Paper studies the theoretical equivalence between implicit and explicit neural networks in high dimensions.
problem Lack of theoretical analysis of implicit and explicit neural networks.
method Examined high-dimensional implicit neural networks and established their equivalence to explicit networks.
result Equivalence between implicit and explicit neural networks in high dimensions.
This paper shows how to train only the implicit layer of overparameterized implicit neural networks.
problem Understanding how the implicit layer contributes to the training of overparameterized implicit neural networks.
method Restricting training to only the implicit layer and analyzing the generalization error for ReLU-activated networks.
result Global convergence is guaranteed even if only the implicit layer is trained, and gradient flow with proper random initialization can achieve small generalization errors.
This paper proves SGD converges to global minimum for over-parameterized ReLU networks.
problem Theoretical understanding of implicit neural networks is limited.
method Gradient flow analysis of ReLU activated implicit neural networks.
result Randomly initialized gradient descent converges to global minimum at a linear rate for square loss function in over-parameterized ReLU networks.
The paper explains implicit regularization in hierarchical tensor factorization and deep CNNs.
problem Understanding implicit regularization in complex neural network architectures.
method Theoretical analysis using dynamical systems to overcome challenges in hierarchy.
result Established implicit regularization towards low hierarchical tensor rank, equivalent to locality in CNNs.
The paper analyzes implicit regularization in tensor factorization using neural networks.
problem Understanding implicit regularization in tensor factorization.
method Dynamical systems perspective and gradient descent analysis.
result Gradient descent induces a form of greedy low tensor rank search.
Gradient descent biases towards stable rank networks for nearly-orthogonal data.
problem Understanding implicit bias in non-smooth neural networks trained by gradient descent.
method Analysis of two-layer ReLU and leaky ReLU networks trained by gradient descent on nearly-orthogonal data.
result Gradient descent biases towards networks with stable rank and uniform margin for nearly-orthogonal data.
Study shows how steepest descent algorithms' geometric margin increases during training.
problem Understanding implicit bias in steepest descent algorithms for neural networks.
method Analysis of steepest descent algorithms with infinitesimal learning rates in homogeneous neural networks.
result Limit points of training trajectories correspond to KKT points of margin-maximization problems.
This study investigates how gradient-based methods bias neural networks trained on high-dimensional data.
problem The implicit biases of gradient-based optimization algorithms in neural networks trained on high-dimensional data.
method Investigation of gradient flow and gradient descent in two-layer fully-connected neural networks with leaky ReLU activations.
result Gradient flow and gradient descent lead to neural networks with low-rank solutions and linear decision boundaries.
Gradient descent on ReLU networks with square loss implicitly favors balanced weights.
problem Understanding implicit regularization in nonlinear neural networks with regression losses.
method Analyzing gradient descent dynamics on ReLU networks with square loss.
result It is impossible to characterize the implicit regularization of ReLU networks with square loss by any explicit function of model parameters.
New framework improves robustness of implicit neural networks.
problem Ill-posedness and convergence instability in implicit neural networks.
method NEMON framework based on contraction theory for ℓ∞ norm, including well-posedness condition, average iteration, and input-output Lipschitz constant regularization. result Improved accuracy and robustness of implicit models with smaller input-output Lipschitz bounds.
Stochastic recurrent neural networks with latent random variables of complex dependency structures have shown to be more successful in modeling sequential data than deterministic deep models. However, the majority of existing methods have limited expressive power due to the Gaussian assumption of latent variables. In t…
Neural networks learn the support of the target function through SGD's implicit regularization effect.
problem Learning the support of the target function in neural networks.
method Investigation of mini-batch SGD's ability to learn the support in the first layer of a neural network.
result Mini-batch SGD effectively learns the support in the first layer by shrinking irrelevant weights, while vanilla GD requires an explicit regularization term.
We propose a novel approach to approximate complex high-dimensional posteriors using neural samplers.
problem Accurately capturing complex, multimodal, and correlated posteriors in high-dimensional spaces.
method Implicit variational inference with neural samplers and local linearisation bounds.
result Our method enables implicit distributions over tens of millions of latent variables, recovering correlations across layers in large Bayesian neural networks.
Over-parameterized neural networks generalize well in practice without any explicit regularization. Although it has not been proven yet, empirical evidence suggests that implicit regularization plays a crucial role in deep learning and prevents the network from overfitting. In this work, we introduce the gradient gap d…
Gradient descent implicitly favors group sparsity in neural networks.
problem Understanding implicit regularization in neural networks for structured sparsity.
method Novel neural reparameterization for diagonally grouped linear networks.
result Gradient descent without explicit regularization biases towards group sparsity.
