Proves limitations of higher-order optimization for convex problems.
problem Limitations of higher-order optimization methods for convex problems.
method Proves polynomial dependence on approximation guarantee and higher-order smoothness parameters.
result Nesterov's accelerated cubic regularization method is nearly tight.
New geometric proof of convex function differentiability and approximation.
problem Second-order differentiability of convex functions and their approximations.
method Elementary geometric approach to prove classical and recent results.
result New proofs of Lusin approximation of convex functions and bodies by C 1 , 1 C^{1,1} C 1 , 1 functions. Flow deforms locally convex curves to curves of constant k-order width.
problem Evolve locally convex curves to curves of constant k-order width.
method Introduced a nonlocal curvature flow to evolve locally convex curves in the plane.
result The flow converges to a smooth, locally convex curve of constant k-order width as time goes to infinity.
Developed a theory of local convexity for second order differential equations on Lie algebroids.
problem Analyzing convexity in differential equations on Lie algebroids.
method Theory development for local convexity of SODEs on Lie algebroids.
result Extensive discussion of homogeneous quadratic SODEs on Lie algebroids.
Study convex embeddability in linear and circular orders, applying to knots.
problem Understanding the quasi-order of convex embeddability in linear and circular orders.
method Combinatorial and descriptive set-theoretic methods applied to arcs and knots.
result Established combinatorial properties and lower bounds for knot complexity.
Established strong geodesic convex functions and their properties.
problem Geodesic convex functions and monotone vector fields on Riemannian manifolds.
method Characterization and relation establishment for strong geodesic convex functions.
result Relation between variational inequality solutions and strict minimizers for multiobjective programming.
New algorithms achieve high probability second-order convergence in non-convex optimization.
problem Stochastic non-convex optimization with high probability second-order convergence.
method Proposed NCG-S updating step and two algorithms.
result First algorithms with high probability second-order convergence and almost linear time complexity.
Study non-standard bi-orders on punctured torus bundles, matching standard ones in key subgroups.
problem Investigate non-standard bi-orders on punctured torus bundles.
method Analyze various bi-orderings and compare them to standard ones formed by the lower central series.
result For every bi-ordering, the largest and second largest proper convex subgroups match those of a standard bi-ordering. Third largest subgroup matches if it exists.
Sharp comparison for sub-Gaussian random variables in convex order.
problem Comparing sub-Gaussian random variables in convex order.
method Proving dominance using moment generating functions and convex functions.
result Sharp comparison established between specific sub-Gaussian random variables.
Second-order methods improve differential privacy in convex optimization.
problem Improving differential privacy in convex optimization.
method Developed a private variant of the regularized cubic Newton method for strongly convex loss functions.
result Achieves quadratic convergence and optimal excess loss for strongly convex loss functions.
Study asks if memory constraints affect optimal convex optimization methods.
problem Characterize the minimax number of queries for convex optimization with memory constraints.
method Analyze first order methods under memory limitations.
result Optimal oracle complexity may be achievable with limited memory.
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.
problem Optimizing decisions in uncertain environments with convex losses.
method Introduces a second-order method for zeroth-order stochastic convex bandits.
result Regret bound of ( 1 + r / d ) [ d 1.5 n + d 3 ] p o l y l o g ( n , d , r ) (1 + r/d)[d^{1.5} \sqrt{n} + d^3] polylog(n, d, r) ( 1 + r / d ) [ d 1.5 n + d 3 ] p o l y l o g ( n , d , r ) . Study on convex ordering in stochastic control for swing contracts, proving value function convexity.
problem Pricing of swing contracts under stochastic dynamics.
method Discrete-time stochastic optimal control problem, convexity propagation, Brownian diffusion model, Stein's formula.
result Value function is convex in underlying asset price, relaxation of convexity assumption for semi-convexity.
Algorithm for exact partitioning of high-order models using convex tensor relaxation.
problem Exact partitioning of high-order models.
method Defining a general class of m m m -degree Homogeneous Polynomial Models, relaxing the high-order combinatorial problem to a convex conic form problem, defining the Carathéodory symmetric tensor cone, and constructing a primal-dual certificate. result The solution of the convex relaxation is correct and provides a statistical upper bound for exact partitioning.
Study first-order locally convex Lie algebroids in Bastiani calculus.
problem Define and study first-order locally convex Lie algebroids.
method Define sheaves of Lie algebroid forms and morphisms, prove category structure, study representations and cohomology.
result First-order locally convex Lie algebroids form a category and have applications in Lie II theorems.
