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1. Kernel and Rich Regimes in Overparametrized Models

   Woodworth, Blake; Gunasekar, Suriya; Lee, Jason D; Moroshko, Edward; Savarese, Pedro; Golan, Itay; Soudry, Daniel; Srebro, Nathan (February 2020, Conference on Learning Theory (COLT))

   A recent line of work studies overparametrized neural networks in the “kernel regime,” i.e. when during training the network behaves as a kernelized linear predictor, and thus, training with gradient descent has the effect of finding the corresponding minimum RKHS norm solution. This stands in contrast to other studies which demonstrate how gradient descent on overparametrized networks can induce rich implicit biases that are not RKHS norms. Building on an observation by \citet{chizat2018note}, we show how the \textbf{\textit{scale of the initialization}} controls the transition between the “kernel” (aka lazy) and “rich” (aka active) regimes and affects generalization properties in multilayer homogeneous models. We provide a complete and detailed analysis for a family of simple depth-D linear networks that exhibit an interesting and meaningful transition between the kernel and rich regimes, and highlight an interesting role for the \emph{width} of the models. We further demonstrate this transition empirically for matrix factorization and multilayer non-linear networks. 

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2. 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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3. 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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4. Pruning neural networks without any data by iteratively conserving synaptic flow

   Tanaka, H.; Kunin, D.; Yamins, Y.; Ganguli, S (December 2020, Advances in neural information processing systems)

   null (Ed.)

   Pruning the parameters of deep neural networks has generated intense interest due to potential savings in time, memory and energy both during training and at test time. Recent works have identified, through an expensive sequence of training and pruning cycles, the existence of winning lottery tickets or sparse trainable subnetworks at initialization. This raises a foundational question: can we identify highly sparse trainable subnetworks at initialization, without ever training, or indeed without ever looking at the data? We provide an affirmative answer to this question through theory driven algorithm design. We first mathematically formulate and experimentally verify a conservation law that explains why existing gradient-based pruning algorithms at initialization suffer from layer-collapse, the premature pruning of an entire layer rendering a network untrainable. This theory also elucidates how layer-collapse can be entirely avoided, motivating a novel pruning algorithm Iterative Synaptic Flow Pruning (SynFlow). This algorithm can be interpreted as preserving the total flow of synaptic strengths through the network at initialization subject to a sparsity constraint. Notably, this algorithm makes no reference to the training data and consistently competes with or outperforms existing state-of-the-art pruning algorithms at initialization over a range of models (VGG and ResNet), datasets (CIFAR-10/100 and Tiny ImageNet), and sparsity constraints (up to 99.99 percent). Thus our data-agnostic pruning algorithm challenges the existing paradigm that, at initialization, data must be used to quantify which synapses are important. 

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5. Bridging the Gap: Unifying the Training and Evaluation of Neural Network Binary Classifiers

   Tsoi, Nathan; Candon, Kate; Li, Deyuan; Milkessa, Yofti; Vázquez, Marynel (October 2022, Advances in neural information processing systems)

   Koyejo, S.; Mohamed, S.; Agarwal, A.; Belgrave, D.; Cho, K.; Oh, A. (Ed.)

   While neural network binary classifiers are often evaluated on metrics such as Accuracy and F1-Score, they are commonly trained with a cross-entropy objective. How can this training-evaluation gap be addressed? While specific techniques have been adopted to optimize certain confusion matrix based metrics, it is challenging or impossible in some cases to generalize the techniques to other metrics. Adversarial learning approaches have also been proposed to optimize networks via confusion matrix based metrics, but they tend to be much slower than common training methods. In this work, we propose a unifying approach to training neural network binary classifiers that combines a differentiable approximation of the Heaviside function with a probabilistic view of the typical confusion matrix values using soft sets. Our theoretical analysis shows the benefit of using our method to optimize for a given evaluation metric, such as F1-Score, with soft sets, and our extensive experiments show the effectiveness of our approach in several domains. 

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