text: In machine learning research, the proximal gradient methods are popular for solving various optimization problems with non-smooth regularization. Inexact proximal gradient methods are extremely important when exactly solving the proximal operator is time-consuming, or the proximal operator does not have an analytic solution. However, existing inexact proximal gradient methods only consider convex problems. The knowledge of inexact proximal gradient methods in the non-convex setting is very limited. Moreover, for some machine learning models, there is still no proposed solver for exactly solving the proximal operator. To address this challenge, in this paper, we first propose three inexact proximal gradient algorithms, including the basic version and Nesterov's accelerated version. After that, we provide the theoretical analysis to the basic and Nesterov's accelerated versions. The theoretical results show that our inexact proximal gradient algorithms can have the same convergence rates as the ones of exact proximal gradient algorithms in the non-convex setting. Finally, we show the applications of our inexact proximal gradient algorithms on three representative non-convex learning problems. All experimental results confirm the superiority of our new inexact proximal gradient algorithms.
  @cite_0: We consider the class of iterative shrinkage-thresholding algorithms (ISTA) for solving linear inverse problems arising in signal image processing. This class of methods, which can be viewed as an extension of the classical gradient algorithm, is attractive due to its simplicity and thus is adequate for solving large-scale problems even with dense matrix data. However, such methods are also known to converge quite slowly. In this paper we present a new fast iterative shrinkage-thresholding algorithm (FISTA) which preserves the computational simplicity of ISTA but with a global rate of convergence which is proven to be significantly better, both theoretically and practically. Initial promising numerical results for wavelet-based image deblurring demonstrate the capabilities of FISTA which is shown to be faster than ISTA by several orders of magnitude.
  @cite_5: Nonconvex and nonsmooth problems have recently received considerable attention in signal image processing, statistics and machine learning. However, solving the nonconvex and nonsmooth optimization problems remains a big challenge. Accelerated proximal gradient (APG) is an excellent method for convex programming. However, it is still unknown whether the usual APG can ensure the convergence to a critical point in nonconvex programming. In this paper, we extend APG for general nonconvex and nonsmooth programs by introducing a monitor that satisfies the sufficient descent property. Accordingly, we propose a monotone APG and a nonmonotone APG. The latter waives the requirement on monotonic reduction of the objective function and needs less computation in each iteration. To the best of our knowledge, we are the first to provide APG-type algorithms for general nonconvex and nonsmooth problems ensuring that every accumulation point is a critical point, and the convergence rates remain O(1 k2) when the problems are convex, in which k is the number of iterations. Numerical results testify to the advantage of our algorithms in speed.
  @cite_10: In this paper, we generalize the well-known Nesterov's accelerated gradient (AG) method, originally designed for convex smooth optimization, to solve nonconvex and possibly stochastic optimization problems. We demonstrate that by properly specifying the stepsize policy, the AG method exhibits the best known rate of convergence for solving general nonconvex smooth optimization problems by using first-order information, similarly to the gradient descent method. We then consider an important class of composite optimization problems and show that the AG method can solve them uniformly, i.e., by using the same aggressive stepsize policy as in the convex case, even if the problem turns out to be nonconvex. We demonstrate that the AG method exhibits an optimal rate of convergence if the composite problem is convex, and improves the best known rate of convergence if the problem is nonconvex. Based on the AG method, we also present new nonconvex stochastic approximation methods and show that they can improve a few existing rates of convergence for nonconvex stochastic optimization. To the best of our knowledge, this is the first time that the convergence of the AG method has been established for solving nonconvex nonlinear programming in the literature.
  @cite_11: This paper provides lower bounds on the convergence rate of Derivative Free Optimization (DFO) with noisy function evaluations, exposing a fundamental and unavoidable gap between the performance of algorithms with access to gradients and those with access to only function evaluations. However, there are situations in which DFO is unavoidable, and for such situations we propose a new DFO algorithm that is proved to be near optimal for the class of strongly convex objective functions. A distinctive feature of the algorithm is that it uses only Boolean-valued function comparisons, rather than function evaluations. This makes the algorithm useful in an even wider range of applications, such as optimization based on paired comparisons from human subjects, for example. We also show that regardless of whether DFO is based on noisy function evaluations or Boolean-valued function comparisons, the convergence rate is the same.
summary: Specifically, for convex problems, @cite_0 proposed basic proximal gradient (PG) method and Nesterov's accelerated proximal gradient (APG) method. They proved that PG has the convergence rate @math , and APG has the convergence rate @math , where @math is the number of iterations. For non-convex problems, @cite_10 considered that only @math could be non-convex, and proved that the convergence rate of APG method is @math . @cite_11 considered that both of @math and @math could be non-convex, and proved the convergence of PG method. @cite_5 considered that both of @math and @math could be non-convex, and proved that the APG algorithm can converge in a finite number of iterations, in a linear rate or a sublinear rate (, @math ) at different conditions. We summarize these exact proximal gradient methods in Table . In addition to the above batch exact proximal gradient methods, there are the stochastic and online proximal gradient methods . Because they are beyond the scope of this paper, we do not review them in this paper.