We investigate adversarial-sample generation methods from a frequency domain perspective and extend standard $l_{\infty}$ Projected Gradient Descent (PGD) to the frequency domain. The resulting method, which we call Spectral Projected Gradient Descent (SPGD), has better success rate compared to PGD during early steps of the method. Adversarially training models using SPGD achieves greater adversarial accuracy compared to PGD when holding the number of attack steps constant. The use of SPGD can, therefore, reduce the overhead of adversarial training when utilizing adversarial generation with a smaller number of steps. However, we also prove that SPGD is equivalent to a variant of the PGD ordinarily used for the $l_{\infty}$ threat model. This PGD variant omits the sign function which is ordinarily applied to the gradient. SPGD can, therefore, be performed without explicitly transforming into the frequency domain. Finally, we visualize the perturbations SPGD generates and find they use both high and low-frequency components, which suggests that removing either high-frequency components or low-frequency components is not an effective defense.

We consider the problem of black-box function optimization over the boolean hypercube. Despite the vast literature on black-box function optimization over continuous domains, not much attention has been paid to learning models for optimization over combinatorial domains until recently. However, the computational complexity of the recently devised algorithms are prohibitive even for moderate numbers of variables; drawing one sample using the existing algorithms is more expensive than a function evaluation for many black-box functions of interest. To address this problem, we propose a computationally efficient model learning algorithm based on multilinear polynomials and exponential weight updates. In the proposed algorithm, we alternate between simulated annealing with respect to the current polynomial representation and updating the weights using monomial experts' advice. Numerical experiments on various datasets in both unconstrained and sum-constrained boolean optimization indicate the competitive performance of the proposed algorithm, while improving the computational time up to several orders of magnitude compared to state-of-the-art algorithms in the literature.

Stochastic gradient descent (SGD) and its variants are mainstream methods to train deep neural networks. Since neural networks are non-convex, more and more works study the dynamic behavior of SGD and the impact to its generalization, especially the escaping efficiency from local minima. However, these works take the over-simplified assumption that the covariance of the noise in SGD is (or can be upper bounded by) constant, although it is actually state-dependent. In this work, we conduct a formal study on the dynamic behavior of SGD with state-dependent noise. Specifically, we show that the covariance of the noise of SGD in the local region of the local minima is a quadratic function of the state. Thus, we propose a novel power-law dynamic with state-dependent diffusion to approximate the dynamic of SGD. We prove that, power-law dynamic can escape from sharp minima exponentially faster than flat minima, while the previous dynamics can only escape sharp minima polynomially faster than flat minima. Our experiments well verified our theoretical results. Inspired by our theory, we propose to add additional state-dependent noise into (large-batch) SGD to further improve its generalization ability. Experiments verify that our method is effective.

This paper investigates the use of nonparametric kernel-regression to obtain a tasksimilarity aware meta-learning algorithm. Our hypothesis is that the use of tasksimilarity helps meta-learning when the available tasks are limited and may contain outlier/ dissimilar tasks. While existing meta-learning approaches implicitly assume the tasks as being similar, it is generally unclear how this task-similarity could be quantified and used in the learning. As a result, most popular metalearning approaches do not actively use the similarity/dissimilarity between the tasks, but rely on availability of huge number of tasks for their working. Our contribution is a novel framework for meta-learning that explicitly uses task-similarity in the form of kernels and an associated meta-learning algorithm. We model the task-specific parameters to belong to a reproducing kernel Hilbert space where the kernel function captures the similarity across tasks. The proposed algorithm iteratively learns a meta-parameter which is used to assign a task-specific descriptor for every task. The task descriptors are then used to quantify the task-similarity through the kernel function. We show how our approach conceptually generalizes the popular meta-learning approaches of model-agnostic meta-learning (MAML) and Meta-stochastic gradient descent (Meta-SGD) approaches. Numerical experiments with regression tasks show that our algorithm outperforms these approaches when the number of tasks is limited, even in the presence of outlier or dissimilar tasks. This supports our hypothesis that task-similarity helps improve the metalearning performance in task-limited and adverse settings.

Meta continual learning algorithms seek to train a model when faced with similar tasks observed in a sequential manner. Despite promising methodological advancements, there is a lack of theoretical frameworks that enable analysis of learning challenges such as generalization and catastrophic forgetting. To that end, we develop a new theoretical approach for meta continual learning~(MCL) where we mathematically model the learning dynamics using dynamic programming, and we establish conditions of optimality for the MCL problem. Moreover, using the theoretical framework, we derive a new dynamic-programming-based MCL method that adopts stochastic-gradient-driven alternating optimization to balance generalization and catastrophic forgetting. We show that, on MCL benchmark data sets, our theoretically grounded method achieves accuracy better than or comparable to that of existing state-of-the-art methods.

