The paper proposes a method to train NNs with a small Lipschitz constant to improve robustness.
problem Neural networks' susceptibility to adversarial perturbations in safety-critical applications.
method The paper introduces a framework to train multi-layer NNs by minimizing their Lipschitz constant, using an optimization scheme based on the Alternating Direction Method of Multipliers.
result The proposed training procedure successfully increases the robustness of neural networks.
LALR adapts learning rate for faster convergence in regression and neural nets.
problem Finding optimal learning rates for faster convergence in regression and neural networks.
method Lipschitz continuity theory applied to Mean Absolute Error and Quantile loss functions.
result Adaptive learning rate policy enables up to 20x faster convergence.
We model how Lipschitz continuity changes during neural network training.
problem Understanding how Lipschitz continuity evolves during training.
method We use a system of stochastic differential equations to capture the dynamics of Lipschitz continuity under SGD.
result We identify three factors driving the evolution of Lipschitz continuity: gradient flow projection, gradient noise, and Hessian projection.
Researchers develop a new framework to control neural network sensitivity.
problem Understanding and controlling the behavior of neural networks.
method Direct parameterization of bi-Lipschitzness in convex neural networks.
result A clear and tight control of neural network sensitivity achieved.
The paper quantifies the regularity of attention operations.
problem Quantifying the regularity of attention operations.
method Proposes a new mathematical framework using measure theory and integral operators.
result Proves attention operation is Lipschitz continuous on compact domains and provides an estimate of its Lipschitz constant.
Develops risk measures on Lipschitz spaces for financial positions.
problem Lack of standard cash-additive methods in Lipschitz spaces.
method Proposes Lipschitz-free space, uses additivity along benchmark-deviation instruments.
result Derives dual representations for convex and coherent risk measures.
This paper tackles safe global optimization of noisy functions with a Lipschitz condition.
problem Safe global maximization of expensive, noisy, Lipschitz functions.
method Develops a δ-Lipschitz framework and two algorithms to ensure safety constraints are met.
result The proposed methods ensure safety constraints are met before evaluating noisy functions.
We develop a method to learn neural network activations with controlled Lipschitz constant.
problem Increase neural network capacity while controlling Lipschitz constant.
method Variational framework to learn activation functions with piecewise-linear constraints.
result Proves existence of solutions with continuous and piecewise-linear activations.
Transfer learning for bandits with latent Lipschitz continuity.
problem Learning to transfer structural information from prior tasks to new tasks.
method Proposes a framework to estimate Lipschitz constant from prior tasks and apply it to new tasks.
result Regret bound close to oracle algorithm with full knowledge of Lipschitz constant under mild assumptions.
LiPopt uses polynomial optimization to estimate neural network Lipschitz constants efficiently.
problem Estimating the Lipschitz constant of neural networks efficiently.
method Sparse polynomial optimization, leveraging network connectivity to reduce complexity.
result Superior estimates of the ℓ ∞ \ell_\infty ℓ ∞ -Lipschitz constant compared to existing methods. RYU framework constructs safe regions for optimization problems.
problem Optimization problems with specific component functions.
method RYU framework for constructing safe regions.
result RYU framework improves upon state-of-the-art methods.
New framework tightens certified robustness gaps in machine learning models.
problem Persistent gap between theoretical certified robustness and empirical accuracy.
method Leverages Lipschitz continuity and novel confidence intervals.
result Improves robust accuracy, compressing the gap between theory and practice.
Framework learns robust control policies from expert demonstrations.
problem Adversarial robustness and closed-loop generalization in feedback control policies.
method Lipschitz-constrained loss minimization for certified robustness and generalization.
result Finite sample bound on policy learning error and robust closed-loop stability.
Paper presents an efficient algorithm for estimating Lipschitz functions from noisy data.
problem Estimating unknown Lipschitz functions from noisy observations.
method Extends max-affine methods to Lipschitz setting using nonlinear feature expansion and adaptive partitioning.
result Achieves minimax convergence rate with respect to intrinsic dimension, up to logarithmic factors.
Stochastic Lipschitz bandit algorithms balance exploration and exploitation, and have been used for a variety of important task domains. In this paper, we present a framework for Lipschitz bandit methods that adaptively learns partitions of context- and arm-space. Due to this flexibility, the algorithm is able to effic…
New method improves optimization algorithms without Lipschitz smoothness.
problem Improving optimization algorithms in the absence of Lipschitz smoothness.
method Dual kernel conditioning (DKC) to provide dual Lipschitz continuity.
result First complexity bounds and iterate convergence for random reshuffling mirror descent.
