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.
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.
3-layer NTK models generalize better than 2-layer models, especially with large input dimensions.
problem Understanding the generalization of overparameterized neural networks.
method Analyzing the 3-layer NTK model's test error and comparing it to 2-layer NTK models.
result 3-layer NTK models have a faster descent in test error with respect to the number of neurons in the second hidden layer.
Recent works have cast some light on the mystery of why deep nets fit any data and generalize despite being very overparametrized. This paper analyzes training and generalization for a simple 2-layer ReLU net with random initialization, and provides the following improvements over recent works: (i) Using a tighter char…
A simple 2-layer linear network outperforms neural networks in learning sparse targets.
problem Learning sparse targets from a sparse input with gradient descent.
method A 2-layer linear network with fully connected input layer and sparse targets.
result The 2-layer linear network achieves a lower expected square loss than neural networks.
SGD efficiently learns the XOR function with near-optimal sample complexity.
problem Learning the XOR function with a 2-layer neural network.
method Minibatch SGD on a 2-layer neural network with ReLU activations, focusing on signal-finding and signal-heavy phases.
result Achieves population error o(1) with dextpolylog(d) samples. We show that there is a simple (approximately radial) function on Rd, expressible by a small 3-layer feedforward neural networks, which cannot be approximated by any 2-layer network, to more than a certain constant accuracy, unless its width is exponential in the dimension. The result holds for virtually all kn…
2L-FUSE enhances feature sparsity through kernel learning.
problem Sparsity and feature selection in regression tasks.
method 2-Layered kernel machines for learning a shape matrix and feature direction identification.
result Minimal yet informative feature sets are identified without losing predictive performance.
Paper identifies key function spaces for ReLU networks based on Fisher information.
problem Understanding the structure of Fisher information matrices in ReLU networks.
method Spectral decomposition of Fisher information matrices, focusing on the first three eigenspaces.
result The first three eigenspaces account for 97.7% of the trace of the Fisher information matrix, corresponding to spherical harmonic functions of order ≤2.
This paper uncovers the low-rank structure of neural network Hessians.
problem Understanding the structure of Hessians in neural networks.
method Proposes a decoupling conjecture to decompose layer-wise Hessians into Kronecker products of smaller matrices.
result Proves the structure of top eigenspaces in 2-layer networks and shows high overlap in top eigenvectors across different models.
This paper explores how enforcing equivariance constraints limits neural network expressivity and proposes compensatory model size increases.
problem The impact of enforcing equivariance constraints on the expressive power of neural networks.
method Examined 2-layer ReLU networks, analyzed boundary hyperplanes and channel vectors, and constructed upper bounds on model size required for compensation.
result Enforcing equivariance constraints reduces the expressive power of neural networks, but this can be compensated by increasing model size.
This study uses continuous-time analysis to understand how momentum affects the optimisation of diagonal linear networks.
problem The effect of momentum on the optimisation trajectory of gradient descent.
method Leveraging a continuous-time approach to analyze momentum gradient descent with step size γ and momentum parameter β.
result Small values of λ help recover sparse solutions in overparametrised regression settings.
Identifies a gradient flow to solve kernel learning problems with noise reduction.
problem Kernel learning problem with Gaussian noise.
method Riemannian gradient flow with continuous Lyapunov functionals.
result Flow reduces noise and finds stationary points.
It is widely observed that deep learning models with learned parameters generalize well, even with much more model parameters than the number of training samples. We systematically investigate the underlying reasons why deep neural networks often generalize well, and reveal the difference between the minima (with the s…
Improves neural network verification by merging abstract domains and Lagrangian methods.
problem Prove provable bounds for neural network outputs given input ranges.
method Uses zonotopes within a Lagrangian decomposition to verify deep neural networks.
result Yields bounds that improve upon existing techniques in both time and tightness.
Overparameterisation can limit the benefits of curriculum learning in neural networks.
problem The ineffectiveness of curriculum learning in deep learning applications.
method Analytical study connecting curriculum learning and overparameterisation in an online learning setting for a 2-layer network in the XOR-like Gaussian Mixture problem.
result High degree of overparameterisation can limit the benefit from curricula.
Curriculum learning helps neural networks learn parities more efficiently.
problem Improving learning efficiency for neural networks on parity targets.
method Using a curriculum learning approach with a mixture of sparse and dense inputs.
result A 2-layer ReLU neural network can learn parities more efficiently than a fully connected network.
