Understanding the loss surface of neural networks is essential for the design of models with predictable performance and their success in applications. Experimental results suggest that sufficiently deep and wide neural networks are not negatively impacted by suboptimal local minima. Despite recent progress, the reason for this outcome is not fully understood. Could deep networks have very few, if at all, suboptimal local optima? or could all of them be equally good? We provide a construction to show that suboptimal local minima (i.e. non-global ones), even though degenerate, exist for fully connected neural networks with sigmoid activation functions. The local minima obtained by our proposed construction belong to a connected set of local solutions that can be escaped from via a non-increasing path on the loss curve. For extremely wide neural networks with two hidden layers, we prove that every suboptimal local minimum belongs to such a connected set. This provides a partial explanation for the successful application of deep neural networks. In addition, we also characterize under what conditions the same construction leads to saddle points instead of local minima for deep neural networks.
To gain better insight into theoretical aspects, some papers consider linear networks, where the activation function is the identity. The classic result by Baldi and Hornik @cite_32 shows that linear two-layer neural networks have a unique global minimum and all other critical values are saddle points. Kawaguchi, @cite_37 , Lu and Kawaguchi @cite_3 and @cite_4 discuss generalizations of @cite_32 to deep linear networks.
@cite_37 @cite_4 @cite_32 @cite_3
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Due to the success of deep learning to solving a variety of challenging machine learning tasks, there is a rising interest in understanding loss functions for training neural networks from a theoretical aspect. Particularly, the properties of critical points and the landscape around them are of importance to determine the convergence performance of optimization algorithms. In this paper, we provide full (necessary and sufficient) characterization of the analytical forms for the critical points (as well as global minimizers) of the square loss functions for various neural networks. We show that the analytical forms of the critical points characterize the values of the corresponding loss functions as well as the necessary and sufficient conditions to achieve global minimum. Furthermore, we exploit the analytical forms of the critical points to characterize the landscape properties for the loss functions of these neural networks. One particular conclusion is that: The loss function of linear networks has no spurious local minimum, while the loss function of one-hidden-layer nonlinear networks with ReLU activation function does have local minimum that is not global minimum. We present a comparative theoretical analysis of representation in artificial neural networks with two extreme architectures, a shallow wide network and a deep narrow network, devised to maximally decouple their representative power due to layer width and network depth. We show that, given a specific activation function, models with comparable VC-dimension are required to guarantee zero error modeling of real functions over a binary input. However, functions that exhibit repeating patterns can be encoded much more efficiently in the deep representation, resulting in significant reduction in complexity. This paper provides some initial theoretical evidence of when and how depth can be extremely effective. In this paper, we prove a conjecture published in 1989 and also partially address an open problem announced at the Conference on Learning Theory (COLT) 2015. For an expected loss function of a deep nonlinear neural network, we prove the following statements under the independence assumption adopted from recent work: 1) the function is non-convex and non-concave, 2) every local minimum is a global minimum, 3) every critical point that is not a global minimum is a saddle point, and 4) the property of saddle points differs for shallow networks (with three layers) and deeper networks (with more than three layers). Moreover, we prove that the same four statements hold for deep linear neural networks with any depth, any widths and no unrealistic assumptions. As a result, we present an instance, for which we can answer to the following question: how difficult to directly train a deep model in theory? It is more difficult than the classical machine learning models (because of the non-convexity), but not too difficult (because of the nonexistence of poor local minima and the property of the saddle points). We note that even though we have advanced the theoretical foundations of deep learning, there is still a gap between theory and practice. Neural networks provide a rich class of high-dimensional, non-convex optimization problems. Despite their non-convexity, gradient-descent methods often successfully optimize these models. This has motivated a recent spur in research attempting to characterize properties of their loss surface that may be responsible for such success. In particular, several authors have noted that appears to act as a remedy against non-convexity. In this paper, we address this phenomenon by studying key topological properties of the loss, such as the presence or absence of "spurious valleys", defined as connected components of sub-level sets that do not include a global minimum. Focusing on a class of two-layer neural networks defined by smooth (but generally non-linear) activation functions, our main contribution is to prove that as soon as the hidden