New conditions prevent gaps in optimal control problems.
problem Preventing gaps in optimal control problems with state constraints.
method Developed new sufficient conditions not relying on convexity.
result Derived bounds for the size of the relaxation gap.
The paper simplifies conditions for optimal paths on manifolds avoiding obstacles.
problem Finding optimal paths on manifolds avoiding obstacles.
method Study of sufficient conditions for optimality on Riemannian manifolds and Lie groups.
result New conditions for optimality are provided in terms of matrix invertibility.
We consider a problem of optimal investment with intermediate consumption in the framework of an incomplete semimartingale model of a financial market. We show that a necessary and sufficient condition for the validity of key assertions of the theory is that the value functions of the primal and dual problems are finit…
Study proves optimal controls for stochastic Volterra equations with singular kernels.
problem Existence of optimal controls for stochastic Volterra equations with singular kernels.
method Sufficient conditions based on integrability and growth hypotheses.
result Existence of optimal relaxed and strict controls under classical convexity assumptions.
We study the error landscape of deep linear and nonlinear neural networks with the squared error loss. Minimizing the loss of a deep linear neural network is a nonconvex problem, and despite recent progress, our understanding of this loss surface is still incomplete. For deep linear networks, we present necessary and s…
A new sufficient condition ensures convergence of Adam and RMSProp in non-convex settings.
problem Adam and RMSProp diverge in convex settings, despite being influential for deep learning.
method Introduces a sufficient condition based on base learning rate and historical second-order moments.
result Guarantees global convergence of generic Adam/RMSProp in non-convex optimization.
This paper optimizes MDP policies for efficient state aggregation.
problem Optimizing policies in aggregated Markov chains while preserving optimal performance.
method Homomorphic mappings to establish optimal policy equivalence and derive performance bounds.
result Developed HPG and EBHPG methods for efficient aggregation and policy optimization.
The ℓ-1 norm based optimization is widely used in signal processing, especially in recent compressed sensing theory. This paper studies the solution path of the ℓ-1 norm penalized least-square problem, whose constrained form is known as Least Absolute Shrinkage and Selection Operator (LASSO). A solution path …
Pushing a little forward an approach proposed by Villani, we are going to prove that in the Riemannian setting the condition ∇2f<g implies that f is c-concave with respect to the quadratic cost as soon as it has a sufficiently small C1-norm. From this, we deduce a sufficient condition for the optimalit…
Paper checks SSC for matrix factorizations using Gurobi.
problem Checking the SSC for various matrix factorizations.
method Formulated as a non-convex quadratic optimization problem over a bounded set, solved with Gurobi.
result SSC can be checked in reasonable time for realistic scenarios.
Researchers found the longest arcs for specific sub-Lorentzian structures.
problem Finding the longest arcs for sub-Lorentzian structures.
method Optimal control problem with unbounded control set and concave cost functional. Sufficient conditions for existence of longest arcs proposed.
result Existence of the longest arcs for left-invariant three-dimensional contact sub-Lorentzian structures proved.
This paper extends Kelly Criterion to include rebalancing frequency for optimal portfolio selection.
problem Optimizing a portfolio with multiple assets and varying rebalancing frequency.
method Using Kelly Criterion, the paper derives necessary and sufficient conditions for the frequency-based Kelly optimal portfolio.
result Proves the necessity and sufficiency of conditions for the frequency-based Kelly optimal portfolio.
Optimal strategies are found for a repeated betting game using diffusion approximation.
problem Finding optimal strategies for a repeated betting game with i.i.d. outcomes.
method Constructing a diffusion approximation of the repeated game and analyzing the wealth share process.
result Necessary and sufficient conditions for the wealth share process to be transient or recurrent are derived.
New analysis shows GMD can converge linearly under PL-like conditions.
problem Establishing linear convergence for generalized mirror descent.
method PL-based analysis for time-dependent mirrors, Taylor-series approach for stochastic GMD.
result Linear convergence of stochastic GMD under PL-like conditions.
Optimizes symplectic matrices under Euclidean and invariant metrics.
problem Finding optimal symplectic matrices under different metrics.
method Necessary and sufficient conditions for critical points, Hessian formula, retraction map.
result Detailed steepest descent and Newton algorithms for optimization.
Study high-dimensional Bayesian linear regression using variational inference.
problem High-dimensional Bayesian linear regression with product priors.
method Non-linear large deviations theory and variational inference.
result Unique optimizer in variational problem governs posterior distribution under separation condition.
