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1. The Price of Implicit Bias in Adversarially Robust Generalization

   Tsilivis, Nikolaos; Frank, Natalie; Srebro, Nathan; Kempe, Julia (April 2025, The International Conference on Learning Representations (ICLR 2025))

   We study the implicit bias of optimization in robust empirical risk minimization (robust ERM) and its connection with robust generalization. In classification settings under adversarial perturbations with linear models, we study what type of regularization should ideally be applied for a given perturbation set to improve (robust) generalization. We then show that the implicit bias of optimization in robust ERM can significantly affect the robustness of the model and identify two ways this can happen; either through the optimization algorithm or the architecture. We verify our predictions in simulations with synthetic data and experimentally study the importance of implicit bias in robust ERM with deep neural networks. 

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2. On Leave-One-Out Conditional Mutual Information For Generalization

   Rammal, R; Achille, A; Golatkar, A; Diggavi, S N; Soatto, S (January 2022, Advances in Neural Information Processing Systems (NeurIPS))

   We derive information theoretic generalization bounds for supervised learning algorithms based on a new measure of leave-one-out conditional mutual information (loo-CMI). Contrary to other CMI bounds, which are black-box bounds that do not exploit the structure of the problem and may be hard to evaluate in practice, our loo-CMI bounds can be computed easily and can be interpreted in connection to other notions such as classical leave-one-out cross-validation, stability of the optimization algorithm, and the geometry of the loss-landscape. It applies both to the output of training algorithms as well as their predictions. We empirically validate the quality of the bound by evaluating its predicted generalization gap in scenarios for deep learning. In particular, our bounds are non-vacuous on large-scale image-classification tasks. 

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3. PAC-Bayes Compression Bounds So Tight That They Can Explain Generalization

   Lotfi, Sanae; Finzi, Marc Anton; Kapoor, Sanyam; Potapczynski, Andres; Goldblum, Micah; Wilson, Andrew Gordon (October 2022, NeurIPS)

   While there has been progress in developing non-vacuous generalization bounds for deep neural networks, these bounds tend to be uninformative about why deep learning works. In this paper, we develop a compression approach based on quantizing neural network parameters in a linear subspace, profoundly improving on previous results to provide state-of-the-art generalization bounds on a variety of tasks, including transfer learning. We use these tight bounds to better understand the role of model size, equivariance, and the implicit biases of optimization, for generalization in deep learning. Notably, we find large models can be compressed to a much greater extent than previously known, encapsulating Occam’s razor. 

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4. On the generalization of learning algorithms that do not converge.

   N. Chandramoorthy, A. Loukas (November 2022, Neural Information Processing Systems (NeurIPS))

   Generalization analyses of deep learning typically assume that the training converges to a fixed point. But, recent results indicate that in practice, the weights of deep neural networks optimized with stochastic gradient descent often oscillate indefinitely. To reduce this discrepancy between theory and practice, this paper focuses on the generalization of neural networks whose training dynamics do not necessarily converge to fixed points. Our main contribution is to propose a notion of statistical algorithmic stability (SAS) that extends classical algorithmic stability to non-convergent algorithms and to study its connection to generalization. This ergodic-theoretic approach leads to new insights when compared to the traditional optimization and learning theory perspectives. We prove that the stability of the time-asymptotic behavior of a learning algorithm relates to its generalization and empirically demonstrate how loss dynamics can provide clues to generalization performance. Our findings provide evidence that networks that “train stably generalize better” even when the training continues indefinitely and the weights do not converge. 

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5. Deep Linear Networks for Matrix Completion—an Infinite Depth Limit

   https://doi.org/10.1137/22M1530653  

   Cohen, Nadav; Menon, Govind; Veraszto, Zsolt (December 2023, SIAM Journal on Applied Dynamical Systems)

   The deep linear network (DLN) is a model for implicit regularization in gradient based optimization of overparametrized learning architectures. Training the DLN corresponds to a Riemannian gradient flow, where the Riemannian metric is defined by the architecture of the network and the loss function is defined by the learning task. We extend this geometric framework, obtaining explicit expressions for the volume form, including the case when the network has infinite depth. We investigate the link between the Riemannian geometry and the training asymptotics for matrix completion with rigorous analysis and numerics. We propose that under small initialization, implicit regularization is a result of bias towards high state space volume. 

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