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Gradient Descent

Gradient Descent for Machine Learning (ML) 101 with Python Tutorial


Gradient descent is one of the most common machine learning algorithms used in neural networks [7], data science, optimization, and machine learning tasks. The gradient descent algorithm and its variants can be found in almost every machine learning model. Gradient descent is a popular optimization method of tuning the parameters in a machine learning model. Its goal is to apply optimization to find the least or minimal error value. It is mostly used to update the parameters of the model -- in this case, parameters refer to coefficients in regression and weights in a neural network.

Householder Dice: A Matrix-Free Algorithm for Simulating Dynamics on Gaussian and Random Orthogonal Ensembles Machine Learning

In the study of large random systems, researchers often need to simulate dynamics in the form of iterated matrix-vector multiplications interspersed with nonlinear operations. Examples include message passing algorithms, gradient descent, and matrix iterative methods for extremal eigenvalue calculations. This paper proposes a new algorithm, named Householder Dice (HD), for simulating such dynamics on several random matrix ensembles with translation-invariant properties. Examples include the Gaussian ensemble, the Haar-distributed random orthogonal ensemble, and their complex-valued counterparts. A "direct" approach to the simulation, where one first generates a dense $n \times n$ matrix from the ensemble, requires at least $\mathcal{O}(n^2)$ resource in space and time. The HD algorithm overcomes this $\mathcal{O}(n^2)$ bottleneck by using the principle of deferred decisions: rather than fixing the entire random matrix in advance, it lets the randomness unfold with the dynamics. Key to this matrix-free construction is an adaptive and recursive construction of (random) Householder reflectors. These orthogonal transformations exploit the group symmetry of the matrix ensembles, while simultaneously maintaining the statistical correlations induced by the dynamics. The memory and computation costs of the HD algorithm are $\mathcal{O}(nT)$ and $\mathcal{O}(nT^2)$, respectively, with $T$ being the number of iterations. When $T \ll n$, which is nearly always the case in practice, the HD algorithm leads to significant reductions in runtime and memory footprint. Numerical results demonstrate the promise of the new algorithm as a new computational tool in the study of high-dimensional random systems.

Code Adam Gradient Descent Optimization From Scratch


Gradient descent is an optimization algorithm that follows the negative gradient of an objective function in order to locate the minimum of the function. A limitation of gradient descent is that a single step size (learning rate) is used for all input variables. Extensions to gradient descent like AdaGrad and RMSProp update the algorithm to use a separate step size for each input variable but may result in a step size that rapidly decreases to very small values. The Adaptive Movement Estimation algorithm, or Adam for short, is an extension to gradient descent and a natural successor to techniques like AdaGrad and RMSProp that automatically adapts a learning rate for each input variable for the objective function and further smooths the search process by using an exponentially decreasing moving average of the gradient to make updates to variables. In this tutorial, you will discover how to develop gradient descent with Adam optimization algorithm from scratch.

Provable Generalization of SGD-trained Neural Networks of Any Width in the Presence of Adversarial Label Noise Machine Learning

We consider a one-hidden-layer leaky ReLU network of arbitrary width trained by stochastic gradient descent following an arbitrary initialization. We prove that stochastic gradient descent (SGD) produces neural networks that have classification accuracy competitive with that of the best halfspace over the distribution for a broad class of distributions that includes log-concave isotropic and hard margin distributions. Equivalently, such networks can generalize when the data distribution is linearly separable but corrupted with adversarial label noise, despite the capacity to overfit. We conduct experiments which suggest that for some distributions our generalization bounds are nearly tight. This is the first result that shows that overparameterized neural networks trained by SGD can generalize when the data is corrupted with adversarial label noise.

Learning with Gradient Descent and Weakly Convex Losses Machine Learning

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.

Solving Min-Max Optimization with Hidden Structure via Gradient Descent Ascent Machine Learning

Many recent AI architectures are inspired by zero-sum games, however, the behavior of their dynamics is still not well understood. Inspired by this, we study standard gradient descent ascent (GDA) dynamics in a specific class of non-convex non-concave zero-sum games, that we call hidden zero-sum games. In this class, players control the inputs of smooth but possibly non-linear functions whose outputs are being applied as inputs to a convex-concave game. Unlike general zero-sum games, these games have a well-defined notion of solution; outcomes that implement the von-Neumann equilibrium of the "hidden" convex-concave game. We prove that if the hidden game is strictly convex-concave then vanilla GDA converges not merely to local Nash, but typically to the von-Neumann solution. If the game lacks strict convexity properties, GDA may fail to converge to any equilibrium, however, by applying standard regularization techniques we can prove convergence to a von-Neumann solution of a slightly perturbed zero-sum game. Our convergence guarantees are non-local, which as far as we know is a first-of-its-kind type of result in non-convex non-concave games. Finally, we discuss connections of our framework with generative adversarial networks.

Beyond Procrustes: Balancing-Free Gradient Descent for Asymmetric Low-Rank Matrix Sensing Machine Learning

Low-rank matrix estimation plays a central role in various applications across science and engineering. Recently, nonconvex formulations based on matrix factorization are provably solved by simple gradient descent algorithms with strong computational and statistical guarantees. However, when the low-rank matrices are asymmetric, existing approaches rely on adding a regularization term to balance the scale of the two matrix factors which in practice can be removed safely without hurting the performance when initialized via the spectral method. In this paper, we provide a theoretical justification to this for the matrix sensing problem, which aims to recover a low-rank matrix from a small number of linear measurements. As long as the measurement ensemble satisfies the restricted isometry property, gradient descent -- in conjunction with spectral initialization -- converges linearly without the need of explicitly promoting balancedness of the factors; in fact, the factors stay balanced automatically throughout the execution of the algorithm. Our analysis is based on analyzing the evolution of a new distance metric that directly accounts for the ambiguity due to invertible transforms, and might be of independent interest.

Linear Representation Meta-Reinforcement Learning for Instant Adaptation Machine Learning

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.

A Comprehensive Study on Optimization Strategies for Gradient Descent In Deep Learning Artificial Intelligence

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

Fairness with Continuous Optimal Transport Machine Learning

Whilst optimal transport (OT) is increasingly being recognized as a powerful and flexible approach for dealing with fairness issues, current OT fairness methods are confined to the use of discrete OT. In this paper, we leverage recent advances from the OT literature to introduce a stochastic-gradient fairness method based on a dual formulation of continuous OT. We show that this method gives superior performance to discrete OT methods when little data is available to solve the OT problem, and similar performance otherwise. We also show that both continuous and discrete OT methods are able to continually adjust the model parameters to adapt to different levels of unfairness that might occur in real-world applications of ML systems.