Purpose. Proton Magnetic Resonance Spectroscopic Imaging (1H-MRSI) provides non-invasive spectral-spatial mapping of metabolism. However, long-standing problems in whole-brain 1H-MRSI are spectral overlap of metabolite peaks with large lipid signal from scalp, and overwhelming water signal that distorts spectra. Fast and effective methods are needed for high-resolution 1H-MRSI to accurately remove lipid and water signals while preserving the metabolite signal. The potential of supervised neural networks for this task remains unexplored, despite their success for other MRSI processing. Methods. We introduce a deep-learning method based on a modified Y-NET network for water and lipid removal in whole-brain 1H-MRSI. The WALINET (WAter and LIpid neural NETwork) was compared to conventional methods such as the state-of-the-art lipid L2 regularization and Hankel-Lanczos singular value decomposition (HLSVD) water suppression. Methods were evaluated on simulated and in-vivo whole-brain MRSI using NMRSE, SNR, CRLB, and FWHM metrics. Results. WALINET is significantly faster and needs 8s for high-resolution whole-brain MRSI, compared to 42 minutes for conventional HLSVD+L2. Quantitative analysis shows WALINET has better performance than HLSVD+L2: 1) more lipid removal with 41% lower NRMSE, 2) better metabolite signal preservation with 71% lower NRMSE in simulated data, 155% higher SNR and 50% lower CRLB in in-vivo data. Metabolic maps obtained by WALINET in healthy subjects and patients show better gray/white-matter contrast with more visible structural details. Conclusions. WALINET has superior performance for nuisance signal removal and metabolite quantification on whole-brain 1H-MRSI compared to conventional state-of-the-art techniques. This represents a new application of deep-learning for MRSI processing, with potential for automated high-throughput workflow.
The remarkable success of large language pretraining and the discovery of scaling laws signify a paradigm shift in machine learning. Notably, the primary objective has evolved from minimizing generalization error to reducing approximation error, and the most effective strategy has transitioned from regularization (in a broad sense) to scaling up models. This raises a critical question: Do the established principles that proved successful in the generalization-centric era remain valid in this new era of scaling? This paper examines several influential regularization-based principles that may no longer hold true in the scaling-centric, large language model (LLM) era. These principles include explicit L2 regularization and implicit regularization through small batch sizes and large learning rates. Additionally, we identify a new phenomenon termed ``scaling law crossover,'' where two scaling curves intersect at a certain scale, implying that methods effective at smaller scales may not generalize to larger ones. Together, these observations highlight two fundamental questions within this new paradigm: $\bullet$ Guiding Principles for Scaling: If regularization is no longer the primary guiding principle for model design, what new principles are emerging to guide scaling? $\bullet$ Model Comparison at Scale: How to reliably and effectively compare models at the scale where only a single experiment is feasible?
This letter presents a high-dimensional analysis of the training dynamics for a single-layer nonlinear contrastive learning model. The empirical distribution of the model weights converges to a deterministic measure governed by a McKean-Vlasov nonlinear partial differential equation (PDE). Under L2 regularization, this PDE reduces to a closed set of low-dimensional ordinary differential equations (ODEs), reflecting the evolution of the model performance during the training process. We analyze the fixed point locations and their stability of the ODEs unveiling several interesting findings. First, only the hidden variable's second moment affects feature learnability at the state with uninformative initialization. Second, higher moments influence the probability of feature selection by controlling the attraction region, rather than affecting local stability. Finally, independent noises added in the data argumentation degrade performance but negatively correlated noise can reduces the variance of gradient estimation yielding better performance. Despite of the simplicity of the analyzed model, it exhibits a rich phenomena of training dynamics, paving a way to understand more complex mechanism behind practical large models.
Prediction of stock prices has been a crucial and challenging task, especially in the case of highly volatile digital currencies such as Bitcoin. This research examineS the potential of using neural network models, namely LSTMs and GRUs, to forecast Bitcoin's price movements. We employ five-fold cross-validation to enhance generalization and utilize L2 regularization to reduce overfitting and noise. Our study demonstrates that the GRUs models offer better accuracy than LSTMs model for predicting Bitcoin's price. Specifically, the GRU model has an MSE of 4.67, while the LSTM model has an MSE of 6.25 when compared to the actual prices in the test set data. This finding indicates that GRU models are better equipped to process sequential data with long-term dependencies, a characteristic of financial time series data such as Bitcoin prices. In summary, our results provide valuable insights into the potential of neural network models for accurate Bitcoin price prediction and emphasize the importance of employing appropriate regularization techniques to enhance model performance.
