A neural network with a single hidden layer can't represent certain multivariable functions.
problem Representing certain multivariable functions with a neural network having only one hidden layer.
method Developed a continuum version of a one-hidden-layer neural network with ReLU activation, and proved constraints on its parameters and second derivative.
result Existence of a smooth binary function that cannot be precisely represented by any such neural network.
This paper explores loss landscapes of sparse neural networks, finding unique characteristics compared to dense networks.
problem Understanding the loss landscape of sparse neural networks, especially one-hidden-layer networks.
method Analyzes sparse networks with dense and sparse final layers, focusing on linear and non-linear models.
result Sparse networks can have no spurious valleys under certain conditions, but spurious valleys and minima can exist for wide sparse networks.
We study how finite Bayesian neural networks adapt their hidden representations.
problem Understanding how finite Bayesian neural networks differ from infinite ones.
method We analyze the asymptotics of learned feature kernels for various network architectures.
result The leading finite-width corrections to feature kernels have a universal form.
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.
We study the problem of learning one-hidden-layer neural networks with Rectified Linear Unit (ReLU) activation function, where the inputs are sampled from standard Gaussian distribution and the outputs are generated from a noisy teacher network. We analyze the performance of gradient descent for training such kind of n…
A method to interpret neural networks by isolating active subnetworks.
problem Interpreting the complex computations of deep neural networks.
method Switched linear projections to decompose network activity.
result Patterns in deactivated neurons are meaningful for network understanding.
New neural network models for functional data.
problem Handling non-linear functional data.
method Functional Direct Neural Network (FDNN) and Functional Basis Neural Network (FBNN) with gradient-based optimization.
result Demonstrated effectiveness in complex functional models.
We present a novel neural network algorithm, the Tensor Switching (TS) network, which generalizes the Rectified Linear Unit (ReLU) nonlinearity to tensor-valued hidden units. The TS network copies its entire input vector to different locations in an expanded representation, with the location determined by its hidden un…
LIFE framework improves model accuracy and interpretability.
problem Achieving high prediction accuracy and interpretability in neural networks.
method Three-step process: subset definition, feature creation, and linear model combination.
result LIFE consistently outperforms other models in prediction accuracy and interpretability.
Deep linear ResNets converge globally with certain transformations.
problem Global convergence of training deep linear ResNets.
method Gradient descent and stochastic gradient descent for training L-hidden-layer linear ResNets. result GD and SGD can converge to global minimum for deep linear ResNets with specific transformations.
New algorithm reduces sample complexity for learning CNNs.
problem Learning one-hidden-layer CNNs with various activation functions.
method Approximate gradient descent algorithm for training CNNs.
result Sample complexity matches information-theoretic lower bound for linear activation functions.
We provide a classification of graphical models according to their representation as subfamilies of exponential families. Undirected graphical models with no hidden variables are linear exponential families (LEFs), directed acyclic graphical models and chain graphs with no hidden variables, including Bayesian networks …
In this paper, we propose a geometric framework to analyze the convergence properties of gradient descent trajectories in the context of linear neural networks. We translate a well-known empirical observation of linear neural nets into a conjecture that we call the \emph{overfitting conjecture} which states that, for a…
Linear recurrent networks explain reinforcement learning performance in partially observable settings.
problem Understanding why linear recurrent networks work in reinforcement learning with partial observability.
method Constructed and studied two linear filters for HMMs and action-controlled HMMs.
result Linear filters serve as sufficient statistics and reduce state ambiguity, explaining empirical reinforcement learning success.
New neural network models for complex functional data analysis.
problem Complex relations between functional predictors and responses.
method Function-on-Function regression models using neural networks with continuous hidden layers.
result Demonstrated power and flexibility in handling complex functional models.
Piecewise linear activations create many spurious local minima in neural networks.
problem Understanding the loss surface of neural networks with piecewise linear activations.
method Proved the existence of infinite spurious local minima and partitioned the loss surface into smooth cells.
result Piecewise linear activations create many spurious local minima that are invariant under a continuous path.
Neural networks can approximate positive homogeneous functions, especially with multiple hidden layers.
problem Approximating positive homogeneous functions with neural networks.
method Using scale-invariant ReLU networks with multiple hidden layers.
result Approximation of positive homogeneous functions is possible with neural networks, especially with two hidden layers.
