Recently there has been significant theoretical progress on understanding the convergence and generalization of gradient-based methods on nonconvex losses with overparameterized models. Nevertheless, many aspects of optimization and generalization and in particular the critical role of small random initialization are not fully understood. In this paper, we take a step towards demystifying this role by proving that small random initialization followed by a few iterations of gradient descent behaves akin to popular spectral methods. We also show that this implicit spectral bias from small random initialization, which is provably more prominent for overparameterized models, also puts the gradient descent iterations on a particular trajectory towards solutions that are not only globally optimal but also generalize well. Concretely, we focus on the problem of reconstructing a low-rank matrix from a few measurements via a natural nonconvex formulation. In this setting, we show that the trajectory of the gradient descent iterations from small random initialization can be approximately decomposed into three phases: (I) a spectral or alignment phase where we show that that the iterates have an implicit spectral bias akin to spectral initialization allowing us to show that at the end of this phase the column space of the iterates and the underlying low-rank matrix are sufficiently aligned, (II) a saddle avoidance/refinement phase where we show that the trajectory of the gradient iterates moves away from certain degenerate saddle points, and (III) a local refinement phase where we show that after avoiding the saddles the iterates converge quickly to the underlying low-rank matrix. Underlying our analysis are insights for the analysis of overparameterized nonconvex optimization schemes that may have implications for computational problems beyond low-rank reconstruction.

A surprising phenomenon in modern machine learning is the ability of a highly overparameterized model to generalize well (small error on the test data) even when it is trained to memorize the training data (zero error on the training data). This has led to an arms race towards increasingly overparameterized models (c.f., deep learning). In this paper, we study an underexplored hidden cost of overparameterization: the fact that overparameterized models may be more vulnerable to privacy attacks, in particular the membership inference attack that predicts the (potentially sensitive) examples used to train a model. We significantly extend the relatively few empirical results on this problem by theoretically proving for an overparameterized linear regression model in the Gaussian data setting that membership inference vulnerability increases with the number of parameters. Moreover, a range of empirical studies indicates that more complex, nonlinear models exhibit the same behavior. Finally, we extend our analysis towards ridge-regularized linear regression and show in the Gaussian data setting that increased regularization also increases membership inference vulnerability in the overparameterized regime.

Decentralized optimization are playing an important role in applications such as training large machine learning models, among others. Despite its superior practical performance, there has been some lack of fundamental understanding about its theoretical properties. In this work, we address the following open research question: To train an overparameterized model over a set of distributed nodes, what is the {\it minimum} communication overhead (in terms of the bits got exchanged) that the system needs to sustain, while still achieving (near) zero training loss? We show that for a class of overparameterized models where the number of parameters $D$ is much larger than the total data samples $N$, the best possible communication complexity is ${\Omega}(N)$, which is independent of the problem dimension $D$. Further, for a few specific overparameterized models (i.e., the linear regression, and certain multi-layer neural network with one wide layer), we develop a set of algorithms which uses certain linear compression followed by adaptive quantization, and show that they achieve dimension independent, and sometimes near optimal, communication complexity. To our knowledge, this is the first time that dimension independent communication complexity has been shown for distributed optimization.

Recently, the benefit of heavily overparameterized models has been observed in machine learning tasks: models with enough capacity to easily cross the \emph{interpolation threshold} improve in generalization error compared to the classical bias-variance tradeoff regime. We demonstrate this behavior for the first time in particle physics data and explore when and where `double descent' appears and under which circumstances overparameterization results in a performance gain.

Machine unlearning is the task of updating a trained model to forget specific training data without retraining from scratch. In this paper, we investigate how unlearning of deep neural networks (DNNs) is affected by the model parameterization level, which corresponds here to the DNN width. We define validation-based tuning for several unlearning methods from the recent literature, and show how these methods perform differently depending on (i) the DNN parameterization level, (ii) the unlearning goal (unlearned data privacy or bias removal), (iii) whether the unlearning method explicitly uses the unlearned examples. Our results show that unlearning excels on overparameterized models, in terms of balancing between generalization and achieving the unlearning goal; although for bias removal this requires the unlearning method to use the unlearned examples. We further elucidate our error-based analysis by measuring how much the unlearning changes the classification decision regions in the proximity of the unlearned examples, and avoids changing them elsewhere. By this we show that the unlearning success for overparameterized models stems from the ability to delicately change the model functionality in small regions in the input space while keeping much of the model functionality unchanged.

Recently there has been significant theoretical progress on understanding the convergence and generalization of gradient-based methods on nonconvex losses with overparameterized models. Nevertheless, many aspects of optimization and generalization and in particular the critical role of small random initialization are not fully understood. In this paper, we take a step towards demystifying this role by proving that small random initialization followed by a few iterations of gradient descent behaves akin to popular spectral methods. We also show that this implicit spectral bias from small random initialization, which is provably more prominent for overparameterized models, also puts the gradient descent iterations on a particular trajectory towards solutions that are not only globally optimal but also generalize well. Concretely, we focus on the problem of reconstructing a low-rank matrix from a few measurements via a natural nonconvex formulation.