New methods reduce constraint violations to certainty in stochastic optimization.
problem Finding a point with certain constraint satisfaction and near-stationarity.
method Single-loop variance-reduced stochastic first-order methods with truncated momentum schemes.
result Achieves strong convergence guarantees for ε ε ε -stochastic stationary points with certain constraint satisfaction. A new method speeds up quantum state estimation.
problem Exponential growth in sample size and dimension for quantum state tomography.
method Stochastic mirror descent with Burg entropy.
result Optimization error vanishes at a O ( ( 1 / t ) d log t ) O (\sqrt{ ( 1 / t ) d \log t }) O ( ( 1/ t ) d log t ) rate. First-order method solves stochastic bilevel optimization with linear constraints.
problem Stochastic bilevel optimization with linear constraints and noise.
method Developed a novel framework using gradient-based techniques and smoothed penalty functions.
result Achieved finite-time convergence guarantees for ( δ , ε ) (δ, ε) ( δ , ε ) -Goldstein stationary points. New first-order algorithm escapes saddle points faster than existing methods.
problem Escaping from saddle points in optimization problems.
method Integrates noise into first-order information to extract negative curvature from Hessian.
result First-order stochastic algorithm achieves almost linear time complexity for finding near second-order stationary points.
New methods solve optimization problems with heavy-tailed noise, improving upon existing complexity bounds.
problem Optimization problems with heavy-tailed noise and weakly average smoothness.
method Normalized stochastic first-order methods with Polyak, multi-extrapolated, and recursive momentum.
result First-order oracle complexity results for finding approximate stochastic stationary points under heavy-tailed noise.
Unified approach for first-order methods with Markovian noise in stochastic optimization and variational inequalities.
problem Stochastic optimization problems with Markovian noise.
method Unified theoretical analysis of first-order gradient methods using randomized batching and multilevel Monte Carlo.
result Optimal (linear) dependence on the mixing time of the noise sequence, eliminating previous limiting assumptions.
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.
Novel BSG method for efficient stochastic optimization.
problem Efficient optimization of non-convex surfaces in stochastic settings.
method Binary search combined with first order gradient optimization.
result BSG produces more promising results and better generalization than other methods.
SPIDER optimizes non-convex problems with reduced gradient computations.
problem Non-convex optimization problems with limited gradient information.
method Stochastic Path-Integrated Differential Estimator (SPIDER) combined with gradient descent.
result SPIDER-SFO and SPIDER-SFO extsuperscript{+} achieve optimal gradient computation costs for non-convex optimization.
The paper analyzes condition numbers for logistic regression to understand first-order methods' performance.
problem Understanding the performance of first-order methods in logistic regression.
method Introducing condition numbers to measure non-separability and separability of data.
result Condition numbers inform the properties and convergence guarantees of first-order methods.
SFLS method finds feasible solutions faster with less data.
problem Efficiently solving SOECs with near-feasibility and near-optimality.
method SFLS method that emphasizes feasibility before convergence.
result SFLS maintains high-probability feasibility at each iteration.
Method solves complex optimization problems with high probability bounds.
problem Nonlinear equality constrained stochastic optimization problems.
method Step-search sequential quadratic programming method.
result High-probability bound on iteration complexity for first-order stationarity.
First-order stochastic methods are the state-of-the-art in large-scale machine learning optimization owing to efficient per-iteration complexity. Second-order methods, while able to provide faster convergence, have been much less explored due to the high cost of computing the second-order information. In this paper we …
First order discretizations of Langevin diffusion can achieve better generalization error with additional smoothness assumptions.
problem Analyzing generalization error for first order discretizations of Langevin diffusion.
method Providing a sufficient smoothness condition to show that first order methods can achieve arbitrarily runtime complexity for a given expected generalization error.
result First order methods can achieve arbitrarily runtime complexity with additional smoothness assumptions.
SVRN accelerates Newton methods by reducing variance and improving performance.
problem Improving the efficiency of Newton methods for large-scale optimization problems.
method Stochastic Variance-Reduced Newton (SVRN) algorithm that accelerates Subsampled Newton and Iterative Hessian Sketch algorithms.
result SVRN accelerates Newton methods by reducing the number of passes over the data, achieving a significant improvement in performance.
A framework for decentralized optimization using first-order methods.
problem Optimization of finite sums over networked nodes.
method Decentralized first-order gradient and stochastic methods.
result General framework for undirected and directed networks.
A new algorithm speeds up machine learning by solving large-scale problems more efficiently.
problem Efficiently solving large-scale machine learning problems with regularization.
method Subsampled proximal Newton-type method that leverages finite sum structure and recent stochastic first-order methods.
result The method achieves faster convergence than state-of-the-art methods for non-smooth regularizers.
