Now we will compile the model using optimizer as stochastic gradient descent, loss as cross-entropy and metrics to measure the performance would be accuracy. After compiling we will train the model and check the performance on validation data. We are taking a batch size of 64 and epochs to be 10.

Emerging applications of machine learning in numerous areas involve continuous gathering of and learning from streams of data. Real-time incorporation of streaming data into the learned models is essential for improved inference in these applications. Further, these applications often involve data that are either inherently gathered at geographically distributed entities or that are intentionally distributed across multiple machines for memory, computational, and/or privacy reasons. Training of models in this distributed, streaming setting requires solving stochastic optimization problems in a collaborative manner over communication links between the physical entities. When the streaming data rate is high compared to the processing capabilities of compute nodes and/or the rate of the communications links, this poses a challenging question: how can one best leverage the incoming data for distributed training under constraints on computing capabilities and/or communications rate? A large body of research has emerged in recent decades to tackle this and related problems. This paper reviews recently developed methods that focus on large-scale distributed stochastic optimization in the compute- and bandwidth-limited regime, with an emphasis on convergence analysis that explicitly accounts for the mismatch between computation, communication and streaming rates. In particular, it focuses on methods that solve: (i) distributed stochastic convex problems, and (ii) distributed principal component analysis, which is a nonconvex problem with geometric structure that permits global convergence. For such methods, the paper discusses recent advances in terms of distributed algorithmic designs when faced with high-rate streaming data. Further, it reviews guarantees underlying these methods, which show there exist regimes in which systems can learn from distributed, streaming data at order-optimal rates.

In this paper, we give explicit descriptions of versions of (Local-) Backtracking Gradient Descent and New Q-Newton's method to the Riemannian setting.Here are some easy to state consequences of results in this paper, where X is a general Riemannian manifold of finite dimension and $f:X\rightarrow \mathbb{R}$ a $C^2$ function which is Morse (that is, all its critical points are non-degenerate). {\bf Theorem.} For random choices of the hyperparameters in the Riemanian Local Backtracking Gradient Descent algorithm and for random choices of the initial point $x_0$, the sequence $\{x_n\}$ constructed by the algorithm either (i) converges to a local minimum of $f$ or (ii) eventually leaves every compact subsets of $X$ (in other words, diverges to infinity on $X$). If $f$ has compact sublevels, then only the former alternative happens. The convergence rate is the same as in the classical paper by Armijo. {\bf Theorem.} Assume that $f$ is $C^3$. For random choices of the hyperparametes in the Riemannian New Q-Newton's method, if the sequence constructed by the algorithm converges, then the limit is a critical point of $f$. We have a local Stable-Center manifold theorem, near saddle points of $f$, for the dynamical system associated to the algorithm. If the limit point is a non-degenerate minimum point, then the rate of convergence is quadratic. If moreover $X$ is an open subset of a Lie group and the initial point $x_0$ is chosen randomly, then we can globally avoid saddle points. As an application, we propose a general method using Riemannian Backtracking GD to find minimum of a function on a bounded ball in a Euclidean space, and do explicit calculations for calculating the smallest eigenvalue of a symmetric square matrix.

We consider the problem of learning the best-fitting single neuron as measured by the expected square loss $\mathbb{E}_{(x,y)\sim \mathcal{D}}[(\sigma(w^\top x)-y)^2]$ over some unknown joint distribution $\mathcal{D}$ by using gradient descent to minimize the empirical risk induced by a set of i.i.d. samples $S\sim \mathcal{D}^n$. The activation function $\sigma$ is an arbitrary Lipschitz and non-decreasing function, making the optimization problem nonconvex and nonsmooth in general, and covers typical neural network activation functions and inverse link functions in the generalized linear model setting. In the agnostic PAC learning setting, where no assumption on the relationship between the labels $y$ and the input $x$ is made, if the optimal population risk is $\mathsf{OPT}$, we show that gradient descent achieves population risk $O(\mathsf{OPT})+\epsilon$ in polynomial time and sample complexity when $\sigma$ is strictly increasing. For the ReLU activation, our population risk guarantee is $O(\mathsf{OPT}^{1/2})+\epsilon$. When labels take the form $y = \sigma(v^\top x) + \xi$ for zero-mean sub-Gaussian noise $\xi$, we show that the population risk guarantees for gradient descent improve to $\mathsf{OPT} + \epsilon$. Our sample complexity and runtime guarantees are (almost) dimension independent, and when $\sigma$ is strictly increasing, require no distributional assumptions beyond boundedness. For ReLU, we show the same results under a nondegeneracy assumption for the marginal distribution of the input.

