In this work we consider the stochastic minimization of nonsmooth convex loss functions, a central problem in machine learning. We propose a novel algorithm called Accelerated Nonsmooth Stochastic Gradient Descent (ANSGD), which exploits the structure of common nonsmooth loss functions to achieve optimal convergence ra…
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48 results for “nonsmooth functions”
Nonsmooth High-Order Averaging Theory with Application to Extremum Seeking Optimization and Controlmath.OC
New theory for nonsmooth systems helps optimize and control complex functions.
problem Optimizing and controlling systems with nonsmooth functions.
method Higher-order averaging theory with nonsmooth near-identity transformation and lexicographic differentiation.
result Closed formula for nonsmooth first and second-order averaging.
PPGD solves nonconvex nonsmooth optimization problems without KL property.
problem Nonconvex and nonsmooth optimization problems in statistics and machine learning.
method Projective Proximal Gradient Descent (PPGD) for solving a class of nonconvex and nonsmooth problems.
result PPGD achieves a fast convergence rate of O(1/k^2) for k ≥ k_0.
Study compares nonsmooth spaces with integrable Ricci bounds.
problem Comparing geometric and functional inequalities on nonsmooth spaces.
method Localization method and one-dimensional comparison estimates.
result Extension of comparison principles to nonsmooth settings.
We analyze stochastic algorithms for optimizing nonconvex, nonsmooth finite-sum problems, where the nonconvex part is smooth and the nonsmooth part is convex. Surprisingly, unlike the smooth case, our knowledge of this fundamental problem is very limited. For example, it is not known whether the proximal stochastic gra…
Regularity results for geodesic X-ray transform on nonsmooth manifolds
problem Geodesic X-ray transform on nonsmooth simple manifolds
method Symbol smoothing arguments and pseudodifferential operators with low regularity symbols
result Improved injectivity results for functions
Analysis of Generalized Bregman Surrogate Algorithms for Nonsmooth Nonconvex Statistical Learningmath.OC
Paper develops algorithms for nonsmooth, nonconvex statistical learning problems.
problem Nonsmooth and nonconvex objectives in statistical learning.
method Bregman-surrogate algorithm framework, including local linear approximation, mirror descent, iterative thresholding, DC programming.
result Global convergence rates for nonconvex and nonsmooth objectives in high dimensions.
New Langevin Monte Carlo algorithms for sampling from nonsmooth distributions.
problem Sampling from distributions with nonsmooth convex composite potentials.
method Leveraging Bregman--Moreau envelopes and proximal operators in mirror descent.
result Efficiency in sampling from nonsmooth distributions, extending existing methods.
We consider a class of nonconvex nonsmooth optimization problems whose objective is the sum of a smooth function and a finite number of nonnegative proper closed possibly nonsmooth functions (whose proximal mappings are easy to compute), some of which are further composed with linear maps. This kind of problems arises …
New methods help escape strict saddle points in nonsmooth optimization.
problem Escaping strict saddle points in nonsmooth optimization.
method An inexact stochastically perturbed gradient method applied to the Moreau envelope.
result A variety of algorithms for nonsmooth optimization can efficiently escape strict saddle points of the Moreau envelope.
We present a stochastic setting for optimization problems with nonsmooth convex separable objective functions over linear equality constraints. To solve such problems, we propose a stochastic Alternating Direction Method of Multipliers (ADMM) algorithm. Our algorithm applies to a more general class of nonsmooth convex …
Stochastic Subgradient Methods with Guaranteed Global Stability in Nonsmooth Nonconvex Optimizationmath.OC
The paper guarantees global stability for stochastic subgradient methods in nonsmooth nonconvex optimization.
problem Minimizing nonsmooth nonconvex functions with convergence guarantees.
method Developed a framework for stochastic subgradient methods with global stability guarantees.
result Iterates are uniformly bounded and asymptotically stabilize around the stable set of the differential inclusion.
Paper develops privacy-preserving federated learning for nonsmooth objectives.
problem Solving nonsmooth objective functions in a privacy-preserving manner.
method Zero-concentrated differential privacy (zCDP) with Gaussian noise, distributed ADMM, and approximation of augmented Lagrangian.
result The algorithm achieves a competitive privacy-accuracy trade-off and converges to the exact solution.
Improved Last-Iterate Convergence of Shuffling Gradient Methods for Nonsmooth Convex Optimizationmath.OC
Improved shuffling gradient methods converge faster for nonsmooth convex optimization.
problem Improving convergence rates for nonsmooth convex optimization problems.
method Analysis of shuffling gradient methods, focusing on Random Reshuffle and Single Shuffle strategies.
result Shuffling gradient methods, particularly Random Reshuffle and Single Shuffle, converge faster than Proximal Gradient Descent for nonsmooth convex optimization.