TRSVR combines SVRG with trust-region for faster optimization.
problem Unconstrained nonconvex optimization problems.
method Adaptive stochastic trust-region method with variance reduction.
result Converges to first-order stationary points with SVRG.
We study distributed optimization algorithms for minimizing the average of convex functions. The applications include empirical risk minimization problems in statistical machine learning where the datasets are large and have to be stored on different machines. We design a distributed stochastic variance reduced gradien…
Paper improves stochastic bilevel optimization methods for highly-smooth problems.
problem Finding ε ε ε -stationary points in stochastic bilevel optimization. method Proposes F 2 {}^2 2 SA- p p p methods using p p p th-order finite differences for hyper-gradient approximation. result Achieves upper complexity bound of i l d e O ( p ε − 4 − p / 2 ) ilde{\mathcal{O}}(p ε^{-4-p/2}) i l d e O ( p ε − 4 − p /2 ) for p p p th-order smooth problems. New lower bounds for bilevel optimization with first-order oracles.
problem Complexity of bilevel optimization with first-order oracles.
method Development of hard instances and proof of lower bounds.
result Nontrivial lower bounds for first-order zero-respecting algorithms.
The paper extends first-order asymptotics for path-dependent derivatives in multiscale stochastic volatility.
problem Analyzing path-dependent derivatives in a multiscale stochastic volatility environment.
method First-order asymptotics analysis using Dupire's functional Ito calculus.
result Market parameters calibrated to vanilla options can price path-dependent derivatives to the same order.
New methods solve complex optimization problems without strong convexity assumptions.
problem Complex bilevel optimization problems with minimax lower-level structures.
method Penalty-based first-order methods for bilevel minimax optimization.
result Achieves ε ε ε -KKT point with improved oracle complexity. This is the first in a series of papers in which we study an efficient approximation scheme for solving the Hamilton-Jacobi-Bellman equation for multi-dimensional problems in stochastic control theory. The method is a combination of a WKB style asymptotic expansion of the value function, which reduces the second order …
Develops first-order methods for average-reward MDPs with strong guarantees.
problem Lack of strong theoretical guarantees for first-order methods in AMDPs.
method Average-reward stochastic policy mirror descent (SPMD) and variance-reduced temporal difference (VRTD) methods.
result Establishes sample complexity results for solving AMDPs.
Study max- and min-stability under first-order stochastic dominance, finding new functional characterizations.
problem Understanding max- and min-stability in stochastic dominance.
method Representation theorem for functionals satisfying max-stability, combining max- and min-stability to define Lambda-quantiles.
result New characterizations of functionals, including Lambda-quantiles, in finance and political science.
Reduces non-convex optimization to finding local minima using gradients.
problem Finding local minima in non-convex optimization problems.
method Reduces non-convex optimization to gradient-based methods.
result Turns various optimization algorithms into local minimum finding methods.
The paper studies the First Order BSPDEs (Backward Stochastic Partial Differential Equations) suggested earlier for a case of multidimensional state domain with a boundary. These equations represent analogs of Hamilton-Jacobi-Bellman equations and allow to construct the value function for stochastic optimal control pro…
Improved first-order algorithm for entropy regularized OT with faster convergence.
problem Solving entropy regularized optimal transport efficiently.
method Accelerated primal-dual stochastic mirror descent algorithm with variance reduction.
result Improved rate from O ~ ( n 2.5 / ε ) \widetilde{O}({n^{2.5}}/ε) O ( n 2.5 / ε ) to O ~ ( n 2 / ε ) \widetilde{O}({n^2}/ε) O ( n 2 / ε ) . A novel distributed method tracks gradients for convex optimization over networks.
problem Distributed optimization of strongly-convex functions over a network.
method S-AB algorithm using auxiliary variables and row/column stochastic weights.
result Linear convergence to a neighborhood of the global minimizer.
Unified framework for analyzing batch updating methods with noisy gradients.
problem Analyzing convergence of batch updating methods with noisy gradients and approximations.
method Unified framework using convergence of stochastic processes.
result Establishes a general theorem for most known convergence results.
SGD's performance improves with critical batch size, minimizing SFO complexity.
problem Optimizing SGD's performance with batch size and learning rate.
method Analysis of SGD using constant and decaying learning rates, focusing on batch size effects.
result SGD with critical batch size minimizes SFO complexity.
