New neural network approximates convex option prices.
problem Approximating prices of options with convex payoffs.
method Input Convex Neural Network (ICNN) architecture, with a scrambling phase.
result Validated convergence and effectiveness in estimating option prices.
Convex dual network improves neural network reconstruction for medical imaging.
problem Non-convex nature of neural networks hinders their use in sensitive applications.
method Introduces a convex duality framework for a two-layer fully-convolutional ReLU denoising network.
result Training neural networks with weight decay regularization induces path sparsity and piecewise linear filtering.
HyCNNs improve convex function learning and optimal transport.
problem Learning and optimizing convex functions efficiently.
method Combining Maxout networks and ICNNs to create a new neural architecture.
result HyCNNs require fewer parameters and outperform existing methods in convex tasks.
Wide and shallow networks approximate convex functions well.
problem Understanding why wide and shallow neural networks perform well.
method Analyzing the epigraph of the input-output map of shallow and wide neural networks.
result The epigraph of the input-output map approximates a convex function.
We solve the optimization of two-layer ReLU networks using convex math.
problem Optimizing two-layer ReLU neural networks.
method Exact characterization of optimal solutions via convex optimization.
result We prove that all globally optimal solutions can be found via convex optimization.
Convex neural networks enforce convex constraints on weights and activations, improving generalization.
problem Improving generalization and reducing overfitting in neural networks.
method Enforce convex constraints on weights and activations, using non-negative weights and non-decreasing convex activation functions.
result Convex neural networks self-regularize, outperforming base architectures and achieving similar performance to convolutional architectures.
Unified framework for analyzing neural networks trained by gradient descent.
problem Lack of generalizable guarantees for neural networks trained by gradient descent.
method Proxy convexity and proxy Polyak-Lojasiewicz inequalities.
result Unified guarantees for neural networks trained by gradient descent.
Presented are two neural network architectures for convex functions, demonstrating competitive performance.
problem Approximating convex functions efficiently and accurately.
method Developed two neural network architectures: one based on linear-by-part representation and the other on cubic splines.
result Cubic ICKAN networks produce results similar to classical ICNNs in solving convex approximation problems.
New regularizers tighten convex relaxation bounds for neural networks.
problem Large gap between certifiable and empirical robustness in neural networks.
method Two regularizers to train neural networks yielding tighter convex relaxation bounds.
result Higher certified accuracy with proposed regularizers.
Develops exact convex optimization formulations for neural networks.
problem Training two-layer neural networks with rectified linear units.
method Uses semi-infinite duality and minimum norm regularization to develop exact convex optimization formulations.
result Shows equivalence of ReLU networks trained with weight decay to block ℓ1 penalized convex models. Novel analysis of neural networks using geometric algebra and convex optimization.
problem Understanding the inner workings of deep neural networks.
method Geometric (Clifford) algebra and convex optimization.
result Optimal weights are given by the wedge product of training samples.
Paper develops exact convex optimization for neural networks with polynomial activations.
problem Training two-layer neural networks with nonlinear polynomial activations.
method Exact convex optimization using semidefinite programming.
result Global optimization of neural networks is polynomial-time computable.
This work shows neural networks can solve non-convex constraints problems.
problem Training neural networks under non-convex constraints.
method Project stochastic gradient descent with no-regret analysis of online learning.
result Overparameterized neural networks achieve near-optimal and near-feasible solutions.
We develop a convex relaxation method for analyzing neural network generalization.
problem Analyzing the generalization of parallel positively homogeneous networks.
method Linking non-convex ERM to a convex optimization problem over prediction functions.
result Achieved generalization bounds with almost linear sample complexity in network width.
Convex optimization refines neural network training, improving model performance and reducing hyperparameter sensitivity.
problem Training deep neural networks using non-convex optimization methods often leads to suboptimal solutions and requires extensive tuning.
method Formulate neural network training as convex programs with regularization terms, leveraging sparse recovery models and semi-infinite programming theory.
result Convex models can achieve global optima and outperform traditional non-convex methods, with improved robustness to hyperparameters.
Neural networks can represent complex piecewise functions efficiently.
problem Representing continuous piecewise affine functions with neural networks.
method Two hidden layers with ReLU activation, O(p) neurons for p pieces. result CPA functions can be represented by a neural network with linear size.
