Probabilistic deep learning uses neural networks and models to handle uncertainty.
problem Handling uncertainty in deep learning models.
method Two approaches: probabilistic neural networks and deep probabilistic models.
result TensorFlow Probability library supports both approaches.
Two new criteria help understand the advantage of deep neural networks.
problem Understanding the advantage of deepening neural networks.
method Proposed two new criteria to evaluate the expressivity of functions computable by deep neural networks.
result Increasing layers is more effective than increasing units in improving the expressivity of deep neural networks.
Novel framework explains generalization in deep neural networks.
problem Understanding and improving generalization in deep neural networks.
method Topological Quantum Neural Networks as the semi-classical limit of Deep Neural Networks.
result Demonstrates that the perceptron, viewed as the semi-classical limit, achieves similar results to standard neural networks without training.
Study of infinitely deep but narrow neural networks using NTK theory.
problem Analyzing the role of depth in deep learning with overparameterized networks.
method Infinite-depth limit analysis of MLP and CNN using Neural Tangent Kernel (NTK) theory.
result Established trainability guarantee for infinitely deep but narrow neural networks.
Study deep maxout networks and their equivalence to Gaussian processes.
problem Understanding neural networks with infinite width.
method Derive equivalence between deep maxout networks and Gaussian processes, characterize maxout kernel, and provide efficient numerical implementation.
result Bayesian inference based on deep maxout network kernel leads to competitive results compared to finite-width counterparts and deep neural network kernels.
Deep neural networks with adversarial training achieve sup-norm convergence for nonparametric regression.
problem Achieving sup-norm convergence for deep neural network estimators in nonparametric regression.
method Developed an adversarial training scheme to address the sup-norm convergence issue.
result Deep neural network estimators achieve optimal sup-norm convergence with the proposed adversarial training.
Deep networks are shown to be equivalent to a new type of kernel chain.
problem Identifying an appropriate function space for deep neural networks.
method Extending Reproducing Kernel Banach Spaces (RKBS) to chain RKBS (cRKBS), which composes kernels rather than functions.
result Any deep neural network function is a neural cRKBS function, and conversely, any neural cRKBS function corresponds to a deep neural network.
Bayesian Neural Networks help quantify uncertainty in deep learning predictions.
problem Uncertainty quantification in deep learning predictions.
method Bayesian statistics applied to neural networks.
result Design, implementation, training, and evaluation of Bayesian Neural Networks.
Generalization bounds derived for neural ODEs and deep residual networks.
problem Understanding the generalization capability of neural ODEs and deep residual networks.
method Lipschitz-based argument and analogy with deep residual networks.
result A generalization bound involving the magnitude of weight matrix differences.
Proposes deep graph persistence to address neural persistence issues in deep learning.
problem Variance of weights and lack of spatial structure in deep neural networks impact neural persistence.
method Extends neural persistence to the whole network, considering interactions between layers.
result Deep graph persistence alleviates variance-related issues and captures persistent paths through the network.
With the growth of deep learning, how to describe deep neural networks unifiedly is becoming an important issue. We first formalize neural networks mathematically with their directed graph representations, and prove a generation theorem about the induced networks of connected directed acyclic graphs. Then, we set up a …
The evolution of a deep neural network trained by the gradient descent can be described by its neural tangent kernel (NTK) as introduced in [20], where it was proven that in the infinite width limit the NTK converges to an explicit limiting kernel and it stays constant during training. The NTK was also implicit in some…
Analysis of over-parameterized neural networks has drawn significant attention in recentyears. It was shown that such systems behave like convex systems under various restrictedsettings, such as for two-level neural networks, and when learning is only restricted locally inthe so-called neural tangent kernel space aroun…
Deep neural networks' infinite-width behavior approximated by Gaussian models.
problem Understanding the behavior of deep neural networks in the limit of infinite width.
method Using the Lindeberg exchange principle to approximate weights by Gaussian random variables.
result Quantitative bounds on the 2-Wasserstein distance between deep neural networks and Gaussian limits.
Deep neural networks with piecewise-polynomial activations can approximate smooth functions and their derivatives.
problem Approximating smooth functions and their derivatives with neural networks.
method Derives the depth, width, and sparsity required for approximation in Hölder norms.
result Deep neural networks with bounded weights can approximate Hölder smooth functions and their derivatives.
