We review basic concepts of convex duality, focusing on the very general and supremely useful Fenchel-Rockafellar duality. We summarize how this duality may be applied to a variety of reinforcement learning (RL) settings, including policy evaluation or optimization, online or offline learning, and discounted or undisco…
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48 results for “Fenchel duality”
The paper defines Fenchel conjugate and biconjugate on Hadamard manifolds.
problem Defining Fenchel conjugate and biconjugate on curved spaces.
method Introduced a new definition of Fenchel conjugate and biconjugate on Hadamard manifolds based on the tangent bundle.
result Developed a Fenchel-Moreau Theorem for geodesically convex functions on Hadamard manifolds.
Paper offers a dual formulation for consumption problem with multiplicative habit.
problem Optimal consumption with multiplicative habit formation.
method Dual formulation using Fenchel's Duality Theorem.
result Strong duality result linking primal and dual controls.
Explicit Recursive Construction of Super-Replication Prices under Proportional Transaction Costsq-fin.MF
New method calculates super-hedging prices with transaction costs.
problem Super-hedging European contingent claims under proportional transaction costs.
method Explicit recursive scheme based on convex duality and Legendre-Fenchel transform.
result Computes super-hedging price and optimal strategy without martingale arguments.
We study the convex duality method for robust utility maximization in the presence of a random endowment. When the underlying price process is a locally bounded semimartingale, we show that the fundamental duality relation holds true for a wide class of utility functions on the whole real line and unbounded random endo…
New method for pricing financial products without no-arbitrage condition.
problem Pricing financial products without relying on no-arbitrage conditions.
method Convex duality and Fenchel conjugate for estimating super-replication cost.
result Endogenous weak no-arbitrage condition (AIP) leads to finite prices.
A new screening rule 'dynamic Sasvi' improves sparse optimization speed.
problem Sparse optimization problem identification.
method Flexible framework based on Fenchel-Rockafellar duality for norm-regularized least squares.
result Dynamic Sasvi can eliminate more features and increase solver speed.
Unified Kantorovich duality for multimarginal optimal transport on Polish spaces.
problem Optimal transport of multiple probability distributions.
method Unified Kantorovich duality theory for multimarginal optimal transport on general Polish product spaces.
result Unified duality theory for multimarginal optimal transport, extending classical two-marginal conjugacy.
Subdifferential theory and the Fenchel conjugate via Busemann functions on Hadamard manifoldsmath.DG
The paper defines subdifferentials on Hadamard manifolds and identifies conditions for Fenchel conjugate equality.
problem Understanding convex analysis on Riemannian manifolds.
method Using Busemann functions to define subdifferentials and investigate Fenchel conjugate equality.
result Identifies conditions for equality in the Fenchel-Young inequality on Hadamard manifolds.
New dual formulation reduces generalization error for ERM-fDR.
problem Generalization error in constrained optimization problems.
method Introduces a dual formulation of ERM-fDR using Legendre-Fenchel transform and implicit function theorem.
result Explicit characterizations of generalization error for algorithms under mild conditions.
For several decades, the no-arbitrage (NA) condition and the martingale measures have played a major role in the financial asset's pricing theory. We propose a new approach for estimating the super-replication cost based on convex duality instead of martingale measures duality: Our prices will be expressed using Fenche…
Reward suffices for convex MDPs, expanding RL to new problems.
problem Capturing goals as convex functions of stationary distribution.
method Reformulated as a min-max game using Fenchel duality.
result Convex MDPs require non-stationary reward functions.
The Stochastic Occupation Kernel (SOCK) Method for Learning Stochastic Differential Equationsstat.ML
A new method estimates SDEs using occupation kernels.
problem Learning multivariate stochastic differential equations (SDEs).
method Two-step procedure: estimate drift, then diffusion. Occupation kernels used in RKHS.
result Validated on simulated and real-world data.
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.
In this paper we study the variational problem associated to support vector regression in Banach function spaces. Using the Fenchel-Rockafellar duality theory, we give explicit formulation of the dual problem as well as of the related optimality conditions. Moreover, we provide a new computational framework for solving…
A new framework for averaging spatio-temporal signals using optimal transport and soft alignments.
problem Averaging complex datasets with time and spatial components.
method Inspired by DTW, OT, and UOT, a new loss function is proposed to address shifts in time, space, and population size.
result The proposed loss function can be used to compute spatio-temporal barycenters efficiently.
Study optimizes option pricing with robust strategies, ensuring consistency with vanilla option prices.
problem Optimizing exotic option pricing with robust strategies.
method Introduces semistatic strategies and robust convex integral functionals on bounded continuous functions.
result Consistent indifference prices with observed vanilla option prices.
Develops a new duality between entropy martingale optimal transport and nonlinear pricing-hedging.
problem Entropy Martingale Optimal Transport problem and its associated optimization problem.
method Combines Entropy Optimal Transport and Martingale Optimal Transport theories, with novel penalization terms and constraints.
result Establishes a nonlinear robust pricing-hedging duality, covering various known robust results.
