the conjugate gradient method to solve the local Nash equilibrium problem in GANs. We give a proof and convergence analysis under mild assumptions showing that the proposed method converges to a local Nash equilibrium with three different learning-rate schedules including a constant learning rate. Furthermore, we demonstrate the convergence of a simple toy problem to a local Nash equilibrium and compare the proposed method with other optimization methods in experiments using real-world data, finding that the proposed method outperforms stochastic gradient descent (SGD) and momentum SGD.

Using Constant Learning Rate of Two Time-Scale Update Rule for Training Generative Adversarial Networks

Previous numerical results have shown that a two time-scale update rule (TTUR) using constant learning rates is practically useful for training generative adversarial networks (GANs). Meanwhile, a theoretical analysis of TTUR to find a stationary local Nash equilibrium of a Nash equilibrium problem with two players, a discriminator and a generator, has been given using decaying learning rates. In this paper, we give a theoretical analysis of TTUR using constant learning rates to bridge the gap between theory and practice. In particular, we show that, for TTUR using constant learning rates, the number of steps needed to find a stationary local Nash equilibrium decreases as the batch size increases. We also provide numerical results to support our theoretical analyzes.

Minimization of Stochastic First-order Oracle Complexity of Adaptive Methods for Nonconvex Optimization

Numerical evaluations have definitively shown that, for deep learning optimizers such as stochastic gradient descent, momentum, and adaptive methods, the number of steps needed to train a deep neural network halves for each doubling of the batch size and that there is a region of diminishing returns beyond the critical batch size. In this paper, we determine the actual critical batch size by using the global minimizer of the stochastic first-order oracle (SFO) complexity of the optimizer. To prove the existence of the actual critical batch size, we set the lower and upper bounds of the SFO complexity and prove that there exist critical batch sizes in the sense of minimizing the lower and upper bounds. This proof implies that, if the SFO complexity fits the lower and upper bounds, then the existence of these critical batch sizes demonstrates the existence of the actual critical batch size. We also discuss the conditions needed for the SFO complexity to fit the lower and upper bounds and provide numerical results that support our theoretical results.

The Number of Steps Needed for Nonconvex Optimization of a Deep Learning Optimizer is a Rational Function of Batch Size

Recently, convergence as well as convergence rate analyses of deep learning optimizers for nonconvex optimization have been widely studied. Meanwhile, numerical evaluations for the optimizers have precisely clarified the relationship between batch size and the number of steps needed for training deep neural networks. The main contribution of this paper is to show theoretically that the number of steps needed for nonconvex optimization of each of the optimizers can be expressed as a rational function of batch size. Having these rational functions leads to two particularly important facts, which were validated numerically in previous studies. The first fact is that there exists an optimal batch size such that the number of steps needed for nonconvex optimization is minimized. This implies that using larger batch sizes than the optimal batch size does not decrease the number of steps needed for nonconvex optimization. The second fact is that the optimal batch size depends on the optimizer. In particular, it is shown theoretically that momentum and Adam-type optimizers can exploit larger optimal batches and further reduce the minimum number of steps needed for nonconvex optimization than can the stochastic gradient descent optimizer.