A common analogy for explaining gradient descent goes like the following: a person is stuck in the mountains during heavy fog, and must navigate their way down. The natural way they will approach this is to look at the slope of the visible ground around them and slowly work their way down the mountain by following the downward slope. This captures the essence of gradient descent, but this analogy always ends up breaking down when we scale to a high dimensional space where we have very little idea what the actual geometry of that space is. Although, in the end it's often not a practical concern because gradient descent seems to work pretty well. But the important question is: how well does gradient descent perform on the actual earth? In a general model gradient descent is used to find weights for a model that minimizes our cost function, which is usually some representation of the errors made by a model over a number of predictions.

A residual network (or ResNet) is a standard deep neural net architecture, with state-of-the-art performance across numerous applications. The main premise of ResNets is that they allow the training of each layer to focus on fitting just the residual of the previous layer's output and the target output. Thus, we should expect that the trained network is no worse than what we can obtain if we remove the residual layers and train a shallower network instead. However, due to the non-convexity of the optimization problem, it is not at all clear that ResNets indeed achieve this behavior, rather than getting stuck at some arbitrarily poor local minimum. In this paper, we rigorously prove that arbitrarily deep, nonlinear ResNets indeed exhibit this behavior, in the sense that the optimization landscape contains no local minima with value above what can be obtained with a linear predictor (namely a 1-layer network). Notably, we show this under minimal or no assumptions on the precise network architecture, data distribution, or loss function used. We also provide a quantitative analysis of second-order stationary points for this problem, and show that with a certain tweak to the architecture, training the network with standard stochastic gradient descent achieves an objective value no worse than any fixed linear predictor.

Multi-layer neural networks are among the most powerful models in machine learning, yet the fundamental reasons for this success defy mathematical understanding. Learning a neural network requires to optimize a non-convex high-dimensional objective (risk function), a problem which is usually attacked using stochastic gradient descent (SGD). Does SGD converge to a global optimum of the risk or only to a local optimum? In the first case, does this happen because local minima are absent, or because SGD somehow avoids them? In the second, why do local minima reached by SGD have good generalization properties? In this paper we consider a simple case, namely two-layers neural networks, and prove that -in a suitable scaling limit- SGD dynamics is captured by a certain non-linear partial differential equation (PDE) that we call distributional dynamics (DD). We then consider several specific examples, and show how DD can be used to prove convergence of SGD to networks with nearlyideal generalization error. This description allows to 'average-out' some of the complexities of the landscape of neural networks, and can be used to prove a general convergence result for noisy SGD.

In many sequential decision making tasks, it is challenging to design reward functions that help an RL agent efficiently learn behavior that is considered good by the agent designer. A number of different formulations of the reward-design problem, or close variants thereof, have been proposed in the literature. In this paper we build on the Optimal Rewards Framework of Singh et.al. that defines the optimal intrinsic reward function as one that when used by an RL agent achieves behavior that optimizes the task-specifying or extrinsic reward function. Previous work in this framework has shown how good intrinsic reward functions can be learned for lookahead search based planning agents. Whether it is possible to learn intrinsic reward functions for learning agents remains an open problem. In this paper we derive a novel algorithm for learning intrinsic rewards for policy-gradient based learning agents. We compare the performance of an augmented agent that uses our algorithm to provide additive intrinsic rewards to an A2C-based policy learner (for Atari games) and a PPO-based policy learner (for Mujoco domains) with a baseline agent that uses the same policy learners but with only extrinsic rewards. Our results show improved performance on most but not all of the domains.

Stochastic gradient methods enable learning probabilistic models from large amounts of data. While large step-sizes (learning rates) have shown to be best for least-squares (e.g., Gaussian noise) once combined with parameter averaging, these are not leading to convergent algorithms in general. In this paper, we consider generalized linear models, that is, conditional models based on exponential families. We propose averaging moment parameters instead of natural parameters for constant-step-size stochastic gradient descent. For finite-dimensional models, we show that this can sometimes (and surprisingly) lead to better predictions than the best linear model. For infinite-dimensional models, we show that it always converges to optimal predictions, while averaging natural parameters never does. We illustrate our findings with simulations on synthetic data and classical benchmarks with many observations.