We analyze the dynamics of training deep ReLU networks and their implications on generalization capability. Using a teacher-student setting, we discovered a novel relationship between the gradient received by hidden student nodes and the activations of teacher nodes for deep ReLU networks. With this relationship and th…
In this paper, we present a novel approach for initializing deep neural networks, i.e., by turning PCA into neural layers. Usually, the initialization of the weights of a deep neural network is done in one of the three following ways: 1) with random values, 2) layer-wise, usually as Deep Belief Network or as auto-encod…
Study shows neural networks learn low frequencies first, proposing solutions.
problem Frequency bias in neural network learning process.
method Developed a PDE to unravel frequency dynamics, used Fourier Features model.
result Appropriate weight initialization can eliminate or control frequency bias.
SGD's training dynamics align with Hessian and gradient spectra in high-dimensional classification tasks.
problem Understanding the spectra of Hessian and gradient matrices in high-dimensional classification tasks.
method Rigorous analysis of SGD dynamics and spectra of Hessian and gradient matrices.
result SGD trajectory and emergent outlier eigenspaces align with a common low-dimensional subspace in multi-class high-dimensional mixtures and neural networks.
Reinforcement learning agents have been mostly developed and evaluated under the assumption that they will operate in a fully autonomous manner -- they will take all actions. In this work, our goal is to develop algorithms that, by learning to switch control between agents, allow existing reinforcement learning agents …
SGD learns neural networks with a complexity measure called leap.
problem Time complexity of SGD learning on neural networks.
method Introduced a complexity measure called leap, proved conjecture for Gaussian data, and showed saddle-to-saddle dynamics.
result Proved a conjecture about the time complexity of learning functions with low-dimensional support.
We study the approximation properties of random ReLU features through their reproducing kernel Hilbert space (RKHS). We first prove a universality theorem for the RKHS induced by random features whose feature maps are of the form of nodes in neural networks. The universality result implies that the random ReLU features…
Understanding the representational power of Restricted Boltzmann Machines (RBMs) with multiple layers is an ill-understood problem and is an area of active research. Motivated from the approach of \emph{Inherent Structure formalism} (Stillinger & Weber, 1982), extensively used in analysing Spin Glasses, we propose a no…
New measure FTC quantifies how much a ReLU network can fine-tune.
problem Analyzing memorization capacity in fine-tuned neural networks.
method Defined Fine-Tuning Capacity (FTC) for additive fine-tuning of ReLU networks.
result Upper and lower bounds on FTC for 2 and 3-layer ReLU networks.
The study analyzes sharpness dynamics in neural networks, revealing mechanisms and conditions.
problem Understanding sharpness in neural network training.
method Fixed point analysis and edge of stability analysis in a simplified 2-layer linear network.
result Reveals mechanisms behind sharpness trends, conditions for edge of stability, and a period-doubling route to chaos.
New neural approach for estimating SEMs with provable convergence.
problem Estimating structural parameters in SEMs.
method Formulated as a min-max game with neural networks, learned using stochastic gradient descent.
result Global convergence in overparametrized regime, improving state-of-the-art.
During the past decade, with the significant progress of computational power as well as ever-rising data availability, deep learning techniques became increasingly popular due to their excellent performance on computer vision problems. The size of the Protein Data Bank has increased more than 15 fold since 1999, which …
New research shows that the dimension gap between intrinsic and ambient dimensions affects adversarial vulnerability of machine learning models.
problem The mystery of adversarial attacks on machine learning models.
method Introducing two types of adversarial attacks and proving their relationship to the dimension gap.
result The dimension gap between intrinsic and ambient dimensions makes clean-trained models more vulnerable to off-manifold adversarial perturbations.
Spiking neural networks (SNNs) have garnered a great amount of interest for supervised and unsupervised learning applications. This paper deals with the problem of training multi-layer feedforward SNNs. The non-linear integrate-and-fire dynamics employed by spiking neurons make it difficult to train SNNs to generate de…
Deep neural networks can grok better than shallow ones, showing multi-stage generalization.
problem Understanding the generalization behavior of deep neural networks.
method Empirical replication and analysis of grokking phenomenon in deep MLP models.
result Deep neural networks exhibit multi-stage generalization, with a secondary surge in test accuracy.