Defines Fenchel-Nielsen coordinates for SL(3,C) representations.
problem No specific problem stated; coordinates defined for a new context.
method Introduced Fenchel-Nielsen coordinates for representations.
result Relates to classical and generalized Fenchel-Nielsen coordinates.
Nonlinear Fenchel Conjugatesmath.FA
Generalizes Fenchel conjugation to nonlinear functions on arbitrary sets.
problem Extending Fenchel conjugation to functions on arbitrary sets without structure.
method Replacing linear test functions with nonlinear ones, investigating properties including biconjugation.
result Derived further results on smooth manifolds and Lie groups, relating to convexity.
This paper concerns the recursive utility maximization problem. We assume that the coefficients of the wealth equation and the recursive utility are concave. Then some interesting and important cases with nonlinear and nonsmooth coefficients satisfy our assumption. After given an equivalent backward formulation of our …
Building upon recent advances in entropy-regularized optimal transport, and upon Fenchel duality between measures and continuous functions , we propose a generalization of the logistic loss that incorporates a metric or cost between classes. Unlike previous attempts to use optimal transport distances for learning, our …
A Fenchel Theorem for the Gauss maps and uniqueness of minimizers of nonlocal curvature energiesmath.CA
The paper proves a Fenchel theorem for Gauss maps and shows circles and disks minimize certain energies.
problem Finding minimizers of nonlocal curvature energies.
method Combining Fenchel-type theorems with geometric analysis techniques.
result Circles and disks minimize specific energy functionals.
Calculates twist in Teichmüller space using cross ratios.
problem Calculating the Fenchel-Nielsen twist in Teichmüller space.
method Using cross ratio coordinates.
result Compact calculation of twist in Teichmüller space.
Introduces Fitzpatrick losses, tighter than Fenchel-Young losses.
problem Improving loss functions for machine learning.
method Introduces Fitzpatrick losses based on the Fitzpatrick function.
result Fitzpatrick losses are tighter than Fenchel-Young losses.
Fenchel-Willmore and Sobolev-type inequalities for submanifolds in non-negatively curved manifoldsmath.DG
Paper connects Fenchel-Willmore and Sobolev inequalities for submanifolds in curved spaces.
problem Developing inequalities for submanifolds in curved spaces.
method Connecting Fenchel-Willmore and logarithmic Sobolev inequalities for mean-convex submanifolds.
result Established extensions of Fenchel-Willmore inequality and derived new Sobolev-type inequalities.
Various Alexandrov-Fenchel type inequalities have appeared and played important roles in convex geometry, matrix theory and complex algebraic geometry. It has been noticed for some time that they share some striking analogies and have intimate relationships. The purpose of this article is to shed new light on this by c…
Regularizing the optimal transport (OT) problem has proven crucial for OT theory to impact the field of machine learning. For instance, it is known that regularizing OT problems with entropy leads to faster computations and better differentiation using the Sinkhorn algorithm, as well as better sample complexity bounds …
Abstract: Poisson bracket on shear coordinates relates to Fenchel-Nielsen bracket on gluing parameters.
problem Relationship between Fenchel-Nielsen coordinates and shear coordinates on Riemann surfaces.
method Explicitly showed the Poisson bracket on shear coordinates induces the Fenchel-Nielsen bracket on gluing parameters.
result Poisson bracket on shear coordinates relates to Fenchel-Nielsen bracket on gluing parameters.
Study flows to analyze sphere quermassintegrals.
problem Analyze quermassintegrals on the sphere.
method Use two types of flows to study quermassintegrals.
result Establish Alexandrov-Fenchel inequalities for the sphere.
Learning Classifiers with Fenchel-Young Losses: Generalized Entropies, Margins, and Algorithmsstat.ML
This paper studies Fenchel-Young losses, a generic way to construct convex loss functions from a regularization function. We analyze their properties in depth, showing that they unify many well-known loss functions and allow to create useful new ones easily. Fenchel-Young losses constructed from a generalized entropy, …
Sharp inequality for submanifolds in manifolds with non-negative Ricci curvature.
problem Establishing a Fenchel-Willmore inequality for submanifolds in manifolds with non-negative Ricci curvature.
method Analyzing submanifolds in manifolds with non-negative intermediate Ricci curvature and Euclidean volume growth.
result Sharp Fenchel-Willmore inequality for submanifolds in manifolds with non-negative intermediate Ricci curvature.