In the problem of optimal investment with utility function defined on (0,∞), we formulate sufficient conditions for the dual optimizer to be a uniformly integrable martingale. Our key requirement consists of the existence of a martingale measure whose density process satisfies the probabilistic Muckenhoupt $(A_p…
Optimal maps, solutions to the optimal transportation problems, are completely determined by the corresponding c-convex potential functions. In this paper, we give simple sufficient conditions for a smooth function to be c-convex when the cost is given by minimizing a Lagrangian action.
We give sufficient conditions on initial and target measures supported on the sphere §n to ensure the solution to the optimal transport problem with the cost ∣x−y∣2/2 is a diffeomorphism.
TREGO improves EGO for global optimization of high-dimensional problems.
problem Efficient Global Optimization struggles with high dimensions and lacks theoretical guarantees.
method TREGO alternates between EGO steps and local steps within a trust region.
result TREGO outperforms EGO and other methods in black-box optimization problems.
Defines new geodesic semilocal E-preinvex functions and studies their properties.
problem Defines new functions to generalize existing convex and preinvex concepts.
method Introduces geodesic semilocal E-preinvex functions and proves their properties.
result Establishes sufficient optimality conditions for nonlinear fractional multiobjective programming.
The paper shows that causal identification is not essential for efficient portfolios, focusing on geometric sufficiency conditions.
problem The necessity of causal identification for efficient portfolios.
method Re-examination of predictive signals and their impact on portfolio efficiency under structural misspecification.
result Efficiency is governed by geometric sufficiency conditions (directional alignment, ranking preservation, and calibration) rather than causal identification.
New binary matrices improve compressed sensing with faster and less storage requirements.
problem Achieving robust sparse recovery with binary measurement matrices.
method Derived bounds and conditions for binary matrices to satisfy the robust null space property (RNSP).
result Binary matrices with girth six are nearly optimal for compressed sensing.
DG algorithms often fail to generalize well in limited domains, highlighting necessary vs. sufficient conditions.
problem DG algorithms fail to consistently outperform ERM in limited domains.
method Examined necessary and sufficient conditions for DG, proposing a subspace alignment method.
result DG methods focus on sufficient conditions, often neglecting necessary conditions, leading to generalization failures.
The paper considers an investment timing problem appearing in real options theory. Present values from an investment project are modeled by general diffusion process. We prove necessary and sufficient conditions under which an optimal investment time is induced by threshold strategy. We study also the conditions of opt…
Sharp risk bounds for early-stopping in Gaussian linear regression are derived.
problem Minimizing in-sample mean squared error in high-dimensional Gaussian linear regression.
method Early-stopped mirror descent (ESMD) with local Gaussian width bounds.
result Sharp risk bounds extend to early-stopped mirror descent for least squares estimator (LSE).
New algorithm SFHC achieves near-optimal costs with predictions for non-convex optimization.
problem Online optimization with non-convex hitting costs and movement costs.
method Synchronized Fixed Horizon Control (SFHC) algorithm with conditions on hitting and movement costs.
result Synchronized Fixed Horizon Control (SFHC) achieves a 1+O(1/w) competitive ratio for near-optimal costs. This paper studies dynamic stochastic optimization problems parametrized by a random variable. Such problems arise in many applications in operations research and mathematical finance. We give sufficient conditions for the existence of solutions and the absence of a duality gap. Our proof uses extended dynamic programm…
Optimal scaling found to depend on operator norm across large models and datasets.
problem Lack of unifying principle for optimal hyperparameter scaling across models and datasets.
method Discovered that optimal scaling is conditioned on the operator norm of the output layer.
result The optimal learning rate/batch size pair (η∗,B∗) consistently has the same operator norm value. Stochastic gradient descent approximates Gaussian process posteriors efficiently.
problem Efficiently sampling from Gaussian process posteriors with limited computational resources.
method Developed stochastic gradient optimization objectives for sampling from Gaussian process posteriors.
result Stochastic gradient descent produces accurate predictive distributions, even in non-convergent cases.
This work improves transferability of rewards inferred from expert demonstrations.
problem Transferability of rewards inferred from expert demonstrations under limited access to the expert's policy.
method Proposed principal angles as a measure of similarity and dissimilarity between transition laws. Established sufficient conditions for transferability under limited access.
result Two key results on sufficient conditions for transferability to any and local changes in transition laws.
