Solves complex Monge-Ampère equation with Hölder continuous boundary data.
problem Complex Monge-Ampère equation with Hölder continuous boundary data.
method Solves the Dirichlet problem for the complex Monge-Ampère equation.
result The solution is Hölder continuous if the boundary data is Hölder continuous.
Study analyzes a new algorithm for complex optimization problems.
problem Stochastic bilevel optimisation problems in continuous-time models.
method Continuous-time, two-timescale stochastic approximation algorithm.
result Obtained weak convergence rate using central limit theorem.
Proves Hölder continuity of complex Monge-Ampère solutions.
problem Global Hölder continuity of solutions to complex Monge-Ampère equation.
method Analyzes Dirichlet problem on strictly pseudoconvex domains or Hermitian manifolds.
result Proves global Hölder continuity of solutions under given conditions.
Study a continuous-time PA problem with private effort and consumption decisions.
problem Continuous-time Principal-Agent problem with private information.
method Proposes a new sufficient condition for solving the agent's problem directly.
result Directly yields a solution to the agent's problem without verification.
Study on Hölder continuity of complex Monge-Ampère solutions on Stein spaces.
problem Understanding continuity of solutions to complex Monge-Ampère equations on Stein spaces.
method Analyzing solutions with Lp densities and Hölder boundary data on Stein spaces with isolated singularities. result Solutions are Hölder continuous outside singular points if boundary data is Hölder continuous.
This paper tackles convex-submodular minimax problems in mixed continuous-discrete domains.
problem Convex-submodular minimax problems in mixed continuous-discrete domains.
method Introduces new notions of optimality and proposes iterative algorithms combining discrete and continuous optimization.
result Characterizes convergence rates, computational complexity, and quality of solutions for convex and monotone-submodular minimax problems.
Proposes hybrid reinforcement learning for both discrete and continuous control problems.
problem Real-world control problems involving both discrete and continuous decision variables.
method Solves hybrid problems by optimizing for discrete and continuous actions simultaneously.
result Efficiently solves hybrid reinforcement learning problems and improves upon expert heuristics.
Generalizes PCA and ICA for continuous-time signals using neural networks.
problem Low-rank decomposition of continuous-time vector-valued signals.
method Implicit neural network framework to learn numerical approximations of PCA and ICA.
result Unified approach to PCA and ICA in continuous domain, enforcing decorrelation and independence.
Researchers find continuous solutions to minimizers in weighted least gradient problems.
problem Existence and regularity of minimizers to weighted least gradient problems.
method Constructing continuous solutions using Sternberg-Williams-Ziemer technique extended to inhomogeneous variations.
result Continuous solutions constructed for minimizers in any dimension n≥2, with level sets being minimal surfaces in a conformal metric.
Bayesian optimization tackles mixed discrete-continuous problems with Gaussian processes.
problem Optimizing problems with both discrete and continuous variables using costly simulations.
method Relaxing discrete variables into continuous latent variables, using Bayesian optimization, and incorporating compatibility constraints with Lagrangians.
result Comparative analysis of different mixed Bayesian optimization approaches.
In this paper we prove the probabilistic continuous complexity conjecture. In continuous complexity theory, this states that the complexity of solving a continuous problem with probability approaching 1 converges (in this limit) to the complexity of solving the same problem in its worst case. We prove the conjecture ho…
The paper studies continuous submodular functions and their optimization.
problem Maximizing continuous submodular functions in poly. time.
method Characterization of continuous submodularity, operations preserving it, and algorithms for constrained maximization.
result Continuous submodularity is equivalent to a weak DR property, leading to continuous DR-submodular functions with the full DR property.
Study on continual learning with Twitter data, developing ConGraD algorithm.
problem Personalized online language learning on a massive scale.
method Developed POLL problem setting, collected Firehose datasets, and introduced ConGraD algorithm.
result ConGraD algorithm outperforms prior continual learning methods on Firehose datasets.
Investment strategy optimization from discrete to continuous models.
problem Optimizing investment strategies and stopping times in both continuous and discrete settings.
method Characterized value functions via quadratic reflected BSDEs for continuous case, discretized BSDEs for discrete case, and derived uniform convergence rates.
result Uniform convergence and rate from discrete to continuous quadratic reflected BSDEs.
Unique solutions found for Plateau problems in smooth and continuous calibrations.
problem Finding unique solutions to the Plateau problem for specific types of currents.
method Boundary regularity theory for area-minimizing currents and unique continuation argument.
result Every compactly supported smoothly or continuously calibrated integral current is the unique solution to the Plateau problem for its boundary data.
