Two classes of methods have been proposed for escaping from saddle points with one using the second-order information carried by the Hessian and the other adding the noise into the first-order information. The existing analysis for algorithms using noise in the first-order information is quite involved and hides the es…
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48 results for “First Order Algorithms”
New algorithm reduces online decision-making regret with efficient LP re-solving and parallel first-order method.
problem Worse regret guarantees and high computational cost of LP-based OLP algorithms.
method Combines LP-based and first-order OLP methods, re-solving LP subproblems periodically and using parallel first-order method.
result Achieves regret, balancing computational efficiency and superior regret guarantee.
Efficient First-Order Contextual Bandits: Prediction, Allocation, and Triangular Discriminationcs.LG
Efficient algorithm for contextual bandits with first-order guarantees.
problem Adapting to low noise in contextual bandits.
method Reduction to online regression with logarithmic loss.
result Optimal and efficient first-order guarantees for contextual bandits.
CEFOL uses deep learning for dynamic programming with recursive utility.
problem Challenges in solving dynamic programming problems with recursive utility.
method Introduces a separate neural network for certainty equivalent, uses first-order optimality conditions to learn value and policy functions.
result CEFOL achieves high accuracy in learning value and policy functions, matching VFI benchmarks.
The filtering-clustering models, including trend filtering and convex clustering, have become an important source of ideas and modeling tools in machine learning and related fields. The statistical guarantee of optimal solutions in these models has been extensively studied yet the investigations on the computational as…
Paper explores second-order optimization in first-order methods, proving and disproving the necessity of square root.
problem Understanding and optimizing first-order optimization methods using second-order information.
method Rigorously proves Nesterov Accelerated Gradient uses past and current gradients to approximate Hessian. Relates adaptive methods to Natural Gradient Descent. Introduces AdaSqrt algorithm to remove square root in denominator.
result New algorithm AdaSqrt comparable to first-order methods on MNIST and beats Adam on CIFAR-10, casting doubt on the necessity of square root.
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.
Efficient RNN algorithm guarantees convergence in online learning.
problem Online nonlinear regression with RNNs.
method First-order training algorithm with convergence guarantee.
result The algorithm converges to optimum network parameters.
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 to .
We analyze stochastic gradient algorithms for optimizing nonconvex problems. In particular, our goal is to find local minima (second-order stationary points) instead of just finding first-order stationary points which may be some bad unstable saddle points. We show that a simple perturbed version of stochastic recursiv…
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.
This paper improves online learning algorithms for LP problems, achieving better regret bounds.
problem Achieving optimal regret bounds in online linear programming.
method Develops a new framework for first-order online learning algorithms under certain error bound conditions.
result First-order learning algorithms achieve regret in continuous support and regret in finite support, improving over .
New algorithms optimize constrained problems faster, avoiding full set optimization.
problem Optimizing constrained problems efficiently and quickly.
method Designing accelerated first-order algorithms that avoid full set optimization.
result Proved convergence to stationary points in nonconvex settings and accelerated rates in convex settings.
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…
In this paper, we consider the problem of prediction with expert advice in dynamic environments. We choose tracking regret as the performance metric and develop two adaptive and efficient algorithms with data-dependent tracking regret bounds. The first algorithm achieves a second-order tracking regret bound, which impr…
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.
A new algorithm for solving constrained convex optimization problems efficiently.
problem Constrained convex optimization problems requiring high accuracy solutions.
method Second-Order Conditional Gradient Sliding (SOCGS) algorithm, using projection-free methods to solve quadratic subproblems inexactly.
result Converges quadratically in primal gap after a finite number of linearly convergent iterations.
Basic Inequalities for First-Order Optimization with Applications to Statistical Risk Analysismath.ST
New inequalities help optimize first-order algorithms for statistical risk analysis.
problem Optimizing first-order iterative algorithms for statistical risk analysis.
method Introducing basic inequalities that connect implicit and explicit regularization.
result The basic inequalities translate the number of iterations into an effective regularization coefficient.
Memory-constrained algorithms need superlinear memory for efficient convex optimization.
problem Efficiently minimizing convex functions with limited memory.
method Analyzing first-order algorithms with superlinear memory constraints.
result Superlinear memory is necessary for optimal performance in convex optimization.
Paper develops fast method for computing optimal transport.
problem Efficient computation of optimal transport distance between distributions.
method Entropy-regularized extragradient method for first-order optimization.
result Achieves state-of-the-art runtime guarantees and good numerical performance.
BMM algorithm improves convergence for nonconvex optimization problems.
problem Constrained nonsmooth nonconvex optimization problems.
method Block majorization-minimization with diminishing radius.
result Improved convergence rate for nonconvex optimization problems.
First-order methods such as stochastic gradient descent (SGD) are currently the standard algorithm for training deep neural networks. Second-order methods, despite their better convergence rate, are rarely used in practice due to the prohibitive computational cost in calculating the second-order information. In this pa…
Unified bounds for iterative algorithms with Gaussian data matrices.
problem Establishing non-asymptotic bounds for iterative algorithms with Gaussian data.
method Explicit coupling between iterates and Gaussian process with deterministic covariance.
result Tight, dimension-free bounds for generalized first-order methods.
