I strongly advise you to read the article linked above. It will set the foundations on the topic, plus some math is already discussed there. To start out, I'll define my dataset -- only three points that are in a linear relationship. I've chosen so few points only because the math will be shorter -- needless to say, the math won't be more complex for longer dataset, it would just be longer, and I don't want to make some stupid arithmetic mistake. Then I'll set coefficients beta 0 and beta 1 to some constant and define the cost function as Sum of Squared Residuals (SSR/SSE).

In this post, we have looked at the batch gradient descent, the need to develop new optimization techniques, and then we briefly discussed how to interpret contour plots. After that, we have looked at behind six different optimization techniques and three different data strategies (batch, mini-batch & stochastic) with an intuitive understanding which helps to know where to use any of these algorithms. In Practice Adam optimizer with mini-batch of sizes 32, 64 and 128 is the default choice, at least for all the image classification tasks which deal with CNN and large sequence to sequence models.

A BSTRACT To analyze deep ReLU network, we adopt a student-teacher setting in which an over-parameterized student network learns from the output of a fixed teacher network of the same depth, with Stochastic Gradient Descent (SGD). First, we prove that when the gradient is zero (or bounded above by a small constant) at every data point in training, a situation called interpolation setting, there exists many-to-one alignment between student and teacher nodes in the lowest layer under mild conditions. This suggests that generalization in unseen dataset is achievable, even the same condition often leads to zero training error. Second, analysis of noisy recovery and training dynamics in 2-layer network shows that strong teacher nodes (with large fan-out weights) are learned first and subtle teacher nodes are left unlearned until late stage of training. As a result, it could take a long time to converge into these small-gradient critical points. Our analysis shows that over-parameterization plays two roles: (1) it is a necessary condition for alignment to happen at the critical points, and (2) in training dynamics, it helps student nodes cover more teacher nodes with fewer iterations. Although networks with even one-hidden layer can fit any function (Hornik et al., 1989), it remains an open question how such networks can generalize to new data. Different from what traditional machine learning theory predicts, empirical evidence (Zhang et al., 2017) shows more parameters in neural network lead to better generalization. How over-parameterization yields strong generalization is an important question for understanding how deep learning works. In this paper, we analyze multi-layer ReLU networks by adopting teacher-student setting. The fixed teacher network provides the output for the student to learn via SGD. The student is over-parameterized (or over-realized): it has more nodes than the teacher. Therefore, there exists student weights whose gradient at every data point is zero. Here, we want to study the inverse problem: With small gradient at every training sample, can the student weights recover the teachers'? If so, then the generalization performance can be guaranteed if the training converges to such critical points. In this paper, we show that this so-called interpolation setting (Ma et al., 2017; Liu & Belkin, 2018; Bassily et al., 2018) leads to alignment: under certain conditions, each teacher node is provably aligned with at least one student node in the lowest layer. The condition is simply that the teacher node is observed by at least one student node, i.e., teacher's ReLU boundary lies in the activation region of that student. Therefore, more over-parameterization increases the probability of teachers being observed and thus being aligned. Furthermore, in 2-layer case, those student nodes that are not aligned with any teacher have zero contribution to the output and can be pruned.

Quantizing large Neural Networks (NN) while maintaining the performance is highly desirable for resource-limited devices due to reduced memory and time complexity. NN quantization is usually formulated as a constrained optimization problem and optimized via a modified version of gradient descent. In this work, by interpreting the continuous parameters (unconstrained) as the dual of the quantized ones, we introduce a Mirror Descent (MD) framework (Bubeck (2015)) for NN quantization. Specifically, we provide conditions on the projections (i.e., mapping from continuous to quantized ones) which would enable us to derive valid mirror maps and in turn the respective MD updates. Furthermore, we discuss a numerically stable implementation of MD by storing an additional set of auxiliary dual variables (continuous). This update is strikingly analogous to the popular Straight Through Estimator (STE) based method which is typically viewed as a "trick" to avoid vanishing gradients issue but here we show that it is an implementation method for MD for certain projections. Our experiments on standard classification datasets (CIFAR-10/100, TinyImageNet) with convolutional and residual architectures show that our MD variants obtain fully-quantized networks with accuracies very close to the floating-point networks.