3D good continuation model explains stereo vision using neurogeometry.
problem Understanding how the brain processes 3D visual correspondence.
method Developed a neurogeometric model involving spatial and orientation disparities.
result Provides insight into neural organization and correspondence problem.
Study on existence and properties of continuous solutions to complex Hessian equations.
problem Existence and properties of continuous solutions to complex Hessian equations.
method Established new capacity estimates and weak stability estimates for the m-Hessian measure. result Existence of continuous solutions to the complex Hessian equation under certain conditions.
Continuous-time algorithms improve online learning performance.
problem Online learning with sequential data and minimizing overall regret.
method Extending discrete-time algorithms to continuous-time models for online linear optimization, adversarial bandit, and adversarial linear bandit.
result Optimal regret bounds are proven for continuous-time settings.
Study robust utility maximization with uncertain continuous semimartingales.
problem Maximizing utility in continuous time under model uncertainty.
method Duality and conjugate problems for logarithmic, exponential, and power utilities.
result Existence of optimal portfolios for various utilities.
New algorithm for aggregate inference in HMMs with continuous observations.
problem Inference in large populations with indistinguishable individuals and continuous measurements.
method Continuous observation collective forward-backward algorithm extending existing discrete case algorithm.
result Efficacy demonstrated through numerical experiments.
Introduces strong equilibrium for time-inconsistent stopping problems in continuous time.
problem Time-inconsistent stopping problems in continuous time.
method Introduces strong equilibrium, compares with existing mild and weak equilibria, and provides an iteration method to construct optimal mild equilibria.
result Optimal mild equilibria are always strong equilibria under certain conditions.
Paper finds unique viscosity solution to complex control problems.
problem Complex stochastic control problems with singular terminal state constraints.
method Establishes existence of unique nonnegative continuous viscosity solution using novel comparison principle.
result Unique viscosity solution to HJB equation for linear-quadratic control problems.
Research on unique continuation principles in medical and seismic imaging.
problem Understanding unique continuation principles for inverse problems.
method Integral geometry and fractional calculus methods applied to various imaging problems.
result Developed new techniques for solving inverse problems with partial data.
ARCADe detects anomalies in a sequence of tasks with limited data.
problem Learning a sequence of anomaly detection tasks with only normal class examples.
method Formulated as a meta-learning problem, ARCADe addresses catastrophic forgetting and overfitting.
result ARCADe outperforms baselines on three datasets.
Researchers created a continuous Markov martingale that mimics Brownian motion but lacks the strong Markov property.
problem Constructing a continuous Markov martingale with Brownian marginals that misses the strong Markov property.
method Developed a new approach to create a continuous Markov martingale that differs from Brownian motion in terms of the strong Markov property.
result A continuous Markov martingale with Brownian marginals that lacks the strong Markov property was successfully constructed.
Root's barrier is continuous and finite under certain conditions.
problem Continuity of the root barrier function.
method Analyzing Skorokhod embedding problem and properties of target measures.
result The barrier function is continuous and finite under specified conditions.
Researchers prove uniqueness and continuity of solution to L_p dual Minkowski problem.
problem Proving uniqueness and continuity of solution to L_p dual Minkowski problem.
method Established new Minkowski-type inequalities related to optimization problem.
result Uniqueness and continuity of solution for general convex bodies when q<p. Tabular Q-Learning with learned state abstractions solves continuous control tasks.
problem Challenging reinforcement learning problems in continuous control.
method Learned state abstraction to transform continuous state-space into discrete.
result Tabular Q-Learning with learned abstractions achieves efficient learning in unseen tasks.
Paper studies continuous prediction with experts' advice using differential equations.
problem Continuous prediction with experts' advice in online learning.
method Continuous-time stochastic calculus and differential equations.
result Improved guarantees for quantile regret with continuous-time algorithm.
The paper proves differentiability of evolution maps in Lie groups.
problem Differentiability of evolution maps in infinite-dimensional Lie groups.
method Showed sequential continuity (Mackey k-continuity) leading to differentiability.
result Differentiability of evolution maps in Ck-semiregular Lie groups. Introduces a continuous version of LWE problem.
problem Hardness of learning mixtures of Gaussians.
method Polynomial-time quantum reduction from CLWE to lattice problems.
result CLWE shares hardness with LWE.