Understanding properties of deep neural networks is an important challenge in deep learning. In this paper, we take a step in this direction by proposing a rigorous way of verifying properties of a popular class of neural networks, Binarized Neural Networks, using the well-developed means of Boolean satisfiability. Our…
Embeds sparsity in deep neural networks, allowing exact zero parameters.
problem Learning sparse structures in deep networks.
method Embeds sparsity into neural network structure, allowing exact zero parameters during training.
result Can learn both structured and unstructured sparsity.
New framework explains deep neural networks using variational spline theory.
problem Understanding functions learned by deep neural networks.
method Developed a variational framework and function space.
result Deep ReLU networks are solutions to regularized data fitting problems over the proposed function space.
Neural networks improve nonparametric regression with measurement errors.
problem Nonparametric regression with measurement errors.
method Proposes a neural network design using FNN, normalizing flow, and inference network.
result Neural network approach is more flexible and superior or comparable to classical methods.
In recent years, deep learning methods applying unsupervised learning to train deep layers of neural networks have achieved remarkable results in numerous fields. In the past, many genetic algorithms based methods have been successfully applied to training neural networks. In this paper, we extend previous work and pro…
The paper explores neural scaling laws for deep operator networks, offering a theoretical foundation.
problem Understanding neural scaling laws in deep operator networks.
method Theoretical analysis of approximation and generalization errors.
result Established a theoretical framework to quantify neural scaling laws for deep operator networks.
New method uses sparse deep neural networks for high-dimensional regression with improved parameter estimation.
problem Improving parameter estimation in high-dimensional sparse regression models.
method Proposes nonparametric estimation of partial derivatives in sparse deep neural networks.
result Established convergence rate of nonparametric estimation of partial derivatives as O(n−1/4). Characterizes deep neural network weight space for adversarial attacks.
problem Poor performance of deep learning models in adversarial examples.
method Characterizes deep neural network solution space using two paradigms.
result Adversarial attacks are less successful against Associative Memory Models.
Deep neural networks and the ENO procedure are both efficient frameworks for approximating rough functions. We prove that at any order, the ENO interpolation procedure can be cast as a deep ReLU neural network. This surprising fact enables the transfer of several desirable properties of the ENO procedure to deep neural…
Speeds up deep neural networks training by 10x using GPU concurrency.
problem Training deep residual neural networks efficiently.
method Layer-wise parallel training with GPU concurrency and Nonlinear Multigrid.
result 10.2x speedup over traditional techniques.
This paper improves deep neural network approximation for fully connected networks, achieving optimal convergence rates.
problem Improving approximation of fully connected deep neural networks for optimal convergence rates.
method Deriving approximation bounds specifically for a narrower fully connected deep neural network.
result Achieves an optimal rate (up to a logarithmic factor) for fully connected deep neural networks.
Langevin algorithms improve training of very deep neural networks, especially for image classification.
problem Training very deep neural networks is challenging due to increased non-linearity and the risk of getting stuck in local minima.
method Comparison of Langevin and non-Langevin algorithms for training deep neural networks, introduction of Layer Langevin algorithm.
result Langevin algorithms, especially Layer Langevin, lead to significant improvements in training deep neural networks, particularly for image classification tasks.
New method proves neural networks can select features consistently.
problem Feature selection for deep neural networks is challenging.
method Adaptive Group Lasso selection procedure with Group Lasso as the base estimator.
result Adaptive Group Lasso is selection-consistent for a wide class of neural networks.
Taking inspiration from biological evolution, we explore the idea of "Can deep neural networks evolve naturally over successive generations into highly efficient deep neural networks?" by introducing the notion of synthesizing new highly efficient, yet powerful deep neural networks over successive generations via an ev…
Lectures on deep learning properties in infinite and large-width networks.
problem Understanding deep neural networks in extreme width conditions.
method Analysis of random deep neural networks, connections to linear models, kernels, and Gaussian processes, perturbative and non-perturbative treatments.
result Properties and behaviors of deep neural networks in the infinite-width limit and large-width regime.
The study examines deep convolutional neural networks and their learning ability.
problem Understanding the learning ability of deep convolutional neural networks (DCNNs).
method Examines DCNNs under both underparameterized and overparameterized settings, using a novel network deepening scheme.
result Establishes the first learning rates of underparameterized DCNNs and shows how adding layers can create interpolating DCNNs with good learning rates.
DeepStreamCE detects new classes in streaming deep neural networks.
problem Detecting new classes in deep neural networks in a streaming environment.
method Uses autoencoder and MCOD stream-based clustering for real-time concept evolution detection.
result DeepStreamCE outperforms OpenMax in identifying concept evolution.
