This work presents a new algorithm for training recurrent neural networks (although ideas are applicable to feedforward networks as well). The algorithm is derived from a theory in nonconvex optimization related to the diffusion equation. The contributions made in this work are two fold. First, we show how some seemingly disconnected mechanisms used in deep learning such as smart initialization, annealed learning rate, layerwise pretraining, and noise injection (as done in dropout and SGD) arise naturally and automatically from this framework, without manually crafting them into the algorithms. Second, we present some preliminary results on comparing the proposed method against SGD. It turns out that the new algorithm can achieve similar level of generalization accuracy of SGD in much fewer number of epochs.
This work relies on smoothing the objective function by convolving it with the Gaussian kernel. We have previously shown that this particular form of smoothing is optimal in a certain sense, by relating Gaussian convolution to a relaxation of the convex envelope. Although connection to the convex envelope is meaningful in the context of c nonconvex objective functions, there are side benefits in smoothing even when the objective function is convex. For example, smoothing a nonsmooth c convex objective function by convolution can improve the convergence rate of stochastic optimization algorithms @cite_12 .
@cite_12
170526737 2962851402 2962774238 2101455165
The continuation method is a popular heuristic in computer vision for nonconvex optimization. The idea is to start from a simplified problem and gradually deform it to the actual task while tracking the solution. It was first used in computer vision under the name of graduated nonconvexity. Since then, it has been utilized explicitly or implicitly in various applications. In fact, state-of-the-art optical flow and shape estimation rely on a form of continuation. Despite its empirical success, there is little theoretical understanding of this method. This work provides some novel insights into this technique. Specifically, there are many ways to choose the initial problem and many ways to progressively deform it to the original task. However, here we show that when this process is constructed by Gaussian smoothing, it is optimal in a specific sense. In fact, we prove that Gaussian smoothing emerges from the best affine approximation to Vese’s nonlinear PDE. The latter PDE evolves any function to its convex envelope, hence providing the optimal convexification. We consider the fundamental problem in nonconvex optimization of efficiently reaching a stationary point. In contrast to the convex case, in the long history of this basic problem, the only known theoretical results on first-order nonconvex optimization remain to be full gradient descent that converges in O(1 e) iterations for smooth objectives, and stochastic gradient descent that converges in O(1 e2) iterations for objectives that are sum of smooth functions. We provide the first improvement in this line of research. Our result is based on the variance reduction trick recently introduced to convex optimization, as well as a brand new analysis of variance reduction that is suitable for non-convex optimization. For objectives that are sum of smooth functions, our first-order minibatch stochastic method converges with an O(1 e) rate, and is faster than full gradient descent by Ω(n1 3). We demonstrate the effectiveness of our methods on empirical risk minimizations with nonconvex loss functions and training neural nets. We consider the linear programming relaxation of an energy minimization problem for Markov Random Fields. The dual objective of this problem can be treated as a concave and unconstrained, but non-smooth function. The idea of smoothing the objective prior to optimization was recently proposed in a series of papers. Some of them suggested the idea to decrease the amount of smoothing (so called temperature) while getting closer to the optimum. However, no theoretical substantiation was provided. We propose an adaptive smoothing diminishing algorithm based on the duality gap between relaxed primal and dual objectives and demonstrate the efficiency of our approach with a smoothed version of Sequential Tree-Reweighted Message Passing (TRW-S) algorithm. The strategy is applicable to other algorithms as well, avoids ad-hoc tuning of the smoothing during iterations, and provably guarantees convergence to the optimum. We introduce a generalization of a deterministic relaxation algorithm for edge-preserving regularization in linear inverse problems. This algorithm transforms the original (possibly nonconvex) optimization problem into a sequence of quadratic optimization problems, and has been shown to converge under certain conditions when the original cost functional being minimized is strictly convex. We prove that our more general algorithm is globally convergent (i.e., converges to a local minimum from any initialization) under less restrictive conditions, even when the original cost functional is nonconvex. We apply this algorithm to tomographic reconstruction from limited-angle data by formulating the problem as one of regularized least-squares optimization. The results demonstrate that the constraint of piecewise smoothness, applied through the use of edge-preserving regularization, can provide excellent limited-angle tomographic reconstructions. Two edge-preserving regularizers-one convex, the other nonconvex-are used in numerous simulations to demonstrate the effectiveness of the algorithm under various limited-angle scenarios, and to explore how factors, such as the choice of error norm, angular sampling rate and amount of noise, affect the reconstruction quality and algorithm performance. These simulation results show that for this application, the nonconvex regularizer produces consistently superior results.