We study the learning performance of gradient descent when the empirical risk is weakly convex, namely, the smallest negative eigenvalue of the empirical risk's Hessian is bounded in magnitude. By showing that this eigenvalue can control the stability of gradient descent, generalisation error bounds are proven that hold under a wider range of step sizes compared to previous work. Out of sample guarantees are then achieved by decomposing the test error into generalisation, optimisation and approximation errors, each of which can be bounded and traded off with respect to algorithmic parameters, sample size and magnitude of this eigenvalue. In the case of a two layer neural network, we demonstrate that the empirical risk can satisfy a notion of local weak convexity, specifically, the Hessian's smallest eigenvalue during training can be controlled by the normalisation of the layers, i.e., network scaling. This allows test error guarantees to then be achieved when the population risk minimiser satisfies a complexity assumption. By trading off the network complexity and scaling, insights are gained into the implicit bias of neural network scaling, which are further supported by experimental findings.
This paper introduces Fast Linearized Adaptive Policy (FLAP), a new meta-reinforcement learning (meta-RL) method that is able to extrapolate well to out-of-distribution tasks without the need to reuse data from training, and adapt almost instantaneously with the need of only a few samples during testing. FLAP builds upon the idea of learning a shared linear representation of the policy so that when adapting to a new task, it suffices to predict a set of linear weights. A separate adapter network is trained simultaneously with the policy such that during adaptation, we can directly use the adapter network to predict these linear weights instead of updating a meta-policy via gradient descent, such as in prior meta-RL methods like MAML, to obtain the new policy. The application of the separate feed-forward network not only speeds up the adaptation run-time significantly, but also generalizes extremely well to very different tasks that prior Meta-RL methods fail to generalize to. Experiments on standard continuous-control meta-RL benchmarks show FLAP presenting significantly stronger performance on out-of-distribution tasks with up to double the average return and up to 8X faster adaptation run-time speeds when compared to prior methods.
One of the most important parts of Artificial Neural Networks is minimizing the loss functions which tells us how good or bad our model is. To minimize these losses we need to tune the weights and biases. Also to calculate the minimum value of a function we need gradient. And to update our weights we need gradient descent. But there are some problems with regular gradient descent ie. it is quite slow and not that accurate. This article aims to give an introduction to optimization strategies to gradient descent. In addition, we shall also discuss the architecture of these algorithms and further optimization of Neural Networks in general
We introduce a hybrid "Modified Genetic Algorithm-Multilevel Stochastic Gradient Descent" (MGA-MSGD) training algorithm that considerably improves accuracy and efficiency of solving 3D mechanical problems described, in strong-form, by PDEs via ANNs (Artificial Neural Networks). This presented approach allows the selection of a number of locations of interest at which the state variables are expected to fulfil the governing equations associated with a physical problem. Unlike classical PDE approximation methods such as finite differences or the finite element method, there is no need to establish and reconstruct the physical field quantity throughout the computational domain in order to predict the mechanical response at specific locations of interest. The basic idea of MGA-MSGD is the manipulation of the learnable parameters' components responsible for the error explosion so that we can train the network with relatively larger learning rates which avoids trapping in local minima. The proposed training approach is less sensitive to the learning rate value, training points density and distribution, and the random initial parameters. The distance function to minimise is where we introduce the PDEs including any physical laws and conditions (so-called, Physics Informed ANN). The Genetic algorithm is modified to be suitable for this type of ANN in which a Coarse-level Stochastic Gradient Descent (CSGD) is exploited to make the decision of the offspring qualification. Employing the presented approach, a considerable improvement in both accuracy and efficiency, compared with standard training algorithms such as classical SGD and Adam optimiser, is observed. The local displacement accuracy is studied and ensured by introducing the results of Finite Element Method (FEM) at sufficiently fine mesh as the reference displacements. A slightly more complex problem is solved ensuring its feasibility.