DEQs and explicit networks are nearly equivalent for Gaussian mixtures.
problem Understanding the equivalence between DEQs and explicit neural networks.
method Random matrix theory and analysis of kernel matrices.
result A shallow explicit network can mimic the kernel of a DEQ.
New model outperforms Neural ODEs while being more efficient.
problem Stable convergence and existence guarantees for implicit-depth models.
method Developed Monotone Operator Equilibrium Network (monDEQ) based on monotone operator theory.
result MonDEQ models outperform Neural ODEs and are more computationally efficient.
Representing shapes as level sets of neural networks has been recently proved to be useful for different shape analysis and reconstruction tasks. So far, such representations were computed using either: (i) pre-computed implicit shape representations; or (ii) loss functions explicitly defined over the neural level sets…
Implicit discourse relation classification is of great challenge due to the lack of connectives as strong linguistic cues, which motivates the use of annotated implicit connectives to improve the recognition. We propose a feature imitation framework in which an implicit relation network is driven to learn from another …
Our paper characterizes how ReLU affects GD's implicit bias in high-dimensional neural networks.
problem Understanding the implicit bias of gradient descent on neural networks.
method Novel primal-dual analysis tracking predictions and coefficients.
result The implicit bias approximates the minimum-ℓ2-norm solution with high probability. Deep residual networks implicitly converge to neural ODEs.
problem Link between discrete and continuous deep learning models.
method Establishing implicit regularization for residual networks towards neural ODEs.
result Deep residual networks initialized as discretizations of neural ODEs converge to such ODEs during training.
Mirror flow in shallow neural networks shows similar implicit bias to gradient flow, with key differences in curvature penalties.
problem Analyzing implicit bias in shallow neural networks with mirror flow.
method Characterization through variational problems and scaled potentials.
result Mirror flow with scaled potentials induces a rich class of biases not captured by RKHS norms.
The study provides a generalization bound for a family of implicit networks.
problem Theoretical understanding of implicit networks' generalization is limited.
method A generalization bound is derived for a family of implicit networks using a covering number argument for Rademacher complexity.
result A theoretical generalization bound is established for implicit networks.
Study reveals how neural network biases align with adversarial attack frequencies.
problem Correlation between neural network biases and adversarial attacks.
method Fourier transform analysis of network implicit bias and adversarial perturbations.
result Network bias and adversarial attack frequencies are highly correlated.
Gradient descent implicitly regularizes neural networks by penalizing large loss gradients.
problem How to optimize deep neural networks without explicit regularization.
method Backward error analysis to calculate implicit gradient regularization and demonstrate its effectiveness empirically.
result Implicit gradient regularization biases gradient descent toward flat minima, improving model robustness and test errors.
iDAD uses neural networks to quickly adapt experiments without likelihoods.
problem Performing adaptive experiments in real-time with implicit models.
method iDAD learns a design policy network upfront to make quick design decisions.
result iDAD can make design decisions in milliseconds, unlike traditional BOED approaches.
We introduce the implicit processes (IPs), a stochastic process that places implicitly defined multivariate distributions over any finite collections of random variables. IPs are therefore highly flexible implicit priors over functions, with examples including data simulators, Bayesian neural networks and non-linear tr…
GNIs induce asymmetric heavy-tailed noise in SGD, affecting network performance.
problem The effect of Gaussian noise injections on SGD dynamics and network performance.
method Developed a Langevin-like SDE driven by asymmetric heavy-tailed noise to model the modified SGD dynamics.
result GNIs induce an implicit bias that varies with noise heaviness and asymmetry, affecting network performance.
The study analyzes implicit biases in neural networks using backward error analysis.
problem Analyzing implicit biases in multitask and continual learning settings.
method Backward error analysis to compute implicit training biases, deriving modified losses with three terms.
result The conflict term, measuring gradient alignment, is a new quantity in continual learning.
In recent years, deep neural networks have yielded state-of-the-art performance on several tasks. Although some recent works have focused on combining deep learning with recommendation, we highlight three issues of existing models. First, these models cannot work on both explicit and implicit feedback, since the networ…
SNN architecture shows gradient descent converges to regularized solution in matrix sensing problems.
problem Understanding implicit regularization in neural networks for matrix sensing.
method Developed Spectral Neural Networks (SNN) for matrix learning problems, rigorously demonstrating implicit regularization.
result Gradient descent converges to the solution of a regularized learning problem in matrix sensing problems.
Implicit deep learning prediction rules generalize the recursive rules of feedforward neural networks. Such rules are based on the solution of a fixed-point equation involving a single vector of hidden features, which is thus only implicitly defined. The implicit framework greatly simplifies the notation of deep learni…
New tensor formulation reveals gradient flow's bias in linear neural networks.
problem Understanding implicit bias in linear neural network training.
method Tensor formulation of neural networks, including fully-connected, diagonal, and convolutional networks.
result Gradient flow on linear tensor networks converges to solutions of specific optimization problems.
