Gradient descent has been a central training principle for artificial neural networks from the early beginnings to today's deep learning networks. The most common implementation is the backpropagation algorithm for training feed-forward neural networks in a supervised fashion. Backpropagation involves computing the gradient of a loss function, with respect to the weights of the network, to update the weights and thus minimize loss. Although the mean square error is often used as a loss function, the general stochastic gradient descent principle does not immediately connect with a specific loss function. Another drawback of backpropagation has been the search for optimal values of two important training parameters, learning rate and momentum weight, which are determined empirically in most systems. The learning rate specifies the step size towards a minimum of the loss function when following the gradient, while the momentum weight considers previous weight changes when updating current weights. Using both parameters in conjunction with each other is generally accepted as a means to improving training, although their specific values do not follow immediately from standard backpropagation theory. This paper proposes a new information-theoretical loss function motivated by neural signal processing in a synapse. The new loss function implies a specific learning rate and momentum weight, leading to empirical parameters often used in practice. The proposed framework also provides a more formal explanation of the momentum term and its smoothing effect on the training process. All results taken together show that loss, learning rate, and momentum are closely connected. To support these theoretical findings, experiments for handwritten digit recognition show the practical usefulness of the proposed loss function and training parameters.

Reliable yet efficient evaluation of generalisation performance of a proposed architecture is crucial to the success of neural architecture search (NAS). Traditional approaches face a variety of limitations: training each architecture to completion is prohibitively expensive, early stopping estimates may correlate poorly with fully trained performance, and model-based estimators require large training sets. Instead, motivated by recent results linking training speed and generalisation with stochastic gradient descent, we propose to estimate the final test performance based on the sum of training losses. Our estimator is inspired by the marginal likelihood, which is used for Bayesian model selection. Our model-free estimator is simple, efficient, and cheap to implement, and does not require hyperparameter-tuning or surrogate training before deployment. We demonstrate empirically that our estimator consistently outperforms other baselines and can achieve a rank correlation of 0.95 with final test accuracy on the NAS-Bench201 dataset within 50 epochs.

Machine learning algorithms with empirical risk minimization are vulnerable to distributional shifts due to the greedy adoption of all the correlations found in training data. Recently, there are robust learning methods aiming at this problem by minimizing the worst-case risk over an uncertainty set. However, they equally treat all covariates to form the uncertainty sets regardless of the stability of their correlations with the target, resulting in the overwhelmingly large set and low confidence of the learner. In this paper, we propose the Invariant Adversarial Learning (IAL) algorithm that leverages heterogeneous data sources to construct a more practical uncertainty set and conduct robustness optimization, where covariates are differentiated according to the stability of their correlations with the target. We theoretically show that our method is tractable for stochastic gradient-based optimization and provide the performance guarantees for our method.

We estimate the Lipschitz constants of the gradient of a deep neural network and the network itself with respect to the full set of parameters. We first develop estimates for a deep feed-forward densely connected network and then, in a more general framework, for all neural networks that can be represented as solutions of controlled ordinary differential equations, where time appears as continuous depth. These estimates can be used to set the step size of stochastic gradient descent methods, which is illustrated for one example method.

Our linear regression has only two predictors (a and b), thus X is a n x 2 matrix (where n is the number of observations and 2 the number of predictors). As you can see, to solve the equation we need to calculate the matrix (X T X) then invert it. In machine learning, the number of observations is often very high as well as the number of predictors. Consequently, this operation is very expensive in terms of calculation and memory. Gradient descent algorithm is an iterative optimization algorithm that allows us to find the solution while keeping the computational complexity low.

We address the problem of Federated Learning (FL) where users are distributed and partitioned into clusters. This setup captures settings where different groups of users have their own objectives (learning tasks) but by aggregating their data with others in the same cluster (same learning task), they can leverage the strength in numbers in order to perform more efficient Federated Learning. We propose a new framework dubbed the Iterative Federated Clustering Algorithm (IFCA), which alternately estimates the cluster identities of the users and optimizes model parameters for the user clusters via gradient descent. We analyze the convergence rate of this algorithm first in a linear model with squared loss and then for generic strongly convex and smooth loss functions. We show that in both settings, with good initialization, IFCA converges at an exponential rate, and discuss the optimality of the statistical error rate. In the experiments, we show that our algorithm can succeed even if we relax the requirements on initialization with random initialization and multiple restarts. We also present experimental results showing that our algorithm is efficient in non-convex problems such as neural networks and outperforms the baselines on several clustered FL benchmarks created based on the MNIST and CIFAR-10 datasets by $5\sim 8\%$.

