Neural Network from Scratch in TensorFlow Create a predict function. Create the main training mechanism and implement gradient descent with automatic differentiation. Apply the neural network model to solve a multi-class classification problem. How to implement a neural network from scratch using TensorFlow. How to solve a multi-class classification problem using the neural network implementation.

Differentially private SGD (DP-SGD) is one of the most popular methods for solving differentially private empirical risk minimization (ERM). Due to its noisy perturbation on each gradient update, the error rate of DP-SGD scales with the ambient dimension $p$, the number of parameters in the model. Such dependence can be problematic for over-parameterized models where $p \gg n$, the number of training samples. Existing lower bounds on private ERM show that such dependence on $p$ is inevitable in the worst case. In this paper, we circumvent the dependence on the ambient dimension by leveraging a low-dimensional structure of gradient space in deep networks---that is, the stochastic gradients for deep nets usually stay in a low dimensional subspace in the training process. We propose Projected DP-SGD that performs noise reduction by projecting the noisy gradients to a low-dimensional subspace, which is given by the top gradient eigenspace on a small public dataset. We provide a general sample complexity analysis on the public dataset for the gradient subspace identification problem and demonstrate that under certain low-dimensional assumptions the public sample complexity only grows logarithmically in $p$. Finally, we provide a theoretical analysis and empirical evaluations to show that our method can substantially improve the accuracy of DP-SGD.

We propose two novel conditional gradient-based methods for solving structured stochastic convex optimization problems with a large number of linear constraints. Instances of this template naturally arise from SDP-relaxations of combinatorial problems, which involve a number of constraints that is polynomial in the problem dimension. The most important feature of our framework is that only a subset of the constraints is processed at each iteration, thus gaining a computational advantage over prior works that require full passes. Our algorithms rely on variance reduction and smoothing used in conjunction with conditional gradient steps, and are accompanied by rigorous convergence guarantees. Preliminary numerical experiments are provided for illustrating the practical performance of the methods.

Gradient descent yields zero training loss in polynomial time for deep neural networks despite non-convex nature of the objective function. The behavior of network in the infinite width limit trained by gradient descent can be described by the Neural Tangent Kernel (NTK) introduced in \cite{Jacot2018Neural}. In this paper, we study dynamics of the NTK for finite width Deep Residual Network (ResNet) using the neural tangent hierarchy (NTH) proposed in \cite{Huang2019Dynamics}. For a ResNet with smooth and Lipschitz activation function, we reduce the requirement on the layer width $m$ with respect to the number of training samples $n$ from quartic to cubic. Our analysis suggests strongly that the particular skip-connection structure of ResNet is the main reason for its triumph over fully-connected network.

Deep learning achieves a high generalization performance in practice, despite the non-convexity of the gradient descent learning problem. Recently, the inductive bias in deep learning has been studied through the characterization of local minima. In this study, we show that the distribution of parameters learned by gradient descent converges to a spectrum of the ridgelet transform based on a ridgelet analysis, which is a wavelet-like analysis developed for neural networks. This convergence is stronger than those shown in previous results, and guarantees the shape of the parameter distribution has been identified with the ridgelet spectrum. In numerical experiments with finite models, we visually confirm the resemblance between the distribution of learned parameters and the ridgelet spectrum. Our study provides a better understanding of the theoretical background of an inductive bias theory based on lazy regimes.

GANs, originally discovered in the context of unsupervised learning, have had far reaching implications to science, engineering, and society. However, training GANs remains challenging (in part) due to the lack of convergent algorithms for nonconvex-nonconcave min-max optimization. In this post, we present a new first-order algorithm for min-max optimization which is particularly suited to GANs. This algorithm is guaranteed to converge to an equilibrium, is competitive in terms of time and memory with gradient descent-ascent and, most importantly, GANs trained using it seem to be stable. Starting with the work of Goodfellow et al., Generative Adversarial Nets (GANs) have become a critical component in various ML systems; for prior posts on GANs, see here for a post on GAN architecture, and here and here for posts which discuss some of the many difficulties arising when training GANs.

