Back-propagation (BP) is the foundation for successfully training deep neural networks. However, BP sometimes has difficulties in propagating a learning signal deep enough effectively, e.g., the vanishing gradient phenomenon. Meanwhile, BP often works well when combining with ``designing tricks'' like orthogonal initialization, batch normalization and skip connection. There is no clear understanding on what is essential to the efficiency of BP. In this paper, we take one step towards clarifying this problem. We view BP as a solution of back-matching propagation which minimizes a sequence of back-matching losses each corresponding to one block of the network. We study the Hessian of the local back-matching loss (local Hessian) and connect it to the efficiency of BP. It turns out that those designing tricks facilitate BP by improving the spectrum of local Hessian. In addition, we can utilize the local Hessian to balance the training pace of each block and design new training algorithms. Based on a scalar approximation of local Hessian, we propose a scale-amended SGD algorithm. We apply it to train neural networks with batch normalization, and achieve favorable results over vanilla SGD. This corroborates the importance of local Hessian from another side.

The backpropagation of error algorithm (BP) is impossible to implement in a real brain. The recent success of deep networks in machine learning and AI, however, has inspired proposals for understanding how the brain might learn across multiple layers, and hence how it might approximate BP. As of yet, none of these proposals have been rigorously evaluated on tasks where BP-guided deep learning has proved critical, or in architectures more structured than simple fully-connected networks. Here we present results on scaling up biologically motivated models of deep learning on datasets which need deep networks with appropriate architectures to achieve good performance. We present results on the MNIST, CIFAR-10, and ImageNet datasets and explore variants of target-propagation (TP) and feedback alignment (FA) algorithms, and explore performance in both fully- and locally-connected architectures. We also introduce weight-transport-free variants of difference target propagation (DTP) modified to remove backpropagation from the penultimate layer. Many of these algorithms perform well for MNIST, but for CIFAR and ImageNet we find that TP and FA variants perform significantly worse than BP, especially for networks composed of locally connected units, opening questions about whether new architectures and algorithms are required to scale these approaches. Our results and implementation details help establish baselines for biologically motivated deep learning schemes going forward.

In this paper, we develop the first one-pass streaming algorithm for submodular maximization that does not evaluate the entire stream even once. By carefully subsampling each element of the data stream, our algorithm enjoys the tightest approximation guarantees in various settings while having the smallest memory footprint and requiring the lowest number of function evaluations. More specifically, for a monotone submodular function and a $p$-matchoid constraint, our randomized algorithm achieves a $4p$ approximation ratio (in expectation) with $O(k)$ memory and $O(km/p)$ queries per element ($k$ is the size of the largest feasible solution and $m$ is the number of matroids used to define the constraint). For the non-monotone case, our approximation ratio increases only slightly to $4p+2-o(1)$. To the best or our knowledge, our algorithm is the first that combines the benefits of streaming and subsampling in a novel way in order to truly scale submodular maximization to massive machine learning problems. To showcase its practicality, we empirically evaluated the performance of our algorithm on a video summarization application and observed that it outperforms the state-of-the-art algorithm by up to fifty-fold while maintaining practically the same utility. We also evaluated the scalability of our algorithm on a large dataset of Uber pick up locations.

It is widely believed that learning good representations is one of the main reasons for the success of deep neural networks. Although highly intuitive, there is a lack of theory and systematic approach quantitatively characterizing what representations do deep neural networks learn. In this work, we move a tiny step towards a theory and better understanding of the representations. Specifically, we study a simpler problem: How similar are the representations learned by two networks with identical architecture but trained from different initializations. We develop a rigorous theory based on the neuron activation subspace match model. The theory gives a complete characterization of the structure of neuron activation subspace matches, where the core concepts are maximum match and simple match which describe the overall and the finest similarity between sets of neurons in two networks respectively. We also propose efficient algorithms to find the maximum match and simple matches. Finally, we conduct extensive experiments using our algorithms. Experimental results suggest that, surprisingly, representations learned by the same convolutional layers of networks trained from different initializations are not as similar as prevalently expected, at least in terms of subspace match.