This work introduces a tensor-based method to perform supervised classification on spatiotemporal data processed in an echo state network. Typically when performing supervised classification tasks on data processed in an echo state network, the entire collection of hidden layer node states from the training dataset is …
Gradient descent with random init solves 1HL NNs in under-param regime.
problem Learning a one-hidden-layer neural network with quadratic activations.
method Provable gradient-based method with random initialization.
result Gradient descent iterates converge to globally optimal model with linear rate.
ReLU activations lead to smoother learning curves compared to sigmoidal activations in neural networks.
problem Comparing the performance of ReLU and sigmoidal activations in neural networks.
method Analytical computation of learning curves in shallow networks with different activation functions.
result ReLU networks exhibit continuous transitions in performance, while sigmoidal networks show discontinuous transitions.
Dropout improves neural network performance by promoting low-rank solutions.
problem Improving neural network generalization through regularization.
method Analyzing Dropout, DropBlock, and DropConnect as regularizers for linear networks and extending to deep networks.
result Dropout, DropBlock, and DropConnect induce low-rank solutions and can be computed in closed form.
The study examines Fisher information matrices and neural tangent kernels for simple ReLU networks with random weights.
problem Understanding the relationship between Fisher information matrices and neural tangent kernels for 2-layer ReLU networks.
method Analyzes Fisher information matrices and neural tangent kernels for 2-layer ReLU networks with random hidden weights, focusing on spectral decomposition and eigenfunctions.
result Obtained an approximation formula for functions represented by 2-layer neural networks.
These notes review six lectures given by Prof. Andrea Montanari on the topic of statistical estimation for linear models. The first two lectures cover the principles of signal recovery from linear measurements in terms of minimax risk. Subsequent lectures demonstrate the application of these principles to several pract…
Learning weights in a spiking neural network with hidden neurons, using local, stable and online rules, to control non-linear body dynamics is an open problem. Here, we employ a supervised scheme, Feedback-based Online Local Learning Of Weights (FOLLOW), to train a network of heterogeneous spiking neurons with hidden l…
In this paper we propose a method to model speaker and session variability and able to generate likelihood ratios using neural networks in an end-to-end phrase dependent speaker verification system. As in Joint Factor Analysis, the model uses tied hidden variables to model speaker and session variability and a MAP adap…
Neural networks and linear systems linked, revealing training loss and kernel limitations.
problem Exploring the training loss and limitations of neural networks and their kernels.
method Drawing connections between neural networks and under-determined linear systems, providing lower bounds, and analyzing gradient descent.
result Zero training loss achievable for neural networks under certain conditions, but not for ReLU kernels.
New proof shows deep neural nets can have sub-optimal local minima.
problem Can over-parameterization eliminate sub-optimal local minima in deep neural networks?
method Counter-example with generic input data and non-linear activation functions.
result Sub-optimal local minima exist in deep neural networks regardless of width.
We present an alternative to the pseudo-inverse method for determining the hidden to output weight values for Extreme Learning Machines performing classification tasks. The method is based on linear discriminant analysis and provides Bayes optimal single point estimates for the weight values.
Background: Statistical mechanics results (Dauphin et al. (2014); Choromanska et al. (2015)) suggest that local minima with high error are exponentially rare in high dimensions. However, to prove low error guarantees for Multilayer Neural Networks (MNNs), previous works so far required either a heavily modified MNN mod…
The paper extends mean field results to three-layer neural networks using SGD.
problem Understanding the dynamics of training three-layer neural networks with SGD.
method Extending mean field results from two-layer networks to three-layer networks with two hidden layers, using non-linear partial differential equations.
result The distributions of weights in the two hidden layers are independent.
A new method learns state and proposal dynamics in state-space models using neural networks.
problem Inference in non-linear state-space models.
method StateMixNN method using neural networks for proposal and transition distributions.
result Significantly improved recovery of hidden state, especially in highly non-linear scenarios.
The study analyzes local minima in ReLU networks and finds low probability of bad local minima.
problem Understanding the existence and probability of local minima in ReLU networks.
method Theoretical analysis combined with linear programming and experiments on MNIST and CIFAR-10 datasets.
result No bad differentiable local minima found almost everywhere in weight space.
