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.
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.
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.
New method for modeling densities on Riemannian manifolds with symmetries.
problem Modeling densities on Riemannian manifolds with known symmetry groups.
method Combining implicit neural layers and optimal transport theory to propose IRCPMs.
result IRCPMs are simpler to incorporate symmetries and less expensive than ODE-flows.
New method improves BLL models for complex datasets.
problem Limited expressive capacity of Gaussian priors in BLL models.
method Combines diffusion techniques and implicit priors for variational learning.
result Enhanced predictive accuracy and uncertainty quantification.
Batch normalization biases linear models towards uniform margins, improving performance in binary classification.
problem Understanding the implicit bias of batch normalization in linear models and neural networks.
method Analyzing gradient descent convergence on linear models and two-layer CNNs with batch normalization.
result Gradient descent with batch normalization in linear models converges to a uniform margin classifier with an exponential convergence rate.
Paper presents an ADMM-based approach to efficiently integrate quadratic programming layers into neural networks.
problem Integrating quadratic programs into neural networks for optimization.
method An ADMM-based network layer architecture for solving quadratic programs efficiently.
result The ADMM layer is approximately an order of magnitude faster than existing methods for medium scaled problems.
Inspired by the adaptation phenomenon of neuronal firing, we propose the regularity normalization (RN) as an unsupervised attention mechanism (UAM) which computes the statistical regularity in the implicit space of neural networks under the Minimum Description Length (MDL) principle. Treating the neural network optimiz…
Polynomial-time convex optimization for CNNs with ReLU activations.
problem Training Convolutional Neural Networks (CNNs) with ReLU activations.
method Developed a convex analytic framework using semi-infinite duality to formulate equivalent convex optimization problems for CNN architectures.
result Proved that two-layer CNNs can be globally optimized via an ℓ2 norm regularized convex program. The paper explores how linear neural networks can overfit without bias when data is well-behaved.
problem Understanding why linear neural networks can generalize well despite fitting noisy data.
method Analyzing two-layer linear neural networks trained with gradient flow, deriving bounds on excess risk.
result The excess risk depends on initialization quality and data covariance matrix properties.
Deep equilibrium models converge globally without explicit computation.
problem Global convergence of deep learning models with implicit layers.
method Analysis of gradient dynamics and proof of convergence rate.
result Deep equilibrium models converge to global optimum at a linear rate.
It remains a puzzle that why deep neural networks (DNNs), with more parameters than samples, often generalize well. An attempt of understanding this puzzle is to discover implicit biases underlying the training process of DNNs, such as the Frequency Principle (F-Principle), i.e., DNNs often fit target functions from lo…
Analysis of gradient descent on wide neural networks reveals strong generalization.
problem Understanding why wide neural networks trained with logistic loss perform well.
method Characterization of gradient flow limits and comparison to max-margin classifier.
result Margin is independent of ambient dimension, leading to strong generalization.
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.
We consider networks, trained via stochastic gradient descent to minimize ℓ2 loss, with the training labels perturbed by independent noise at each iteration. We characterize the behavior of the training dynamics near any parameter vector that achieves zero training error, in terms of an implicit regularization te…
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.
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.
New algorithm improves dataset distillation with 108% improvement on ImageNet.
problem Improving dataset distillation performance.
method Reparameterization and convexification of implicit gradients (RCIG).
result Establishes new state-of-the-art on various dataset distillation tasks.
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…
Two-layer neural networks learn efficiently using kernel methods in mean-field analysis.
problem Feature learning ability of two-layer neural networks in the mean-field regime.
method Mean-field analysis through kernel methods, focusing on dynamics of the first layer's kernel.
result Two-layer neural networks can learn a union of multiple reproducing kernel Hilbert spaces more efficiently than kernel methods.
Bayesian neural networks simplified with input augmentation.
problem Uncertainty in deep learning models.
method Layer-wise input augmentation to induce uncertainty distributions.
result State-of-the-art performance in uncertainty representation.
End-to-end model predicts multiagent trajectories using game theory and neural nets.
problem Predicting trajectories of interacting agents in complex scenarios.
method Hybrid neural net with game-theoretic reasoning, using implicit layers to map preferences to Nash equilibria.
result Trains an interpretable model that predicts future trajectories and transfers to decision making.
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.
Proposes EM-HRNN model for better hierarchical language representation.
problem Learning hierarchical representations in natural language processing.
method Introduces latent indicator layer and EM algorithm for training.
result EM-HRNN model outperforms other RNN-based models in document classification.
Algorithmic approaches endow deep learning systems with implicit bias that helps them generalize even in over-parametrized settings. In this paper, we focus on understanding such a bias induced in learning through dropout, a popular technique to avoid overfitting in deep learning. For single hidden-layer linear neural …
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.
