356,933 research outputs found
On the Global Convergence of Derivative Free Methods for Unconstrained Optimization.
In this paper, starting from the study of the common elements that some globally
convergent direct search methods share, a general convergence theory is established for unconstrained
minimization methods employing only function values. The introduced convergence conditions are
useful for developing and analyzing new derivative-free algorithms with guaranteed global convergence.
As examples, we describe three new algorithms which combine pattern and line search approaches
Convergence and stability of line search methods for unconstrained optimization.
This paper explores the stability of general line search methods in the sense of Lyapunov, for minimizing a smooth nonlinear function. In particular we give sufficient conditions for a line search method to be globally asymptotical stable. Our analysis suggests that the proposed sufficient conditions for asymptotical stability is equivalent to the Zoutendijk-type conditions in conventional global convergence analysis
Don't be so Monotone: Relaxing Stochastic Line Search in Over-Parameterized Models
Recent works have shown that line search methods can speed up Stochastic
Gradient Descent (SGD) and Adam in modern over-parameterized settings. However,
existing line searches may take steps that are smaller than necessary since
they require a monotone decrease of the (mini-)batch objective function. We
explore nonmonotone line search methods to relax this condition and possibly
accept larger step sizes. Despite the lack of a monotonic decrease, we prove
the same fast rates of convergence as in the monotone case. Our experiments
show that nonmonotone methods improve the speed of convergence and
generalization properties of SGD/Adam even beyond the previous monotone line
searches. We propose a POlyak NOnmonotone Stochastic (PoNoS) method, obtained
by combining a nonmonotone line search with a Polyak initial step size.
Furthermore, we develop a new resetting technique that in the majority of the
iterations reduces the amount of backtracks to zero while still maintaining a
large initial step size. To the best of our knowledge, a first runtime
comparison shows that the epoch-wise advantage of line-search-based methods
gets reflected in the overall computational time
Forward-backward truncated Newton methods for convex composite optimization
This paper proposes two proximal Newton-CG methods for convex nonsmooth
optimization problems in composite form. The algorithms are based on a a
reformulation of the original nonsmooth problem as the unconstrained
minimization of a continuously differentiable function, namely the
forward-backward envelope (FBE). The first algorithm is based on a standard
line search strategy, whereas the second one combines the global efficiency
estimates of the corresponding first-order methods, while achieving fast
asymptotic convergence rates. Furthermore, they are computationally attractive
since each Newton iteration requires the approximate solution of a linear
system of usually small dimension
A New Class of Non-Quest-Newton Methods and Their Global Convergence with Goldstein Line Search
In this paper, on the basis of the DFP method a class of non-quasi-Newton methods is presented. Under some condition the global convergence property of these methods with Goldstein line search on uniformly convex objective function is proved
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