Bayesian linear networks reveal optimal depth and width trade-offs.
problem Understanding how depth, width, and dataset size affect model quality in linear networks.
method Zero noise Bayesian inference with Gaussian weight priors and mean squared error.
result Optimal predictions at infinite depth and maximized Bayesian model evidence at infinite depth.
We consider deep linear networks with arbitrary convex differentiable loss. We provide a short and elementary proof of the fact that all local minima are global minima if the hidden layers are either 1) at least as wide as the input layer, or 2) at least as wide as the output layer. This result is the strongest possibl…
We analyze deep neural networks using convex duality to reveal hidden layer structures.
problem Understanding the structure of deep neural networks.
method Introducing a convex analytic framework to characterize hidden layer weights.
result Optimal hidden layer weights align with previous layers via duality.
A new model improves recurrent neural networks' ability to memorize long sequences.
problem Improving recurrent neural networks' ability to memorize long sequences and extract task-relevant features.
method Proposes a Linear Memory Network with an encoding-based memorization component and a specialized training algorithm.
result Improves the final performance of recurrent neural networks when memorizing long sequences is necessary.
Supervised learning frequently boils down to determining hidden and bright parameters in a parameterized hypothesis space based on finite input-output samples. The hidden parameters determine the attributions of hidden predictors or the nonlinear mechanism of an estimator, while the bright parameters characterize how h…
Study shows directional convergence for neural networks under spherical symmetry.
problem Learning linear predictors with neural networks under spherically symmetric data.
method Analysis of gradient flow and gradient descent for two-layer and deep linear networks.
result Directional convergence guarantees with exact convergence rate for specific network architectures.
Many widely studied graphical models with latent variables lead to nontrivial constraints on the distribution of the observed variables. Inspired by the Bell inequalities in quantum mechanics, we refer to any linear inequality whose violation rules out some latent variable model as a "hidden variable test" for that mod…
Gradient descent proves global convergence for deep networks with a single wide layer.
problem Proving global convergence of gradient descent for deep ReLU networks.
method Simplified proof using a single wide layer, leveraging ReLU's Lipschitz property.
result Gradient descent converges globally for networks with a single wide layer.
In this paper, we consider regression problems with one-hidden-layer neural networks (1NNs). We distill some properties of activation functions that lead to local strong convexity in the neighborhood of the ground-truth parameters for the 1NN squared-loss objective. Most popular nonlinear activation function…
Generative model initializes 2-layer network weights for small datasets.
problem Approximating functions with 2-layer networks using small datasets and gradient-based training.
method Initialize hidden weights with a learned proposal distribution parameterized as a deep generative model. Refine with gradient-based post-processing and regularization.
result Demonstrates effectiveness of the approach with numerical examples.
Study on generalisation in random feature learning and hidden manifold models.
problem Generalisation in high-dimensional learning problems.
method Replica method from statistical physics for asymptotic generalisation performance.
result Closed-form expression for generalisation performance in various high-dimensional settings.
Paper studies shallow ReLU networks' approximation rates for Hölder functions.
problem Understanding shallow ReLU networks' efficiency in approximating Hölder functions.
method Analyzes rates of uniform approximation by ReLU shallow neural networks with m hidden neurons. result Shows ReLU shallow neural networks can uniformly approximate Hölder functions with rates close to optimal.
ReLU networks trained with MILPs match deep learning accuracy.
problem Training deep neural networks efficiently.
method Iterative training with Mixed Integer Linear Programs (MILPs).
result ReLU networks can be trained with MILPs achieving similar accuracy to deep learning methods.
A new optimizer for deep learning improves accuracy and reduces training time.
problem Training deep neural networks for classification tasks.
method Hybrid Newton/Gradient Descent (NGD) method exploiting convexity of cross-entropy loss.
result Improves validation error and provides qualitative differences in hidden layer basis functions.
Upper bounds on fixed points in PWL neural networks with hyperplane analysis.
problem Analyzing the number of fixed points in neural networks with PWL activation.
method Hyperplane arrangements to bound the number of fixed points.
result Upper bounds on the number of fixed points for PWL networks, showing exponential growth in layers.
We propose and analyze a new family of algorithms for training neural networks with ReLU activations. Our algorithms are based on the technique of alternating minimization: estimating the activation patterns of each ReLU for all given samples, interleaved with weight updates via a least-squares step. The main focus of …