Zero-shot learning (ZSL) is made possible by learning a projection function between a feature space and a semantic space (e.g., an attribute space). Key to ZSL is thus to learn a projection that is robust against the often large domain gap between the seen and unseen class domains. In this work, this is achieved by unseen class data synthesis and robust projection function learning. Specifically, a novel semantic data synthesis strategy is proposed, by which semantic class prototypes (e.g., attribute vectors) are used to simply perturb seen class data for generating unseen class ones. As in any data synthesis hallucination approach, there are ambiguities and uncertainties on how well the synthesised data can capture the targeted unseen class data distribution. To cope with this, the second contribution of this work is a novel projection learning model termed competitive bidirectional projection learning (BPL) designed to best utilise the ambiguous synthesised data. Specifically, we assume that each synthesised data point can belong to any unseen class; and the most likely two class candidates are exploited to learn a robust projection function in a competitive fashion. As a third contribution, we show that the proposed ZSL model can be easily extended to few-shot learning (FSL) by again exploiting semantic (class prototype guided) feature synthesis and competitive BPL. Extensive experiments show that our model achieves the state-of-the-art results on both problems.
Gradient-Based Optimisation A hybrid gradient descent ascent algorithm is developed to train our competitive BPL model, inspired by generalised competitive learning @cite_39 @cite_62 . This formulation yields a mix of min-min and max-min optimisation problems. Solving max-min (or min-min) problems is non-trivial, and many complicated optimisation methods have been developed such as entropy-based aggregate method @cite_65 and projected Lagrangian @cite_29 . In this paper, we propose a gradient-based optimisation algorithm which differs from existing optimisation methods in that: the gradient descent and ascent strategies are combined in a unified framework, resulting in an extremely efficient solver with rigorous theoretic algorithm analysis provided.
@cite_29 @cite_62 @cite_65 @cite_39
2769394111 2963487351 2570983198 1189174436
Nesterov's accelerated gradient descent (AGD), an instance of the general family of "momentum methods", provably achieves faster convergence rate than gradient descent (GD) in the convex setting. However, whether these methods are superior to GD in the nonconvex setting remains open. This paper studies a simple variant of AGD, and shows that it escapes saddle points and finds a second-order stationary point in @math iterations, faster than the @math iterations required by GD. To the best of our knowledge, this is the first Hessian-free algorithm to find a second-order stationary point faster than GD, and also the first single-loop algorithm with a faster rate than GD even in the setting of finding a first-order stationary point. Our analysis is based on two key ideas: (1) the use of a simple Hamiltonian function, inspired by a continuous-time perspective, which AGD monotonically decreases per step even for nonconvex functions, and (2) a novel framework called improve or localize, which is useful for tracking the long-term behavior of gradient-based optimization algorithms. We believe that these techniques may deepen our understanding of both acceleration algorithms and nonconvex optimization. Nesterov's accelerated gradient descent (AGD), an instance of the general family of momentum methods,'' provably achieves faster convergence rate than gradient descent (GD) in the convex setting. While these methods are widely used in modern nonconvex applications, including training of deep neural networks, whether they are provably superior to GD in the nonconvex setting remains open. This paper studies a simple variant of Nesterov's AGD, and shows that it escapes saddle points and finds a second-order stationary point in ( O (1 ^ 7 4 ) ) iterations, matching the best known convergence rate, which is faster than the ( O (1 ^ 2 ) ) iterations required by GD. To the best of our knowledge, this is the first direct acceleration (single-loop) algorithm that is provably faster than GD in general nonconvex setting---all previous nonconvex accelerated algorithms rely on more complex mechanisms such as nested loops and proximal terms. Our analysis is based on two key ideas: (1) the use of a simple Hamiltonian function, inspired by a continuous-time perspective, which AGD monotonically decreases on each step even for nonconvex functions, and (2) a novel framework called improve or localize, which is useful for tracking the long-term behavior of gradient-based optimization algorithms. We believe that these techniques may deepen our understanding of both acceleration algorithms and nonconvex optimization. We analyze a fast incremental aggregated gradient method for optimizing nonconvex problems of the form minΣ i ƒ i (x). Specifically, we analyze the SAGA algorithm within an Incremental First-order Oracle framework, and show that it converges to a stationary point provably faster than both gradient descent and stochastic gradient descent. We also discuss a Polyak's special class of nonconvex problems for which SAGA converges at a linear rate to the global optimum. Finally, we analyze the practically valuable regularized and minibatch variants of SAGA. To our knowledge, this paper presents the first analysis of fast convergence for an incremental aggregated gradient method for nonconvex problems. We propose a simple, scalable, and fast gradient descent algorithm to optimize a nonconvex objective for the rank minimization problem and a closely related family of semidefinite programs. With @math random measurements of a positive semidefinite @math matrix of rank @math and condition number @math , our method is guaranteed to converge linearly to the global optimum.