Gradient descent dynamics in neural networks show quenching and activation phases.
problem Understanding training dynamics in neural networks.
method Numerical and phenomenological study of gradient descent algorithm for two-layer neural networks.
result Gradient descent dynamics exhibit quenching and activation phases in under-parametrized networks.
In this paper, we consider one dimensional (shallow) ReLU neural networks in which weights are chosen randomly and only the terminal layer is trained. First, we mathematically show that for such networks L2-regularized regression corresponds in function space to regularizing the estimate's second derivative for fairly …
Early alignment in neural networks leads to sparse representations but hinders convergence.
problem The implicit bias of gradient descent during early training phases.
method Quantitative description of early alignment phase in small initialisation, one hidden layer networks.
result Early alignment induces a sparse representation but also hinders convergence to global minima.
The paper generalizes equivariant neural networks on homogeneous spaces to the non-linear setting.
problem Equivariant neural networks on homogeneous spaces.
method Deriving generalized steerability constraints for non-linear equivariant layers.
result The universality of the derived construction for non-linear equivariant layers.
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.
In this work, we contribute a new multi-layer neural network architecture named ONCF to perform collaborative filtering. The idea is to use an outer product to explicitly model the pairwise correlations between the dimensions of the embedding space. In contrast to existing neural recommender models that combine user em…
Paper explains neural collapse in neural networks using a new model.
problem Understanding neural collapse in neural networks during training.
method Introducing the unconstrained layer-peeled model (ULPM) to prove gradient flow convergence to critical points of a minimum-norm separation problem.
result Proves that all critical points are strict saddle points except the global minimizers exhibiting neural collapse.
MSLs use parallelizable root-finding for efficient ODE and PDE solutions.
problem Efficiently solving initial value problems for ODEs and PDEs.
method Leveraging time-parallel methods, MSLs use parallelizable root-finding algorithms.
result MSLs offer significant speedups in NFEs and inference time.
Study shows trained neural networks can overfit without bias or variance issues.
problem Understanding overfitting in trained two-layer ReLU networks.
method Analysis of gradient flow in the neural tangent kernel regime, decomposition of excess risk.
result Trained networks can overfit benignly without bias or variance issues.
A new layer learns abstract relations from graph structure using finite-state automata.
problem Learning abstract relations from graph structure for program analysis.
method Relaxing the problem into learning finite-state automata policies on a graph-based POMDP and training these policies using implicit differentiation.
result GFSA layer finds shortcuts in grid-world graphs and reproduces simple static analyses on Python programs.
We introduce a new deep convolutional neural network, CrescendoNet, by stacking simple building blocks without residual connections. Each Crescendo block contains independent convolution paths with increased depths. The numbers of convolution layers and parameters are only increased linearly in Crescendo blocks. In exp…
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.
Batch Normalization (BN) improves both convergence and generalization in training neural networks. This work understands these phenomena theoretically. We analyze BN by using a basic block of neural networks, consisting of a kernel layer, a BN layer, and a nonlinear activation function. This basic network helps us unde…
When optimizing over-parameterized models, such as deep neural networks, a large set of parameters can achieve zero training error. In such cases, the choice of the optimization algorithm and its respective hyper-parameters introduces biases that will lead to convergence to specific minimizers of the objective. Consequ…
This paper presents OptNet, a network architecture that integrates optimization problems (here, specifically in the form of quadratic programs) as individual layers in larger end-to-end trainable deep networks. These layers encode constraints and complex dependencies between the hidden states that traditional convoluti…
DVIP improves on IP-based methods by using IPs as priors over latent functions.
problem Limited expressiveness of IP-based models, especially in function space.
method Proposes DVIP, a multi-layer generalization of IPs, and scalable variational inference.
result DVIP outperforms previous IP-based methods and deep GPs in regression and classification tasks.
Gradient descent converges to a global minimum in nonlinear ReLU implicit networks with linear width.
problem Understanding convergence of gradient methods in nonlinear, infinitely deep ReLU networks.
method Introduced a scaling constant to ensure well-posedness of the equilibrium equation, proving convergence to a global minimum for linear width networks.
result Gradient descent converges to a global minimum at a linear rate for nonlinear ReLU implicit networks with linear width.
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.
Study reveals biases in gradient descent for GLNs, improving neural network performance.
problem Understanding and improving the inductive biases of deep neural networks.
method Derive infinite-time training limit of gated linear networks and generalize to other networks.
result Theoretical framework captures key inductive biases of ReLU networks.
Generalization is essential for deep learning. In contrast to previous works claiming that Deep Neural Networks (DNNs) have an implicit regularization implemented by the stochastic gradient descent, we demonstrate explicitly Bayesian regularizations in a specific category of DNNs, i.e., Convolutional Neural Networks (C…
Improved portfolio optimization reduces sensitivity to neural network initialization.
problem High sensitivity to neural network initialization in portfolio optimization.
method Robust end-to-end framework for risk budgeting portfolios.
result Enhanced stability in portfolio optimization without compromising performance.