arXiv research

New geometric proof of convex function differentiability and approximation.

Flow deforms locally convex curves to curves of constant k-order width.

Developed a theory of local convexity for second order differential equations on Lie algebroids.

Study convex embeddability in linear and circular orders, applying to knots.

State-of-the-art methods in convex and non-convex optimization employ higher-order derivative information, either implicitly or explicitly. We explore the limitations of higher-order optimization and prove that even for convex optimization, a polynomial dependence on the approximation guarantee and higher-order smoothn…

Study non-standard bi-orders on punctured torus bundles, matching standard ones in key subgroups.

Sharp comparison for sub-Gaussian random variables in convex order.

Second-order methods improve differential privacy in convex optimization.

Geodesic convexity generalizes the notion of (vector space) convexity to nonlinear metric spaces. But unlike convex optimization, geodesically convex (g-convex) optimization is much less developed. In this paper we contribute to the understanding of g-convex optimization by developing iteration complexity analysis for …

New algorithm reduces regret in stochastic bandit convex optimization.

Study on convex ordering in stochastic control for swing contracts, proving value function convexity.

Study first-order locally convex Lie algebroids in Bastiani calculus.

Strong geodesic convex function and strong monotone vector field of order $m$ on Riemannian manifolds have been established. A characterization of strong geodesic convex function of order $m$ for the continuously differentiable functions has been discussed. The relation between the solution of a new variational inequal…

First order methods can take extremely long to find global minima of non-convex functions.

Expands learning paradigm to stochastic orders using Choquet-Toland distance and Variational Dominance Criterion.

New algorithms optimize convex functions with high-order derivatives.

In this paper, we study stochastic non-convex optimization with non-convex random functions. Recent studies on non-convex optimization revolve around establishing second-order convergence, i.e., converging to a nearly second-order optimal stationary points. However, existing results on stochastic non-convex optimizatio…

Optimal algorithms for online convex optimization with random order.

Paper optimizes portfolio selection with ICX order constraints.

Book covers tools for zeroth-order convex optimisation.

New characterization of second-order stochastic dominance with applications in risk management.

The Lebesgue property (order-continuity) of a monotone convex function on a solid vector space of measurable functions is characterized in terms of (1) the weak inf-compactness of the conjugate function on the order-continuous dual space, (2) the attainment of the supremum in the dual representation by order-continuous…

Lower bounds for higher-order methods in non-convex optimization.

Paper finds optimal transport measures for arbitrage strategies.

New algorithm optimizes convex functions with noisy evaluations in one dimension.

Second-order guarantees for federated learning algorithms.

We consider the problem of stochastic comparison of general Garch-like processes, for different parameters and different distributions of the innovations. We identify several stochastic orders that are propagated from the innovations to the Garch process itself, and discuss their interpretations. We focus on the convex…

This paper studies quasar-convex functions to improve optimization methods.

We note that known methods achieving the optimal oracle complexity for first order convex optimization require quadratic memory, and ask whether this is necessary, and more broadly seek to characterize the minimax number of first order queries required to optimize a convex Lipschitz function subject to a memory constra…

Optimal transport theory characterizes convex order between probability measures.

Optimized method tackles convex optimization with heavy-tailed noise.

In this paper we propose an algorithm for exact partitioning of high-order models. We define a general class of $m$-degree Homogeneous Polynomial Models, which subsumes several examples motivated from prior literature. Exact partitioning can be formulated as a tensor optimization problem. We relax this high-order combi…

Zeroth-order optimization methods lack inherent privacy guarantees.

New memory-query tradeoffs for convex optimization algorithms.

The paper extends Strassen's theorem to include biased martingales for American options.

Improved regret bound for adversarial bandit convex optimisation.

Exact partitioning of high-order planted models achieved through convex optimization.

We consider empirical risk minimization of linear predictors with convex loss functions. Such problems can be reformulated as convex-concave saddle point problems, and thus are well suitable for primal-dual first-order algorithms. However, primal-dual algorithms often require explicit strongly convex regularization in …

Geometric optics describes wave behavior near convex obstacles.

We consider the minimization of submodular functions subject to ordering constraints. We show that this optimization problem can be cast as a convex optimization problem on a space of uni-dimensional measures, with ordering constraints corresponding to first-order stochastic dominance. We propose new discretization sch…

We propose a method for zeroth order stochastic convex optimization that attains the suboptimality rate of $\tilde{\mathcal{O}}(n^{7}T^{-1/2})$ after $T$ queries for a convex bounded function $f:{\mathbb R}^n\to{\mathbb R}$. The method is based on a random walk (the \emph{Ball Walk}) on the epigraph of the function. Th…

It has often been stated that, within the class of continuous stochastic volatility models calibrated to vanillas, the price of a VIX future is maximized by the Dupire local volatility model. In this article we prove that this statement is incorrect: we build a continuous stochastic volatility model in which a VIX futu…

New methods for convex optimization with locally Lipschitz gradient, achieving faster convergence.

Local invertibility of higher order tensor transforms on compact manifolds.

Improved Local SGD convergence for general convex objectives with bounded second-order heterogeneity.

New algorithm finds approximate stationary points in non-convex optimization.

Paper establishes tight lower bounds for minimizing certain smooth and convex functions.

The paper extends inequalities for convex bodies to higher dimensions and various norms.