Mini-batch stochastic gradient descent (SGD) approximates the gradient of an objective function with the average gradient of some batch of constant size. While small batch sizes can yield high-variance gradient estimates that prevent the model from learning a good model, large batches may require more data and computational effort. This work presents a method to change the batch size adaptively with model quality. We show that our method requires the same number of model updates as full-batch gradient descent while requiring the same total number of gradient computations as SGD. While this method requires evaluating the objective function, we present a passive approximation that eliminates this constraint and improves computational efficiency. We provide extensive experiments illustrating that our methods require far fewer model updates without increasing the total amount of computation.

When samples have internal structure, we often see a mismatch between the objective optimized during training and the model's goal during inference. For example, in sequence-to-sequence modeling we are interested in high-quality translated sentences, but training typically uses maximum likelihood at the word level. Learning to recognize individual faces from group photos, each captioned with the correct but unordered list of people in it, is another example where a mismatch between training and inference objectives occurs. In both cases, the natural training-time loss would involve a combinatorial problem -- dynamic programming-based global sequence alignment and weighted bipartite graph matching, respectively -- but solutions to combinatorial problems are not differentiable with respect to their input parameters, so surrogate, differentiable losses are used instead. Here, we show how to perform gradient descent over combinatorial optimization algorithms that involve continuous parameters, for example edge weights, and can be efficiently expressed as integer, linear, or mixed-integer linear programs. We demonstrate usefulness of gradient descent over combinatorial optimization in sequence-to-sequence modeling using differentiable encoder-decoder architecture with softmax or Gumbel-softmax, and in weakly supervised learning involving a convolutional, residual feed-forward network for image classification.

Within the context of empirical risk minimization, see Raginsky, Rakhlin, and Telgarsky (2017), we are concerned with a non-asymptotic analysis of sampling algorithms used in optimization. In particular, we obtain non-asymptotic error bounds for a popular class of algorithms called Stochastic Gradient Langevin Dynamics (SGLD). These results are derived in appropriate Wasserstein distances in the absence of the log-concavity of the target distribution. More precisely, the local Lipschitzness of the stochastic gradient $H(\theta, x)$ is assumed, and furthermore, the dissipativity and convexity at infinity condition are relaxed by removing the uniform dependence in $x$.

We will first import libraries as NumPy, matplotlib, pyplot and derivative function. Then with a NumPy function – linspace() we define our variable \(w \) domain between 1.0 and 5.0 and 100 points. Also we define alpha which will represent learning rate. Next, we will define our \(y \) ( in our case \(J(w) \)) and plot to see a convex function, we will use \((w-3) 2 \). So we can see that we plotted our convex function as an example.

Large-scale machine learning models are often trained by parallel stochastic gradient descent algorithms. However, the communication cost of gradient aggregation and model synchronization between the master and worker nodes becomes the major obstacle for efficient learning as the number of workers and the dimension of the model increase. In this paper, we propose DORE, a DOuble REsidual compression stochastic gradient descent algorithm, to reduce over $95\%$ of the overall communication such that the obstacle can be immensely mitigated. Our theoretical analyses demonstrate that the proposed strategy has superior convergence properties for both strongly convex and nonconvex objective functions. The experimental results validate that DORE achieves the best communication efficiency while maintaining similar model accuracy and convergence speed in comparison with start-of-the-art baselines.

Many tasks in modern machine learning can be formulated as finding equilibria in \emph{sequential} games. In particular, two-player zero-sum sequential games, also known as minimax optimization, have received growing interest. It is tempting to apply gradient descent to solve minimax optimization given its popularity and success in supervised learning. However, it has been noted that naive application of gradient descent fails to find some local minimax and can converge to non-local-minimax points. In this paper, we propose \emph{Follow-the-Ridge} (FR), a novel algorithm that provably converges to and only converges to local minimax. We show theoretically that the algorithm addresses the notorious rotational behaviour of gradient dynamics, and is compatible with preconditioning and \emph{positive} momentum. Empirically, FR solves toy minimax problems and improves the convergence of GAN training compared to the recent minimax optimization algorithms.