Paper refines RNN training by analyzing smoothness and attractors.
problem Exploding and vanishing gradients in RNNs.
method Refined concept of exploding gradients using cost function smoothness.
result RNNs need to learn attractors to fully use their power.
New neural network approach mitigates vanishing/exploding gradients.
problem Vanishing and exploding gradients in neural networks.
method Gaussian-Poincaré normalized functions and orthogonal weight matrices.
result High-dimensional probability theory shows gradients disappear with high probability in wide neural networks.
TSGO optimizes gradients in tensor networks to avoid vanishing/exploding issues.
problem Gradient vanishing and exploding problems in deep learning models.
method TSGO rotates parameters towards gradient direction in tangent space of normalized state.
result TSGO naturally determines learning rate based on angle between parameters and gradient.
Blind Descent avoids gradient issues, using a different learning approach.
problem Gradient issues like exploding and vanishing gradients.
method Does not use gradients to guide learning; instead, it is a more fundamental learning process.
result Gradient descent is a specific case of Blind Descent.
PIPPS solves deep learning's exploding gradient problem by reparameterization gradients.
problem Exploding gradients in deep learning and model-based RL.
method Develops PIPPS framework, a flexible policy search method robust to chaos-like gradients.
result PIPPS improves over reparameterization gradients by up to 10^6 times.
BatchNorm helps BNNs avoid exploding gradients during training.
problem Difficulty in training Binary Neural Networks (BNNs).
method Theoretical study and numerical experiments on the role of BatchNorm in BNNs.
result BatchNorm is crucial for avoiding exploding gradients in BNNs.
This work tackles exploding inverses in INNs, revealing and mitigating their numerical non-invertibility.
problem Exploding inverses in INNs cause numerical non-invertibility, leading to failures in various tasks.
method Derived bi-Lipschitz properties of INN building blocks, proposed regularizers for local invertibility, and stable INN designs for global invertibility.
result Bi-Lipschitz properties and stable INN designs are crucial for addressing numerical non-invertibility.
Maxout networks study gradients and propose initialization strategies.
problem Complexity in input-output Jacobian distribution complicates stable parameter initialization.
method Obtained bounds on moments of gradients and formulated initialization strategies.
result Parameter initialization strategies improve training of deep maxout networks.
ERNNs evolve hidden states on an ODE's equilibrium manifold to mitigate vanishing and exploding gradients.
problem Vanishing and exploding gradients in RNNs.
method Develop a novel family of RNNs (ERNNs) that evolve hidden states on the equilibrium manifold of an ODE.
result ERNNs achieve state-of-the-art accuracy with 3-10x speedups and 1.5-3x model size reduction.
Volume-preserving neural networks prevent gradient issues.
problem Vanishing and exploding gradients in deep neural networks.
method A new neural network architecture with volume-preserving sublayers.
result Volume-preserving neural networks maintain gradient stability.
ADMMiRNN solves RNN training issues with stable convergence.
problem Training RNN with stable convergence and avoiding gradient issues.
method Built ADMMiRNN framework on unfolded RNN, providing novel update rules and theoretical analysis.
result ADMMiRNN achieves convergent results and outperforms baselines.
A new RNN model tackles long-time dependencies with fast, invertible, and memory-efficient hidden states.
problem Challenges in processing sequential inputs with long-time dependencies in RNNs.
method A novel RNN architecture based on a Hamiltonian system of oscillators.
result The proposed RNN mitigates exploding and vanishing gradient problems, providing state-of-the-art performance.
LTM tackles long sequence language modeling by avoiding vanishing and exploding gradients.
problem Language models struggle with long sequences due to gradient issues.
method Introduces Long Term Memory network (LTM) that scales memory and weights input, avoiding overfitting.
result LTM achieves state-of-the-art perplexity results with fewer cells than previous models.
ENRNN uses eigenvalue normalization for short-term memory in RNNs.
problem Vanishing/exploding gradient problem and long-term dependency modeling.
method Eigenvalue normalization of recurrent matrix to simulate short-term memory.
result ENRNN outperforms existing RNN variants in experiments.
Fixup replaces normalization in deep networks, achieving similar stability and performance.
problem The effectiveness of normalization layers in deep neural networks.
method Fixed-update initialization (Fixup) to solve exploding and vanishing gradient problems.
result Residual networks trained with Fixup achieve state-of-the-art performance without normalization.