layer size matches the dimension of the reproducing space, defined as the linear functional space generated by the activations, no spurious valleys exist, thus allowing the existence of descent directions. Our setup includes smooth activations such as polynomials, both in the empirical and population risk, and generic activations in the empirical risk case. Local minima and plateaus pose a serious problem in learning of neural networks. We investigate the hierarchical geometric structure of the parameter space of three-layer perceptrons in order to show the existence of local minima and plateaus. It is proved that a critical point of the model with H−1 hidden units always gives many critical points of the model with H hidden units. These critical points consist of many lines in the parameter space, which can cause plateaus in learning of neural networks. Based on this result, we prove that a point in the critical lines corresponding to the global minimum of the smaller model can be a local minimum or a saddle point of the larger model. We give a necessary and sufficient condition for this, and show that this kind of local minima exist as a line segment if any. The results are universal in the sense that they do not require special properties of the target, loss functions and activation functions, but only use the hierarchical structure of the model. 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. We prove that neural networks with a single hidden layer are capable of providing an optimal order of approximation for functions assumed to possess a given number of derivatives, if the activation function evaluated by each principal element satisfies certain technical conditions. Under these conditions, it is also possible to construct networks that provide a geometric order of approximation for analytic target functions. The permissible activation functions include the squashing function (1-e-x)-1 as well as a variety of radial basis functions. Our proofs are constructive. The weights and thresholds of our networks are chosen independently of the target function; we give explicit formulas for the coefficients as simple, continuous, linear functionals of the target function. With the increasing interest in deeper understanding of the loss surface of many non-convex deep models, this paper presents a unifying framework to study the local global optima equivalence of the optimization problems arising from training of such non-convex models. Using the "local openness" property of the underlying training models, we provide simple sufficient conditions under which any local optimum of the resulting optimization problem is globally optimal. We first completely characterize the local openness of matrix multiplication mapping in its range. Then we use our characterization to: 1) show that every local optimum of two layer linear networks is globally optimal. Unlike many existing results in the literature, our result requires no assumption on the target data matrix Y, and input data matrix X. 2) develop almost complete characterization of the local global optima equivalence of multi-layer linear neural networks. We provide various counterexamples to show the necessity of each of our assumptions. 3) show global local optima equivalence of non-linear deep models having certain pyramidal structure. Unlike some existing works, our result requires no assumption on the differentiability of the activation functions and can go beyond "full-rank" cases. One of the mysteries in the success of neural networks is randomly initialized first order methods like gradient descent can achieve zero training loss even though the objective function is non-convex and non-smooth. This paper demystifies this surprising phenomenon for two-layer fully connected ReLU activated neural networks. For an @math hidden node shallow neural network with ReLU activation and @math training data, we show as long as @math is large enough and no two inputs are parallel, randomly initialized gradient descent converges to a globally optimal solution at a linear convergence rate for the quadratic loss function. Our analysis relies on the following observation: over-parameterization and random initialization jointly restrict every weight vector to be close to its initialization for all iterations, which allows us to exploit a strong convexity-like property to show that gradient descent converges at a global linear rate to the global optimum. We believe these insights are also useful in analyzing deep models and other first order methods. In this paper we evaluate the quality of the activation layers of a convolutional neural network (CNN) for the gen- eration of object proposals. We generate hypotheses in a sliding-window fashion over different activation layers and show that the final convolutional layers can find the object of interest with high recall but poor localization due to the coarseness of the feature maps. Instead, the first layers of the network can better localize the object of interest but with a reduced recall. Based on this observation we design a method for proposing object locations that is based on CNN features and that combines the best of both worlds. We build an inverse cascade that, going from the final to the initial convolutional layers of the CNN, selects the most promising object locations and refines their boxes in a coarse-to-fine manner. The method is efficient, because i) it uses the same features extracted for detection, ii) it aggregates features using integral images, and iii) it avoids a dense evaluation of the proposals due to the inverse coarse-to-fine cascade. The method is also accurate; it outperforms most of the previously