In supervised learning, given a sample of n pairs of inputs and outputs, the goal is to estimate a function to be used to predict future outputs based on observing only the corresponding inputs. The quality of an estimate is often measured in terms of the mean-squared prediction error, in which case the regression function is optimal. Since the properties of the function to be estimated are not known a priori, nonparametric techniques, that can adapt their complexity to the problem at hand, are often key to good results. Kernel methods [15, 36] are probably the most common nonparametric approaches to learning. They are based on choosing a reproducing kernel Hilbert space (RKHS) as the hypothesis space in the design of learning algorithms. A classical learning algorithm using kernel methods to perform learning tasks is kernel ridge regression (KRR), which is based on minimizing the sum of a data-fitting term and an explicit penalty term. The penalty term is used for regularization, and controls the complexity of the solution, preventing overfitting. The statistical properties of KRR have been studied extensively, see e.g.

We analyze the learning properties of the stochastic gradient method when multiple passes over the data and mini-batches are allowed. We study how regularization properties are controlled by the step-size, the number of passes and the mini-batch size. In particular, we consider the square loss and show that for a universal step-size choice, the number of passes acts as a regularization parameter, and optimal finite sample bounds can be achieved by early-stopping. Moreover, we show that larger step-sizes are allowed when considering mini-batches. Our analysis is based on a unifying approach, encompassing both batch and stochastic gradient methods as special cases. As a byproduct, we derive optimal convergence results for batch gradient methods (even in the non-attainable cases).

Noname manuscript No. (will be inserted by the editor) Abstract In this paper, we propose an implicit gradient descent algorithm for the classic k-means problem. The implicit gradient step or backward Euler is solved via stochastic fixed-point iteration, in which we randomly sample a mini-batch gradient in every iteration. It is the average of the fixed-point trajectory that is carried over to the next gradient step. We draw connections between the proposed stochastic backward Euler and the recent entropy stochastic gradient descent (Entropy-SGD) for improving the training of deep neural networks. Numerical experiments on various synthetic and real datasets show that the proposed algorithm finds the global minimum (or its neighborhood) with high probability, when given the correct number of clusters. The method provides better clustering results compared to k-means algorithms in the sense that it decreased the objective function (the cluster) and is much more robust to initialization.

Saddle points have long been regarded as a major obstacle for non-convex optimization over continuous spaces. It is well understood that in many applications of interest, the number of saddle points significantly outnumber the number of local minima, which is especially problematic when the solutions associated with worst-case saddle points are considerably worse than those associated with worst-case local minima [12, 14, 34]. Moreover, it is not hard to construct examples where a worst-case initialization of gradient descent (or other first-order methods) provably converge to saddle points [30, Section 1.2.3]. The main message of our paper is that, under very mild regularity conditions, saddle points have little effect on the asymptotic behavior of first-order methods. Building on tools from the theory of dynamical systems, we generalize recent analysis of gradient descent [24, 33] to establish that a wide variety of first-order methods -- including gradient descent, proximal point algorithm, block coordinate descent, mirror descent -- avoid so-called "strict" saddle points for almost all initializations; that is, saddle points where the Hessian of the objective function admits at least one direction of negative curvature (see Definition 1). Our results provide a unified theoretical framework for analyzing the asymptotic behavior of a wide variety of classic optimization heuristics in non-convex optimization. Furthermore, we believe that furthering our understanding of the behavior and geometry of deterministic optimization techniques with random initialization can serve in the development of stochastic algorithms which improve upon their deterministic counterparts and achieve strong convergence-rate results; indeed, such insights have already led to significant improves in modifying gradient descent to navigate saddle-point geometry [15, 21]. This paper significantly extends upon the special case of gradient descent dynamics developed in the conference proceedings of the authors [24, 33].

Deep neural networks (DNNs) have become increasingly important due to their excellent empirical performance on a wide range of problems. However, regularization is generally achieved by indirect means, largely due to the complex set of functions defined by a network and the difficulty in measuring function complexity. There exists no method in the literature for additive regularization based on a norm of the function, as is classically considered in statistical learning theory. In this work, we propose sampling-based approximations to weighted function norms as regularizers for deep neural networks. We provide, to the best of our knowledge, the first proof in the literature of the NP-hardness of computing function norms of DNNs, motivating the necessity of a stochastic optimization strategy. Based on our proposed regularization scheme, stability-based bounds yield a $\mathcal{O}(N^{-\frac{1}{2}})$ generalization error for our proposed regularizer when applied to convex function sets. We demonstrate broad conditions for the convergence of stochastic gradient descent on our objective, including for non-convex function sets such as those defined by DNNs. Finally, we empirically validate the improved performance of the proposed regularization strategy for both convex function sets as well as DNNs on real-world classification and segmentation tasks.

