Twitter users will have seen the proliferation of "I have a joke" tweets in their feed over the past few days. The AI community produced some gems so we've collected a selection here for your amusement. I have a reinforcement learning joke, but not sure it's rewarding. I have a stochastic gradient descent joke but the punchline isn't on this saddle point https://t.co/B7GM2tmz5Z I have a deep learning joke but it has a lot of layers to it.

We study the long time behavior of an underdamped mean-field Langevin (MFL) equation, and provide a general convergence as well as an exponential convergence rate result under different conditions. The results on the MFL equation can be applied to study the convergence of the Hamiltonian gradient descent algorithm for the overparametrized optimization. We then provide a numerical example of the algorithm to train a generative adversarial networks (GAN).

The scalability of Distributed Stochastic Gradient Descent (SGD) is today limited by communication bottlenecks. We propose a novel SGD variant: Communication-efficient SGD with Error Reset, or CSER. The key idea in CSER is first a new technique called "error reset" that adapts arbitrary compressors for SGD, producing bifurcated local models with periodic reset of resulting local residual errors. Second we introduce partial synchronization for both the gradients and the models, leveraging advantages from them. We prove the convergence of CSER for smooth non-convex problems. Empirical results show that when combined with highly aggressive compressors, the CSER algorithms: i) cause no loss of accuracy, and ii) accelerate the training by nearly $10\times$ for CIFAR-100, and by $4.5\times$ for ImageNet.

Structured non-convex learning problems, for which critical points have favorable statistical properties, arise frequently in statistical machine learning. Algorithmic convergence and statistical estimation rates are well-understood for such problems. However, quantifying the uncertainty associated with the underlying training algorithm is not well-studied in the non-convex setting. In order to address this shortcoming, in this work, we establish an asymptotic normality result for the constant step size stochastic gradient descent (SGD) algorithm--a widely used algorithm in practice. Specifically, based on the relationship between SGD and Markov Chains [DDB19], we show that the average of SGD iterates is asymptotically normally distributed around the expected value of their unique invariant distribution, as long as the non-convex and non-smooth objective function satisfies a dissipativity property. We also characterize the bias between this expected value and the critical points of the objective function under various local regularity conditions. Together, the above two results could be leveraged to construct confidence intervals for non-convex problems that are trained using the SGD algorithm.

Physics-informed neural networks (PINNs) have lately received great attention thanks to their flexibility in tackling a wide range of forward and inverse problems involving partial differential equations. However, despite their noticeable empirical success, little is known about how such constrained neural networks behave during their training via gradient descent. More importantly, even less is known about why such models sometimes fail to train at all. In this work, we aim to investigate these questions through the lens of the Neural Tangent Kernel (NTK); a kernel that captures the behavior of fully-connected neural networks in the infinite width limit during training via gradient descent. Specifically, we derive the NTK of PINNs and prove that, under appropriate conditions, it converges to a deterministic kernel that stays constant during training in the infinite-width limit. This allows us to analyze the training dynamics of PINNs through the lens of their limiting NTK and find a remarkable discrepancy in the convergence rate of the different loss components contributing to the total training error. To address this fundamental pathology, we propose a novel gradient descent algorithm that utilizes the eigenvalues of the NTK to adaptively calibrate the convergence rate of the total training error. Finally, we perform a series of numerical experiments to verify the correctness of our theory and the practical effectiveness of the proposed algorithms. The data and code accompanying this manuscript are publicly available at \url{https://github.com/PredictiveIntelligenceLab/PINNsNTK}.

Stochastic Gradient Descent (SGD) and its variants are the most used algorithms in machine learning applications. In particular, SGD with adaptive learning rates and momentum is the industry standard to train deep networks. Despite the enormous success of these methods, our theoretical understanding of these variants in the nonconvex setting is not complete, with most of the results only proving convergence in expectation and with strong assumptions on the stochastic gradients. In this paper, we present a high probability analysis for adaptive and momentum algorithms, under weak assumptions on the function, stochastic gradients, and learning rates. We use it to prove for the first time the convergence of the gradients to zero in high probability in the smooth nonconvex setting for Delayed AdaGrad with momentum.

Stochastic gradient descent (SGD) and its variants have been the dominating optimization methods in machine learning. Compared with small batch training, SGD with large batch training can better utilize the computational power of current multi-core systems like GPUs and can reduce the number of communication rounds in distributed training. Hence, SGD with large batch training has attracted more and more attention. However, existing empirical results show that large batch training typically leads to a drop of generalization accuracy. As a result, large batch training has also become a challenging topic. In this paper, we propose a novel method, called stochastic normalized gradient descent with momentum (SNGM), for large batch training. We theoretically prove that compared to momentum SGD (MSGD) which is one of the most widely used variants of SGD, SNGM can adopt a larger batch size to converge to the $\epsilon$-stationary point with the same computation complexity (total number of gradient computation). Empirical results on deep learning also show that SNGM can achieve the state-of-the-art accuracy with a large batch size.

We propose Multi-Level Local SGD, a distributed gradient method for learning a smooth, non-convex objective in a heterogeneous multi-level network. Our network model consists of a set of disjoint sub-networks, with a single hub and multiple worker nodes; further, worker nodes may have different operating rates. The hubs exchange information with one another via a connected, but not necessarily complete communication network. In our algorithm, sub-networks execute a distributed SGD algorithm, using a hub-and-spoke paradigm, and the hubs periodically average their models with neighboring hubs. We first provide a unified mathematical framework that describes the Multi-Level Local SGD algorithm. We then present a theoretical analysis of the algorithm; our analysis shows the dependence of the convergence error on the worker node heterogeneity, hub network topology, and the number of local, sub-network, and global iterations. We back up our theoretical results via simulation-based experiments using both convex and non-convex objectives.

It has been observed that design choices of neural networks are often crucial for their successful optimization. In this article, we therefore discuss the question if it is always possible to redesign a neural network so that it trains well with gradient descent. This yields the following universality result: If, for a given network, there is any algorithm that can find good network weights for a classification task, then there exists an extension of this network that reproduces these weights and the corresponding forward output by mere gradient descent training. The construction is not intended for practical computations, but it provides some orientation on the possibilities of meta-learning and related approaches.

In this paper, we present a novel stochastic optimization method, which uses the binary search technique with first order gradient based optimization method, called Binary Search Gradient Optimization (BSG) or BiGrad. In this optimization setup, a non-convex surface is treated as a set of convex surfaces. In BSG, at first, a region is defined, assuming region is convex. If region is not convex, then the algorithm leaves the region very fast and defines a new one, otherwise, it tries to converge at the optimal point of the region. In BSG, core purpose of binary search is to decide, whether region is convex or not in logarithmic time, whereas, first order gradient based method is primarily applied, to define a new region. In this paper, Adam is used as a first order gradient based method, nevertheless, other methods of this class may also be considered. In deep neural network setup, it handles the problem of vanishing and exploding gradient efficiently. We evaluate BSG on the MNIST handwritten digit, IMDB, and CIFAR10 data set, using logistic regression and deep neural networks. We produce more promising results as compared to other first order gradient based optimization methods. Furthermore, proposed algorithm generalizes significantly better on unseen data as compared to other methods.