proposed object proposals approaches and when plugged into a CNN-based detector produces state-of-the- art detection performance. We study the error landscape of deep linear and nonlinear neural networks with the squared error loss. Minimizing the loss of a deep linear neural network is a nonconvex problem, and despite recent progress, our understanding of this loss surface is still incomplete. For deep linear networks, we present necessary and sufficient conditions for a critical point of the risk function to be a global minimum. Surprisingly, our conditions provide an efficiently checkable test for global optimality, while such tests are typically intractable in nonconvex optimization. We further extend these results to deep nonlinear neural networks and prove similar sufficient conditions for global optimality, albeit in a more limited function space setting. Several recently proposed architectures of neural networks such as ResNeXt, Inception, Xception, SqueezeNet and Wide ResNet are based on the designing idea of having multiple branches and have demonstrated improved performance in many applications. We show that one cause for such success is due to the fact that the multi-branch architecture is less non-convex in terms of duality gap. The duality gap measures the degree of intrinsic non-convexity of an optimization problem: smaller gap in relative value implies lower degree of intrinsic non-convexity. The challenge is to quantitatively measure the duality gap of highly non-convex problems such as deep neural networks. In this work, we provide strong guarantees of this quantity for two classes of network architectures. For the neural networks with arbitrary activation functions, multi-branch architecture and a variant of hinge loss, we show that the duality gap of both population and empirical risks shrinks to zero as the number of branches increases. This result sheds light on better understanding the power of over-parametrization where increasing the network width tends to make the loss surface less non-convex. For the neural networks with linear activation function and @math loss, we show that the duality gap of empirical risk is zero. Our two results work for arbitrary depths and adversarial data, while the analytical techniques might be of independent interest to non-convex optimization more broadly. Experiments on both synthetic and real-world datasets validate our results. Deep neural networks (DNNs) have demonstrated dominating performance in many fields; since AlexNet, networks used in practice are going wider and deeper. On the theoretical side, a long line of works has been focusing on training neural networks with one hidden layer. The theory of multi-layer networks remains largely unsettled. In this work, we prove why stochastic gradient descent (SGD) can find @math on the training objective of DNNs in @math . We only make two assumptions: the inputs are non-degenerate and the network is over-parameterized. The latter means the network width is sufficiently large: @math in @math , the number of layers and in @math , the number of samples. Our key technique is to derive that, in a sufficiently large neighborhood of the random initialization, the optimization landscape is almost-convex and semi-smooth even with ReLU activations. This implies an equivalence between over-parameterized neural networks and neural tangent kernel (NTK) in the finite (and polynomial) width setting. As concrete examples, starting from randomly initialized weights, we prove that SGD can attain 100 training accuracy in classification tasks, or minimize regression loss in linear convergence speed, with running time polynomial in @math . Our theory applies to the widely-used but non-smooth ReLU activation, and to any smooth and possibly non-convex loss functions. In terms of network architectures, our theory at least applies to fully-connected neural networks, convolutional neural networks (CNN), and residual neural networks (ResNet). This paper designs a high-performance deep convolutional network (DeepID2+) for face recognition. It is learned with the identification-verification supervisory signal. By increasing the dimension of hidden representations and adding supervision to early convolutional layers, DeepID2+ achieves new state-of-the-art on LFW and YouTube Faces benchmarks. Through empirical studies, we have discovered three properties of its deep neural activations critical for the high performance: sparsity, selectiveness and robustness. (1) It is observed that neural activations are moderately sparse. Moderate sparsity maximizes the discriminative power of the deep net as well as the distance between images. It is surprising that DeepID2+ still can achieve high recognition accuracy even after the neural responses are binarized. (2) Its neurons in higher layers are highly selective to identities and identity-related attributes. We can identify different subsets of neurons which are either constantly excited or inhibited when different identities or attributes are present. Although DeepID2+ is not taught to distinguish attributes during training, it has implicitly learned such high-level concepts. (3) It is much more robust to occlusions, although occlusion patterns are not included in the training set. This paper designs a high-performance deep convolutional network (DeepID2+) for face recognition. It is learned with the identification-verification supervisory signal. By increasing the dimension of hidden representations and adding supervision to early convolutional layers, DeepID2+ achieves new state-of-the-art on LFW and YouTube Faces benchmarks. Through empirical studies, we have discovered three properties of