Although remarkable successful in practice, training generative adversarial networks(GANs) is still quite difficult and iteratively prone to cyclic behaviors, as GANs need to solve a non-convex non-concave min-max game using a gradient descent ascent (GDA) method. Motivated by the ideas of simultaneous centripetal acceleration (SCA) and modified predictive methods (MPM), we propose a novel predictive projection centripetal acceleration (PPCA) methods to alleviate the cyclic behaviors. Besides, under suitable assumptions, we show that the difference between the signed vector of partial derivatives at t + 1 and t is orthogonal to the signed vector of partial derivatives at t for GDA, and the last-iterate exponential convergence on the bilinear game. Finally, numerical simulations are conducted by PPCA in GANs setting, and the results illustrate the effectiveness of our approach.

Continual learning (CL) is a setting in which an agent has to learn from an incoming stream of data during its entire lifetime. Although major advances have been made in the field, one recurring problem which remains unsolved is that of Catastrophic Forgetting (CF). While the issue has been extensively studied empirically, little attention has been paid from a theoretical angle. In this paper, we show that the impact of CF increases as two tasks increasingly align. We introduce a measure of task similarity called the NTK overlap matrix which is at the core of CF. We analyze common projected gradient algorithms and demonstrate how they mitigate forgetting. Then, we propose a variant of Orthogonal Gradient Descent (OGD) which leverages structure of the data through Principal Component Analysis (PCA). Experiments support our theoretical findings and show how our method reduces CF on classical CL datasets.

Overparametrized neural networks have infinitely many solutions that achieve zero training error, and such global minima have different generalization performance. Moreover, training a neural network is a high-dimensional nonconvex problem, which is typically intractable to solve. However, the success of deep learning indicates that first-order methods such as gradient descent or stochastic gradient descent (GD/SGD) not only (a) succeed in finding global minima, but also (b) are biased towards solutions that generalize well, which largely has remained a mystery in the literature. To explain part (a) of the phenomenon, there is a growing literature studying the convergence of GD/SGD on overparametrized neural networks (e.g., Du et al. (2018a,b); Allen-Zhu et al. (2018); Zou et al. (2018); Jacot et al. (2018); Oymak and Soltanolkotabi (2020), and many more). There are also convergence results that focus on linear networks, without nonlinear activations (Bartlett et al., 2018; Arora et al., 2019a; Wu et al., 2019; Du and Hu, 2019; Hu et al., 2020). These results typically focus on the convergence of loss, hence do not address which of the many global minima is reached.

We introduce a notion of usable information contained in the representation learned by a deep network, and use it to study how optimal representations for the task emerge during training, and how they adapt to different tasks. We use this to characterize the transient dynamics of deep neural networks on perceptual decision-making tasks inspired by neuroscience literature. In particular, we show that both the random initialization and the implicit regularization from Stochastic Gradient Descent play an important role in learning minimal sufficient representations for the task. If the network is not randomly initialized, we show that the training may not recover an optimal representation, increasing the chance of overfitting. An important open question for the theory of deep learning is why highly overparametrized neural networks learn solutions that generalize well even though models can in principle memorize the entire training set. Some have speculated that neural networks learn minimal but sufficient representations of the input through implicit regularization of Stochastic Gradient Descent (SGD) (Shwartz-Ziv & Tishby, 2017; Achille & Soatto, 2018), and that the minimality of the representations relates to generalizability. Followup work has disputed some of these claims (Saxe et al., 2018), leading to an ongoing debate on the optimality of representations and how they are learned during training. Here, we design a simple task to empirically study how representations are formed during training, and how implicit regularization from SGD and initializations affect the resulting representations in deep networks. We then validate these results on a variant of an MNIST classification task to assess how SGD affects the minimality of representations. Investigations into the optimality of representations have typically used information-theoretic reasoning.

The problem of learning to generalize to unseen classes during training, known as few-shot classification, has attracted considerable attention. Initialization based methods, such as the gradient-based model agnostic meta-learning (MAML), tackle the few-shot learning problem by "learning to fine-tune". The goal of these approaches is to learn proper model initialization, so that the classifiers for new classes can be learned from a few labeled examples with a small number of gradient update steps. Few shot meta-learning is well-known with its fast-adapted capability and accuracy generalization onto unseen tasks. Learning fairly with unbiased outcomes is another significant hallmark of human intelligence, which is rarely touched in few-shot meta-learning. In this work, we propose a Primal-Dual Fair Meta-learning framework, namely PDFM, which learns to train fair machine learning models using only a few examples based on data from related tasks. The key idea is to learn a good initialization of a fair model's primal and dual parameters so that it can adapt to a new fair learning task via a few gradient update steps. Instead of manually tuning the dual parameters as hyperparameters via a grid search, PDFM optimizes the initialization of the primal and dual parameters jointly for fair meta-learning via a subgradient primal-dual approach. We further instantiate examples of bias controlling using mean difference and decision boundary covariance as fairness constraints to each task for supervised regression and classification, respectively. We demonstrate the versatility of our proposed approach by applying our approach to various real-world datasets. Our experiments show substantial improvements over the best prior work for this setting.