Faster Acceleration for Steepest Descent

• URL: http://arxiv.org/abs/2409.19200v1
• Date: Sat, 28 Sep 2024 01:21:03 GMT
• Title: Faster Acceleration for Steepest Descent
• Authors: Site Bai, Brian Bullins,
• Abstract summary: We propose a new accelerated first-order method for convex optimization under non-Euclidean smoothness assumptions. For $ell_p$ norm problems in $d$ dimensions, our method provides an complexity improvement of up to $O(d1-frac2p)$ in terms of calls to a first-order oracle.
• Score: 6.972653925522813
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose a new accelerated first-order method for convex optimization under non-Euclidean smoothness assumptions. In contrast to standard acceleration techniques, our approach uses primal-dual iterate sequences taken with respect to differing norms, which are then coupled using an implicitly determined interpolation parameter. For $\ell_p$ norm smooth problems in $d$ dimensions, our method provides an iteration complexity improvement of up to $O(d^{1-\frac{2}{p}})$ in terms of calls to a first-order oracle, thereby allowing us to circumvent long-standing barriers in accelerated non-Euclidean steepest descent.

Related papers

• Non-Euclidean High-Order Smooth Convex Optimization [9.940728137241214]
  We develop algorithms for the optimization of convex objectives that have H"older continuous $q$-th derivatives. Our algorithms work for general norms under mild conditions, including the $ell_p$-settings for $1leq pleq infty$.
  arXiv  Detail & Related papers  (2024-11-13T19:22:34Z)
• Methods for Convex $(L_0,L_1)$-Smooth Optimization: Clipping, Acceleration, and Adaptivity [50.25258834153574]
  We focus on the class of (strongly) convex $(L0)$-smooth functions and derive new convergence guarantees for several existing methods. In particular, we derive improved convergence rates for Gradient Descent with smoothnessed Gradient Clipping and for Gradient Descent with Polyak Stepsizes.
  arXiv  Detail & Related papers  (2024-09-23T13:11:37Z)
• First-Order Methods for Linearly Constrained Bilevel Optimization [38.19659447295665]
  We present first-order linearly constrained optimization methods for high-level Hessian computations. For linear inequality constraints, we attain $(delta,epsilon)$-Goldstein stationarity in $widetildeO(ddelta-1 epsilon-3)$ gradient oracle calls.
  arXiv  Detail & Related papers  (2024-06-18T16:41:21Z)
• Strictly Low Rank Constraint Optimization -- An Asymptotically $\mathcal{O}(\frac{1}{t^2})$ Method [5.770309971945476]
  We propose a class of non-text and non-smooth problems with textitrank regularization to promote sparsity in optimal solution. We show that our algorithms are able to achieve a singular convergence of $Ofrac(t2)$, which is exactly same as Nesterov's optimal convergence for first-order methods on smooth convex problems.
  arXiv  Detail & Related papers  (2023-07-04T16:55:41Z)
• Accelerated Quasi-Newton Proximal Extragradient: Faster Rate for Smooth Convex Optimization [26.328847475942894]
  We prove that our method can achieve a convergence rate of $Obigl(minfrac1k2, fracsqrtdlog kk2.5bigr)$, where $d$ is the problem dimension and $k$ is the number of iterations. To the best of our knowledge, this result is the first to demonstrate a provable gain of a quasi-Newton-type method over Nesterov's accelerated gradient.
  arXiv  Detail & Related papers  (2023-06-03T23:31:27Z)
• Pseudonorm Approachability and Applications to Regret Minimization [73.54127663296906]
  We convert high-dimensional $ell_infty$-approachability problems to low-dimensional pseudonorm approachability problems. We develop an algorithmic theory of pseudonorm approachability, analogous to previous work on approachability for $ell$ and other norms.
  arXiv  Detail & Related papers  (2023-02-03T03:19:14Z)
• Accelerated First-Order Optimization under Nonlinear Constraints [73.2273449996098]
  We exploit between first-order algorithms for constrained optimization and non-smooth systems to design a new class of accelerated first-order algorithms. An important property of these algorithms is that constraints are expressed in terms of velocities instead of sparse variables.
  arXiv  Detail & Related papers  (2023-02-01T08:50:48Z)
• Second-order optimization with lazy Hessians [55.51077907483634]