its deep neural activations critical for the high performance: sparsity, selectiveness and robustness. (1) It is observed that neural activations are moderately sparse. Moderate sparsity maximizes the discriminative power of the deep net as well as the distance between images. It is surprising that DeepID2+ still can achieve high recognition accuracy even after the neural responses are binarized. (2) Its neurons in higher layers are highly selective to identities and identity-related attributes. We can identify different subsets of neurons which are either constantly excited or inhibited when different identities or attributes are present. Although DeepID2+ is not taught to distinguish attributes during training, it has implicitly learned such high-level concepts. (3) It is much more robust to occlusions, although occlusion patterns are not included in the training set. 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. Finding minimum distortion of adversarial examples and thus certifying robustness in neural networks classifiers is known to be a challenging problem. Nevertheless, recently it has been shown to be possible to give a non-trivial certified lower bound of minimum distortion, and some recent progress has been made towards this direction by exploiting the piece-wise linear nature of ReLU activations. However, a generic robustness certification for activation functions still remains largely unexplored. To address this issue, in this paper we introduce CROWN, a general framework to certify robustness of neural networks with general activation functions. The novelty in our algorithm consists of bounding a given activation function with linear and quadratic functions, hence allowing it to tackle general activation functions including but not limited to the four popular choices: ReLU, tanh, sigmoid and arctan. In addition, we facilitate the search for a tighter certified lower bound by selecting appropriate surrogates for each neuron activation. Experimental results show that CROWN on ReLU networks can notably improve the certified lower bounds compared to the current state-of-the-art algorithm Fast-Lin, while having comparable computational efficiency. Furthermore, CROWN also demonstrates its effectiveness and flexibility on networks with general activation functions, including tanh, sigmoid and arctan. To the best of our knowledge, CROWN is the first framework that can efficiently certify non-trivial robustness for general activation functions in neural networks. We study the complexity of functions computable by deep feedforward neural networks with piecewise linear activations in terms of the symmetries and the number of linear regions that they have. Deep networks are able to sequentially map portions of each layer's input-space to the same output. In this way, deep models compute functions that react equally to complicated patterns of different inputs. The compositional structure of these functions enables them to re-use pieces of computation exponentially often in terms of the network's depth. This paper investigates the complexity of such compositional maps and contributes new theoretical results regarding the advantage of depth for neural networks with piecewise linear activation functions. In particular, our analysis is not specific to a single family of models, and as an example, we employ it for rectifier and maxout networks. We improve complexity bounds from pre-existing work and investigate the behavior of units in higher layers. In this paper, we study the problem of learning a shallow artificial neural network that best fits a training data set. We study this problem in the over-parameterized regime where the numbers of observations are fewer than the number of parameters in the model. We show that with the quadratic activations, the optimization landscape of training, such shallow neural networks, has certain favorable characteristics that allow globally optimal models to be found efficiently using a variety of local search heuristics. This result holds for an arbitrary training data of input output pairs. For differentiable activation functions, we also show that gradient descent, when suitably initialized, converges at a linear rate to a globally optimal model. This result focuses on a realizable model where the inputs are chosen i.i.d. from a Gaussian distribution and the labels are generated according to planted weight coefficients. In this paper we evaluate the quality of the activation layers of a convolutional neural network (CNN) for the generation of object proposals. We generate hypotheses in a sliding-window fashion over different activation layers and show that the final convolutional layers can find the object of interest with high recall but poor localization due to the coarseness of the feature maps. Instead, the first layers of the network can better localize the object of interest but with a reduced recall. Based on this observation we design a method for proposing object locations that is based on CNN features and that combines the best of both worlds. We build an inverse cascade that, going from the final to the initial convolutional layers of the CNN, selects the most promising object locations and refines their boxes in a coarse-to-fine manner. The method is efficient, because i) it uses the same features extracted for detection, ii) it aggregates features using integral images, and iii) it avoids a dense evaluation of the proposals due to the inverse coarse-to-fine cascade. The method is also accurate, it outperforms most of the previously proposed object proposals approaches and when plugged into a CNN-based detector produces state-of-the-art detection performance.