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.
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.
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.
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.
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.
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.
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.
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.
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.
Proposes a differentiable LSE-ICNN for modeling multi-well potentials.
problem Modeling multi-well potentials in various scientific domains.
method Log-sum-exponential (LSE) mixture of input convex neural network (ICNN) modes.
result Smooth surrogate that retains convexity within basins and allows gradient-based learning.
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.
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…
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 certifies neural network robustness under random input noise.
problem Certifying neural network robustness against random input noise.
method Chance-constrained optimization problem reformulated with input-output samples, convex conditions developed.
result Proposed method certifies robustness against various input noise regimes over larger uncertainty regions.
Paper proposes a new model for better engine control.
problem Optimal control problems are non-convex and hard to solve online.
method Combines Hammerstein-Wiener model with input convex neural networks.
result Optimal control problems are effectively solvable due to convexity and partial linearity.
SOC-ICNN expands neural network representational capacity by using conic optimization.
problem Restrictive representational capacity of ReLU-based ICNNs.
method Proposes SOC-ICNN architecture that uses Second-Order Cone Programming.
result SOC-ICNN strictly expands representational space without increasing complexity.
We give a new algorithm for learning a two-layer neural network under a general class of input distributions. Assuming there is a ground-truth two-layer network y=Aσ(Wx)+ξ, where A,W are weight matrices, ξ represents noise, and the number of neurons in the hidden layer is no larger than the input or outp…
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.
A new method lifts training of input-convex neural networks to avoid dead weights and plateaued loss.
problem Training input-convex neural networks with non-negative weights.
method Introduces a hypernetwork that emits non-negative weights from a summary of the input batch, adding stochasticity to soften the loss landscape.
result The lift method achieves lower test loss than projected gradient descent and direct softplus reparametrization.
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.
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.
Scalable algorithm for computing Wasserstein-2 barycenters without bias.
problem Computing Wasserstein-2 barycenters efficiently and accurately.
method Input convex neural networks and cycle-consistency regularization.
result Our approach avoids introducing bias and does not require minimax optimization.
Safe offline RL for chemical reactors using input convex neural networks.
problem Safe control of exothermic polymerization reactors using historical data.
method Gymnasium-compatible simulation, behaviour cloning, implicit Q-learning, input convex neural networks (PICNNs).
result Offline RL with convex action correction outperforms traditional control approaches.
Training neural networks is a challenging non-convex optimization problem, and backpropagation or gradient descent can get stuck in spurious local optima. We propose a novel algorithm based on tensor decomposition for guaranteed training of two-layer neural networks. We provide risk bounds for our proposed method, with…
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.
Neural optimal transport improves multivariate conformal prediction.
problem Multivariate quantile regression challenges and existing methods ignore joint distribution geometry.
method Combines neural optimal transport with amortized optimization for efficient training and faster inference.
result Constructs tighter and more informative predictive regions for multivariate conformal prediction.
We apply neural nets with ReLU gates in online reinforcement learning. Our goal is to train these networks in an incremental manner, without the computationally expensive experience replay. By studying how individual neural nodes behave in online training, we recognize that the global nature of ReLU gates can cause und…
This study explains gradient flow dynamics in neural networks for small initialisation.
problem Understanding the training dynamics of neural networks for small initialisation.
method Analysis of gradient flow dynamics for one-hidden layer ReLU networks with orthogonal inputs.
result Gradient flow converges to zero loss and characterizes implicit bias towards minimum variation norm.
Study probabilistic safety of BNNs under adversarial attacks.
problem Evaluate vulnerability of BNNs to adversarial attacks.
method Relaxation techniques from non-convex optimization to compute probabilistic safety bounds.
result Certify probabilistic safety of BNNs with millions of parameters.
Deep neural networks favor symmetric structures, enabling multilevel symmetries.
problem Understanding and optimizing deep neural networks.
method Formulating DNN training as convex Lasso problems with geometric algebra.
result Deep networks inherently favor symmetric structures, enabling multilevel symmetries.
Optimizes functionals on probability space using ICNNs.
problem Optimizing functionals on the space of probabilities with high-dimensional convex functions.
method Proposes an approach using input-convex neural networks (ICNNs) to approximate the JKO scheme.
result Demonstrates feasibility and validity in approximating solutions of PDEs and molecular discovery.
