Pedro H. P. Savarese

Mor Shpigel Nacson, Jason D. Lee, Suriya Gunasekar et al.

We provide a detailed study on the implicit bias of gradient descent when optimizing loss functions with strictly monotone tails, such as the logistic loss, over separable datasets. We look at two basic questions: (a) what are the conditions on the tail of the loss function under which gradient descent converges in the direction of the $L_2$ maximum-margin separator? (b) how does the rate of margin convergence depend on the tail of the loss function and the choice of the step size? We show that for a large family of super-polynomial tailed losses, gradient descent iterates on linear networks of any depth converge in the direction of $L_2$ maximum-margin solution, while this does not hold for losses with heavier tails. Within this family, for simple linear models we show that the optimal rates with fixed step size is indeed obtained for the commonly used exponentially tailed losses such as logistic loss. However, with a fixed step size the optimal convergence rate is extremely slow as $1/\log(t)$, as also proved in Soudry et al. (2018). For linear models with exponential loss, we further prove that the convergence rate could be improved to $\log (t) /\sqrt{t}$ by using aggressive step sizes that compensates for the rapidly vanishing gradients. Numerical results suggest this method might be useful for deep networks.

Pedro H. P. Savarese, Mayank Kakodkar, Bruno Ribeiro

We propose a Las Vegas transformation of Markov Chain Monte Carlo (MCMC) estimators of Restricted Boltzmann Machines (RBMs). We denote our approach Markov Chain Las Vegas (MCLV). MCLV gives statistical guarantees in exchange for random running times. MCLV uses a stopping set built from the training data and has maximum number of Markov chain steps K (referred as MCLV-K). We present a MCLV-K gradient estimator (LVS-K) for RBMs and explore the correspondence and differences between LVS-K and Contrastive Divergence (CD-K), with LVS-K significantly outperforming CD-K training RBMs over the MNIST dataset, indicating MCLV to be a promising direction in learning generative models.

Leonardo F. R. Ribeiro, Pedro H. P. Savarese, Daniel R. Figueiredo

Structural identity is a concept of symmetry in which network nodes are identified according to the network structure and their relationship to other nodes. Structural identity has been studied in theory and practice over the past decades, but only recently has it been addressed with representational learning techniques. This work presents struc2vec, a novel and flexible framework for learning latent representations for the structural identity of nodes. struc2vec uses a hierarchy to measure node similarity at different scales, and constructs a multilayer graph to encode structural similarities and generate structural context for nodes. Numerical experiments indicate that state-of-the-art techniques for learning node representations fail in capturing stronger notions of structural identity, while struc2vec exhibits much superior performance in this task, as it overcomes limitations of prior approaches. As a consequence, numerical experiments indicate that struc2vec improves performance on classification tasks that depend more on structural identity.

Pedro H. P. Savarese, Leonardo O. Mazza, Daniel R. Figueiredo

We propose a new layer design by adding a linear gating mechanism to shortcut connections. By using a scalar parameter to control each gate, we provide a way to learn identity mappings by optimizing only one parameter. We build upon the motivation behind Residual Networks, where a layer is reformulated in order to make learning identity mappings less problematic to the optimizer. The augmentation introduces only one extra parameter per layer, and provides easier optimization by making degeneration into identity mappings simpler. We propose a new model, the Gated Residual Network, which is the result when augmenting Residual Networks. Experimental results show that augmenting layers provides better optimization, increased performance, and more layer independence. We evaluate our method on MNIST using fully-connected networks, showing empirical indications that our augmentation facilitates the optimization of deep models, and that it provides high tolerance to full layer removal: the model retains over 90% of its performance even after half of its layers have been randomly removed. We also evaluate our model on CIFAR-10 and CIFAR-100 using Wide Gated ResNets, achieving 3.65% and 18.27% error, respectively.