We present a new method and system, called DeepZ, for certifying neural network robustness based on abstract interpretation. Compared to state-of-the-art automated verifiers for neural networks, DeepZ: (i) handles ReLU, Tanh and Sigmoid activation functions, (ii) supports feedforward and convolutional architectures, (iii) is significantly more scalable and precise, and (iv) and is sound with respect to floating point arithmetic. These benefits are due to carefully designed approximations tailored to the setting of neural networks. As an example, DeepZ achieves a verification accuracy of 97% on a large network with 88,500 hidden units under $L_{\infty}$ attack with $\epsilon = 0.1$ with an average runtime of 133 seconds.

We identify and study two common failure modes for early training in deep ReLU nets. For each, we give a rigorous proof of when it occurs and how to avoid it, for fully connected, convolutional, and residual architectures. We show that the first failure mode, exploding or vanishing mean activation length, can be avoided by initializing weights from a symmetric distribution with variance 2/fan-in and, for ResNets, by correctly scaling the residual modules. We prove that the second failure mode, exponentially large variance of activation length, never occurs in residual nets once the first failure mode is avoided. In contrast, for fully connected nets, we prove that this failure mode can happen and is avoided by keeping constant the sum of the reciprocals of layer widths. We demonstrate empirically the effectiveness of our theoretical results in predicting when networks are able to start training. In particular, we note that many popular initializations fail our criteria, whereas correct initialization and architecture allows much deeper networks to be trained.

We present the very first robust Bayesian Online Changepoint Detection algorithm through General Bayesian Inference (GBI) with $\beta$-divergences. The resulting inference procedure is doubly robust for both the predictive and the changepoint (CP) posterior, with linear time and constant space complexity. We provide a construction for exponential models and demonstrate it on the Bayesian Linear Regression model. In so doing, we make two additional contributions: Firstly, we make GBI scalable using Structural Variational approximations that are exact as $\beta \to 0$. Secondly, we give a principled way of choosing the divergence parameter $\beta$ by minimizing expected predictive loss on-line. Reducing False Discovery Rates of \CPs from up to 99\% to 0\% on real world data, this offers the state of the art.

We study the implicit regularization imposed by gradient descent for learning multi-layer homogeneous functions including feed-forward fully connected and convolutional deep neural networks with linear, ReLU or Leaky ReLU activation. We rigorously prove that gradient flow (i.e. gradient descent with infinitesimal step size) effectively enforces the differences between squared norms across different layers to remain invariant without any explicit regularization. This result implies that if the weights are initially small, gradient flow automatically balances the magnitudes of all layers. Using a discretization argument, we analyze gradient descent with positive step size for the non-convex low-rank asymmetric matrix factorization problem without any regularization. Inspired by our findings for gradient flow, we prove that gradient descent with step sizes $\eta_t=O(t^{−(1/2+\delta)}) (0<\delta\le1/2)$ automatically balances two low-rank factors and converges to a bounded global optimum. Furthermore, for rank-1 asymmetric matrix factorization we give a finer analysis showing gradient descent with constant step size converges to the global minimum at a globally linear rate. We believe that the idea of examining the invariance imposed by first order algorithms in learning homogeneous models could serve as a fundamental building block for studying optimization for learning deep models.

We consider the problem of maximizing a submodular function when given access to its approximate version. Submodular functions are heavily studied in a wide variety of disciplines, since they are used to model many real world phenomena, and are amenable to optimization. However, there are many cases in which the phenomena we observe is only approximately submodular and the approximation guarantees cease to hold. We describe a technique which we call the sampled mean approximation that yields strong guarantees for maximization of submodular functions from approximate surrogates under cardinality and intersection of matroid constraints. In particular, we show tight guarantees for maximization under a cardinality constraint and 1/(1+P) approximation under intersection of P matroids.

Model-free reinforcement learning aims to offer off-the-shelf solutions for controlling dynamical systems without requiring models of the system dynamics. We introduce a model-free random search algorithm for training static, linear policies for continuous control problems. Common evaluation methodology shows that our method matches state-of-the-art sample efficiency on the benchmark MuJoCo locomotion tasks. Nonetheless, more rigorous evaluation reveals that the assessment of performance on these benchmarks is optimistic. We evaluate the performance of our method over hundreds of random seeds and many different hyperparameter configurations for each benchmark task. This extensive evaluation is possible because of the small computational footprint of our method. Our simulations highlight a high variability in performance in these benchmark tasks, indicating that commonly used estimations of sample efficiency do not adequately evaluate the performance of RL algorithms. Our results stress the need for new baselines, benchmarks and evaluation methodology for RL algorithms.