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We propose a neural network approach for solving highdimensional optimal control problems. In particular, we focus on multiagent control problems with obstacle and collision avoidance. These problems immediately become highdimensional, even for moderate phasespace dimensions per agent. Our approach fuses the Pontryagin Maximum Principle and HamiltonJacobiBellman (HJB) approaches and parameterizes the value function with a neural network. Our approach yields controls in a feedback form for quick calculation and robustness to moderate disturbances to the system. We train our model using the objective function and optimality conditions of the control problem. Therefore, our training algorithm neither involves a data generationmore »

Training activation quantized neural networks involves minimizing a piecewise constant function whose gradient vanishes almost everywhere, which is undesirable for the standard backpropagation or chain rule. An empirical way around this issue is to use a straightthrough estimator (STE) (Bengio et al., 2013) in the backward pass only, so that the “gradient” through the modified chain rule becomes nontrivial. Since this unusual “gradient” is certainly not the gradient of loss function, the following question arises: why searching in its negative direction minimizes the training loss? In this paper, we provide the theoretical justification of the concept of STE by answeringmore »

Quantized deep neural networks (QDNNs) are attractive due to their much lower memory storage and faster inference speed than their regular fullprecision counterparts. To maintain the same performance level especially at low bitwidths, QDNNs must be retrained. Their training involves piecewise constant activation functions and discrete weights; hence, mathematical challenges arise. We introduce the notion of coarse gradient and propose the blended coarse gradient descent (BCGD) algorithm, for training fully quantized neural networks. Coarse gradient is generally not a gradient of any function but an artificial ascent direction. The weight update of BCGD goes by coarse gradient correction of amore »

We propose a new algorithm to approximate the Earth Mover's distance (EMD). Our main idea is motivated by the theory of optimal transport, in which EMD can be reformulated as a familiar L1 type minimization. We use a regularization which gives us a unique solution for this L1 type problem. The new regularized minimization is very similar to problems which have been solved in the fields of compressed sensing and image processing, where several fast methods are available. In this paper, we adopt a primaldual algorithm designed there, which uses very simple updates at each iteration and is shown tomore »

We propose BinaryRelax, a simple twophase algorithm, for training deep neural networks with quantized weights. The set constraint that characterizes the quantization of weights is not imposed until the late stage of training, and a sequence of pseudo quantized weights is maintained. Specifically, we relax the hard constraint into a continuous regularizer via Moreau envelope, which turns out to be the squared Euclidean distance to the set of quantized weights. The pseudo quantized weights are obtained by linearly interpolating between the float weights and their quantizations. A continuation strategy is adopted to push the weights towards the quantized state bymore »