In this work, we introduce a novel approach to regularization in multivariable regression problems. Our regularizer, called DLoss, penalises differences between the model's derivatives and derivatives of the data generating function as estimated from the training data. We call these estimated derivatives data derivatives. The goal of our method is to align the model to the data, not only in terms of target values but also in terms of the derivatives involved. To estimate data derivatives, we select (from the training data) 2-tuples of input-value pairs, using either nearest neighbour or random, selection. On synthetic and real datasets, we evaluate the effectiveness of adding DLoss, with different weights, to the standard mean squared error loss. The experimental results show that with DLoss (using nearest neighbour selection) we obtain, on average, the best rank with respect to MSE on validation data sets, compared to no regularization, L2 regularization, and Dropout.
In the wake of the global spread of monkeypox, accurate disease recognition has become crucial. This study introduces an improved SE-InceptionV3 model, embedding the SENet module and incorporating L2 regularization into the InceptionV3 framework to enhance monkeypox disease detection. Utilizing the Kaggle monkeypox dataset, which includes images of monkeypox and similar skin conditions, our model demonstrates a noteworthy accuracy of 96.71% on the test set, outperforming conventional methods and deep learning models. The SENet modules channel attention mechanism significantly elevates feature representation, while L2 regularization ensures robust generalization. Extensive experiments validate the models superiority in precision, recall, and F1 score, highlighting its effectiveness in differentiating monkeypox lesions in diverse and complex cases. The study not only provides insights into the application of advanced CNN architectures in medical diagnostics but also opens avenues for further research in model optimization and hyperparameter tuning for enhanced disease recognition. https://github.com/jzc777/SE-inceptionV3-L2
Although Multi Armed Bandit (MAB) on one hand and the policy gradient approach on the other hand are among the most used frameworks of Reinforcement Learning, the theoretical properties of the policy gradient algorithm used for MAB have not been given enough attention. We investigate in this work the convergence of such a procedure for the situation when a $L2$ regularization term is present jointly with the 'softmax' parametrization. We prove convergence under appropriate technical hypotheses and test numerically the procedure including situations beyond the theoretical setting. The tests show that a time dependent regularized procedure can improve over the canonical approach especially when the initial guess is far from the solution.
In supervised machine learning, feature selection plays a very important role by potentially enhancing explainability and performance as measured by computing time and accuracy-related metrics. In this paper, we investigate a method for feature selection based on the well-known L1 and L2 regularization strategies associated with logistic regression (LR). It is well known that the learned coefficients, which serve as weights, can be used to rank the features. Our approach is to synthesize the findings of L1 and L2 regularization. For our experiment, we chose the CIC-IDS2018 dataset owing partly to its size and also to the existence of two problematic classes that are hard to separate. We report first with the exclusion of one of them and then with its inclusion. We ranked features first with L1 and then with L2, and then compared logistic regression with L1 (LR+L1) against that with L2 (LR+L2) by varying the sizes of the feature sets for each of the two rankings. We found no significant difference in accuracy between the two methods once the feature set is selected. We chose a synthesis, i.e., only those features that were present in both the sets obtained from L1 and that from L2, and experimented with it on more complex models like Decision Tree and Random Forest and observed that the accuracy was very close in spite of the small size of the feature set. Additionally, we also report on the standard metrics: accuracy, precision, recall, and f1-score.
Logistic regression is a ubiquitous method for probabilistic classification. However, the effectiveness of logistic regression depends upon careful and relatively computationally expensive tuning, especially for the regularisation hyperparameter, and especially in the context of high-dimensional data. We present a prevalidated ridge regression model that closely matches logistic regression in terms of classification error and log-loss, particularly for high-dimensional data, while being significantly more computationally efficient and having effectively no hyperparameters beyond regularisation. We scale the coefficients of the model so as to minimise log-loss for a set of prevalidated predictions derived from the estimated leave-one-out cross-validation error. This exploits quantities already computed in the course of fitting the ridge regression model in order to find the scaling parameter with nominal additional computational expense.
Reverse engineering deep ReLU networks is a critical problem in understanding the complex behavior and interpretability of neural networks. In this research, we present a novel method for reconstructing deep ReLU networks by leveraging convex optimization techniques and a sampling-based approach. Our method begins by sampling points in the input space and querying the black box model to obtain the corresponding hyperplanes. We then define a convex optimization problem with carefully chosen constraints and conditions to guarantee its convexity. The objective function is designed to minimize the discrepancy between the reconstructed networks output and the target models output, subject to the constraints. We employ gradient descent to optimize the objective function, incorporating L1 or L2 regularization as needed to encourage sparse or smooth solutions. Our research contributes to the growing body of work on reverse engineering deep ReLU networks and paves the way for new advancements in neural network interpretability and security.