Improves energy efficiency of neuromorphic hardware by optimizing memory organization and encoding schemes.
problem Energy inefficiency in neuromorphic hardware, especially in digital accelerators.
method Synthesized controller and memory for different encoding schemes, introduced functional encoding for structured connectivity.
result Functional encoding offers a 58% reduction in energy for weight updates in convolutional layers.
New findings on how neural networks generalize with varying dimensions.
problem Understanding how neural networks generalize with different dimensions and noise levels.
method Study of 2-layer ReLU NNs in a classification setting, proving transitions in overfitting types.
result The type of overfitting transitions from tempered to benign as input dimension increases.
Inspired by chaotic firing of neurons in the brain, we propose ChaosNet -- a novel chaos based artificial neural network architecture for classification tasks. ChaosNet is built using layers of neurons, each of which is a 1D chaotic map known as the Generalized Luroth Series (GLS) which has been shown in earlier works …
We study the design of computationally efficient algorithms with provable guarantees, that are robust to adversarial (test time) perturbations. While there has been an proliferation of recent work on this topic due to its connections to test time robustness of deep networks, there is limited theoretical understanding o…
Often, more time is spent on finding a model that works well, rather than tuning the model and working directly with the dataset. Our research began as an attempt to improve upon a simple Recurrent Neural Network for answering "simple" first-order questions (QA-RNN), developed by Ferhan Ture and Oliver Jojic, from Comc…
The enormous inference cost of deep neural networks can be scaled down by network compression. Pruning is one of the predominant approaches used for deep network compression. However, existing pruning techniques have one or more of the following limitations: 1) Additional energy cost on top of the compute heavy trainin…
Mixup helps learn both features in multi-view classification problems.
problem Learning both features in multi-view classification problems.
method Training with a specific instantiation of Mixup.
result Training with Mixup leads to learning both features for every class.
Long Short-Term Memory (LSTM) is a well-known method used widely on sequence learning and time series prediction. In this paper we deployed stacked LSTM model in an application of weather forecasting. We propose a 2-layer spatio-temporal stacked LSTM model which consists of independent LSTM models per location in the f…
We propose a fast second-order method that can be used as a drop-in replacement for current deep learning solvers. Compared to stochastic gradient descent (SGD), it only requires two additional forward-mode automatic differentiation operations per iteration, which has a computational cost comparable to two standard for…
One way to recognise an object is to study how the echo has been shaped during the interaction with the target. Wideband sonar allows the study of the energy distribution for a large range of frequencies. The frequency distribution contains information about an object, including its inner structure. This information is…
The paper connects flatness to generalization in learning multi-index models with neural networks.
problem Understanding the generalization of non-convex neural networks using flatness measures.
method Analyzes 2-layer non-convex homogeneous neural networks and their connection to multi-index models.
result Flattest interpolators achieve small population loss and generalize well, establishing a direct link between flatness and generalization.
The study classifies tilings of the sphere by congruent quadrilaterals.
problem Classifying edge-to-edge tilings of the sphere by congruent quadrilaterals.
method Classification of tilings into three classes based on geometric data and parameters.
result Three classes of tilings are identified: 2-layer earth map tilings, quadrilateral subdivisions of the octahedron, and 3-layer earth map tilings.
GCN and GPCA are mathematically connected, leading to improved node classification performance.
problem Improving node classification performance in semi-supervised settings.
method Established a mathematical connection between GCN and GPCA, demonstrating their equivalence and using this to design an effective initialization strategy.
result GPCA paired with a simple MLP achieves similar or better performance than GCN on semi-supervised node classification tasks.
New insights into adversarial vulnerability linked to manifold separability issues.
problem Adversarial vulnerability in machine learning models.
method Characterized data distribution as a low-dimensional manifold, focusing on on/off manifold directions and the impact of first-order vs. second-order optimization methods.
result First-order optimization methods lead to poor convergence in the off-manifold direction, causing adversarial vulnerability in inseparable datasets.
The paper proves universality in optimization problems with i.i.d. random vectors.
problem Optimization problems with i.i.d. random vectors and their projections.
method Proves universality of empirical risk minimization under specific conditions.
result The minimum value of the optimization problem is universal and depends only on the mean and covariance of the random vectors.
Hierarchical Modular Reinforcement Learning (HMRL), consists of 2 layered learning where Profit Sharing works to plan a prey position in the higher layer and Q-learning method trains the state-actions to the target in the lower layer. In this paper, we expanded HMRL to multi-target problem to take the distance between …