AntisymmetricRNN improves RNN trainability without extra computation.
problem Difficulty in learning long-term dependencies in RNNs.
method Connecting RNNs to ordinary differential equations and proposing AntisymmetricRNN.
result AntisymmetricRNN captures long-term dependencies more predictably and efficiently.
A simple gating mechanism improves deep learning convergence.
problem Vanishing or exploding gradients in deep networks.
method Introducing a zero-initialized parameter to each residual connection.
result Training deep networks (up to 120 layers) with fast convergence and better performance.
Gradient flossing stabilizes RNN training by controlling Lyapunov exponents.
problem Gradient instability in RNNs leading to exploding and vanishing gradients.
method Regularizing Lyapunov exponents through backpropagation using differentiable linear algebra.
result Gradient flossing improves RNN training success rate and convergence speed.
Hamiltonian RNN controls hidden states gradient for long-term dependencies.
problem Challenges in learning long-term dependencies in RNNs.
method Symplectic discretization of Hamiltonian system to control gradient.
result Hamiltonian RNN outperforms other RNNs without hyperparameter optimization.
Novel BSG method for efficient stochastic optimization.
problem Efficient optimization of non-convex surfaces in stochastic settings.
method Binary search combined with first order gradient optimization.
result BSG produces more promising results and better generalization than other methods.
Vanishing nodes cause hidden nodes to behave similarly, complicating deep neural network training.
problem Complicating deep neural network training due to hidden nodes behaving similarly.
method Introduced vanishing node indicator (VNI) to quantify node redundancy and correlated behavior.
result The effective number of nodes vanishes to one as VNI increases, indicating a challenging training scenario.
Stable ResNet stabilizes gradients in deep networks.
problem Gradient vanishing and exploding in deep ResNet architectures.
method Introducing Stable ResNet architectures with gradient stabilization and infinite depth expressivity.
result Stable ResNet maintains gradient stability and expressivity in deep networks.
Non-normal RNNs outperform orthogonal ones in sequential tasks.
problem Vanishing/exploding gradients in RNNs training.
method Investigate non-normal RNNs with non-normal recurrent connectivity matrix.
result Non-normal RNNs outperform orthogonal ones in various benchmarks.
This paper extends de Rham theory of smooth manifolds to exploded manifolds. Included are versions of Stokes' theorem, De Rham cohomology, Poincare duality, and integration along the fiber. The resulting cohomology theory is used to define Gromov Witten invariants of exploded manifolds in a separate paper.
RNNs struggle with chaotic dynamics due to exploding gradients, but we found a way to optimize training.
problem Challenging training of RNNs with chaotic dynamics due to exploding gradients.
method Relating loss gradients to Lyapunov spectrum to optimize training on chaotic data.
result RNNs with chaotic dynamics always have diverging gradients, while stable ones have bounded gradients.
New RNN model handles long-term dependencies in irregularly-sampled time series.
problem Handling long-term dependencies in irregularly-sampled time series data.
method Designing ODE-LSTMs that separate memory from continuous-time state.
result ODE-LSTMs outperform other RNN-based models on non-uniformly sampled data with long-term dependencies.
Analysis shows BN prevents gradient vanishing/explosion in residual networks.
problem Gradient vanishing/explosion problem in residual networks.
method Mathematical analysis of BN and residual network training.
result BN confounds gradient variance, preventing vanishing/explosion.
Vanishing and exploding gradients are two of the main obstacles in training deep neural networks, especially in capturing long range dependencies in recurrent neural networks~(RNNs). In this paper, we present an efficient parametrization of the transition matrix of an RNN that allows us to stabilize the gradients that …
This paper analyzes challenges and solutions in deep learning optimization.
problem Gradient vanishing and exploding issues in deep learning.
method Improvement of gradient flow and constraints on Lipschitz constant.
result Enhanced understanding of Jacobian matrices and Lipschitz constants in deep learning modules.
We give a rigorous analysis of the statistical behavior of gradients in a randomly initialized fully connected network N with ReLU activations. Our results show that the empirical variance of the squares of the entries in the input-output Jacobian of N is exponential in a simple architecture-dependent constant beta, gi…
A new method reduces variance in training early-stage rankers for large-scale search systems.
problem Training early-stage rankers for large-scale search systems is challenging due to exploding variance in policy gradient methods.
method Proposes credit-assigned policy gradient (CA-PG) to mitigate variance in training early-stage rankers.
result CA-PG significantly reduces variance in training early-stage rankers compared to vanilla policy gradient.