Bayesian optimisation presents a sample-efficient methodology for global optimisation. Within this framework, a crucial performance-determining subroutine is the maximisation of the acquisition function, a task complicated by the fact that acquisition functions tend to be non-convex and thus nontrivial to optimise. In this paper, we undertake a comprehensive empirical study of approaches to maximise the acquisition function. Additionally, by deriving novel, yet mathematically equivalent, compositional forms for popular acquisition functions, we recast the maximisation task as a compositional optimisation problem, allowing us to benefit from the extensive literature in this field. We highlight the empirical advantages of the compositional approach to acquisition function maximisation across 3958 individual experiments comprising synthetic optimisation tasks as well as tasks from Bayesmark. Given the generality of the acquisition function maximisation subroutine, we posit that the adoption of compositional optimisers has the potential to yield performance improvements across all domains in which Bayesian optimisation is currently being applied.
In this article, we will discuss regularization and optimization techniques that are used by programmers to build a more robust and generalized neural network. We will study the most effective regularization techniques like L1, L2, Early Stopping, and Drop out which help for model generalization. We will take a deeper look at different optimization techniques like Batch Gradient Descent, Stochastic Gradient Descent, AdaGrad, and AdaDelta for better convergence of the neural networks. Overfitting and underfitting are the most common problems that programmers face while working with deep learning models. A model that is well generalized to data is considered to be an optimal fit for the data.
Deep learning neural network models are fit on training data using the stochastic gradient descent optimization algorithm. Updates to the weights of the model are made, using the backpropagation of error algorithm. The combination of the optimization and weight update algorithm was carefully chosen and is the most efficient approach known to fit neural networks. Nevertheless, it is possible to use alternate optimization algorithms to fit a neural network model to a training dataset. This can be a useful exercise to learn more about how neural networks function and the central nature of optimization in applied machine learning. It may also be required for neural networks with unconventional model architectures and non-differentiable transfer functions.
The success of deep learning models has led to a lot of recent interest in understanding the properties of "interpolating" neural network models, that achieve (near-)zero training loss [Zha 17a; Bel 19]. One aspect of understanding these models is to theoretically characterize how first-order gradient methods (with appropriate random initialization) seem to reliably find interpolating solutions to non-convex optimization problems. In this paper, we show that, under two sets of conditions, training fixed-width two-layer networks with gradient descent drives the logistic loss to zero. The networks have smooth "Huberized" ReLUs [Tat 20, see (1) and Figure 1] and the output weights are not trained. The first result only requires the assumption that the initial loss is small, but does not require any assumption about either the width of the network or the number of samples. It guarantees that if the initial loss is small then gradient descent drives the logistic loss to zero. For our second result we assume that the inputs come from four clusters, two per class, and that the clusters corresponding to the opposite labels are appropriately separated. Under these assumptions, we show that random Gaussian initialization along with a single step of gradient descent is enough to guarantee that the loss reduces sufficiently that the first result applies. A few proof ideas that facilitate our results are as follows: under our first set of assumptions, when the loss is small, we show that the negative gradient aligns well with the parameter vector. 1
At a high level, training supervised machine learning models involves a few easy steps: feeding data to your model, computing loss based on the differences between predictions and ground truth, and using loss to improve the model with an optimizer. For example, it's possible to choose multiple optimizers – ranging from traditional Stochastic Gradient Descent to adaptive optimizers, which are also very common today. Say that you settle for the first – Stochastic Gradient Descent (SGD). Likely, in your deep learning framework, you'll see that the learning rate is a parameter that can be configured, with a default value that is preconfigured most of the times. Now, what is this learning rate? Why do we need them?
The large communication cost for exchanging gradients between different nodes significantly limits the scalability of distributed training for large-scale learning models. Motivated by this observation, there has been significant recent interest in techniques that reduce the communication cost of distributed Stochastic Gradient Descent (SGD), with gradient sparsification techniques such as top-k and random-k shown to be particularly effective. The same observation has also motivated a separate line of work in distributed statistical estimation theory focusing on the impact of communication constraints on the estimation efficiency of different statistical models. The primary goal of this paper is to connect these two research lines and demonstrate how statistical estimation models and their analysis can lead to new insights in the design of communication-efficient training techniques. We propose a simple statistical estimation model for the stochastic gradients which captures the sparsity and skewness of their distribution. The statistically optimal communication scheme arising from the analysis of this model leads to a new sparsification technique for SGD, which concatenates random-k and top-k, considered separately in the prior literature. We show through extensive experiments on both image and language domains with CIFAR-10, ImageNet, and Penn Treebank datasets that the concatenated application of these two sparsification methods consistently and significantly outperforms either method applied alone.