Federated learning platforms are gaining popularity. One of the major benefits is to mitigate the privacy risks as the learning of algorithms can be achieved without collecting or sharing data. While federated learning (i.e., many based on stochastic gradient algorithms) has shown great promise, there are still many challenging problems in protecting privacy, especially during the process of gradients update and exchange. This paper presents the first gradient-free federated learning framework called GRAFFL for learning a Bayesian generative model based on approximate Bayesian computation. Unlike conventional federated learning algorithms based on gradients, our framework does not require to disassemble a model (i.e., to linear components) or to perturb data (or encryption of data for aggregation) to preserve privacy. Instead, this framework uses implicit information derived from each participating institution to learn posterior distributions of parameters. The implicit information is summary statistics derived from SuffiAE that is a neural network developed in this study to create compressed and linearly separable representations thereby protecting sensitive information from leakage. As a sufficient dimensionality reduction technique, this is proved to provide sufficient summary statistics. We propose the GRAFFL-based Bayesian Gaussian mixture model to serve as a proof-of-concept of the framework. Using several datasets, we demonstrated the feasibility and usefulness of our model in terms of privacy protection and prediction performance (i.e., close to an ideal setting). The trained model as a quasi-global model can generate informative samples involving information from other institutions and enhances data analysis of each institution.

The theory and practice of stochastic optimization has focused on stochastic gradient descent (SGD) in recent years, retaining the basic first-order stochastic nature of SGD while aiming to improve it via mechanisms such as averaging, momentum, and variance reduction. Improvement can be measured along various dimensions, however, and it has proved difficult to achieve improvements both in terms of nonasymptotic measures of convergence rate and asymptotic measures of distributional tightness. In this work, we consider first-order stochastic optimization from a general statistical point of view, motivating a specific form of recursive averaging of past stochastic gradients. The resulting algorithm, which we refer to as \emph{Recursive One-Over-T SGD} (ROOT-SGD), matches the state-of-the-art convergence rate among online variance-reduced stochastic approximation methods. Moreover, under slightly stronger distributional assumptions, the rescaled last-iterate of ROOT-SGD converges to a zero-mean Gaussian distribution that achieves near-optimal covariance.

We tackle the problem of predicting the number of optimization steps that a pre-trained deep network needs to converge to a given value of the loss function. To do so, we leverage the fact that the training dynamics of a deep network during fine-tuning are well approximated by those of a linearized model. This allows us to approximate the training loss and accuracy at any point during training by solving a low-dimensional Stochastic Differential Equation (SDE) in function space. Using this result, we are able to predict the time it takes for Stochastic Gradient Descent (SGD) to fine-tune a model to a given loss without having to perform any training. In our experiments, we are able to predict training time of a ResNet within a 20% error margin on a variety of datasets and hyper-parameters, at a 30 to 45-fold reduction in cost compared to actual training. We also discuss how to further reduce the computational and memory cost of our method, and in particular we show that by exploiting the spectral properties of the gradients' matrix it is possible predict training time on a large dataset while processing only a subset of the samples.

Convergence detection of iterative stochastic optimization methods is of great practical interest. This paper considers stochastic gradient descent (SGD) with a constant learning rate and momentum. We show that there exists a transient phase in which iterates move towards a region of interest, and a stationary phase in which iterates remain bounded in that region around a minimum point. We construct a statistical diagnostic test for convergence to the stationary phase using the inner product between successive gradients and demonstrate that the proposed diagnostic works well. We theoretically and empirically characterize how momentum can affect the test statistic of the diagnostic, and how the test statistic captures a relatively sparse signal within the gradients in convergence. Finally, we demonstrate an application to automatically tune the learning rate by reducing it each time stationarity is detected, and show the procedure is robust to mis-specified initial rates.

The gradient descent is the simplest optimisation algorithm represented by gradient dynamics in a potential. When the input data is finite, gradient descent dynamics fluctuates due to the finite size effects, and is called stochastic gradient descent. In this paper, we study stability of stochastic gradient descent dynamics from the viewpoint of dynamical systems theory. Learning is characterised as nonautonomous dynamics driven by uncertain input from the external, and as multi-scale dynamics which consists of slow memory dynamics and fast system dynamics. When the uncertain input sequences are modelled by stochastic processes, dynamics of learning is described by a random dynamical system. In contrast to the traditional Fokker-Planck approaches [5, 15], the random dynamical system approaches enable the study not only of stationary distributions and global statistics, but also of the pathwise structure of stochastic dynamics. Based on nonautonomous and random dynamical system theory, it is possible to analyse stability and bifurcation in machine learning.

We describe an alternative learning method for neural networks, which we call Blind Descent. By design, Blind Descent does not face problems like exploding or vanishing gradients. In Blind Descent, gradients are not used to guide the learning process. In this paper, we present Blind Descent as a more fundamental learning process compared to gradient descent. We also show that gradient descent can be seen as a specific case of the Blind Descent algorithm. We also train two neural network architectures, a multilayer perceptron and a convolutional neural network, using the most general Blind Descent algorithm to demonstrate a proof of concept.