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1. Understanding Self-Supervised Learning Dynamics without Contrastive Pairs

   Tian, Y.; Chen, X.; Ganguli, S. (April 2021, International conference on machine learning)

   null (Ed.)

   While contrastive approaches of self-supervised learning (SSL) learn representations by minimizing the distance between two augmented views of the same data point (positive pairs) and maximizing views from different data points (negative pairs), recent \emph{non-contrastive} SSL (e.g., BYOL and SimSiam) show remarkable performance {\it without} negative pairs, with an extra learnable predictor and a stop-gradient operation. A fundamental question rises: why they do not collapse into trivial representation? In this paper, we answer this question via a simple theoretical study and propose a novel approach, \ourmethod{}, that \emph{directly} sets the linear predictor based on the statistics of its inputs, rather than trained with gradient update. On ImageNet, it performs comparably with more complex two-layer non-linear predictors that employ BatchNorm and outperforms linear predictor by 2.5 in 300-epoch training (and 5 in 60-epoch). \ourmethod{} is motivated by our theoretical study of the nonlinear learning dynamics of non-contrastive SSL in simple linear networks. Our study yields conceptual insights into how non-contrastive SSL methods learn, how they avoid representational collapse, and how multiple factors, like predictor networks, stop-gradients, exponential moving averages, and weight decay all come into play. Our simple theory recapitulates the results of real-world ablation studies in both STL-10 and ImageNet. Code is released. 

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2. The capacity of feedforward neural networks.

   Baldi, P.; Vershynin, R. (January 2019, Neural networks)

   A long standing open problem in the theory of neural networks is the development of quantitative methods to estimate and compare the capabilities of different architectures. Here we define the capacity of an architecture by the binary logarithm of the number of functions it can compute, as the synaptic weights are varied. The capacity provides an upper bound on the number of bits that can be extracted from the training data and stored in the architecture during learning. We study the capacity of layered, fully-connected, architectures of linear threshold neurons with L layers and show that in essence the capacity is given by a cubic polynomial in the layer sizes. In proving the main result, we also develop new techniques (multiplexing, enrichment, and stacking) as well as new bounds on the capacity of finite sets. We use the main result to identify architectures with maximal or minimal capacity under a number of natural constraints. This leads to the notion of structural regularization for deep architectures. While in general, everything else being equal, shallow networks compute more functions than deep networks, the functions computed by deep networks are more regular and “interesting". 

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3. Learning Fixed Points of Recurrent Neural Networks by Reparameterizing the Network Model

   https://doi.org/10.1162/neco_a_01681  

   Zhu, Vicky; Rosenbaum, Robert (July 2024, Neural Computation)

   In computational neuroscience, recurrent neural networks are widely used to model neural activity and learning. In many studies, fixed points of recurrent neural networks are used to model neural responses to static or slowly changing stimuli, such as visual cortical responses to static visual stimuli. These applications raise the question of how to train the weights in a recurrent neural network to minimize a loss function evaluated on fixed points. In parallel, training fixed points is a central topic in the study of deep equilibrium models in machine learning. A natural approach is to use gradient descent on the Euclidean space of weights. We show that this approach can lead to poor learning performance due in part to singularities that arise in the loss surface. We use a reparameterization of the recurrent network model to derive two alternative learning rules that produce more robust learning dynamics. We demonstrate that these learning rules avoid singularities and learn more effectively than standard gradient descent. The new learning rules can be interpreted as steepest descent and gradient descent, respectively, under a non-Euclidean metric on the space of recurrent weights. Our results question the common, implicit assumption that learning in the brain should be expected to follow the negative Euclidean gradient of synaptic weights. 

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4. Compositional sparsity of learnable functions

   https://doi.org/10.1090/bull/1820  

   Poggio, Tomaso; Fraser, Maia (July 2024, Bulletin of the American Mathematical Society)

   Neural networks have demonstrated impressive success in various domains, raising the question of what fundamental principles underlie the effectiveness of the best AI systems and quite possibly of human intelligence. This perspective argues that compositional sparsity, or the property that a compositional function have “few” constituent functions, each depending on only a small subset of inputs, is a key principle underlying successful learning architectures. Surprisingly, all functions that are efficiently Turing computable have a compositional sparse representation. Furthermore, deep networks that are also sparse can exploit this general property to avoid the “curse of dimensionality”. This framework suggests interesting implications about the role that machine learning may play in mathematics. 

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5. 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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