Paper introduces ICGNs to model convex gradients.
problem Modeling convex gradients efficiently.
method Integrates Jacobian-vector product in a neural network.
result Single layer ICGN outperforms single layer ICNN in fitting.
Neural networks solve copositive programs, revealing insights into training problems.
problem Training two-layer vector-output ReLU neural networks.
method Convex analysis and copositive programming.
result Neural networks solve copositive programs, providing insights into training problems.
New method approximates neural network training for robustness.
problem Training robust neural networks with adversarial input perturbations.
method Stochastic convex optimization approach to adversarial training.
result Method achieves better adversarial robustness and performance.
Neural nets solve electric field in non-convex microfluidic devices.
problem Solving differential equations in non-convex geometries.
method Neural network approximation of electric potential and field.
result Deep neural networks outperform shallow networks in accuracy.
Neural networks solve the Dirichlet problem for Monge-Ampère equations.
problem Solving the Dirichlet problem for the Monge-Ampère equation.
method Using deep input convex neural networks to find the unique convex solution.
result Deep input convex neural networks can solve the Monge-Ampère Dirichlet problem.
Convex geometry explains optimal neural network parameters.
problem Understanding optimal parameters in over-parameterized neural networks.
method Convex geometry, extreme points, linear spline interpolation, kernel matrix, cutting-plane algorithm.
result Optimal network parameters can be characterized as interpretable closed-form formulas.
Analysis of over-parameterized neural networks has drawn significant attention in recentyears. It was shown that such systems behave like convex systems under various restrictedsettings, such as for two-level neural networks, and when learning is only restricted locally inthe so-called neural tangent kernel space aroun…
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.
This work connects neural network training to convex optimization via the NTK.
problem Understanding and optimizing neural network training via convex programs.
method Interpreting gated ReLU network as MKL, showing NTK equivalence, and improving weights.
result The NTK cannot perform better than the optimal MKL kernel on the training set.
New saddle network architectures preserve convex-concave geometry in optimization problems.
problem Optimization models with convex x and concave y components.
method Structured separable decomposition and saddle network architectures.
result Proven one-dimensional approximation theorem and high accuracy on various test functions.
Recent studies show overparameterized neural networks behave like convex systems.
problem Understanding the behavior of overparameterized neural networks.
method Analysis of two-layer neural networks, focusing on restricted settings and neural tangent kernel space.
result Overparameterized neural networks behave like convex systems under certain conditions.
Path regularization reveals convex optimization in deep ReLU networks.
problem Understanding the optimization landscape of deep neural networks.
method Introducing path regularization to make the training problem convex and sparsity-inducing.
result Path regularized parallel ReLU networks are a parsimonious convex model in high dimensions.
Improved neural network robustness certification through tighter convex relaxations.
problem Certifying neural network robustness to perturbed and adversarial inputs.
method Exploiting ReLU network structure, novel partition-based certification procedure.
result Tightens existing linear programming relaxations to achieve zero relaxation error asymptotically.
New method trains quantized neural networks to global optimality.
problem Training optimal quantized neural networks is intractable due to combinatorial non-convex optimization.
method Convex optimization strategy using hidden convexity, semidefinite lifting, and Grothendieck's identity.
result Quantized NN problems can be solved to global optimality in polynomial-time.
Sig-Splines model uses signatures and splines for time series data, achieving universality and convexity.
problem Creating a generative model for multivariate time series data.
method Combines linear transformations and signature transforms into a neural spline flow.
result Achieves universality and introduces convexity in model parameters.
Optimization algorithms help overparameterized neural networks achieve high performance.
problem Understanding the convergence of optimization algorithms on overparameterized neural networks.
method Analyzing a broad class of optimization algorithms using dynamical systems and finite over-parameterized neural networks with ReLU activation.
result The Heavy Ball method converges to global minimum at a linear rate, while NAG converges sublinearly.
Deep neural nets on 1-D data are convex Lasso models with reflection features.
problem Training neural networks on 1-D data.
method Proving equivalence to convex Lasso problems with discrete, explicitly defined dictionary matrices.
result Reflection features in neural networks with certain activations.