The paper bounds the excess risk of deep neural networks for weakly dependent processes.
problem Learning with weakly dependent data using deep neural networks.
method Approximation of smooth functions by deep neural networks and a bound on excess risk.
result The excess risk bound for deep learning under weak dependence is close to O(n−1/2) for sufficiently smooth functions. Wide deep neural networks are easy to optimize without constraints.
problem Optimizing wide deep neural networks.
method Analysis of optimization landscapes and empirical-risk minimization.
result Wide neural networks have no confined points, making optimization easier.
Deep neural networks approximate functions in shift-invariant spaces with controlled error.
problem Approximating functions in shift-invariant spaces with neural networks.
method Using deep ReLU neural networks, estimating approximation error bounds based on network width and depth.
result Deep neural networks achieve optimal approximation rates for Sobolev spaces up to a logarithmic factor.
The paper studies how regularization parameters affect sparsity in deep neural networks.
problem Reducing the complexity of deep neural networks by promoting sparsity.
method Derives ℓ1-norm sparsity-promoting models, characterizes sparsity levels, and develops algorithms for selecting optimal regularization parameters. result Developed algorithms to select regularization parameters for desired sparsity levels in neural networks.
Analyzes minima of deep linear networks with weight decay.
problem Understanding the loss landscape of deep neural networks.
method Analytical solutions for global minima with weight decay and stochastic neurons.
result The origin is a special point with qualitatively different minima in networks with more than 1 hidden layer.
Gradient descent methods for deep ReLU networks achieve optimal generalization rates.
problem Generalization of gradient descent methods for deep neural networks
method Establishing minimax-optimal rates for GD and SGD with deep ReLU networks
result Gradient descent methods for deep ReLU networks achieve optimal generalization rates
MaxDropout removes most active neurons to prevent overfitting in deep neural networks.
problem Preventing overfitting in deep neural networks.
method MaxDropout removes the most active neurons in each hidden layer to enforce sparsity and prevent overfitting.
result MaxDropout improves neural network performance in image classification and achieves comparable results to existing regularizers.
Deep learning using multi-layer neural networks (NNs) architecture manifests superb power in modern machine learning systems. The trained Deep Neural Networks (DNNs) are typically large. The question we would like to address is whether it is possible to simplify the NN during training process to achieve a reasonable pe…
Deep learning has been widely applied and brought breakthroughs in speech recognition, computer vision, and many other domains. The involved deep neural network architectures and computational issues have been well studied in machine learning. But there lacks a theoretical foundation for understanding the approximation…
Theoretical analysis of deep neural networks for time series data.
problem Theoretical development for deep neural networks on temporally dependent observations is lacking.
method Established non-asymptotic bounds for prediction error of deep neural networks under mixing-type assumptions.
result Deep neural networks can model non-linear time series data with additional logarithmic factors due to dependence.
Deep neural network with l_1-regularization achieves nearly optimal risk bounds.
problem Achieving optimal risk bounds in deep learning.
method Empirical risk minimization with l_1-regularization.
result Adaptively nearly-minimax risk bound across various function classes.
Langevin algorithms enhance training of deep neural networks for stochastic control problems.
problem Training acceleration for deep neural networks in stochastic control problems.
method Application of Langevin algorithms to minimize the loss of deep neural networks in stochastic control problems.
result Langevin algorithms improve training on various stochastic control problems.
Study of deep neural networks using finite-time Lyapunov exponents.
problem Understanding the geometric structures in input space formed by deep neural networks.
method Analogy with dynamical systems, computing finite-time Lyapunov exponents.
result Ridges of large positive exponents divide input space into regions associated with different classes.
Deep neural networks can solve optimal stopping problems without dimensionality issues.
problem Optimal stopping problems in high-dimensional state spaces.
method Established a general framework for deep ReLU neural networks to approximate value functions and continuation values.
result Deep neural networks can approximate value functions and continuation values with error at most ε of size κd^q ε^(-r).
Deep quantum neural networks applied to finance for efficient risk management.
problem Efficiently solving numerical problems in finance, especially risk management.
method Application of deep quantum neural networks to finance, focusing on implied volatilities, option prices, and Greeks.
result Deep quantum neural networks can compute Greeks analytically and efficiently solve financial numerical problems.
Despite their increasing popularity and success in a variety of supervised learning problems, deep neural networks are extremely hard to interpret and debug: Given and already trained Deep Neural Net, and a set of test inputs, how can we gain insight into how those inputs interact with different layers of the neural ne…