New algorithms improve CCA with stochastic approximation.
problem Efficiently compute canonical correlation analysis.
method Inexact MSG and MEG algorithms for CCA.
result Achieves ε-suboptimality in poly(1/ε) iterations.
New adaptive step-size method for convex optimization without tuning.
problem Optimizing convex functions efficiently with stochastic gradients.
method Adapted Adaptive Gradient Descent Without Descent to stochastic setting.
result Stochastic gradient descent converges under various assumptions.
Unified analysis of first-order methods for smooth games using IQCs.
problem Certify convergence rates of first-order methods for smooth and strongly-monotone games.
method Adapted integral quadratic constraints (IQCs) to study first-order methods and derive tight upper bounds of convergence rates.
result First global convergence rate for the negative momentum method with O ( κ 1.5 ) \mathcal{O}(κ^{1.5}) O ( κ 1.5 ) iteration complexity. New algorithms optimize without knowing problem parameters.
problem Optimizing large-scale problems without knowing key parameters.
method Combining mirror descent with dual averaging techniques.
result Converges without prior knowledge of problem parameters.
The paper calculates option prices using Mellin transform for stochastic volatility models.
problem Calculating prices for path-dependent options under stochastic volatility.
method Asymptotic approach and Mellin transform for deriving closed-form formulas.
result Derives closed-form formulas for option prices with first-order approximation.
New optimization method speeds up learning from data.
problem Efficiently optimizing large datasets for machine learning.
method Minibatch stochastic variance reduced proximal iterations.
result Improved convergence speed for quadratic objectives.
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 …
Two new methods solve nonsmooth optimization on Riemannian Stiefel manifold.
problem Optimization over nonsmooth, non-differentiable functions on Riemannian manifolds.
method R-ProxSGD and R-ProxSPB, generalizing proximal SGD and SpiderBoost.
result R-ProxSPB finds ε-stationary points with IFO complexity of Ø(ε^(-3)) in online and Ø(n + √nε^(-2)) in finite-sum cases.
A new method helps escape saddle points in non-convex optimization.
problem Escaping saddle points in non-convex optimization problems.
method CNC-SCSG method using a separate SGD step to help escape from strict saddle points.
result The method converges to a second-order stationary point with a rate of O ( ε − 2 l o g ( 1 / ε ) ) O(ε^{-2} log(1/ε)) O ( ε − 2 l o g ( 1/ ε )) . Paper develops a TR-SSQP method for noisy optimization with heavy-tailed noise.
problem Optimization problems with stochastic objectives and heavy-tailed noise.
method Trust-Region Stochastic Sequential Quadratic Programming (TR-SSQP) method.
result Achieves high-probability first-order and second-order stationarity bounds for heavy-tailed noise.
A new method reduces the complexity of decentralized optimization.
problem Decentralized stochastic non-convex optimization over a network.
method GT-HSGD, a hybrid variance-reduced method.
result Achieves an oracle complexity of O(n^(-1)ε^(-3)) for small ε.
Paper proposes an algorithm to solve complex minimax problems efficiently.
problem Stochastic nonconvex-concave minimax problems in various fields.
method Accelerated first-order regularized momentum descent ascent algorithm (FORMDA).
result Achieves best-known complexity bound of i l d e O ( ε − 6.5 ) ilde{\mathcal{O}}(\varepsilon ^{-6.5}) i l d e O ( ε − 6.5 ) for single-loop algorithms. SSRGD finds local minima in nonconvex problems with simple gradient updates.
problem Finding local minima in nonconvex optimization problems.
method Simple perturbed stochastic recursive gradient descent (SSRGD).
result SSRGD finds ( ε , δ ) (ε,δ) ( ε , δ ) -second-order stationary points efficiently. Paper proposes FONE for efficient distributed estimation and inference.
problem Efficient distributed estimation and inference for non-differentiable convex losses.
method Proposes a multi-round distributed estimation procedure using a First-Order Newton-type Estimator (FONE).
result FONE efficiently estimates Σ − 1 w Σ^{-1} w Σ − 1 w for non-differentiable losses, facilitating inference. Consider the stochastic composition optimization problem where the objective is a composition of two expected-value functions. We propose a new stochastic first-order method, namely the accelerated stochastic compositional proximal gradient (ASC-PG) method, which updates based on queries to the sampling oracle using tw…
Proposes a new method for optimizing large-scale models using Nyström approximation of the Hessian.
problem Optimizing non-convex functions like deep learning models using second-order methods.
method Nyström-approximated curvature for stochastic optimization of large-scale empirical risk minimization.
result The proposed method achieves performance competitive with state-of-the-art first-order and stochastic quasi-Newton methods.