Randomized Geometric Algebra for Convex Neural Networks Optimizes Transfer Learning.
problem Training neural networks to global optimality via convex optimization.
method Randomized algorithms in Clifford's Geometric Algebra for hypercomplex vector spaces.
result Convex optimization and geometric algebra improve LLMs' robustness and reliability in transfer learning.
RedEx improves neural network optimization with convex optimization guarantees.
problem Difficult optimization of neural networks.
method RedEx architecture using convex optimization with semi-definite constraints.
result RedEx can efficiently learn functions fixed methods cannot.
Several recently proposed architectures of neural networks such as ResNeXt, Inception, Xception, SqueezeNet and Wide ResNet are based on the designing idea of having multiple branches and have demonstrated improved performance in many applications. We show that one cause for such success is due to the fact that the mul…
Wasserstein GANs are shown to have hidden convexity, enabling exact solutions with convex optimization.
problem Non-convex and non-concave optimization in GANs.
method Convex duality analysis of Wasserstein GANs with two-layer neural network discriminators.
result Wasserstein GANs can be solved exactly with convex optimization under certain conditions.
SGD converges with positive probability for non-convex deep neural networks under specific conditions.
problem Convergence of SGD for non-convex deep neural networks.
method Established local convergence with positive probability under local Łojasiewicz condition and additional structural assumption.
result SGD converges with positive probability for non-convex deep neural networks under specific conditions.
Learning rate annealing helps even in convex problems, improving generalization.
problem Improving generalization in neural networks, especially convex problems.
method Learning rate annealing schedule (large initial, then small learning rate).
result Gradient descent can reach minima with better generalization using learning rate annealing.
This study uses neural networks to solve interpolation problems with sparse, infinitely wide layers.
problem Exact data interpolation using sparse, infinitely wide neural networks.
method Atomic norm framework to derive convex hulls and equivalent convex formulations.
result Simple characterizations of convex hulls for different constraints on network weights and biases.
New method trains neural networks with threshold activation functions efficiently.
problem Training neural networks with threshold activation functions is challenging due to zero gradients.
method We study weight decay regularized training problems of deep neural networks with threshold activations, showing they can be formulated as convex optimization problems.
result Regularized deep threshold network training problems can be formulated as standard convex optimization problems, paralleling the LASSO method.
In this paper, we present a novel and principled approach to learn the optimal transport between two distributions, from samples. Guided by the optimal transport theory, we learn the optimal Kantorovich potential which induces the optimal transport map. This involves learning two convex functions, by solving a novel mi…
New weight initialisation for ICNNs accelerates learning and improves generalization.
problem Lack of effective initialisation strategies for ICNNs due to their unique weight and activation properties.
method Derived a principled weight initialisation by generalizing signal propagation theory for ICNNs with non-negative weights.
result Principled initialisation effectively accelerates learning and leads to better generalization in ICNNs.
The paper develops mixed-integer formulations for neural networks using partitioning.
problem Optimizing trained ReLU neural networks with balanced model size and tightness.
method Partitioning node inputs into groups, forming the convex hull via disjunctive programming.
result The proposed formulations outperform existing ones, especially with fewer partitions.
New method improves neural network verification by considering multivariate input space of ReLU neurons.
problem Improving the effectiveness of neural network verification algorithms.
method A new tightened convex relaxation for ReLU neurons considering multivariate input space.
result Our convex relaxation is significantly stronger than the commonly used univariate-input relaxation.
Alternative neural network training using monotone variational inequality.
problem Training neural networks efficiently and with guarantees.
method Using monotone variational inequality to solve non-convex problems efficiently.
result Our approach leads to fast convergence and competitive performance compared to traditional methods.
A scalable algorithm approximates Wasserstein Barycenters using neural networks.
problem Representing the weighted mean of probability distributions in high dimensions.
method Input Convex Neural Networks (ICNNs) for Kantorovich dual formulation of Wasserstein-2 distance.
result Generative model representation of the Barycenter with infinite samples.
Almost all local minima in neural networks are strongly convex.
problem The prevalence of strongly convex neighborhoods around local minima in neural network optimization landscapes.
method Rigorous analysis of shallow neural networks with analytic activation functions, dividing parameter space into efficient and redundant domains.
result For shallow neural networks on the efficient domain, almost all local minima are strongly convex.