Graph-based framework for provably robust adversarial training.
problem Adversarial robustness of machine learning models.
method Formulates adversarial robustness as loss minimization with a Lipschitz constraint, using graph-based discretization and primal-dual algorithms.
result Establishes a connection between elliptic operators and adversarial learning, and proves fundamental lower bounds on adversarial sensitivity.
Unified framework for analyzing graph neural operators converging to graph limits.
problem Analyzing convergence of graph neural operators to graph limits.
method Develops a unified spectral framework for graph neural operators under various graphon assumptions.
result Unified framework enables direct comparison of convergence rates and tradeoffs.
Paper presents a method to accurately estimate Lipschitz constants for DNNs.
problem Estimating Lipschitz constants for deep neural networks is crucial for robustness and stability analysis.
method Convex optimization framework using quadratic constraints to solve SDP for accurate and efficient estimation.
result Our method provides the most accurate Lipschitz bounds compared to existing methods.
We know SGAN may have a risk of gradient vanishing. A significant improvement is WGAN, with the help of 1-Lipschitz constraint on discriminator to prevent from gradient vanishing. Is there any GAN having no gradient vanishing and no 1-Lipschitz constraint on discriminator? We do find one, called GAN-QP. To construct a …
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 ∞ \ell_{\infty} 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.
SRNF framework extends surface distance to Lipschitz surfaces.
problem Defining a distance metric for unparametrized surfaces.
method Square Root Normal Fields (SRNF) and Wasserstein Fisher Rao (WFR) metric.
result SRNF distance on Lipschitz surfaces is equivalent to WFR metric.
Mathematical framework for understanding attention in neural networks.
problem Lack of theoretical understanding of attention in neural networks.
method Proposes a measure-theoretic model of attention and interprets self-attention as a system of self-interacting particles.
result Shows that attention is Lipschitz-continuous under suitable assumptions.
New framework enhances neural network robustness against adversarial attacks.
problem Vulnerability of deep neural networks to small perturbations.
method Integrates Lipschitz constraint using optimal transport and hinge regularization.
result Proposes a new loss function that certifies adversarial robustness.
Develops a new framework for temporal anchoring in deep embedding spaces.
problem Temporal anchoring in deep embedding spaces, especially drift and convergence issues.
method Operator-theoretic framework with drift maps and event-indexed blocks, proving convergence theorems and equivalence theorems.
result Proves convergence theorems and equivalence theorems for the proposed framework.
MLDL preserves manifold geometry in vector transformations.
problem Geometric deterioration in neural network transformations.
method Locally isometric smoothness (LIS) and Markov random field (MRF) encoding.
result Enhanced vector transformations into well-behaved metric homeomorphisms.
This paper presents a novel nonmyopic adaptive Gaussian process planning (GPP) framework endowed with a general class of Lipschitz continuous reward functions that can unify some active learning/sensing and Bayesian optimization criteria and offer practitioners some flexibility to specify their desired choices for defi…
ECPv2 optimizes Lipschitz functions efficiently and scalably.
problem Global optimization of Lipschitz-continuous functions with unknown Lipschitz constants.
method Adapting the Every Call is Precious (ECP) framework, ECPv2 introduces adaptive lower bounds, Worst-m memory, and random projections to reduce computational cost and improve acceptance regions.
result ECPv2 retains ECP's no-regret guarantees with optimal finite-time bounds and expands the acceptance region with high probability.
We show that stochastic interpolation flow maps are Lipschitz with a sharp constant.
problem High dimensional sampling and transport problems.
method Investigating stochastic interpolation flow for generating data samples.
result Stochastic interpolation flow maps are Lipschitz with a sharp constant matching optimal transport maps.
Proves continuum limits of Lipschitz learning using Γ-convergence.
problem Semi-supervised learning with graph-based methods and continuum limits of p p p -Laplacian learning. method Proves continuum limits of Lipschitz learning using Γ-convergence.
result Proves Γ Γ Γ -convergence in the L ∞ L^\infty L ∞ -topology to the supremum norm of the gradient. Unified framework for risk-aware policy learning in contextual bandits.
problem Optimizing decision rules in high-stakes domains with adverse outcomes.
method Distributional framework for Lipschitz-continuous risk functionals, with novel empirical concentration inequalities.
result Data-dependent suboptimality bounds with an i l d e O ( 1 / n ) ilde{\mathcal{O}}(1/\sqrt{n}) i l d e O ( 1/ n ) rate, matching risk-neutral offline policy optimization. Proposes DMOC for more nuanced neural network robustness.
problem Lipschitz continuity is too coarse for nuanced data-dependent behavior.
method Data-driven, architecture-agnostic framework based on DMOC.
result DMOC provides a finer notion of robustness relative to data distribution.
