Neural networks trained to solve modular arithmetic tasks exhibit grokking, a phenomenon where the test accuracy starts improving long after the model achieves 100% training accuracy in the training process. It is often taken as an example of "emergence", where model ability manifests sharply through a phase transition. In this work, we show that the phenomenon of grokking is not specific to neural networks nor to gradient descent-based optimization. Specifically, we show that this phenomenon occurs when learning modular arithmetic with Recursive Feature Machines (RFM), an iterative algorithm that uses the Average Gradient Outer Product (AGOP) to enable task-specific feature learning with general machine learning models. When used in conjunction with kernel machines, iterating RFM results in a fast transition from random, near zero, test accuracy to perfect test accuracy. This transition cannot be predicted from the training loss, which is identically zero, nor from the test loss, which remains constant in initial iterations. Instead, as we show, the transition is completely determined by feature learning: RFM gradually learns block-circulant features to solve modular arithmetic. Paralleling the results for RFM, we show that neural networks that solve modular arithmetic also learn block-circulant features. Furthermore, we present theoretical evidence that RFM uses such block-circulant features to implement the Fourier Multiplication Algorithm, which prior work posited as the generalizing solution neural networks learn on these tasks. Our results demonstrate that emergence can result purely from learning task-relevant features and is not specific to neural architectures nor gradient descent-based optimization methods. Furthermore, our work provides more evidence for AGOP as a key mechanism for feature learning in neural networks.

Soil organic matter (SOM) content is a crucial indicator for assessing soil fertility and serves as a key factor in sustaining a viable agricultural system. With the continuous advancement and improvement of remote sensing technology, hyperspectral imagery has been employed for the monitoring of SOM. While the numerous bands reveal finer details within the spectral features, this also brings information redundancy and noise interference. Currently, the dimensionality reduction methods designed for hyperspectral imagery encounter difficulties in achieving optimal band combinations. As a result, swiftly and accurately capturing the spectral features of SOM becomes a challenging task. In this article, aiming to address the inefficiency and instability in hyperspectral feature selection, we propose a metaheuristic-based algorithm—the improved weighted marine predators algorithm (IWMPA)—for hyperspectral feature selection. Specifically, we simulated the process of hyperspectral feature selection using the foraging strategy of marine predators. We employed prior weight coefficients and reverse learning operations in the initialization phase to accelerate the convergence of the population and introduced mutation operations into the phase of development to prevent the occurrence of local optima traps. We employed the IWMPA feature selection method to establish SOM estimation models within the research area of Yitong Manchu Autonomous County in China. The results demonstrated that the hyperspectral features selected using the IWMPA approach yield favorable outcomes in the SOM estimation models. Specifically, in the best-performing regression model of this study, R2 on the test set was 0.7225. These experimental results suggest that, in comparison to the existing methods, the proposed IWMPA method is more adept at capturing the spectral features of SOM.

While neural networks can be approximated by linear models as their width increases, certain properties of wide neural networks cannot be captured by linear models. In this work we show that recently proposed Neural Quadratic Models can exhibit the "catapult phase" Lewkowycz et al. (2020) that arises when training such models with large learning rates. We then empirically show that the behaviour of quadratic models parallels that of neural networks in generalization, especially in the catapult phase regime. Our analysis further demonstrates that quadratic models are an effective tool for analysis of neural networks.

In this paper, we first present an explanation regarding the common occurrence of spikes in the training loss when neural networks are trained with stochastic gradient descent (SGD). We provide evidence that the spikes in the training loss of SGD are "catapults", an optimization phenomenon originally observed in GD with large learning rates in Lewkowycz et al. (2020). We empirically show that these catapults occur in a low-dimensional subspace spanned by the top eigenvectors of the tangent kernel, for both GD and SGD. Second, we posit an explanation for how catapults lead to better generalization by demonstrating that catapults increase feature learning by increasing alignment with the Average Gradient Outer Product (AGOP) of the true predictor. Furthermore, we demonstrate that a smaller batch size in SGD induces a larger number of catapults, thereby improving AGOP alignment and test performance.

In this paper we study the problem of lower bounding the minimum eigenvalue of the neural tangent kernel (NTK) at initialization, an important quantity for the theoretical analysis of training in neural networks. We consider feedforward neural networks with smooth activation functions. Without any distributional assumptions on the input, we present a novel result: we show that for suitable initialization variance, $\widetilde{\Omega}(n)$ width, where $n$ is the number of training samples, suffices to ensure that the NTK at initialization is positive definite, improving prior results for smooth activations under our setting. Prior to our work, the sufficiency of linear width has only been shown either for networks with ReLU activation functions, and sublinear width has been shown for smooth networks but with additional conditions on the distribution of the data. The technical challenge in the analysis stems from the layerwise inhomogeneity of smooth activation functions and we handle the challenge using {\em generalized} Hermite series expansion of such activations.

In this paper we show that feedforward neural networks corresponding to arbitrary directed acyclic graphs undergo transition to linearity as their ``width'' approaches infinity. The width of these general networks is characterized by the minimum in-degree of their neurons, except for the input and first layers. Our results identify the mathematical structure underlying transition to linearity and generalize a number of recent works aimed at characterizing transition to linearity or constancy of the Neural Tangent Kernel for standard architectures.

The goal of this work is to shed light on the remarkable phenomenon of "transition to linearity" of certain neural networks as their width approaches infinity. We show that the "transition to linearity'' of the model and, equivalently, constancy of the (neural) tangent kernel (NTK) result from the scaling properties of the norm of the Hessian matrix of the network as a function of the network width. We present a general framework for understanding the constancy of the tangent kernel via Hessian scaling applicable to the standard classes of neural networks. Our analysis provides a new perspective on the phenomenon of constant tangent kernel, which is different from the widely accepted "lazy training''. Furthermore, we show that the "transition to linearity" is not a general property of wide neural networks and does not hold when the last layer of the network is non-linear. It is also not necessary for successful optimization by gradient descent.