First order methods can take extremely long to find global minima of non-convex functions.
problem Finding global minimizers of non-convex functions.
method Designing a family of non-convex functions and using statistical lower bounds for parameter estimation.
result First order methods can take exponential time to converge to a global minimizer.
Expands learning paradigm to stochastic orders using Choquet-Toland distance and Variational Dominance Criterion.
problem Learning high-dimensional distributions with stochastic orders.
method Introduces Choquet-Toland distance and Variational Dominance Criterion, uses input convex maxout networks (ICMNs).
result Proposes surrogates for Choquet-Toland distance and Variational Dominance Criterion with parametric rates.
New algorithms solve non-convex isotonic regression problems efficiently.
problem Minimizing submodular functions with ordering constraints.
method Discretization schemes leading to zero-th, first, or higher order oracles for efficient optimization.
result Non-convex loss functions can be robust to outliers and still lead to efficient optimization.
New algorithms exploit data's strong convexity for fast linear convergence without explicit regularization.
problem Empirical risk minimization with convex loss functions.
method Primal-dual first-order algorithms that exploit data's strong convexity.
result Adaptive primal-dual algorithms achieve linear convergence without explicit regularization.
New algorithms optimize convex functions with high-order derivatives.
problem Optimizing convex functions with high-order derivatives under various norms.
method Developed a non-Euclidean inexact accelerated proximal point method using an inexact uniformly convex regularizer.
result Showed nearly optimal algorithms for high dimensions in the black-box oracle model for ℓ p \ell_p ℓ p -settings and all q ≥ 1 q \geq 1 q ≥ 1 . Optimal algorithms for online convex optimization with random order.
problem Online convex optimization with random order and non-convex loss functions.
method Stochastic gradient descent and algorithmic stability analysis.
result Achieves optimal bounds and significantly outperforms previous methods.
Paper optimizes portfolio selection with ICX order constraints.
problem Minimizing portfolio variance with ICX order constraints.
method Optimal and efficient portfolios are derived in closed form.
result Closed-form solutions for optimal and efficient portfolios.
Book covers tools for zeroth-order convex optimisation.
problem Zeroth-order convex optimisation.
method Cutting plane methods, interior point methods, continuous exponential weights, gradient descent, online Newton step.
result Improved existing bounds and algorithms.
New characterization of second-order stochastic dominance with applications in risk management.
problem Characterizing second-order stochastic dominance.
method Properties of Expected Shortfall risk measures.
result New interpretation and proof techniques for second-order stochastic dominance.
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…
Paper explores closedness properties of convex sets in rearrangement invariant spaces.
problem Closedness properties of law-invariant convex sets in rearrangement invariant spaces.
method Analyzes equivalence of different closedness types in rearrangement invariant spaces.
result Order closedness, σ ( X , X n ∼ ) σ(\mathcal{X},\mathcal{X}_n^\sim) σ ( X , X n ∼ ) -closedness and σ ( X , L ∞ ) σ(\mathcal{X},L^\infty) σ ( X , L ∞ ) -closedness of a law-invariant convex set are equivalent. Lower bounds for higher-order methods in non-convex optimization.
problem Proving lower bounds for higher-order methods in smooth non-convex finite-sum optimization.
method Analyzing deterministic and randomized algorithms, proposing a new smoothness assumption.
result Proves optimal lower bounds for simulating pth-order regularized methods on the whole function.
The paper defines projections for Wasserstein distances to preserve convex order in probability measures.
problem Designing sampling techniques to preserve convex order in probability measures.
method Defining projections for Wasserstein distances and solving optimization problems.
result The projections do not depend on ρ in dimension 1 and their quantile functions are explicit.
New model shows VIX futures are more expensive than local volatility model suggests.
problem VIX futures pricing under local volatility model is incorrect.
method Developed a continuous stochastic volatility model to show VIX futures are more expensive than local volatility model.
result Inversion of convex ordering between local and stochastic variances observed in SPX market for short maturities.
Paper finds optimal transport measures for arbitrage strategies.
problem Link between convex order and arbitrage strategies.
method Develops algorithms and models for finding optimal transport measures.
result Constructs a model-independent arbitrage strategy.