Develops a mathematical model for automatic differentiation in machine learning.
problem Current automatic differentiation lacks a simple mathematical model for machine learning.
method Articulates relationships between program differentiation and nonsmooth functions, provides a class of functions and nonsmooth calculus.
result Shows how nonsmooth calculus applies to stochastic approximation methods and evidence of artificial critical points.
Riemannian Stochastic Proximal Gradient Methods for Nonsmooth Optimization over the Stiefel Manifoldmath.OC
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.
We consider in this paper a class of composite optimization problems whose objective function is given by the summation of a general smooth and nonsmooth component, together with a relatively simple nonsmooth term. We present a new class of first-order methods, namely the gradient sliding algorithms, which can skip the…
In this paper, we consider the problem of minimizing the average of a large number of nonsmooth and convex functions. Such problems often arise in typical machine learning problems as empirical risk minimization, but are computationally very challenging. We develop and analyze a new algorithm that achieves robust linea…
New method solves nonsmooth low-rank matrix optimization problems efficiently.
problem Nonsmooth and low-rank matrix optimization problems in statistics and machine learning.
method Low-rank Extragradient Method with warm-start initialization.
result The extragradient method converges to an optimal solution with rate and requires only two low-rank SVDs per iteration.
New algorithm for fast nonsmooth optimization with applications in image processing and machine learning.
problem Minimizing the sum of three convex functions with specific properties.
method PDDY algorithm, based on Davis-Yin splitting in a primal-dual product space.
result Sublinear and linear convergence rates in various scenarios, including strong convexity.
Frank-Wolfe methods (FW) have gained significant interest in the machine learning community due to its ability to efficiently solve large problems that admit a sparse structure (e.g. sparse vectors and low-rank matrices). However the performance of the existing FW method hinges on the quality of the linear approximatio…
Study on numerical reliability of AD for MaxPool in neural nets.
problem Reliability of automatic differentiation for nonsmooth operations like MaxPool.
method Investigation across precision levels and architectures on various datasets.
result Lower norms of nonsmooth Jacobians help maintain stable learning.
We introduce a proximal subdifferential and develop a calculus for nonsmooth functions defined on any Riemannian manifold . We give several applications of this theory, concerning: 1) differentiability and geometrical properties of the distance function to a closed subset of ; 2) solvability and implicit func…
Many machine learning techniques sacrifice convenient computational structures to gain estimation robustness and modeling flexibility. However, by exploring the modeling structures, we find these "sacrifices" do not always require more computational efforts. To shed light on such a "free-lunch" phenomenon, we study the…
This paper concerns dictionary learning, i.e., sparse coding, a fundamental representation learning problem. We show that a subgradient descent algorithm, with random initialization, can provably recover orthogonal dictionaries on a natural nonsmooth, nonconvex minimization formulation of the problem, under mi…
We study the dual formulation of the utility maximization problem in incomplete markets when the utility function is finitely valued on the whole real line. We extend the existing results in this literature in two directions. First, we allow for nonsmooth utility functions, so as to include the shortfall minimization p…
Unified Lagrangian-based methods for nonsmooth nonconvex optimization.
problem Minimizing nonsmooth nonconvex functions with constraints.
method Developed a unified framework for Lagrangian-based methods using subgradient updates.
result Global convergence guarantees for the proposed framework under mild conditions.
Asynchronous Delay-Aware Accelerated Proximal Coordinate Descent for Nonconvex Nonsmooth Problemsmath.OC
Nonconvex and nonsmooth problems have recently attracted considerable attention in machine learning. However, developing efficient methods for the nonconvex and nonsmooth optimization problems with certain performance guarantee remains a challenge. Proximal coordinate descent (PCD) has been widely used for solving opti…
Study on Adam-family methods for nonsmooth optimization with convergence guarantees.
problem Training nonsmooth neural networks with convergence guarantees.
method Two-timescale updating scheme and stochastic subgradient methods with gradient clipping.
result Convergence guarantees for various Adam-family methods in training nonsmooth neural networks.
Nonsmoothness in Machine Learning: specific structure, proximal identification, and applicationsmath.OC
This paper tackles nonsmooth optimization in machine learning.
problem Nonsmoothness in machine learning optimization problems.
method Identifying specific structures and leveraging them for practical applications.
result Compression, acceleration, and dimension reduction are possible with nonsmooth optimization.
In this paper, we consider the convergence of an abstract inexact nonconvex and nonsmooth algorithm. We promise a pseudo sufficient descent condition and a pseudo relative error condition, which are both related to an auxiliary sequence, for the algorithm; and a continuity condition is assumed to hold. In fact, a lot o…
Study on nonsmooth contractive SA with constant stepsize and Q-learning.
problem Understanding convergence and bias in nonsmooth contractive SA with different noise types.
method Proposed prelimit coupling technique for steady-state convergence and derived asymptotic bias.
result Asymptotic bias of nonsmooth SA is proportional to the square root of the stepsize.