  We analyze Newton's lazy Hessian updates for solving general possibly non-linear optimization problems. We reuse a previously seen Hessian iteration while computing new gradients at each step of the method.
  arXiv  Detail & Related papers  (2022-12-01T18:58:26Z)
• Extra-Newton: A First Approach to Noise-Adaptive Accelerated Second-Order Methods [57.050204432302195]
  This work proposes a universal and adaptive second-order method for minimizing second-order smooth, convex functions. Our algorithm achieves $O(sigma / sqrtT)$ convergence when the oracle feedback is with variance $sigma2$, and improves its convergence to $O( 1 / T3)$ with deterministic oracles.
  arXiv  Detail & Related papers  (2022-11-03T14:12:51Z)
• Explicit Second-Order Min-Max Optimization Methods with Optimal Convergence Guarantee [86.05440220344755]
  We propose and analyze inexact regularized Newton-type methods for finding a global saddle point of emphcon unconstrained min-max optimization problems. We show that the proposed methods generate iterates that remain within a bounded set and that the iterations converge to an $epsilon$-saddle point within $O(epsilon-2/3)$ in terms of a restricted function.
  arXiv  Detail & Related papers  (2022-10-23T21:24:37Z)
• Alternating Differentiation for Optimization Layers [133.2668019610731]
  We develop a new framework, named Alternating Differentiation (Alt-Diff), that differentiates optimization problems. We show that Alt-Diff substantially decreases the dimensions of the Jacobian matrix especially for optimization with large-scale constraints. We also propose to truncate Alt-Diff to further accelerate the computational speed.
  arXiv  Detail & Related papers  (2022-10-03T11:32:13Z)
• Accelerated SGD for Non-Strongly-Convex Least Squares [14.010916616909743]
  We consider approximation for the least squares regression problem in the non-strongly convex setting. We present the first practical algorithm that achieves the optimal prediction error rates in terms of dependence on the noise of the problem.
  arXiv  Detail & Related papers  (2022-03-03T14:39:33Z)
• Lifted Primal-Dual Method for Bilinearly Coupled Smooth Minimax Optimization [47.27237492375659]
  We study the bilinearly coupled minimax problem: $min_x max_y f(x) + ytop A x - h(y)$, where $f$ and $h$ are both strongly convex smooth functions. No known first-order algorithms have hitherto achieved the lower complexity bound of $Omega(sqrtfracL_xmu_x + frac|A|sqrtmu_x,mu_y) log(frac1vareps
  arXiv  Detail & Related papers  (2022-01-19T05:56:19Z)
• Halpern-Type Accelerated and Splitting Algorithms For Monotone Inclusions [14.445131747570962]
  We develop a new type of accelerated algorithms to solve some classes of maximally monotone equations. Our methods rely on a so-called Halpern-type fixed-point iteration in [32], and recently exploited by a number of researchers.
  arXiv  Detail & Related papers  (2021-10-15T15:26:32Z)
• Projection Efficient Subgradient Method and Optimal Nonsmooth Frank-Wolfe Method [54.93433440034386]
  We find a feasible $epsilon$-suboptimal solution using only $O(epsilon-1)$ PO calls and optimal $O(epsilon-2)$ FO calls. Our experiments confirm that these methods achieve significant speedups over the state-of-the-art, for a problem with costly PO and LMO calls.
  arXiv  Detail & Related papers  (2020-10-05T08:16:56Z)
• Second-Order Information in Non-Convex Stochastic Optimization: Power and Limitations [54.42518331209581]
  We find an algorithm which finds. epsilon$-approximate stationary point (with $|nabla F(x)|le epsilon$) using. $(epsilon,gamma)$surimate random random points. Our lower bounds here are novel even in the noiseless case.
  arXiv  Detail & Related papers  (2020-06-24T04:41:43Z)
• Convergence of adaptive algorithms for weakly convex constrained optimization [59.36386973876765]
  We prove the $mathcaltilde O(t-1/4)$ rate of convergence for the norm of the gradient of Moreau envelope. Our analysis works with mini-batch size of $1$, constant first and second order moment parameters, and possibly smooth optimization domains.
  arXiv  Detail & Related papers  (2020-06-11T17:43:19Z)
• A Simple Convergence Proof of Adam and Adagrad [74.24716715922759]
  We show a proof of convergence between the Adam Adagrad and $O(d(N)/st)$ algorithms. Adam converges with the same convergence $O(d(N)/st)$ when used with the default parameters.
  arXiv  Detail & Related papers  (2020-03-05T01:56:17Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.