In this paper, we propose and analyze zeroth-order stochastic approximation algorithms for nonconvex and convex optimization, with a focus on addressing constrained optimization, high-dimensional setting and saddle-point avoiding. To handle constrained optimization, we first propose generalizations of the conditional g…
In this paper, we study optimization methods consisting of iteratively minimizing surrogates of an objective function. By proposing several algorithmic variants and simple convergence analyses, we make two main contributions. First, we provide a unified viewpoint for several first-order optimization techniques such as …
We establish that first-order methods avoid saddle points for almost all initializations. Our results apply to a wide variety of first-order methods, including gradient descent, block coordinate descent, mirror descent and variants thereof. The connecting thread is that such algorithms can be studied from a dynamical s…
New algorithm for safer machine learning with different testing and training distributions.
problem Challenges in modern machine learning where training and testing distributions differ.
method First-order optimization algorithm for superquantile-based learning.
result Promising numerical results show the approach's effectiveness.
Local search heuristics for non-convex optimizations are popular in applied machine learning. However, in general it is hard to guarantee that such algorithms even converge to a local minimum, due to the existence of complicated saddle point structures in high dimensions. Many functions have degenerate saddle points su…
Generalizes Hamiltonian theory for variational problems, applied to first order gravity.
problem Formulating Hamiltonian field theory for variational problems of general nature.
method Introduces a generalized Hamiltonian formalism without requiring a Hamiltonian section.
result Develops a novel multisymplectic Hamiltonian field theory for first order gravity.
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 …
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.
SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimatormath.OC
In this paper, we propose a new technique named \textit{Stochastic Path-Integrated Differential EstimatoR} (SPIDER), which can be used to track many deterministic quantities of interest with significantly reduced computational cost. We apply SPIDER to two tasks, namely the stochastic first-order and zeroth-order method…
Enhances clustering performance with a novel high-order Laplacian matrix.
problem Limited representation capability and insufficient information exploitation in multi-view spectral clustering.
method Proposes a multi-view spectral clustering algorithm that learns a high-order optimal neighborhood Laplacian matrix.
result Improves clustering performance through enhanced representation capacity of the learned optimal Laplacian matrix.
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 …
A new method for faster optimization of noisy functions.
problem Optimizing noisy functions efficiently.
method A universal and adaptive second-order method for convex functions.
result Achieves convergence for stochastic oracles and for deterministic oracles.
We propose a reduction for non-convex optimization that can (1) turn an stationary-point finding algorithm into an local-minimum finding one, and (2) replace the Hessian-vector product computations with only gradient computations. It works both in the stochastic and the deterministic settings, without hurting the algor…
Improved zeroth-order algorithms for nonconvex optimization with reduced complexity and improved performance.
problem Designing efficient zeroth-order algorithms for nonconvex optimization with reduced function query complexities and improved convergence rates.
method Proposed new algorithms ZO-SVRG-Coord-Rand and ZO-SPIDER-Coord, developed new analyses, and addressed issues of function query complexities and stepsize generation.
result New algorithms outperform existing methods in terms of function query complexities and convergence rates.
Novel methods for accelerating optimization in complex bilevel and minimax problems.
problem Optimization challenges in bilevel and minimax problems, especially when strong convexity assumptions are not met.
method Accelerated fully first-order methods for Bilevel Optimization (BLO) and Minimax Optimization (NCSC).
result State-of-the-art complexity for finding approximate second-order stationary points in BLO and NCSC.
Recent applications that arise in machine learning have surged significant interest in solving min-max saddle point games. This problem has been extensively studied in the convex-concave regime for which a global equilibrium solution can be computed efficiently. In this paper, we study the problem in the non-convex reg…
A new algorithm solves bilevel optimization with linear constraints.
problem Solving bilevel optimization problems with coupled linear constraints.
method Penalty and augmented Lagrangian methods reformulate the problem; a single-loop, first-order algorithm proposed.
result Improved convergence rates compared to prior methods.
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…
Cochran defined the nth-order integral Alexander module of a knot in the three sphere as the first homology group of the knot's (n+1)th-iterated abelian cover. The case n=0 gives the classical Alexander module (and polynomial). After a localization, one can get a finitely presented module over a principal ideal domain,…
Combines neural networks and probabilistic graphical models for efficient higher-order inference.
problem Lack of efficient higher-order relational information in graph neural networks and probabilistic graphical models.
method Derives efficient approximate sum-product loopy belief propagation for higher-order PGMs, embeds into neural network, proposes methods for constructing higher-order factors.
result Substantially outperforms state-of-the-art k-order graph neural networks in molecular datasets.
In this paper, we implement the Stochastic Damped LBFGS (SdLBFGS) for stochastic non-convex optimization. We make two important modifications to the original SdLBFGS algorithm. First, by initializing the Hessian at each step using an identity matrix, the algorithm converges better than original algorithm. Second, by pe…
In this paper, we consider first-order convergence theory and algorithms for solving a class of non-convex non-concave min-max saddle-point problems, whose objective function is weakly convex in the variables of minimization and weakly concave in the variables of maximization. It has many important applications in mach…
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 convergence rate.
result Achieved the optimal convergence rate, closing the gap in one dimension.
We present a predictor-corrector framework, called PicCoLO, that can transform a first-order model-free reinforcement or imitation learning algorithm into a new hybrid method that leverages predictive models to accelerate policy learning. The new "PicCoLOed" algorithm optimizes a policy by recursively repeating two ste…
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.