We develop a fast, tractable technique called Net-Trim for simplifying a trained neural network. The method is a convex post-processing module, which prunes (sparsifies) a trained network layer by layer, while preserving the internal responses. We present a comprehensive analysis of Net-Trim from both the algorithmic a…
We propose a novel method for computing exact pointwise robustness of deep neural networks for all convex ℓp norms. Our algorithm, GeoCert, finds the largest ℓp ball centered at an input point x0, within which the output class of a given neural network with ReLU nonlinearities remains unchanged. We relat…
Paper provides efficient robustness certificates for neural networks.
problem Ensuring neural networks are robust against adversarial attacks.
method Two-step approach: 1) Efficient convex optimization for robustness certificates with bounded Hessian eigenvalues, 2) Curvature-based regularization during training.
result Significantly higher certified robust accuracy achieved compared to existing methods.
Although deep learning has shown its powerful performance in many applications, the mathematical principles behind neural networks are still mysterious. In this paper, we consider the problem of learning a one-hidden-layer neural network with quadratic activations. We focus on the under-parameterized regime where the n…
A neural network learns a convex regularizer for better image reconstruction.
problem Improving image reconstruction in inverse problems.
method Adversarial training of a data-adaptive ICNN as a convex regularizer.
result The convex regularizer leads to better convergence and error reduction in image reconstruction.
We consider deep linear networks with arbitrary convex differentiable loss. We provide a short and elementary proof of the fact that all local minima are global minima if the hidden layers are either 1) at least as wide as the input layer, or 2) at least as wide as the output layer. This result is the strongest possibl…
In this paper, we consider parameter recovery for non-overlapping convolutional neural networks (CNNs) with multiple kernels. We show that when the inputs follow Gaussian distribution and the sample size is sufficiently large, the squared loss of such CNNs is locally strongly convex in a basin of attraction…
PEREGRiNN verifies safety of ReLU NNs by penalizing relaxation in a greedy manner.
problem Formal verification of safety specifications for ReLU NNs.
method Uses a relaxed convex program to verify polytopic input/output constraints, penalizing relaxation and forcing largest relaxations to early layers.
result Significantly faster and more properties verified compared to other approaches.
Neural network implementation of Brenier's polar factorization for vector fields.
problem Implementing Brenier's polar factorization theorem for vector fields using neural networks.
method Parameterizing the convex function u as an input convex neural network and estimating the measure-preserving map M. result Practical neural implementation of Brenier's polar factorization theorem.
Our objective is to estimate the unknown compositional input from its output response through an unknown system after estimating the inverse of the original system with a training set. The proposed methods using artificial neural networks (ANNs) can compete with the optimal bounds for linear systems, where convex optim…
This paper bounds the Lipschitz constants of neural networks and their gradients.
problem Estimating the Lipschitz constant of complex models like neural networks.
method Local upper and lower bounds on Lipschitz constants computed with respect to network parameters.
result It is impossible to derive global upper bounds for the Lipschitz constants of neural networks.
We present an approach for designing correct-by-construction neural networks (and other machine learning models) that are guaranteed to be consistent with a collection of input-output specifications before, during, and after algorithm training. Our method involves designing a constrained predictor for each set of compa…
We study model recovery for data classification, where the training labels are generated from a one-hidden-layer neural network with sigmoid activations, also known as a single-layer feedforward network, and the goal is to recover the weights of the neural network. We consider two network models, the fully-connected ne…
This paper develops a framework for training and evaluating neural networks for MPC.
problem Lack of a general framework for characterizing learning approaches in MPC.
method Developed a framework using PyTorch and CVXPY, incorporating hit-and-run sampling for efficient training data generation.
result Proposed metrics for validating neural network-based MPC approaches.
MQF2 forecasts multivariate quantiles globally.
problem Forecasting multi-horizon dependencies with error accumulation.
method Multivariate quantile function using input-convex neural networks.
result MQF2 avoids quantile crossing and captures time dependency. Over-parameterization makes optimization easier for simple neural networks, even with minor extra neurons.
problem Understanding the impact of over-parameterization on optimization landscapes of shallow neural networks.
method Analyzing a simple ReLU neural network with Gaussian inputs, focusing on optimization properties and landscape changes.
result Over-parameterization makes the objective function one-point strongly convex in most directions, aiding optimization.
One of the mysteries in the success of neural networks is randomly initialized first order methods like gradient descent can achieve zero training loss even though the objective function is non-convex and non-smooth. This paper demystifies this surprising phenomenon for two-layer fully connected ReLU activated neural n…