In Continual Learning settings, deep neural networks are prone to Catastrophic Forgetting. Orthogonal Gradient Descent was proposed to tackle the challenge. However, no theoretical guarantees have been proven yet. We present a theoretical framework to study Continual Learning algorithms in the Neural Tangent Kernel regime. This framework comprises closed form expression of the model through tasks and proxies for Transfer Learning, generalisation and tasks similarity. In this framework, we prove that OGD is robust to Catastrophic Forgetting then derive the first generalisation bound for SGD and OGD for Continual Learning. Finally, we study the limits of this framework in practice for OGD and highlight the importance of the Neural Tangent Kernel variation for Continual Learning with OGD. Continual Learning is a setting in which an agent is exposed to multiples tasks sequentially (Kirkpatrick et al., 2016). The core challenge lies in the ability of the agent to learn the new tasks while retaining the knowledge acquired from previous tasks. Too much plasticity (Nguyen et al., 2018) will lead to catastrophic forgetting, which means the degradation of the ability of the agent to perform the past tasks (McCloskey & Cohen 1989, Ratcliff 1990, Goodfellow et al. 2014). On the other hand, too much stability will hinder the agent from adapting to new tasks. While there is a large literature on Continual Learning (Parisi et al., 2019), few works have addressed the problem from a theoretical perspective. Recently, Jacot et al. (2018) established the connection between overparameterized neural networks and kernel methods by introducing the Neural Tangent Kernel (NTK).

We are interested in gradient-based Explicit Generative Modeling where samples can be derived from iterative gradient updates based on an estimate of the score function of the data distribution. Recent advances in Stochastic Gradient Langevin Dynamics (SGLD) demonstrates impressive results with energy-based models on high-dimensional and complex data distributions. Stein Variational Gradient Descent (SVGD) is a deterministic sampling algorithm that iteratively transports a set of particles to approximate a given distribution, based on functional gradient descent that decreases the KL divergence. SVGD has promising results on several Bayesian inference applications. However, applying SVGD on high dimensional problems is still under-explored. The goal of this work is to study high dimensional inference with SVGD. We first identify key challenges in practical kernel SVGD inference in high-dimension. We propose noise conditional kernel SVGD (NCK-SVGD), that works in tandem with the recently introduced Noise Conditional Score Network estimator. NCK is crucial for successful inference with SVGD in high dimension, as it adapts the kernel to the noise level of the score estimate. As we anneal the noise, NCK-SVGD targets the real data distribution. We then extend the annealed SVGD with an entropic regularization. We show that this offers a flexible control between sample quality and diversity, and verify it empirically by precision and recall evaluations. The NCK-SVGD produces samples comparable to GANs and annealed SGLD on computer vision benchmarks, including MNIST and CIFAR-10.

We study the data deletion problem for convex models. By leveraging techniques from convex optimization and reservoir sampling, we give the first data deletion algorithms that are able to handle an arbitrarily long sequence of adversarial updates while promising both per-deletion run-time and steady-state error that do not grow with the length of the update sequence. We also introduce several new conceptual distinctions: for example, we can ask that after a deletion, the entire state maintained by the optimization algorithm is statistically indistinguishable from the state that would have resulted had we retrained, or we can ask for the weaker condition that only the observable output is statistically indistinguishable from the observable output that would have resulted from retraining. We are able to give more efficient deletion algorithms under this weaker deletion criterion.

One fundamental problem when solving inverse problems is how to find regularization parameters. This article considers solving this problem using data-driven bilevel optimization, i.e. we consider the adaptive learning of the regularization parameter from data by means of optimization. This approach can be interpreted as solving an empirical risk minimization problem, and we analyze its performance in the large data sample size limit for general nonlinear problems. To reduce the associated computational cost, online numerical schemes are derived using the stochastic gradient method. We prove convergence of these numerical schemes under suitable assumptions on the forward problem. Numerical experiments are presented illustrating the theoretical results and demonstrating the applicability and efficiency of the proposed approaches for various linear and nonlinear inverse problems, including Darcy flow, the eikonal equation, and an image denoising example.