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1. Dropout: Explicit Forms and Capacity Control

   Arora, Raman; Bartlett, Peter; Mianjy, Poorya; Srebro, Nathan (July 2021, Proceedings of Machine Learning Research)

   null (Ed.)

   We investigate the capacity control provided by dropout in various machine learning problems. First, we study dropout for matrix completion, where it induces a distribution-dependent regularizer that equals the weighted trace-norm of the product of the factors. In deep learning, we show that the distribution-dependent regularizer due to dropout directly controls the Rademacher complexity of the underlying class of deep neural networks. These developments enable us to give concrete generalization error bounds for the dropout algorithm in both matrix completion as well as training deep neural networks. 

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2. Simple, fast, and flexible framework for matrix completion with infinite width neural networks

   https://doi.org/10.1073/pnas.2115064119  

   Radhakrishnan, Adityanarayanan; Stefanakis, George; Belkin, Mikhail; Uhler, Caroline (April 2022, Proceedings of the National Academy of Sciences)

   Matrix completion problems arise in many applications including recommendation systems, computer vision, and genomics. Increasingly larger neural networks have been successful in many of these applications but at considerable computational costs. Remarkably, taking the width of a neural network to infinity allows for improved computational performance. In this work, we develop an infinite width neural network framework for matrix completion that is simple, fast, and flexible. Simplicity and speed come from the connection between the infinite width limit of neural networks and kernels known as neural tangent kernels (NTK). In particular, we derive the NTK for fully connected and convolutional neural networks for matrix completion. The flexibility stems from a feature prior, which allows encoding relationships between coordinates of the target matrix, akin to semisupervised learning. The effectiveness of our framework is demonstrated through competitive results for virtual drug screening and image inpainting/reconstruction. We also provide an implementation in Python to make our framework accessible on standard hardware to a broad audience. 

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3. Deep Augmentation: Dropout as Augmentation for Self-Supervised Learning

   Gabrielsson, Rickard; Wang, Tongzhou; Baradad, Manel; Solomon, Justin (May 2025, Transactions on machine learning research)

   Despite dropout’s ubiquity in machine learning, its effectiveness as a form of data augmentation remains under-explored. We address two key questions: (i) When is dropout effective as an augmentation strategy? (ii) Is dropout uniquely effective under these conditions? To explore these questions, we propose Deep Augmentation, a network- and modality-agnostic method that applies dropout or PCA transformations to targeted layers in neural networks. Through extensive experiments on contrastive learning tasks in NLP, computer vision, and graph learning, we find that uniformly applying dropout across layers does not consistently improve performance. Instead, dropout proves most beneficial in deeper layers and can be matched by alternative augmentations (e.g., PCA). We also show that a stop-gradient operation is critical for ensuring dropout functions effectively as an augmentation, and that performance trends invert when moving from contrastive tasks to supervised tasks. Our analysis suggests that Deep Augmentation helps mitigate inter-layer co-adaptation---a notable issue in self-supervised learning due to the absence of labeled data. Drawing on these insights, we outline a procedure for selecting the optimal augmentation layer and demonstrate that Deep Augmentation can outperform traditional input-level augmentations. This simple yet powerful approach can be seamlessly integrated into a wide range of architectures and modalities, yielding notable gains in both performance and generalization. 

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4. Stochastic synapses as resource for efficient deep learning machines

   https://doi.org/10.1109/IEDM.2017.8268368  

   Neftci, Emre (December 2017, 2017 IEEE International Electron Devices Meeting (IEDM))

   Synaptic unreliability was shown to be a robust and sufficient mechanism for inducing the stochasticity in biological and artificial neural network models. Previous work demonstrated multiplicative noise (also called dropconnect) as a powerful regularizer during training. Here, we show that always-on stochasticity at networks connections is a sufficient resource for deep learning machines when combined with simple threshold non-linearities. Furthermore, the resulting activity function exhibits a self-normalizing property that reflects a recently proposed “Weight Normalization” technique, itself fulfilling many of the features of batch normalization in an online fashion. Normalization of activities during training can speed up convergence by preventing so-called internal covariate shift caused by changes in the distribution of inputs as the parameters of the previous layers are trained. Collectively, our findings can improve performance of deep learning machines with fixed point representations and argue in favor of stochastic nanodevices as primitives for efficient deep learning machines with online and embedded learning capabilities. 

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5. Scalable and Sustainable Deep Learning via Randomized Hashing

   https://doi.org/10.1145/3097983.3098035  

   Spring, Ryan; Shrivastava, Anshumali (January 2017, Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining)

   Current deep learning architectures are growing larger in order to learn from complex datasets. These architectures require giant matrix multiplication operations to train millions of parameters. Conversely, there is another growing trend to bring deep learning to low-power, embedded devices. The matrix operations, associated with the training and testing of deep networks, are very expensive from a computational and energy standpoint. We present a novel hashing-based technique to drastically reduce the amount of computation needed to train and test neural networks. Our approach combines two recent ideas, Adaptive Dropout and Randomized Hashing for Maximum Inner Product Search (MIPS), to select the nodes with the highest activations efficiently. Our new algorithm for deep learning reduces the overall computational cost of the forward and backward propagation steps by operating on significantly fewer nodes. As a consequence, our algorithm uses only 5% of the total multiplications, while keeping within 1% of the accuracy of the original model on average. A unique property of the proposed hashing-based back-propagation is that the updates are always sparse. Due to the sparse gradient updates, our algorithm is ideally suited for asynchronous, parallel training, leading to near-linear speedup, as the number of cores increases. We demonstrate the scalability and sustainability (energy efficiency) of our proposed algorithm via rigorous experimental evaluations on several datasets. 

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