Neural Information Processing Systems http://nips.cc/

In this work, we consider the problem of matrix sensing over graphs (MSoG). As a general case of matrix completion and matrix sensing problems, the MSoG problem has not been analyzed in the literature and the existing results cannot be directly applied to the MSoG problem. This work provides the first theoretical results on the optimization landscape of the MSoG problem. More specifically, we propose a new condition, named the Ω-RIP condition, to characterize the optimization complexity of the problem. In addition, with an improved regularizer of the incoherence, we prove that the strict saddle property holds for the MSoG problem with high probability under the incoherence condition and the Ω-RIP condition, which guarantees the polynomial-time global convergence of saddleavoiding methods. Compared with state-of-the-art results, the bounds in this work are tight up to a constant. Besides the theoretical guarantees, we numerically illustrate the close relation between the Ω-RIP condition and the optimization complexity.

How does our motor system solve the problem of anticipatory control in spite of a wide spectrum of response dynamics from different musculo-skeletal systems, transport delays as well as response latencies throughout the central nervous system? To a great extent, our highly-skilled motor responses are a result of a reactive feedback system, originating in the brain-stem and spinal cord, combined with a feed-forward anticipatory system, that is adaptively fine-tuned by sensory experience and originates in the cerebellum. Based on that interaction we design the counterfactual predictive control (CFPC) architecture, an anticipatory adaptive motor control scheme in which a feed-forward module, based on the cerebellum, steers an error feedback controller with counterfactual error signals. Those are signals that trigger reactions as actual errors would, but that do not code for any current or forthcoming errors. In order to determine the optimal learning strategy, we derive a novel learning rule for the feed-forward module that involves an eligibility trace and operates at the synaptic level. In particular, our eligibility trace provides a mechanism beyond co-incidence detection in that it convolves a history of prior synaptic inputs with error signals. In the context of cerebellar physiology, this solution implies that Purkinje cell synapses should generate eligibility traces using a forward model of the system being controlled. From an engineering perspective, CFPC provides a general-purpose anticipatory control architecture equipped with a learning rule that exploits the full dynamics of the closed-loop system.

This paper investigates a class of stochastic bilevel optimization problems where the upper-level function is nonconvex with potentially unbounded smoothness and the lower-level problem is strongly convex. These problems have significant applications in sequential data learning, such as text classification using recurrent neural networks. The unbounded smoothness is characterized by the smoothness constant of the upper-level function scaling linearly with the gradient norm, lacking a uniform upper bound. Existing state-of-the-art algorithms require $\widetilde{O}(\epsilon^{-4})$ oracle calls of stochastic gradient or Hessian/Jacobian-vector product to find an $\epsilon$-stationary point. However, it remains unclear if we can further improve the convergence rate when the assumptions for the function in the population level also hold for each random realization almost surely (e.g., Lipschitzness of each realization of the stochastic gradient).

Neural Information Processing Systems http://nips.cc/

Our nonconvex optimization formulation solves for a filter h on the unit sphere that produces sparse outputyi ~h.

In Section B, we provide some preliminaries. In Section C, we provide sparsity analysis. We show convergence analysis in Section D. In Section E, we show how to combine the sparsity, convergence, running time all together. In Section F, we show correlation between sparsity and spectral gap of Hessian in neural tangent kernel. In Section G, we discuss how to generalize our result to quantum setting.

Deep neural networks have achieved impressive performance in many areas. Designing a fast and provable method for training neural networks is a fundamental question in machine learning. The classical training method requires paying Ω(mnd) cost for both forward computation and backward computation, where m is the width of the neural network, and we are given n training points in d-dimensional space.