Summary Provably Faster Gradient Descent via Long Steps arxiv.org
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The text introduces a novel analysis technique for gradient descent, utilizing long step sizes to achieve improved convergence rates and providing a table of step size patterns that lead to faster convergence.
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Key Points
- The work presents a new analysis technique for gradient descent that establishes faster convergence rates.
- The use of long step sizes in gradient descent algorithms can improve convergence rates.
- Nonconstant stepsize policies, including periodic long steps, may increase the objective value in the short term but lead to faster convergence.
- The concept of strong convexity and the growth bound condition play a role in achieving faster convergence in gradient descent algorithms.
- The analysis of optimal accelerated and subgradient methods takes into account the decreasing distance to a minimizer.
Summaries
36 word summary
This work presents a new analysis technique for gradient descent, using long step sizes, to achieve faster convergence rates. It includes a table of step size patterns that result in faster convergence, using inequalities and equalities.
47 word summary
This work introduces a new analysis technique for gradient descent that achieves faster convergence rates. It discusses the use of long step sizes in gradient descent algorithms and presents a table showing different step size patterns that result in faster convergence. The analysis involves inequalities and equalities
428 word summary
This work presents a new analysis technique for gradient descent that establishes faster convergence rates than previous proofs. The theory allows for nonconstant stepsize policies, including periodic long steps that may increase the objective value in the short term but lead to faster convergence in the
The document discusses the use of long step sizes in gradient descent algorithms to improve convergence rates. It presents a table showing different step size patterns that result in faster convergence, with each pattern proven using a semidefinite programming solution certificate. The analysis of non
The given text excerpt discusses the use of long steps in gradient descent algorithms to improve their efficiency. The authors present inequalities and equalities that describe the behavior of gradient descent on convex and smooth functions. They introduce nonnegative multipliers and combine these inequalities to
The text excerpt discusses the convergence rate of gradient descent algorithms and the conditions under which faster convergence can be achieved. It introduces the concept of strong convexity and the growth bound condition for optimization functions. The excerpt also mentions the straightforward stepsize pattern and its
The text discusses a method for faster gradient descent in optimization problems. It introduces a problem formulation (3.1) and then presents a relaxation of this problem to an SDP (3.2). The text explains that under certain conditions, the QC
The text excerpt discusses the concept of straightforward stepsize patterns in gradient descent algorithms. The main result, Theorem 3.1, states that if a stepsize pattern satisfies certain conditions, it is considered straightforward. The proof of this theorem involves checking
The text discusses the use of long stepsize patterns in gradient descent optimization algorithms. The authors present a theorem that proves the convergence rate of gradient descent with a specific stepsize pattern. They provide numerical evidence and conjecture about the longest straightforward stepsize patterns
The focus of the research is on the decrease of the objective gap in gradient descent methods. The analysis of optimal accelerated and subgradient methods also takes into account the decreasing distance to a minimizer. Finding more specific Lyapunovs and stepsize
The summary includes a list of references cited in the document. These references are from various publications related to optimization methods. The publications cover topics such as the design and analysis of first-order methods, branch-and-bound performance estimation programming, optimized first-order methods for
The excerpt includes a list of references to various scientific papers and publications. It also mentions a computer-generated certificate proving a specific rate for a pattern of length 7. The certificate includes a matrix with values and calculations verifying their feasibility.