We provide a model-free pricing-hedging duality in continuous time. For a frictionless market consisting of d risky assets with continuous price trajectories, we show that the purely analytic problem of finding the minimal superhedging price of a path dependent European option has the same value as the purely probabi…
This paper examines challenges and solutions for solving variational inequalities.
problem Stability issues in solving variational inequalities, especially in multi-objective scenarios.
method Continuous-time analysis to understand and improve stability of algorithms.
result Understanding continuous-time dynamics can help in designing more stable algorithms for variational inequalities.
Efficient deep policy gradient method for continuous-time control problems.
problem Optimal control in continuous time with fine time discretization.
method Multi-scale deep policy gradient method with varying time discretization.
result Targeted efficiency in computational resources achieved through multi-scale approach.
We classify hyperbolic monopoles with continuous symmetries and construct new examples.
problem Classifying and constructing hyperbolic monopoles with continuous symmetries.
method Developed a Structure Theorem and used representation theory to simplify the problem.
result Found constraints on structure groups and constructed novel spherically symmetric Sp(n) hyperbolic monopoles. CGNNs use wavelets for continuous function generation in infinite-dimensional spaces.
problem Generating continuous functions in infinite-dimensional spaces for applications like inverse problems.
method Inspired by DCGAN, CGNNs use wavelet multiresolution analysis with convolutional and nonlinear layers.
result CGNNs can be injective under certain conditions on filters and nonlinearity, leading to Lipschitz stability estimates.
We prove continuity of a controlled SDE solution in Skorokhod's M1 and J1 topologies and also uniformly, in probability, as a non-linear functional of the control strategy. The functional comes from a finance problem to model price impact of a large investor in an illiquid market. We show that M1-continuity is…
Derives an approximation algorithm for continuous submodular maximization without derivative information.
problem Maximizing a continuous submodular function with only function values and no derivative information.
method Black-box Continuous Greedy algorithm for DR-submodular functions, extended to stochastic setting.
result Achieves a (1−1/e)OPT−ε approximation guarantee with O(d/ε3) function evaluations. Study capacity constraints in continual learning with a simple model.
problem Understanding optimal resource allocation for agents with limited memory and compute resources.
method Analyzes a capacity-constrained linear-quadratic-Gaussian (LQG) sequential prediction problem and demonstrates optimal capacity allocation strategies.
result Derives a solution to the capacity-constrained LQG sequential prediction problem and shows how to optimally allocate capacity across sub-problems in the steady state.
In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally Hölder continuous. In [39], F. H. Lin proposed a challenge problem: Can the Hölder continuity …
Research shows continual learning challenges in confounded datasets.
problem Challenges in mitigating confounders in continual learning settings.
method Formal description of continual confounders, construction of ConCon dataset.
result Standard continual learning methods fail to ignore confounders.
Paper tackles risk-sensitive impulse control for continuous-time processes.
problem Risk-sensitive impulse control for continuous-time Feller-Markov processes.
method Probabilistic approach to solve Bellman equation and construct optimal strategy.
result Optimal strategy approximated by dyadic impulse strategies.
We extend an offline evaluation method for continuous-armed bandits.
problem Evaluating policies in continuous-armed bandits with limited data.
method Extend existing offline evaluation method for continuous-armed bandits.
result Our method provides a consistent ranking of policies.
Stochastic gradient descent in continuous time (SGDCT) provides a computationally efficient method for the statistical learning of continuous-time models, which are widely used in science, engineering, and finance. The SGDCT algorithm follows a (noisy) descent direction along a continuous stream of data. SGDCT performs…
CAQL tackles continuous action maximization in Q-learning.
problem Maximizing continuous actions in Q-learning.
method Developed CAQL algorithms using plug-and-play optimizers and MIP for optimal max-Q.
result CAQL outperforms policy-based methods in heavily constrained environments.
Novel mean estimation method under user-level differential privacy reduces noise in continual mean estimates.
problem Maintaining accurate running mean estimates under user-level differential privacy.
method Developed a novel mean estimation specific factorization under approximate differential privacy.
result Achieved asymptotically lower mean-squared error bounds in continual mean estimation.
New algorithm approximates continuous Wasserstein barycenters efficiently.
problem Computing Wasserstein barycenters for continuous distributions.
method Stochastic algorithm using dual potentials and stochastic gradient descent.
result Efficient online approximation of continuous Wasserstein barycenters.
A new method optimises problems with both continuous and categorical inputs.
problem Optimising black-box problems with mixed continuous and categorical inputs.
method Continuous and Categorical Bayesian Optimisation (CoCaBO) combining multi-armed bandits and Bayesian optimisation.
result CoCaBO outperforms existing methods on synthetic and real-world tasks.