IGNN captures long-range graph dependencies using fixed-point equations.
problem Limited GNN ability to capture long-range graph dependencies.
method Fixed-point equilibrium equations involving implicitly defined state vectors, leveraging Perron-Frobenius theory and projected gradient descent.
result IGNN consistently captures long-range dependencies and outperforms state-of-the-art GNNs.
Deep tensor factorization benefits from implicit regularization with polynomial growth.
problem Tensor factorization's implicit regularization effect in deep networks is not well understood.
method Investigated the implicit regularization in deep tensor factorization, showing polynomial growth.
result Implicit regularization in deep tensor factorization grows polynomially with depth, improving estimation accuracy and convergence.
New proof shows norms can't explain deep learning's implicit regularization.
problem Understanding the implicit regularization in deep learning.
method Mathematical proof on matrix factorization problems.
result Implicit regularization drives norms towards infinity, suggesting rank minimization is key.
Graphs of neural networks are represented to preserve symmetry, improving performance across various tasks.
problem Lack of equivariance in neural network representations of other neural networks.
method Represent neural networks as computational graphs and use graph neural networks to preserve permutation symmetry.
result Single model encodes diverse neural architectures, outperforming state-of-the-art methods.
The paper analyzes how gradient descent implicitly regularizes solutions in overparameterized neural networks, revealing depth-dependent regularization effects.
problem Understanding implicit regularization in overparameterized linear neural networks for regression problems.
method Analyzing the approximation error between gradient flow limit points and ℓ1-minimization solutions, deriving tight upper and lower bounds. result The approximation error decreases linearly for D≥3 and at a slower rate for D=2, linked to null space property constants. FA algorithm provides convergence guarantees for deep linear networks.
problem Training efficiency and convergence of deep neural networks.
method Theoretical analysis of Feedback Alignment (FA) algorithm for deep linear networks.
result Certain initializations lead to implicit anti-regularization, affecting learning effectiveness.
Arguably the biggest challenge in applying neural networks is tuning the hyperparameters, in particular the learning rate. The sensitivity to the learning rate is due to the reliance on backpropagation to train the network. In this paper we present the first application of Implicit Stochastic Gradient Descent (ISGD) to…
In this paper, we revisit implicit regularization from the ground up using notions from dynamical systems and invariant subspaces of Morse functions. The key contributions are a new criterion for implicit regularization---a leading contender to explain the generalization power of deep models such as neural networks---a…
Study improves understanding and performance of FA learning rules in neural networks.
problem Lack of theoretical understanding and limited applications of Feedback Alignment (FA) methods.
method Introduces a unified framework linking synaptic weight changes to implicit regularization, providing convergence conditions and empirical evidence.
result Better alignment can enhance FA performance on complex multi-class tasks.
Neural networks help create summary statistics for complex models.
problem Creating summary statistics for models with intractable likelihood functions.
method Infomax learning with neural networks to maximize mutual information.
result Improves performance of approximate Bayesian computation and neural likelihood methods.
Modality-agnostic compression improves across diverse data types.
problem Efficiently compressing data across multiple modalities.
method Functional view of data, Implicit Neural Representation (INR), modality-agnostic latent representations, variational compression.
result Improved performance compared to existing methods, especially for diverse modalities.
Gradient descent training of neural networks leads to solutions close to natural cubic splines.
problem Understanding the implicit bias of gradient descent in neural networks.
method Analysis of gradient descent training for wide neural networks, focusing on the curvature penalty and initialization schemes.
result The solutions of gradient descent training are polyharmonic splines for certain initialization schemes.
Neural network has attracted great attention for a long time and many researchers are devoted to improve the effectiveness of neural network training algorithms. Though stochastic gradient descent (SGD) and other explicit gradient-based methods are widely adopted, there are still many challenges such as gradient vanish…
The paper explores how the depth of neural networks affects their ability to represent data accurately.
problem Understanding the implicit bias and rank of neural networks with large depth.
method Analyzing the convergence of representation cost to a notion of rank as network depth increases, and investigating conditions for recovering the true rank of data.
result There is a range of network depths where the true rank of data is recovered, and this affects the topology of class boundaries.
We reparametrize ReLU NNs as splines to understand their learning dynamics.
problem Understanding the learning dynamics and inductive bias of neural networks.
method Reparametrize ReLU NNs as continuous piecewise linear splines to study learning dynamics.
result Standard weight initializations yield very flat functions, leading to strength and type of implicit regularization.