Within the PAC-Bayesian framework, the Gibbs classifier (defined on a posterior $Q$) and the corresponding $Q$-weighted majority vote classifier are commonly used to analyze the generalization performance. However, there exists a notable lack in theoretical research exploring the certified robustness of majority vote classifier and its interplay with generalization. In this study, we develop a generalization error bound that possesses a certified robust radius for the smoothed majority vote classifier (i.e., the $Q$-weighted majority vote classifier with smoothed inputs); In other words, the generalization bound holds under any data perturbation within the certified robust radius. As a byproduct, we find that the underpinnings of both the generalization bound and the certified robust radius draw, in part, upon weight spectral norm, which thereby inspires the adoption of spectral regularization in smooth training to boost certified robustness. Utilizing the dimension-independent property of spherical Gaussian inputs in smooth training, we propose a novel and inexpensive spectral regularizer to enhance the smoothed majority vote classifier. In addition to the theoretical contribution, a set of empirical results is provided to substantiate the effectiveness of our proposed method.

Loss of plasticity is a phenomenon where neural networks become more difficult to train during the course of learning. Continual learning algorithms seek to mitigate this effect by sustaining good predictive performance while maintaining network trainability. We develop new techniques for improving continual learning by first reconsidering how initialization can ensure trainability during early phases of learning. From this perspective, we derive new regularization strategies for continual learning that ensure beneficial initialization properties are better maintained throughout training. In particular, we investigate two new regularization techniques for continual learning: (i) Wasserstein regularization toward the initial weight distribution, which is less restrictive than regularizing toward initial weights; and (ii) regularizing weight matrix singular values, which directly ensures gradient diversity is maintained throughout training. We present an experimental analysis that shows these alternative regularizers can improve continual learning performance across a range of supervised learning tasks and model architectures. The alternative regularizers prove to be less sensitive to hyperparameters while demonstrating better training in individual tasks, sustaining trainability as new tasks arrive, and achieving better generalization performance.

We study the problem of diverse feature selection in linear regression: selecting a small subset of diverse features that can predict a given objective. Diversity is useful for several reasons such as interpretability, robustness to noise, etc. We propose several spectral regularizers that capture a notion of diversity of features and show that these are all submodular set functions. These regularizers, when added to the objective function for linear regression, result in approximately submodular functions, which can then be maximized approximately by efficient greedy and local search algorithms, with provable guarantees. We compare our algorithms to traditional greedy and \ell_1 -regularization schemes and show that we obtain a more diverse set of features that result in the regression problem being stable under perturbations.

Backpropagation-optimized artificial neural networks, while precise, lack robustness, leading to unforeseen behaviors that affect their safety. Biological neural systems do solve some of these issues already. Thus, understanding the biological mechanisms of robustness is an important step towards building trustworthy and safe systems. Unlike artificial models, biological neurons adjust connectivity based on neighboring cell activity. Robustness in neural representations is hypothesized to correlate with the smoothness of the encoding manifold. Recent work suggests power law covariance spectra, which were observed studying the primary visual cortex of mice, to be indicative of a balanced trade-off between accuracy and robustness in representations. Here, we show that unsupervised local learning models with winner takes all dynamics learn such power law representations, providing upcoming studies a mechanistic model with that characteristic. Our research aims to understand the interplay between geometry, spectral properties, robustness, and expressivity in neural representations. Hence, we study the link between representation smoothness and spectrum by using weight, Jacobian and spectral regularization while assessing performance and adversarial robustness. Our work serves as a foundation for future research into the mechanisms underlying power law spectra and optimally smooth encodings in both biological and artificial systems. The insights gained may elucidate the mechanisms that realize robust neural networks in mammalian brains and inform the development of more stable and reliable artificial systems.

