We propose interpretable deep Gaussian Processes (GPs) that combine the expressiveness of deep Neural Networks (NNs) with quantified uncertainty of deep GPs. Our approach is based on approximating deep GP as a GP, which allows explicit, analytic forms for compositions of a wide variety of kernels. Consequently, our approach admits interpretation as both NNs with specified activation functions and as a variational approximation to deep GPs. We provide general recipes for deriving the effective kernels for deep GPs of two, three, or infinitely many layers, composed of homogeneous or heterogeneous kernels. Results illustrate the expressiveness of our effective kernels through samples from the prior and inference on simulated data and demonstrate advantages of interpretability by analysis of analytic forms, drawing relations and equivalences across kernels, and a priori identification of non-pathological regimes of hyperparameter space.
Williams @cite_18 was first to derive analytic kernels of one-layer neural networks, using sigmoidal and Gaussian activation functions. Later, Cho and Saul @cite_14 extended analytic results to polynomial activation functions and further made extension to deep models. Daniely al @cite_20 extended these results for homogeneous deep networks with 2nd hermite, step, and exponential activation functions and Poole al @cite_5 to @math activation functions. Our approach differs by focusing on deep GPs for Bayesian inference, and allowing heterogeneous kernels.
@cite_5 @cite_18 @cite_14 @cite_20
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In this paper, we consider regression problems with one-hidden-layer neural networks (1NNs). We distill some properties of activation functions that lead to @math in the neighborhood of the ground-truth parameters for the 1NN squared-loss objective. Most popular nonlinear activation functions satisfy the distilled properties, including rectified linear units (ReLUs), leaky ReLUs, squared ReLUs and sigmoids. For activation functions that are also smooth, we show @math guarantees of gradient descent under a resampling rule. For homogeneous activations, we show tensor methods are able to initialize the parameters to fall into the local strong convexity region. As a result, tensor initialization followed by gradient descent is guaranteed to recover the ground truth with sample complexity @math and computational complexity @math for smooth homogeneous activations with high probability, where @math is the dimension of the input, @math ( @math ) is the number of hidden nodes, @math is a conditioning property of the ground-truth parameter matrix between the input layer and the hidden layer, @math is the targeted precision and @math is the number of samples. To the best of our knowledge, this is the first work that provides recovery guarantees for 1NNs with both sample complexity and computational complexity @math in the input dimension and @math in the precision. In this paper, we consider parameter recovery for non-overlapping convolutional neural networks (CNNs) with multiple kernels. We show that when the inputs follow Gaussian distribution and the sample size is sufficiently large, the squared loss of such CNNs is @math in a basin of attraction near the global optima for most popular activation functions, like ReLU, Leaky ReLU, Squared ReLU, Sigmoid and Tanh. The required sample complexity is proportional to the dimension of the input and polynomial in the number of kernels and a condition number of the parameters. We also show that tensor methods are able to initialize the parameters to the local strong convex region. Hence, for most smooth activations, gradient descent following tensor initialization is guaranteed to converge to the global optimal with time that is linear in input dimension, logarithmic in precision and polynomial in other factors. To the best of our knowledge, this is the first work that provides recovery guarantees for CNNs with multiple kernels under polynomial sample and computational complexities. We study the improper learning of multi-layer neural networks. Suppose that the neural network to be learned has @math hidden layers and that the @math -norm of the incoming weights of any neuron is bounded by @math . We present a kernel-based method, such that with probability at least @math , it learns a predictor whose generalization error is at most @math worse than that of the neural network. The sample complexity and the time complexity of the presented method are polynomial in the input dimension and in @math , where @math is a function depending on @math and on the activation function, independent of the number of neurons. The algorithm applies to both sigmoid-like activation functions and ReLU-like activation functions. It implies that any sufficiently sparse neural network is learnable in polynomial time. We show that the output of a (residual) convolutional neural network (CNN) with an appropriate prior over the weights and biases is a Gaussian process (GP) in the limit of infinitely many convolutional filters, extending similar results for dense networks. For a CNN, the equivalent kernel can be computed exactly and, unlike "deep kernels", has very few parameters: only the hyperparameters of the original CNN. Further, we show that this kernel has two properties that allow it to be computed efficiently; the cost of evaluating the kernel for a pair of images is similar to a single forward pass through the original CNN with only one filter per layer. The kernel equivalent to a 32-layer ResNet obtains 0.84 classification error on MNIST, a new record for GPs with a comparable number of parameters.