Given a real-valued function defined on the Heisenberg group, we provide a definition of abstract convexity and Fenchel transform that takes into account the sub-Riemannian structure of the group. In our main result, we prove that, likewise the Euclidean case, a convex function can be characterized via its iterated Fen…
Alexandrov-Fenchel inequalities for convex anisotropic capillary hypersurfaces in the half-spacemath.DG
Paper extends capillary convex body results to anisotropic setting with Alexandrov-Fenchel inequalities.
problem Extending capillary convex body results to anisotropic setting.
method Developed theory for anisotropic capillary convex bodies in half-space and established Alexandrov-Fenchel inequality for mixed volumes.
result Established a general Alexandrov-Fenchel inequality for mixed volumes of anisotropic capillary convex bodies, weakening and extending previous results.
Paper proves a generalized Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary.
problem Proving a generalized Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary.
method Using a locally constrained nonlinear curvature flow to preserve the -th quermassintegral and decrease the -th quermassintegral.
result Obtained the Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in .
The paper computes Fenchel-Nielsen coordinates for fixed points of cyclic actions on Teichmüller space.
problem Computing Fenchel-Nielsen coordinates for cyclic actions on Teichmüller space.
method Developed algorithms to describe Fenchel-Nielsen coordinates of fixed points of cyclic subgroups of Mod(S_g) on Teich(S_g).
result Computed Fenchel-Nielsen coordinates for cyclic subgroups of orders 10, 8, and 4 in Mod(S_2).
Given a topological orientable surface of finite or infinite type equipped with a pair of pants decomposition and given a base complex structure on , there is an associated deformation space of complex structures on , which we call the Fenchel-Nielsen Teichmüller space associated to the pair $(\…
Topological proof of Weil-Petersson symplectic form using Fenchel-Nielsen coordinates.
problem Proving Wolpert's formula for the Weil-Petersson symplectic form.
method Introducing a cell decomposition and groupoid cocycle on a surface to represent points in Teichmüller space.
result Topological proof of Wolpert's formula for the Weil-Petersson symplectic form.
We extend the classical Aleksandrov-Fenchel inequality for mixed volumes to functionals arising naturally in hermitian integral geometry. As a consequence, we obtain Brunn-Minkowski and isoperimetric inequalities for hermitian quermassintegrals.
Study flow on de Sitter space for convex hypersurfaces.
problem Behavior of locally constrained inverse curvature flow in de Sitter space.
method Analyze flow on de Sitter space with specific initial conditions and inequalities.
result Derive Alexandrov-Fenchel type inequalities.
Risk-Sensitive Deep RL: Variance-Constrained Actor-Critic Provably Finds Globally Optimal Policycs.LG
This work tackles risk-sensitive deep RL by optimizing policies with variance constraints.
problem Risk and aleatoric uncertainty in deep reinforcement learning.
method Lagrangian and Fenchel dualities to transform the problem into an unconstrained saddle-point policy optimization problem, and an actor-critic algorithm to iteratively update policy, Lagrange multiplier, and Fenchel dual variable.
result The proposed actor-critic algorithm finds a globally optimal policy at a sublinear rate.
We find a monotone quantity along the inverse mean curvature flow and use it to prove an Alexandrov-Fenchel-type inequality for strictly convex hypersurfaces in the -dimensional sphere, .
Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundarymath.DG
Paper solves inequalities for capillary hypersurfaces in half-spaces.
problem Finding inequalities for convex capillary hypersurfaces in half-spaces.
method Introduced quermassintegrals and constructed a new locally constrained curvature flow to prove convergence to spherical caps.
result Obtained Alexandrov-Fenchel inequalities for convex capillary hypersurfaces.
In this paper, we solve various isoperimetric problems for the quermassintegrals and the curvature integrals in the hyperbolic space $\H^n$, by using quermassintegral preserving curvature flows. As a byproduct, we obtain hyperbolic Alexandrov-Fenchel inequalities.
We develop Fenchel-Nielsen coordinates for representations of surface groups into Sp(2n,R) with maximal Toledo invariant. Analogous to classical Fenchel-Nielsen coordinates on the Teichmüller space they consist of a parametrization of representations of the fundamental group of a pair of pants and a careful investigati…
New inequalities for convex hypersurfaces in various spaces.
problem Deriving inequalities for hypersurfaces under convex weight.
method Sharp weighted Alexandrov-Fenchel and Minkowski inequalities for smooth, closed hypersurfaces in Euclidean, spherical, and hyperbolic spaces.
result Incorporates convex, non-decreasing positive functions as weights, yielding a broad family of geometric inequalities.
The article proves inequalities for capillary hypersurfaces in hyperbolic space.
problem Proving inequalities for capillary hypersurfaces in hyperbolic space.
method Constructing a new locally constrained inverse curvature flow.
result Obtained Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in hyperbolic space.
Study of star-shaped hypersurfaces with capillary boundary using constrained mean curvature flow.
problem Understanding the evolution of hypersurfaces with capillary boundaries.
method Locally constrained mean curvature flow for star-shaped hypersurfaces in the half-space.
result Established new Alexandrov-Fenchel inequalities for convex hypersurfaces with capillary boundary.