In this paper we consider the problem of learning from data the support of a probability distribution when the distribution {\em does not} have a density (with respect to some reference measure). We propose a new class of regularized spectral estimators based on a new notion of reproducing kernel Hilbert space, which we call {\em completely regular''}. Completely regular kernels allow to capture the relevant geometric and topological properties of an arbitrary probability space. In particular, they are the key ingredient to prove the universal consistency of the spectral estimators and in this respect they are the analogue of universal kernels for supervised problems. Numerical experiments show that spectral estimators compare favorably to state of the art machine learning algorithms for density support estimation.

Various studies have shown the advantages of using Machine Learning (ML) techniques for analog and digital IC design automation and optimization. Data scarcity is still an issue for electronic designs, while training highly accurate ML models. This work proposes generating and evaluating artificial data using generative adversarial networks (GANs) for circuit data to aid and improve the accuracy of ML models trained with a small training data set. The training data is obtained by various simulations in the Cadence Virtuoso, HSPICE, and Microcap design environment with TSMC 180nm and 22nm CMOS technology nodes. The artificial data is generated and tested for an appropriate set of analog and digital circuits. The experimental results show that the proposed artificial data generation significantly improves ML models and reduces the percentage error by more than 50\% of the original percentage error, which were previously trained with insufficient data. Furthermore, this research aims to contribute to the extensive application of AI/ML in the field of VLSI design and technology by relieving the training data availability-related challenges.

Various forms of regularization in learning tasks strive for different notions of simplicity. This paper presents a spectral regularization technique, which attaches a unique inductive bias to sequence modeling based on an intuitive concept of simplicity defined in the Chomsky hierarchy. From fundamental connections between Hankel matrices and regular grammars, we propose to use the trace norm of the Hankel matrix, the tightest convex relaxation of its rank, as the spectral regularizer. To cope with the fact that the Hankel matrix is bi-infinite, we propose an unbiased stochastic estimator for its trace norm. Ultimately, we demonstrate experimental results on Tomita grammars, which exhibit the potential benefits of spectral regularization and validate the proposed stochastic estimator.

Data-driven machine learning models are being increasingly employed in several important inference problems in biology, chemistry, and physics which require learning over combinatorial spaces. Recent empirical evidence (see, e.g., [1], [2], [3]) suggests that regularizing the spectral representation of such models improves their generalization power when labeled data is scarce. However, despite these empirical studies, the theoretical underpinning of when and how spectral regularization enables improved generalization is poorly understood. In this paper, we focus on learning pseudo-Boolean functions and demonstrate that regularizing the empirical mean squared error by the L_1 norm of the spectral transform of the learned function reshapes the loss landscape and allows for data-frugal learning, under a restricted secant condition on the learner's empirical error measured against the ground truth function. Under a weaker quadratic growth condition, we show that stationary points which also approximately interpolate the training data points achieve statistically optimal generalization performance. Complementing our theory, we empirically demonstrate that running gradient descent on the regularized loss results in a better generalization performance compared to baseline algorithms in several data-scarce real-world problems.

Abstract reasoning and logic inference are difficult problems for neural networks, yet essential to their applicability in highly structured domains. In this work we demonstrate that a well known technique such as spectral regularization can significantly boost the capabilities of a neural learner. We introduce the Neural Abstract Reasoner (NAR), a memory augmented architecture capable of learning and using abstract rules. We show that, when trained with spectral regularization, NAR achieves $78.8\%$ accuracy on the Abstraction and Reasoning Corpus, improving performance 4 times over the best known human hand-crafted symbolic solvers. We provide some intuition for the effects of spectral regularization in the domain of abstract reasoning based on theoretical generalization bounds and Solomonoff's theory of inductive inference.

In this paper we consider the problem of learning from data the support of a probability distribution when the distribution {\em does not} have a density (with respect to some reference measure). We propose a new class of regularized spectral estimators based on a new notion of reproducing kernel Hilbert space, which we call {\em completely regular''}. Completely regular kernels allow to capture the relevant geometric and topological properties of an arbitrary probability space. In particular, they are the key ingredient to prove the universal consistency of the spectral estimators and in this respect they are the analogue of universal kernels for supervised problems. Numerical experiments show that spectral estimators compare favorably to state of the art machine learning algorithms for density support estimation.