Computational topology is a vibrant contemporary subfield and this article integrates knot theory and mathematical visualization. Previous work on computer graphics developed a sequence of smooth knots that were shown to converge point wise to a piecewise linear (PL) approximant. This is extended to isotopic convergenc…
Optimal insurance strategy for maximizing RDEU under various premium principles.
problem Maximizing a risk-averse individual's RDEU with insurance priced by a distortion-deviation principle.
method Proved necessary and sufficient conditions for the optimal solution, considered ambiguity orders, and analyzed specific examples.
result Conditions for no insurance or deductible insurance to be optimal.
Exact recovery method for community detection in Gaussian mixtures with dependent noise.
problem Community detection in Gaussian mixtures with dependent and heterogeneous noise.
method Maximum likelihood estimator (MLE) for constrained quadratic optimization problem, using Σ-whitened separation and local inequalities. result Sharp exact-recovery threshold and no-gap mechanism in the unknown-size setting.
Establishes a condition for multiclass classification-calibration of Gamma-Phi losses.
problem Ensuring classification-calibration of multiclass Gamma-Phi losses.
method Develops a general sufficient condition for classification-calibration of Gamma-Phi losses.
result Proves the first family of nonconvex multiclass surrogate losses for which classification-calibration has been fully justified.
The aim of this work consists in the study of the optimal investment strategy for a behavioural investor, whose preference towards risk is described by both a probability distortion and an S-shaped utility function. Within a continuous-time financial market framework and assuming that asset prices are modelled by semim…
Study abelianization of Lie algebroids and groupoids, providing conditions for existence.
problem Existence conditions for abelianization of Lie algebroids and groupoids.
method Investigation of abelianization for Lie algebroids and groupoids, providing necessary and sufficient conditions.
result Necessary and sufficient conditions for the existence of abelianization in both Lie algebroids and groupoids.
For an infinite-horizon continuous-time optimal stopping problem under non-exponential discounting, we look for an optimal equilibrium, which generates larger values than any other equilibrium does on the entire state space. When the discount function is log sub-additive and the state process is one-dimensional, an opt…
Proves sufficient condition for 2D orbifolds to be good.
problem Characterizing 2D orbifolds as good.
method Analyzes orbifold fundamental groups for goodness.
result Connected 2D orbifolds with infinite orbifold fundamental group are good.
Investigates optimal execution under time-varying liquidity, preventing price manipulation.
problem Optimal execution with time-varying liquidity impacts and price manipulation prevention.
method Almgren-Chriss framework, deterministic time variation, well-posedness, second-order conditions, price manipulation prevention.
result Sufficient conditions for a unique solution and prevention of price manipulation.
New findings on flatness for specific driftless systems.
problem Determining flatness for driftless systems with m inputs and 2m or 2m-1 states.
method Using pure prolongation, the paper presents new sufficient conditions for flatness.
result The conditions proposed broaden the class of recognized flat systems.
We consider the high-dimensional discriminant analysis problem. For this problem, different methods have been proposed and justified by establishing exact convergence rates for the classification risk, as well as the l2 convergence results to the discriminative rule. However, sharp theoretical analysis for the variable…
This paper introduces SS-MAMP to address convergence issues in AMP algorithms.
problem Convergence issues in AMP algorithms for signal reconstruction.
method Proposes SS-MAMP algorithm framework for right-unitarily invariant sensing matrices and Lipschitz-continuous local processors.
result Covariance matrices of SS-MAMP are L-banded and convergent, ensuring optimal convergence.
We propose a stochastic variance reduced optimization algorithm for solving sparse learning problems with cardinality constraints. Sufficient conditions are provided, under which the proposed algorithm enjoys strong linear convergence guarantees and optimal estimation accuracy in high dimensions. We further extend the …
We consider the pricing of American put options in a model-independent setting: that is, we do not assume that asset prices behave according to a given model, but aim to draw conclusions that hold in any model. We incorporate market information by supposing that the prices of European options are known. In this setting…
Paper proves min-vol NMF robust to noise under expanded condition.
problem Robustness of min-vol NMF to noise.
method Proved robustness under expanded sufficiently scattered condition.
result Proves min-vol NMF identifies groundtruth factors in noise.
This study is aimed at answering the famous question of how the approximation errors at each iteration of Approximate Dynamic Programming (ADP) affect the quality of the final results considering the fact that errors at each iteration affect the next iteration. To this goal, convergence of Value Iteration scheme of ADP…
Continuous-time mean-variance portfolio selection model with nonlinear wealth equations and bankruptcy prohibition is investigated by the dual method. A necessary and sufficient condition which the optimal terminal wealth satisfies is obtained through a terminal perturbation technique. It is also shown that the optimal…