Common regularization algorithms for linear regression, such as LASSO and Ridge regression, rely on a regularization hyperparameter that balances the tradeoff between minimizing the fitting error and the norm of the learned model coefficients. As this hyperparameter is scalar, it can be easily selected via random or grid search optimizing a cross-validation criterion. However, using a scalar hyperparameter limits the algorithm's flexibility and potential for better generalization. In this paper, we address the problem of linear regression with l2-regularization, where a different regularization hyperparameter is associated with each input variable. We optimize these hyperparameters using a gradient-based approach, wherein the gradient of a cross-validation criterion with respect to the regularization hyperparameters is computed analytically through matrix differential calculus. Additionally, we introduce two strategies tailored for sparse model learning problems aiming at reducing the risk of overfitting to the validation data. Numerical examples demonstrate that our multi-hyperparameter regularization approach outperforms LASSO, Ridge, and Elastic Net regression. Moreover, the analytical computation of the gradient proves to be more efficient in terms of computational time compared to automatic differentiation, especially when handling a large number of input variables. Application to the identification of over-parameterized Linear Parameter-Varying models is also presented.
Transformer neural networks can exhibit a surprising capacity for in-context learning (ICL) despite not being explicitly trained for it. Prior work has provided a deeper understanding of how ICL emerges in transformers, e.g. through the lens of mechanistic interpretability, Bayesian inference, or by examining the distributional properties of training data. However, in each of these cases, ICL is treated largely as a persistent phenomenon; namely, once ICL emerges, it is assumed to persist asymptotically. Here, we show that the emergence of ICL during transformer training is, in fact, often transient. We train transformers on synthetic data designed so that both ICL and in-weights learning (IWL) strategies can lead to correct predictions. We find that ICL first emerges, then disappears and gives way to IWL, all while the training loss decreases, indicating an asymptotic preference for IWL. The transient nature of ICL is observed in transformers across a range of model sizes and datasets, raising the question of how much to "overtrain" transformers when seeking compact, cheaper-to-run models. We find that L2 regularization may offer a path to more persistent ICL that removes the need for early stopping based on ICL-style validation tasks. Finally, we present initial evidence that ICL transience may be caused by competition between ICL and IWL circuits.
Sparse regression and feature extraction are the cornerstones of knowledge discovery from massive data. Their goal is to discover interpretable and predictive models that provide simple relationships among scientific variables. While the statistical tools for model discovery are well established in the context of linear regression, their generalization to nonlinear regression in material modeling is highly problem-specific and insufficiently understood. Here we explore the potential of neural networks for automatic model discovery and induce sparsity by a hybrid approach that combines two strategies: regularization and physical constraints. We integrate the concept of Lp regularization for subset selection with constitutive neural networks that leverage our domain knowledge in kinematics and thermodynamics. We train our networks with both, synthetic and real data, and perform several thousand discovery runs to infer common guidelines and trends: L2 regularization or ridge regression is unsuitable for model discovery; L1 regularization or lasso promotes sparsity, but induces strong bias; only L0 regularization allows us to transparently fine-tune the trade-off between interpretability and predictability, simplicity and accuracy, and bias and variance. With these insights, we demonstrate that Lp regularized constitutive neural networks can simultaneously discover both, interpretable models and physically meaningful parameters. We anticipate that our findings will generalize to alternative discovery techniques such as sparse and symbolic regression, and to other domains such as biology, chemistry, or medicine. Our ability to automatically discover material models from data could have tremendous applications in generative material design and open new opportunities to manipulate matter, alter properties of existing materials, and discover new materials with user-defined properties.
Model sparsification in deep learning promotes simpler, more interpretable models with fewer parameters. This not only reduces the model's memory footprint and computational needs but also shortens inference time. This work focuses on creating sparse models optimized for multiple tasks with fewer parameters. These parsimonious models also possess the potential to match or outperform dense models in terms of performance. In this work, we introduce channel-wise l1/l2 group sparsity in the shared convolutional layers parameters (or weights) of the multi-task learning model. This approach facilitates the removal of extraneous groups i.e., channels (due to l1 regularization) and also imposes a penalty on the weights, further enhancing the learning efficiency for all tasks (due to l2 regularization). We analyzed the results of group sparsity in both single-task and multi-task settings on two widely-used Multi-Task Learning (MTL) datasets: NYU-v2 and CelebAMask-HQ. On both datasets, which consist of three different computer vision tasks each, multi-task models with approximately 70% sparsity outperform their dense equivalents. We also investigate how changing the degree of sparsification influences the model's performance, the overall sparsity percentage, the patterns of sparsity, and the inference time.