As learning algorithms are increasingly deployed in markets and other competitive environments, understanding their dynamics is becoming increasingly important. We study the limiting behavior of competitive agents employing gradient-based learning algorithms. Specifically, we introduce a general framework for competitive gradient-based learning that encompasses a wide breadth of learning algorithms including policy gradient reinforcement learning, gradient based bandits, and certain online convex optimization algorithms. We show that unlike the single agent case, gradient learning schemes in competitive settings do not necessarily correspond to gradient flows and, hence, it is possible for limiting behaviors like periodic orbits to exist. We introduce a new class of games, Morse-Smale games, that correspond to gradient-like flows. We provide guarantees that competitive gradient-based learning algorithms (both in the full information and gradient-free settings) avoid linearly unstable critical points (i.e. strict saddle points and unstable limit cycles). Since generic local Nash equilibria are not unstable critical points---that is, in a formal mathematical sense, almost all Nash equilibria are not strict saddles---these results imply that gradient-based learning almost surely does not get stuck at critical points that do not correspond to Nash equilibria. For Morse-Smale games, we show that competitive gradient learning converges to linearly stable cycles (which includes stable Nash equilibria) almost surely. Finally, we specialize these results to commonly used multi-agent learning algorithms and provide illustrative examples that demonstrate the wide range of limiting behaviors competitive gradient learning exhibits.

Emerging technologies and applications including Internet of Things (IoT), social networking, and crowd-sourcing generate large amounts of data at the network edge. Machine learning models are often built from the collected data, to enable the detection, classification, and prediction of future events. Due to bandwidth, storage, and privacy concerns, it is often impractical to send all the data to a centralized location. In this paper, we consider the problem of learning model parameters from data distributed across multiple edge nodes, without sending raw data to a centralized place. Our focus is on a generic class of machine learning models that are trained using gradient-descent based approaches. We analyze the convergence rate of distributed gradient descent from a theoretical point of view, based on which we propose a control algorithm that determines the best trade-off between local update and global parameter aggregation to minimize the loss function under a given resource budget. The performance of the proposed algorithm is evaluated via extensive experiments with real datasets, both on a networked prototype system and in a larger-scale simulated environment. The experimentation results show that our proposed approach performs near to the optimum with various machine learning models and different data distributions.

We investigate the effect of explicitly enforcing the Lipschitz continuity of neural networks. Our main hypothesis is that constraining the Lipschitz constant of a networks will have a regularising effect. To this end, we provide a simple technique for computing the Lipschitz constant of a feed forward neural network composed of commonly used layer types. This technique is then utilised to formulate training a Lipschitz continuous neural network as a constrained optimisation problem, which can be easily solved using projected stochastic gradient methods. Our evaluation study shows that, in isolation, our method performs comparatively to state-of-the-art regularisation techniques. Moreover, when combined with existing approaches to regularising neural networks the performance gains are cumulative.

With the development of big data technology, Gradient Boosting Decision Tree, i.e. GBDT, becomes one of the most important machine learning algorithms for its accurate output. However, the training process of GBDT needs a lot of computational resources and time. In order to accelerate the training process of GBDT, the asynchronous parallel sampling gradient boosting decision tree, abbr. asynch-SGBDT is proposed in this paper. Via introducing sampling, we adapt the numerical optimization process of traditional GBDT training process into stochastic optimization process and use asynchronous parallel stochastic gradient descent to accelerate the GBDT training process. Meanwhile, the theoretical analysis of asynch-SGBDT is provided by us in this paper. Experimental results show that GBDT training process could be accelerated by asynch-SGBDT. Our asynchronous parallel strategy achieves an almost linear speedup, especially for high-dimensional sparse datasets.

Anomaly Detection has several important applications. In this paper, our focus is on detecting anomalies in seller-reviewer data using tensor decomposition. While tensor-decomposition is mostly unsupervised, we formulate Bayesian semi-supervised tensor decomposition to take advantage of sparse labeled data. In addition, we use Polya-Gamma data augmentation for the semi-supervised Bayesian tensor decomposition. Finally, we show that the Polya-Gamma formulation simplifies calculation of the Fisher information matrix for partial natural gradient learning. Our experimental results show that our semi-supervised approach outperforms state of the art unsupervised baselines. And that the partial natural gradient learning outperforms stochastic gradient learning and Online-EM with sufficient statistics.