New algorithm optimizes convex functions with noisy evaluations in one dimension.
problem Optimizing convex functions with noisy zero-order evaluations in one dimension.
method Proposed a computationally efficient algorithm achieving O ( 1 / T ) O(1/\sqrt{T}) O ( 1/ T ) convergence rate. result Achieved the optimal O ( 1 / T ) O(1/\sqrt{T}) O ( 1/ T ) convergence rate, closing the gap in one dimension. Second-order guarantees for federated learning algorithms.
problem Non-convex optimization in federated learning with saddle-points as bottlenecks.
method Drawing on recent results on second-order optimality in centralized and decentralized settings, establish second-order guarantees for federated learning algorithms.
result Established 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.
problem Improving optimization methods for non-convex functions.
method Study of first order methods for quasar-convex functions.
result Proves complexity upper bounds similar to convex functions.
Optimal transport theory characterizes convex order between probability measures.
problem Characterizing convex order between probability measures using optimal transport.
method Quantitative bounds on optimal transport, infimum of functionals over 1-Lipschitz functions.
result Two measures are in convex order if and only if a specific cost functional inequality holds.
New algorithm tackles risk-aware learning problems efficiently.
problem Risk-aware learning with mean-semideviation objective.
method Zeroth-order compositional stochastic optimization algorithm.
result Algorithm converges to optimal solutions with explicit rates.
Optimized method tackles convex optimization with heavy-tailed noise.
problem Convex optimization problems with noisy gradients.
method Vanilla stochastic proximal subgradient method without gradient clipping or normalization.
result Achieves optimal complexity for various convex optimization types under heavy-tailed noise.
Zeroth-order optimization methods lack inherent privacy guarantees.
problem Ensuring differential privacy in zeroth-order optimization methods.
method Analyzing ZO-GD with and without random initialization for convex and strongly convex objectives.
result ZO-GD is not differentially private for strongly convex objectives and can have superlinear privacy loss.
New memory-query tradeoffs for convex optimization algorithms.
problem Optimizing memory usage for convex optimization algorithms.
method Analyzing randomized first-order algorithms for minimizing convex functions.
result Cutting plane methods are optimal in terms of memory and query complexity.
The paper extends Strassen's theorem to include biased martingales for American options.
problem Existence of martingales for arbitrage-free prices of American options.
method Derives an extension of Strassen's theorem linking biased martingales to strengthened convex order.
result Characterizes the strengthened convex order through integrals with respect to compensated Poisson processes.
Convex relaxations improve CNNs with fixed weights.
problem Improving CNNs with fixed weights.
method Convex relaxations for CNNs with fixed weights using second order cone programs.
result The relaxation recovers the global minimum under a planted model assumption.
Improved regret bound for adversarial bandit convex optimisation.
problem Minimizing regret in zeroth-order adversarial bandit convex optimisation.
method Identifying an improved exploratory distribution for convex functions.
result Proved minimax regret bound of O ( d 2.5 n log ( n ) ) O(d^{2.5} \sqrt{n} \log(n)) O ( d 2.5 n log ( n )) . Study contact partial order on non-compact manifolds.
problem Understanding contact partial order on non-compact manifolds.
method Analyzing orbits of adjoint action on Lie algebras of contactomorphism groups.
result Remnants of contact partial order on orbits of adjoint action.
Exact partitioning of high-order planted models achieved through convex optimization.
problem Efficiently partitioning hypergraphs generated by high-order planted models.
method Solving a computationally efficient convex optimization problem with a tensor nuclear norm constraint.
result Exact recovery of true underlying cluster structures with high probability.
Geometric optics describes wave behavior near convex obstacles.
problem Wave behavior near convex obstacles.
method Geometric optics in L 2 L^2 L 2 and H 1 H^1 H 1 spaces. result Oscillations transport along grazing rays to any order.
We propose a method for zeroth order stochastic convex optimization that attains the suboptimality rate of O ~ ( n 7 T − 1 / 2 ) \tilde{\mathcal{O}}(n^{7}T^{-1/2}) O ~ ( n 7 T − 1/2 ) after T T T queries for a convex bounded function f : R n → R f:{\mathbb R}^n\to{\mathbb R} f : R n → R . The method is based on a random walk (the \emph{Ball Walk}) on the epigraph of the function. Th…
New methods for convex optimization with locally Lipschitz gradient, achieving faster convergence.
problem Optimization problems with locally Lipschitz continuous gradient.
method Accelerated proximal gradient (APG) methods and proximal augmented Lagrangian method.
result Achieved faster convergence rates for convex optimization problems with locally Lipschitz gradient.