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In this paper, we investigate the popular deep learning optimization routine, Adam, from the perspective of statistical moments. While Adam is an adaptive lower-order moment based (of the stochastic gradient) method, we propose an extension namely, HAdam, which uses higher order moments of the stochastic gradient. Our analysis and experiments reveal that certain higher-order moments of the stochastic gradient are able to achieve better performance compared to the vanilla Adam algorithm. We also provide some analysis of HAdam related to odd and even moments to explain some intriguing and seemingly non-intuitive empirical results.
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Adaptive optimization algorithms, such as Adam and RMSprop, have shown better optimization performance than stochastic gradient descent (SGD) in some scenarios. However, recent studies show that they often lead to worse generalization performance than SGD, especially for training deep neural networks (DNNs). In this work, we identify the reasons that Adam generalizes worse than SGD, and develop a variant of Adam to eliminate the generalization gap. The proposed method, normalized direction-preserving Adam (ND-Adam), enables more precise control of the direction and step size for updating weight vectors, leading to significantly improved generalization performance. Following a similar rationale, we further improve the generalization performance in classification tasks by regularizing the softmax logits. By bridging the gap between SGD and Adam, we also hope to shed light on why certain optimization algorithms generalize better than others.
Graph neural network models have been extensively used to learn node representations for graph structured data in an end-to-end setting. These models often rely on localized first order approximations of spectral graph convolutions and hence are unable to capture higher-order relational information between nodes. Probabilistic Graphical Models form another class of models that provide rich flexibility in incorporating such relational information but are limited by inefficient approximate inference algorithms at higher order. In this paper, we propose to combine these approaches to learn better node and graph representations. First, we derive an efficient approximate sum-product loopy belief propagation inference algorithm for higher-order PGMs. We then embed the message passing updates into a neural network to provide the inductive bias of the inference algorithm in end-to-end learning. This gives us a model that is flexible enough to accommodate domain knowledge while maintaining the computational advantage. We further propose methods for constructing higher-order factors that are conditioned on node and edge features and share parameters wherever necessary. Our experimental evaluation shows that our model indeed captures higher-order information, substantially outperforming state-of-the-art $k$-order graph neural networks in molecular datasets.
Randomized smoothing is a recently proposed defense against adversarial attacks that has achieved SOTA provable robustness against $ell_2$ perturbations. A number of publications have extended the guarantees to other metrics, such as $ell_1$ or $ell_infty$, by using different smoothing measures. Although the current framework has been shown to yield near-optimal $ell_p$ radii, the total safety region certified by the current framework can be arbitrarily small compared to the optimal. In this work, we propose a framework to improve the certified safety region for these smoothed classifiers without changing the underlying smoothing scheme. The theoretical contributions are as follows: 1) We generalize the certification for randomized smoothing by reformulating certified radius calculation as a nested optimization problem over a class of functions. 2) We provide a method to calculate the certified safety region using $0^{th}$-order and $1^{st}$-order information for Gaussian-smoothed classifiers. We also provide a framework that generalizes the calculation for certification using higher-order information. 3) We design efficient, high-confidence estimators for the relevant statistics of the first-order information. Combining the theoretical contribution 2) and 3) allows us to certify safety region that are significantly larger than the ones provided by the current methods. On CIFAR10 and Imagenet datasets, the new regions certified by our approach achieve significant improvements on general $ell_1$ certified radii and on the $ell_2$ certified radii for color-space attacks ($ell_2$ restricted to 1 channel) while also achieving smaller improvements on the general $ell_2$ certified radii. Our framework can also provide a way to circumvent the current impossibility results on achieving higher magnitude of certified radii without requiring the use of data-dependent smoothing techniques.
Adam is a widely used optimization method for training deep learning models. It computes individual adaptive learning rates for different parameters. In this paper, we propose a generalization of Adam, called Adambs, that allows us to also adapt to different training examples based on their importance in the models convergence. To achieve this, we maintain a distribution over all examples, selecting a mini-batch in each iteration by sampling according to this distribution, which we update using a multi-armed bandit algorithm. This ensures that examples that are more beneficial to the model training are sampled with higher probabilities. We theoretically show that Adambs improves the convergence rate of Adam---$O(sqrt{frac{log n}{T} })$ instead of $O(sqrt{frac{n}{T}})$ in some cases. Experiments on various models and datasets demonstrate Adambss fast convergence in practice.
A calculation method for higher-order moments of physical quantities, including magnetization and energy, based on the higher-order tensor renormalization group is proposed. The physical observables are represented by impurity tensors. A systematic summation scheme provides coarse-grained tensors including multiple impurities. Our method is compared with the Monte Carlo method on the two-dimensional Potts model. While the nature of the transition of the $q$-state Potts model has been known for a long time owing to the analytical arguments, a clear numerical confirmation has been difficult due to extremely long correlation length in the weakly first-order transitions, e.g., for $q=5$. A jump of the Binder ratio precisely determines the transition temperature. The finite-size scaling analysis provides critical exponents and distinguishes the weakly first-order and the continuous transitions.
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