Optimizes privacy-preserving optimization for heavy-tailed data.
problem Privacy-preserving optimization with heavy-tailed gradients.
method Pure ε-differential privacy framework for Lipschitz extensions.
result Minimax optimal excess-risk rate for pure ε-DP heavy-tailed SCO.
Neural networks are dense among Lipschitz functions with fixed Lipschitz constant.
problem Characterizing neural network approximations to Lipschitz functions.
method Analyzing L L L -Lipschitz neural networks and their density in L L L -Lipschitz functions. result One layer neural networks are dense in the set of all L L L -Lipschitz functions. Paper introduces risk assessment for contextual bandits without experiments.
problem Evaluate policies using logged data in context bandits.
method Lipschitz risk functionals and Off-Policy Risk Assessment (OPRA) framework.
result OPRA provides finite sample guarantees for various risk estimates.
The paper proves new comparison theorems for sub-Laplacian in foliations with minimal leaves.
problem Proving comparison theorems for sub-Laplacian in Riemannian foliations with minimal leaves.
method Using Riemannian foliations with minimal leaves, the paper proves comparison theorems for the sub-Laplacian.
result The comparison theorems yield a Bonnet-Myers type theorem, stochastic completeness, and Lipschitz regularization property for the sub-Riemannian semigroup.
Revisits shallow neural networks using Lipschitz norms and measures.
problem Existence and compactness of minimizers in neural network formulations.
method Mean field parametrization, signed measures, duality pairings, Kantorovich-Rubinstein norms.
result Compactness results and uniform large data limits for empirical risk minimization.
Constructs a map with prescribed local Lipschitz constants on a subset of a manifold.
problem Creating a Lipschitz map with specific local Lipschitz constants on a subset of a manifold.
method Constructs a Lipschitz map that matches a given map on a subset and has a local Lipschitz constant defined by a continuous function.
result A Lipschitz map can be constructed with a local Lipschitz constant prescribed by a continuous function.
Enhances ENet's prediction accuracy while maintaining uncertainty estimation.
problem Gradient shrinkage problem in ENet's loss function.
method Proposes a multi-task learning framework with a modified MSE loss function.
result Improves ENet's prediction accuracy without losing uncertainty estimation.
Maps between certain Lipschitz manifolds are isometries if they preserve volume.
problem Volume preservation and isometry conditions for Lipschitz manifolds.
method Volume-preserving 1-Lipschitz maps from integral currents onto infinitesimally Euclidean Lipschitz manifolds.
result Volume-preserving maps are isometries under given conditions.
Improved robustness for deep neural networks with tighter bounds and attacks.
problem Loose upper bounds and prohibitive computation in existing adversarial robustness methods.
method Primal approach with exact Lipschitz certificates for ReLU networks and modern architectures, and novel Wasserstein Distributional Attacks.
result Tighter upper bounds and greater flexibility in attack points compared to existing methods.
Enhances robustness of deep neural networks with randomized smoothing.
problem Improving robustness of deep neural networks against noisy inputs and adversarial attacks.
method Introduces a variance-margin trade-off approach to increase certified robust radius using pre-trained models.
result Significant improvement in certified accuracy compared to state-of-the-art methods.
Abstract: Lipschitz homeomorphisms are deformed using Perelman's methods.
problem Deformation of Lipschitz homeomorphisms
method Lipschitz analogues of Siebenmann's and Perelman's homeomorphism theory
result Lipschitz stability theorem and gluing theorem
Deep neural networks can approximate rough functions with high accuracy.
problem Approximating rough functions with neural networks.
method Proved that ENO interpolation can be cast as a deep ReLU neural network, transferring ENO's high-order accuracy.
result Deep neural networks can achieve high-order accuracy in approximating Lipschitz functions.
Deep neural operators learn complex probabilistic models efficiently.
problem Learning complex probabilistic models with global Lipschitz conditions.
method Deep neural-operator framework under global Lipschitz conditions.
result Explicit network-size bounds for universal approximation of probabilistic models.
We compute the local Lipschitz constant of ReLU networks precisely.
problem Estimating the local Lipschitz constant of ReLU networks is hard.
method We use a novel approach involving the generalized Jacobian and backpropagation.
result We provide an algorithm to compute the exact Lipschitz constant of ReLU networks.
Paper introduces a new framework to improve sample efficiency in POMDPs learning.
problem Challenges in off-policy evaluation for POMDPs, especially with hidden states.
method Exploits the metric structure of belief space to relax coverage assumptions.
result Unified analysis technique yields tighter error bounds and sample efficiency improvements.
We introduce a new approximation of f f f -divergences for machine learning.
problem Variational representations of f f f -divergences for machine learning. method Definition and analysis of Moreau-Yosida approximation of f f f -divergences with the Wasserstein-1 metric. result Generalization and relaxation of hard Lipschitz constraints in f f f -divergences.