Channel normalization prevents vanishing gradients in convolutional neural networks.
problem Vanishing gradients in convolutional neural networks during optimization.
method Channel normalization, which centers and normalizes each channel individually.
result Channel normalization avoids vanishing gradients, enabling efficient optimization.
Scaling ResNets requires careful consideration of the layer depth and output scaling factors.
problem Avoiding vanishing or exploding gradients in deep ResNets as depth increases.
method Probabilistic analysis and continuous-time limit interpretation of ResNets.
result The optimal scaling factor is α L = 1 L α_L = \frac{1}{\sqrt{L}} α L = L 1 for standard i.i.d. initializations. New method bounds singular values of convolutional kernels to stabilize gradients.
problem Stable gradients in convolutional neural networks.
method Frobenius norm regularization for convolutional kernels.
result Bounded singular values of transformation matrices.
Notes for a short lecture series, covering exploded manifolds, the moduli stack of curves in exploded manifolds, and a tropical gluing formula for Gromov-Witten invariants: a gluing formula providing a degeneration formula for Gromov-Witten invariants in normal-crossing degenerations. I gave the original lecture series…
AuON is a linear-time optimizer that improves upon Muon's performance without approximate orthogonal matrices.
problem High memory and computational costs of orthogonal momentum updates.
method AuON uses normalized nonlinear scaling and a 'emergency brake' to handle exploding attention logits.
result AuON achieves strong performance without approximate orthogonal matrices, preserving structural alignment and reconditioning.
Novel method improves training RNNs by accelerating gradient descent.
problem Vanishing and exploding gradient problems in RNNs training.
method Adaptive stochastic Nesterov accelerated quasi-Newton method.
result Improved performance in training RNNs with low per-iteration cost.
Gradient control plays an important role in feed-forward networks applied to various computer vision tasks. Previous work has shown that Recurrent Highway Networks minimize the problem of vanishing or exploding gradients. They achieve this by setting the eigenvalues of the temporal Jacobian to 1 across the time steps. …
Residual Network (ResNet) is the state-of-the-art architecture that realizes successful training of really deep neural network. It is also known that good weight initialization of neural network avoids problem of vanishing/exploding gradients. In this paper, simplified models of ResNets are analyzed. We argue that good…
One of the difficulties of training deep neural networks is caused by improper scaling between layers. Scaling issues introduce exploding / gradient problems, and have typically been addressed by careful scale-preserving initialization. We investigate the value of preserving scale, or isometry, beyond the initial weigh…
A long-standing obstacle to progress in deep learning is the problem of vanishing and exploding gradients. Although, the problem has largely been overcome via carefully constructed initializations and batch normalization, architectures incorporating skip-connections such as highway and resnets perform much better than …
We present a gluing formula for Gromov-Witten invariants in the case of a triple product. This gluing formula is a simple case of a much more general gluing formula proved and stated using exploded manifolds. We present this simple case because it is relatively easy to explain without any knowledge of exploded manifold…
It is a known fact that training recurrent neural networks for tasks that have long term dependencies is challenging. One of the main reasons is the vanishing or exploding gradient problem, which prevents gradient information from propagating to early layers. In this paper we propose a simple recurrent architecture, th…
A novel RNN model with shuffled hidden states.
problem Improving recurrent neural network performance.
method Permuting hidden states and applying learned functions.
result Competitive results compared to literature baselines.
A new RNN model based on coupled oscillators mitigates gradient issues.
problem Gradient vanishing and exploding issues in RNNs.
method Time-discretization of a system of second-order ODEs modeling coupled oscillators.
result The model maintains bounded gradients, leading to stable learning of long-term dependencies.
Proposes a method to constrain singular values of convolutional kernels in neural networks.
problem Avoiding exploding/vanishing gradient problems and improving generalizability in neural networks.
method Introduces a penalty function to constrain singular values of convolutional kernels around 1, and derives an algorithm for optimization.
result Demonstrates the effectiveness of the method through numerical examples.
Proposes methods to add constraints to neural networks to improve stability and generalization.
problem Improving stability and generalization of neural networks.
method Constraint-based regularization using stochastic gradient Langevin dynamics.
result Constraints help stabilize and improve the robustness of deep neural networks.
Deep neural networks' Jacobian spectrum becomes well-conditioned with orthogonal weights.
problem Understanding and handling the Jacobian spectrum of deep neural networks.
method Applying free probability theory to show almost sure asymptotic freeness of Jacobians in the wide limit.
result Layer-wise Jacobians of deep neural networks with orthogonal weights are almost surely asymptotically free.