In continual learning, plasticity refers to the ability of an agent to quickly adapt to new information. Neural networks are known to lose plasticity when processing non-stationary data streams. In this paper, we propose L2 Init, a simple approach for maintaining plasticity by incorporating in the loss function L2 regularization toward initial parameters. This is very similar to standard L2 regularization (L2), the only difference being that L2 regularizes toward the origin. L2 Init is simple to implement and requires selecting only a single hyper-parameter. The motivation for this method is the same as that of methods that reset neurons or parameter values. Intuitively, when recent losses are insensitive to particular parameters, these parameters should drift toward their initial values. This prepares parameters to adapt quickly to new tasks. On problems representative of different types of nonstationarity in continual supervised learning, we demonstrate that L2 Init most consistently mitigates plasticity loss compared to previously proposed approaches.
Modern deep-learning systems are specialized to problem settings in which training occurs once and then never again, as opposed to continual-learning settings in which training occurs continually. If deep-learning systems are applied in a continual learning setting, then it is well known that they may fail to remember earlier examples. More fundamental, but less well known, is that they may also lose their ability to learn on new examples, a phenomenon called loss of plasticity. We provide direct demonstrations of loss of plasticity using the MNIST and ImageNet datasets repurposed for continual learning as sequences of tasks. In ImageNet, binary classification performance dropped from 89% accuracy on an early task down to 77%, about the level of a linear network, on the 2000th task. Loss of plasticity occurred with a wide range of deep network architectures, optimizers, activation functions, batch normalization, dropout, but was substantially eased by L2-regularization, particularly when combined with weight perturbation. Further, we introduce a new algorithm -- continual backpropagation -- which slightly modifies conventional backpropagation to reinitialize a small fraction of less-used units after each example and appears to maintain plasticity indefinitely.
We investigate the statistical behavior of gradient descent iterates with dropout in the linear regression model. In particular, non-asymptotic bounds for the convergence of expectations and covariance matrices of the iterates are derived. The results shed more light on the widely cited connection between dropout and l2-regularization in the linear model. We indicate a more subtle relationship, owing to interactions between the gradient descent dynamics and the additional randomness induced by dropout. Further, we study a simplified variant of dropout which does not have a regularizing effect and converges to the least squares estimator
This study investigates how weight decay affects the update behavior of individual neurons in deep neural networks through a combination of applied analysis and experimentation. Weight decay can cause the expected magnitude and angular updates of a neuron's weight vector to converge to a steady state we call rotational equilibrium. These states can be highly homogeneous, effectively balancing the average rotation -- a proxy for the effective learning rate -- across different layers and neurons. Our work analyzes these dynamics across optimizers like Adam, Lion, and SGD with momentum, offering a new simple perspective on training that elucidates the efficacy of widely used but poorly understood methods in deep learning. We demonstrate how balanced rotation plays a key role in the effectiveness of normalization like Weight Standardization, as well as that of AdamW over Adam with L2-regularization. Finally, we show that explicitly controlling the rotation provides the benefits of weight decay while substantially reducing the need for learning rate warmup.
Linear autoencoder models learn an item-to-item weight matrix via convex optimization with L2 regularization and zero-diagonal constraints. Despite their simplicity, they have shown remarkable performance compared to sophisticated non-linear models. This paper aims to theoretically understand the properties of two terms in linear autoencoders. Through the lens of singular value decomposition (SVD) and principal component analysis (PCA), it is revealed that L2 regularization enhances the impact of high-ranked PCs. Meanwhile, zero-diagonal constraints reduce the impact of low-ranked PCs, leading to performance degradation for unpopular items. Inspired by this analysis, we propose simple-yet-effective linear autoencoder models using diagonal inequality constraints, called Relaxed Linear AutoEncoder (RLAE) and Relaxed Denoising Linear AutoEncoder (RDLAE). We prove that they generalize linear autoencoders by adjusting the degree of diagonal constraints. Experimental results demonstrate that our models are comparable or superior to state-of-the-art linear and non-linear models on six benchmark datasets; they significantly improve the accuracy of long-tail items. These results also support our theoretical insights on regularization and diagonal constraints in linear autoencoders.
We have implemented a deep learning model with L2 regularization and trained it on Electromyography (EMG) data. The data comprises of EMG signals collected from control group, myopathy and ALS patients. Our proposed deep neural network consists of eight layers; five fully connected, two batch normalization and one dropout layers. The data is divided into training and testing sections by subsequently dividing the training data into sub-training and validation sections. Having implemented this model, an accuracy of 99 percent is achieved on the test data set. The model was able to distinguishes the normal cases (control group) from the others at a precision of 100 percent and classify the myopathy and ALS with high accuracy of 97.4 and 98.2 percents, respectively. Thus we believe that, this highly improved classification accuracies will be beneficial for their use in the clinical diagnosis of neuromuscular disorders.