2,171 research outputs found

    Every Local Minimum Value is the Global Minimum Value of Induced Model in Non-convex Machine Learning

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    For nonconvex optimization in machine learning, this article proves that every local minimum achieves the globally optimal value of the perturbable gradient basis model at any differentiable point. As a result, nonconvex machine learning is theoretically as supported as convex machine learning with a handcrafted basis in terms of the loss at differentiable local minima, except in the case when a preference is given to the handcrafted basis over the perturbable gradient basis. The proofs of these results are derived under mild assumptions. Accordingly, the proven results are directly applicable to many machine learning models, including practical deep neural networks, without any modification of practical methods. Furthermore, as special cases of our general results, this article improves or complements several state-of-the-art theoretical results on deep neural networks, deep residual networks, and overparameterized deep neural networks with a unified proof technique and novel geometric insights. A special case of our results also contributes to the theoretical foundation of representation learning.Comment: Neural computation, MIT pres

    Effect of Depth and Width on Local Minima in Deep Learning

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    In this paper, we analyze the effects of depth and width on the quality of local minima, without strong over-parameterization and simplification assumptions in the literature. Without any simplification assumption, for deep nonlinear neural networks with the squared loss, we theoretically show that the quality of local minima tends to improve towards the global minimum value as depth and width increase. Furthermore, with a locally-induced structure on deep nonlinear neural networks, the values of local minima of neural networks are theoretically proven to be no worse than the globally optimal values of corresponding classical machine learning models. We empirically support our theoretical observation with a synthetic dataset as well as MNIST, CIFAR-10 and SVHN datasets. When compared to previous studies with strong over-parameterization assumptions, the results in this paper do not require over-parameterization, and instead show the gradual effects of over-parameterization as consequences of general results

    Universal Convexification via Risk-Aversion

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    We develop a framework for convexifying a fairly general class of optimization problems. Under additional assumptions, we analyze the suboptimality of the solution to the convexified problem relative to the original nonconvex problem and prove additive approximation guarantees. We then develop algorithms based on stochastic gradient methods to solve the resulting optimization problems and show bounds on convergence rates. %We show a simple application of this framework to supervised learning, where one can perform integration explicitly and can use standard (non-stochastic) optimization algorithms with better convergence guarantees. We then extend this framework to apply to a general class of discrete-time dynamical systems. In this context, our convexification approach falls under the well-studied paradigm of risk-sensitive Markov Decision Processes. We derive the first known model-based and model-free policy gradient optimization algorithms with guaranteed convergence to the optimal solution. Finally, we present numerical results validating our formulation in different applications
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