Multiview representation learning is very popular for latent factor analysis. It naturally arises in many data analysis, machine learning, and information retrieval applications to model dependent structures among multiple data sources. For computational convenience, existing approaches usually formulate the multiview representation learning as convex optimization problems, where global optima can be obtained by certain algorithms in polynomial time. However, many evidences have corroborated that heuristic nonconvex approaches also have good empirical computational performance and convergence to the global optima, although there is a lack of theoretical justification. Such a gap between theory and practice motivates us to study a nonconvex formulation for multiview representation learning, which can be efficiently solved by a simple stochastic gradient descent (SGD) algorithm. We first illustrate the geometry of the nonconvex formulation; Then by characterizing the dynamics of the approximate limiting process, we establish global rates of convergence to the global optima. Numerical experiments are provided to support our theory.

This paper proposes Deep Hyperalignment (DHA) as a regularized, deep extension, scalable Hyperalignment (HA) method, which is well-suited for applying functional alignment to fMRI datasets with nonlinearity, high-dimensionality (broad ROI), and a large number of subjects. Unlink previous methods, DHA is not limited by a restricted fixed kernel function. Further, it uses a parametric approach, rank-$m$ Singular Value Decomposition (SVD), and stochastic gradient descent for optimization. Therefore, DHA has a suitable time complexity for large datasets, and DHA does not require the training data when it computes the functional alignment for a new subject. Experimental studies on multi-subject fMRI analysis confirm that the DHA method achieves superior performance to other state-of-the-art HA algorithms.

The first two algorithms you mention (Nelder-Mead and Simulated Annealing) are generally considered pretty much obsolete in optimization circles, as there are much better alternatives which are both more reliable and less costly. Genetic algorithms covers a wide range, and some of these can be reasonable. However, in the broader class of derivative-free optimization algorithms, there are many which are significantly better than these "classics", as this has been an active area of research in recent decades. So, might some of these newer approaches be reasonable for deep learning? This is a nice paper which has many interesting insights into recent techniques.

We perform an average case analysis of the generalization dynamics of large neural networks trained using gradient descent. We study the practically-relevant "high-dimensional" regime where the number of free parameters in the network is on the order of or even larger than the number of examples in the dataset. Using random matrix theory and exact solutions in linear models, we derive the generalization error and training error dynamics of learning and analyze how they depend on the dimensionality of data and signal to noise ratio of the learning problem. We find that the dynamics of gradient descent learning naturally protect against overtraining and overfitting in large networks. Overtraining is worst at intermediate network sizes, when the effective number of free parameters equals the number of samples, and thus can be reduced by making a network smaller or larger. Additionally, in the high-dimensional regime, low generalization error requires starting with small initial weights. We then turn to non-linear neural networks, and show that making networks very large does not harm their generalization performance. On the contrary, it can in fact reduce overtraining, even without early stopping or regularization of any sort. We identify two novel phenomena underlying this behavior in overcomplete models: first, there is a frozen subspace of the weights in which no learning occurs under gradient descent; and second, the statistical properties of the high-dimensional regime yield better-conditioned input correlations which protect against overtraining. We demonstrate that naive application of worst-case theories such as Rademacher complexity are inaccurate in predicting the generalization performance of deep neural networks, and derive an alternative bound which incorporates the frozen subspace and conditioning effects and qualitatively matches the behavior observed in simulation.

In this paper, we study the online learning algorithm without explicit regularization terms. This algorithm is essentially a stochastic gradient descent scheme in a reproducing kernel Hilbert space (RKHS). The polynomially decaying step size in each iteration can play a role of regularization to ensure the generalization ability of online learning algorithm. We develop a novel capacity dependent analysis on the performance of the last iterate of online learning algorithm. The contribution of this paper is two-fold. First, our nice analysis can lead to the convergence rate in the standard mean square distance which is the best so far. Second, we establish, for the first time, the strong convergence of the last iterate with polynomially decaying step sizes in the RKHS norm. We demonstrate that the theoretical analysis established in this paper fully exploits the fine structure of the underlying RKHS, and thus can lead to sharp error estimates of online learning algorithm.