We maximize the expected utility of terminal wealth in an incomplete market where there are cone constraints on the investor's portfolio process and the utility function is not assumed to be strictly concave or differentiable. We establish the existence of the optimal solutions to the primal and dual problems and their…
Much of the vast literature on the integral during the last two centuries concerns extending the class of integrable functions. In contrast, our viewpoint is akin to that taken by Hassler Whitney [{\it Geometric integration theory}, Princeton Univ. Press, Princeton, NJ, 1957] and by geometric measure theorists because …
New stability bounds for SGD on nonsmooth convex losses.
problem Understanding stability of SGD on nonsmooth convex losses.
method Sharp upper and lower bounds for SGD and full-batch GD on nonsmooth convex losses.
result SGD can be less stable but still useful for generalization bounds.
Efficient Distributed Learning over Decentralized Networks with Convoluted Support Vector Machinestat.ML
Paper tackles efficient SVM classification over decentralized networks.
problem Efficiently classifying high-dimensional data over decentralized networks.
method Convolution-based smoothing technique for nonsmooth hinge loss function, combined with an efficient ADMM algorithm.
result Provable linear convergence of the ADMM algorithm and near-optimal statistical convergence of the sparse estimator.
Improved method reduces projection calls for nonsmooth convex optimization.
problem Optimizing nonsmooth convex functions with convex constraints.
method MOPES and MOLES methods combining Moreau-Yosida smoothing and accelerated first-order schemes.
result Achieves -suboptimality with significantly fewer projection calls.
New adaptive methods solve weakly convex stochastic optimization problems.
problem Solving weakly convex stochastic optimization problems.
method Adaptive first and zeroth-order methods using exponential moving averages.
result Established non-asymptotic convergence rates for nonsmooth and nonconvex problems.
Cyclic coordinate descent identifies models in finite time and converges linearly.
problem Model identification in composite nonsmooth optimization problems.
method Cyclic coordinate descent for a wide class of functions.
result Explicit local linear convergence rates for coordinate descent.
The paper relaxes assumptions for analyzing stochastic optimization algorithms.
problem Analyzing the convergence of stochastic gradient algorithms under weaker variance assumptions.
method Building on and extending a connection to the Halpern iteration, the paper analyzes algorithms for convex nonsmooth optimization and min-max problems.
result Rates for optimality measures are obtained without requiring boundedness of the feasible set for problems beyond simple constrained optimization.
A new method solves convex optimization problems on manifolds efficiently.
problem Optimization on Hadamard manifolds with convex objectives.
method Intrinsic Riemannian proximal gradient method.
result Sublinear and linear convergence rates for convex and strongly convex problems, respectively.
Proposes BMME for optimizing nonsmooth nonconvex problems with block structure.
problem Optimizing nonsmooth nonconvex problems with block structure.
method Block Alternating Bregman Majorization Minimization with Extrapolation (BMME).
result Subsequential convergence to a first-order stationary point under mild assumptions, global convergence under stronger conditions.
Develops a fast algorithm for high-dimensional LASSO penalized quantile regression.
problem Computational challenges in high-dimensional penalized quantile regression.
method Pathwise coordinate descent algorithm to solve exact coordinatewise minimum of the nonsmooth loss function.
result Algorithm runs faster than existing alternatives and maintains estimation accuracy.
We consider optimization problems over the Stiefel manifold whose objective function is the summation of a smooth function and a nonsmooth function. Existing methods for solving this kind of problems can be classified into three classes. Algorithms in the first class rely on information of the subgradients of the objec…
We construct some nonsmoothable actions of Z2 * Z2 on spin four-manifolds by using an equivariant version of Furuta' s 10/8inequality. The examples satisfy following property: any proper subgroup of Z2 * Z2 is smoothable for some smooth structure.
In regularized risk minimization, the associated optimization problem becomes particularly difficult when both the loss and regularizer are nonsmooth. Existing approaches either have slow or unclear convergence properties, are restricted to limited problem subclasses, or require careful setting of a smoothing parameter…
We construct a nonsmoothable Z\times Z-action on the connected sum of an Enriques surface and S^2\times S^2, such that each of generators is smoothable. We also construct a nonsmoothable self-homeomorphism on an Enriques surface.
As surrogate functions of -norm, many nonconvex penalty functions have been proposed to enhance the sparse vector recovery. It is easy to extend these nonconvex penalty functions on singular values of a matrix to enhance low-rank matrix recovery. However, different from convex optimization, solving the nonconvex l…