The extrapolation strategy raised by Nesterov, which can accelerate the convergence rate of gradient descent methods by orders of magnitude when dealing with smooth convex objective, has led to tremendous success in training machine learning tasks. In this article, the convergence of individual iterates of projected subgradient (PSG) methods for nonsmooth convex optimization problems is theoretically studied based on Nesterov's extrapolation, which we name individual convergence. We prove that Nesterov's extrapolation has the strength to make the individual convergence of PSG optimal for nonsmooth problems. In light of this consideration, a direct modification of the subgradient evaluation suffices to achieve optimal individual convergence for strongly convex problems, which can be regarded as making an interesting step toward the open question about stochastic gradient descent (SGD) posed by Shamir. Furthermore, we give an extension of the derived algorithms to solve regularized learning tasks with nonsmooth losses in stochastic settings. Compared with other state-of-the-art nonsmooth methods, the derived algorithms can serve as an alternative to the basic SGD especially in coping with machine learning problems, where an individual output is needed to guarantee the regularization structure while keeping an optimal rate of convergence. Typically, our method is applicable as an efficient tool for solving large-scale $l$1-regularized hinge-loss learning problems. Several comparison experiments demonstrate that our individual output not only achieves an optimal convergence rate but also guarantees better sparsity than the averaged solution.

We consider the contextual bandit problem, where a player sequentially makes decisions based on past observations to maximize the cumulative reward. Although many algorithms have been proposed for contextual bandit, most of them rely on finding the maximum likelihood estimator at each iteration, which requires $O(t)$ time at the $t$-th iteration and are memory inefficient. A natural way to resolve this problem is to apply online stochastic gradient descent (SGD) so that the per-step time and memory complexity can be reduced to constant with respect to $t$, but a contextual bandit policy based on online SGD updates that balances exploration and exploitation has remained elusive. In this work, we show that online SGD can be applied to the generalized linear bandit problem. The proposed SGD-TS algorithm, which uses a single-step SGD update to exploit past information and uses Thompson Sampling for exploration, achieves $\tilde{O}(\sqrt{dT})$ regret with the total time complexity that scales linearly in $T$ and $d$, where $T$ is the total number of rounds and $d$ is the number of features. Experimental results show that SGD-TS consistently outperforms existing algorithms on both synthetic and real datasets.

In this paper, we introduce proximal gradient temporal difference learning, which provides a principled way of designing and analyzing true stochastic gradient temporal difference learning algorithms. We show how gradient TD (GTD) reinforcement learning methods can be formally derived, not by starting from their original objective functions, as previously attempted, but rather from a primal-dual saddle-point objective function. We also conduct a saddle-point error analysis to obtain finite-sample bounds on their performance. Previous analyses of this class of algorithms use stochastic approximation techniques to prove asymptotic convergence, and do not provide any finite-sample analysis. We also propose an accelerated algorithm, called GTD2-MP, that uses proximal ``mirror maps'' to yield an improved convergence rate. The results of our theoretical analysis imply that the GTD family of algorithms are comparable and may indeed be preferred over existing least squares TD methods for off-policy learning, due to their linear complexity. We provide experimental results showing the improved performance of our accelerated gradient TD methods.

Deep neural networks is today one of the most popular choices in classification, regression and function approximation. However, the training of such deep networks is far from trivial as there are often millions of parameters to tune. Typically, one use some optimization method that hopefully converges towards some minimum. The most popular and successful methods are based on gradient descent. In this paper, another optimization method, Frank-Wolfe optimization, is applied to a small deep network and compared to gradient descent. Although the optimization does converge, it does so slowly and not close to the speed of gradient descent. Further, in a stochastic setting, the optimization becomes very unstable and does not seem to converge unless one uses a line search approach.