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1. From Optimization Dynamics to Generalization Bounds via {\L}ojasiewicz Gradient Inequality

   Fusheng Liu; Haizhao Yang; Soufiane Hayou; Qianxiao Li (October 2022, Transactions on machine learning research)

   Yiming Ying (Ed.)

   Optimization and generalization are two essential aspects of statistical machine learning. In this paper, we propose a framework to connect optimization with generalization by analyz- ing the generalization error based on the optimization trajectory under the gradient flow algorithm. The key ingredient of this framework is the Uniform-LGI, a property that is generally satisfied when training machine learning models. Leveraging the Uniform-LGI, we first derive convergence rates for gradient flow algorithm, then we give generalization bounds for a large class of machine learning models. We further apply our framework to three distinct machine learning models: linear regression, kernel regression, and two-layer neural networks. Through our approach, we obtain generalization estimates that match or extend previous results. 

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2. Deep-n-Cheap: An Automated Efficient and Extensible Search Framework for Cost-Effective Deep Learning

   Dey, Sourya; Babakniya, Sara; Kanala, Saikrishna C; Paolieri, Marco; Golubchik, Leana; Beerel, Peter A; Chugg, Keith M (May 2021, SN computer science)

   null (Ed.)

   Artificial neural networks (NNs) in deep learning systems are critical drivers of emerging technologies such as computer vision, text classification, and natural language processing. Fundamental to their success is the development of accurate and efficient NN models. In this article, we report our work on Deep-n-Cheap—an open-source automated machine learning (AutoML) search framework for deep learning models. The search includes both architecture and training hyperparameters and supports convolutional neural networks and multi-layer perceptrons, applicable to multiple domains. Our framework is targeted for deployment on both benchmark and custom datasets, and as a result, offers a greater degree of search space customizability as compared to a more limited search over only pre-existing models from literature. We also introduce the technique of ‘search transfer’, which demonstrates the generalization capabilities of the models found by our framework to multiple datasets. Deep-n-Cheap includes a user-customizable complexity penalty which trades off performance with training time or number of parameters. Specifically, our framework can find models with performance comparable to state-of-the- art while taking 1–2 orders of magnitude less time to train than models from other AutoML and model search frameworks. Additionally, we investigate and develop insight into the search process that should aid future development of deep learning models. 

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3. Understanding Dynamics of Nonlinear Representation Learning and Its Application

   https://doi.org/10.1162/neco_a_01483  

   Kawaguchi, Kenji; Zhang, Linjun; Deng, Zhun (March 2022, Neural Computation)

   Abstract Representations of the world environment play a crucial role in artificial intelligence. It is often inefficient to conduct reasoning and inference directly in the space of raw sensory representations, such as pixel values of images. Representation learning allows us to automatically discover suitable representations from raw sensory data. For example, given raw sensory data, a deep neural network learns nonlinear representations at its hidden layers, which are subsequently used for classification (or regression) at its output layer. This happens implicitly during training through minimizing a supervised or unsupervised loss. In this letter, we study the dynamics of such implicit nonlinear representation learning. We identify a pair of a new assumption and a novel condition, called the on-model structure assumption and the data architecture alignment condition. Under the on-model structure assumption, the data architecture alignment condition is shown to be sufficient for the global convergence and necessary for global optimality. Moreover, our theory explains how and when increasing network size does and does not improve the training behaviors in the practical regime. Our results provide practical guidance for designing a model structure; for example, the on-model structure assumption can be used as a justification for using a particular model structure instead of others. As an application, we then derive a new training framework, which satisfies the data architecture alignment condition without assuming it by automatically modifying any given training algorithm dependent on data and architecture. Given a standard training algorithm, the framework running its modified version is empirically shown to maintain competitive (practical) test performances while providing global convergence guarantees for deep residual neural networks with convolutions, skip connections, and batch normalization with standard benchmark data sets, including MNIST, CIFAR-10, CIFAR-100, Semeion, KMNIST, and SVHN. 

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4. What Makes Freezing Layers in Deep Neural Networks Effective? A Linear Separability Perspective

   Coil, Collin; Cheney, Nick (June 2025, AutoML Conference 2025)

   Freezing layers in deep neural networks has been shown to enhance generalization and accelerate training, yet the underlying mechanisms remain unclear. This paper investigates the impact of frozen layers from the perspective of linear separability, examining how untrained, randomly initialized layers influence feature representations and model performance. Using multilayer perceptrons trained on MNIST, CIFAR-10, and CIFAR-100, we systematically analyze the effects freezing layers and network architecture. While prior work attributes the benefits of frozen layers to Cover’s theorem, which suggests that nonlinear transformations improve linear separability, we find that this explanation is insufficient. Instead, our results indicate that the observed improvements in generalization and convergence stem from other mechanisms. We hypothesize that freezing may have similar effects to other regularization techniques and that it may smooth the loss landscape to facilitate training. Furthermore, we identify key architectural factors---such as network overparameterization and use of skip connections---that modulate the effectiveness of frozen layers. These findings offer new insights into the conditions under which freezing layers can optimize deep learning performance, informing future work on neural architecture search. 

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5. Measuring Generalization with Optimal Transport

   Chuang, C; Mroueh, Y; Greenewald, K.; Torralba, A.; Jegelka, S. (December 2021, Advances in neural information processing systems)

   Understanding the generalization of deep neural networks is one of the most important tasks in deep learning. Although much progress has been made, theoretical error bounds still often behave disparately from empirical observations. In this work, we develop margin-based generalization bounds, where the margins are normalized with optimal transport costs between independent random subsets sampled from the training distribution. In particular, the optimal transport cost can be interpreted as a generalization of variance which captures the structural properties of the learned feature space. Our bounds robustly predict the generalization error, given training data and network parameters, on large scale datasets. Theoretically, we demonstrate that the concentration and separation of features play crucial roles in generalization, supporting empirical results in the literature. The code is available at https://github.com/chingyaoc/kV-Margin. 

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