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1. Implicit Regularization Leads to Benign Overfitting for Sparse Linear Regression

   Zhou, Mo; Ge, Rong (January 2023, International Conference on Machine Learning)

   In deep learning, often the training process nds an interpolator (a solution with 0 training loss), but the test loss is still low. This phenomenon, known as benign overfitting, is a major mystery that received a lot of recent attention. One common mechanism for benign overfitting is implicit regularization, where the training process leads to additional properties for the interpolator, often characterized by minimizing certain norms. However, even for a simple sparse linear regression problem y = Ax+ noise with sparse x , neither minimum l_1 orl_`2 norm interpolator gives the optimal test loss. In this work, we give a different parametrization of the model which leads to a new implicit regularization effect that combines the benefit of l_1 and l_2 interpolators. We show that training our new model via gradient descent leads to an interpolator with near-optimal test loss. Our result is based on careful analysis of the training dynamics and provides another example of implicit regularization effect that goes beyond norm minimization. 

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2. Usable Information and Evolution of Optimal Representations During Training

   Kleinman, Michael; Achille, Alessandro; Idnani, Daksh; Kao, Jonathan C (January 2021, International Conference on Learning Representations (ICLR))

   null (Ed.)

   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. We show that the implicit regularization coming from training with Stochastic Gradient Descent with a high learning-rate and small batch size plays an important role in learning minimal sufficient representations for the task. In the process of arriving at a minimal sufficient representation, we find that the content of the representation changes dynamically during training. In particular, we find that semantically meaningful but ultimately irrelevant information is encoded in the early transient dynamics of training, before being later discarded. In addition, we evaluate how perturbing the initial part of training impacts the learning dynamics and the resulting representations. We show these effects on both perceptual decision-making tasks inspired by neuroscience literature, as well as on standard image classification tasks. 

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3. Implicit regularization for deep neural networks driven by an Orstein-Uhlenbeck like process

   Blanc, Guy; Gupta, Neha; Valiant, Gregory; Valiant, Paul (January 2020, 33rd Annual Conference on Learning Theory (COLT))

   We consider networks, trained via stochastic gradient descent to minimize L2 loss, with the training labels perturbed by independent noise at each iteration. We characterize the behavior of the training dynamics near any parameter vector that achieves zero training error, in terms of an implicit regularization term corresponding to the sum over the datapoints, of the squared L2 of the gradient of the model with respect to the parameter vector, evaluated at each data point. This holds for networks of any connectivity, width,depth, and choice of activation function. We interpret this implicit regularization term for three simple settings: matrix sensing, two layer ReLU networks trained on one-dimensional data, and two layer networks with sigmoid activations trained on a single datapoint. For these settings, we show why this new and general implicit regularization effect drives the networks towards "simple" models. 

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4. Implicit Bias of Gradient Descent for Mean Squared Error Regression with Two-Layer Wide Neural Networks

   Jin, Hui; Montufar, Guido (April 2023, Journal of machine learning research)

   We investigate gradient descent training of wide neural networks and the corresponding implicit bias in function space. For univariate regression, we show that the solution of training a width-n shallow ReLU network is within n−1/2 of the function which fits the training data and whose difference from the initial function has the smallest 2-norm of the second derivative weighted by a curvature penalty that depends on the probability distribution that is used to initialize the network parameters. We compute the curvature penalty function explicitly for various common initialization procedures. For instance, asymmetric initialization with a uniform distribution yields a constant curvature penalty, and thence the solution function is the natural cubic spline interpolation of the training data. For stochastic gradient descent we obtain the same implicit bias result. We obtain a similar result for different activation functions. For multivariate regression we show an analogous result, whereby the second derivative is replaced by the Radon transform of a fractional Laplacian. For initialization schemes that yield a constant penalty function, the solutions are polyharmonic splines. Moreover, we show that the training trajectories are captured by trajectories of smoothing splines with decreasing regularization strength. 

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5. Implicit Bias of Gradient Descent for Mean Squared Error Regression with Two-Layer Wide Neural Networks

   Jin, Hui; Montufar, Guido (April 2023, Journal of Machine Learning Research)

   We investigate gradient descent training of wide neural networks and the corresponding implicit bias in function space. For univariate regression, we show that the solution of training a width-n shallow ReLU network is within n1/2 of the function which fits the training data and whose difference from the initial function has the smallest 2-norm of the second derivative weighted by a curvature penalty that depends on the probability distribution that is used to initialize the network parameters. We compute the curvature penalty function explicitly for various common initialization procedures. For instance, asymmetric initialization with a uniform distribution yields a constant curvature penalty, and thence the solution function is the natural cubic spline interpolation of the training data. For stochastic gradient descent we obtain the same implicit bias result. We obtain a similar result for different activation functions. For multivariate regression we show an analogous result, whereby the second derivative is replaced by the Radon transform of a fractional Laplacian. For initialization schemes that yield a constant penalty function, the solutions are polyharmonic splines. Moreover, we show that the training trajectories are captured by trajectories of smoothing splines with decreasing regularization strength. 

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