One LEGO at a Time Explaining the Math of how Neural Networks Learn with Implementation from…


A topic that is not always explained in-depth, despite its intuitive and modular nature, is the backpropagation technique responsible for updating trainable parameters. Let's build a neural network from scratch to see the internal functioning of a neural network using LEGO pieces as a modular analogy, one brick at a time. The above figure depicts some of the Math used for training a neural network. We will make sense of this during this article. At this point, these operations only compute a general linear system, which doesn't have the capacity to model non-linear interactions.

Options as responses: Grounding behavioural hierarchies in multi-agent RL Artificial Intelligence

We propose a novel hierarchical agent architecture for multi-agent reinforcement learning with concealed information. The hierarchy is grounded in the concealed information about other players, which resolves "the chicken or the egg" nature of option discovery. We factorise the value function over a latent representation of the concealed information and then re-use this latent space to factorise the policy into options. Low-level policies (options) are trained to respond to particular states of other agents grouped by the latent representation, while the top level (meta-policy) learns to infer the latent representation from its own observation thereby to select the right option. This grounding facilitates credit assignment across the levels of hierarchy. We show that this helps generalisation---performance against a held-out set of pre-trained competitors, while training in self- or population-play---and resolution of social dilemmas in self-play.

An Introduction to Variational Autoencoders Machine Learning

Variational autoencoders provide a principled framework for learning deep latent-variable models and corresponding inference models. In this work, we provide an introduction to variational autoencoders and some important extensions.

Distributed Training with Heterogeneous Data: Bridging Median- and Mean-Based Algorithms Machine Learning

Recently, there is a growing interest in the study of median-based algorithms for distributed non-convex optimization. Two prominent such algorithms include signSGD with majority vote, an effective approach for communication reduction via 1-bit compression on the local gradients, and medianSGD, an algorithm recently proposed to ensure robustness against Byzantine workers. The convergence analyses for these algorithms critically rely on the assumption that all the distributed data are drawn iid from the same distribution. However, in applications such as Federated Learning, the data across different nodes or machines can be inherently heterogeneous, which violates such an iid assumption. This work analyzes signSGD and medianSGD in distributed settings with heterogeneous data. We show that these algorithms are non-convergent whenever there is some disparity between the expected median and mean over the local gradients. To overcome this gap, we provide a novel gradient correction mechanism that perturbs the local gradients with noise, together with a series results that provable close the gap between mean and median of the gradients. The proposed methods largely preserve nice properties of these methods, such as the low per-iteration communication complexity of signSGD, and further enjoy global convergence to stationary solutions. Our perturbation technique can be of independent interest when one wishes to estimate mean through a median estimator.

Failures detection at directional drilling using real-time analogues search Machine Learning

One of the main challenges in the construction of oil and gas wells is the need to detect and avoid abnormal situations, which can lead to accidents. Accidents have some indicators that help to find them during the drilling process. In this article, we present a data-driven model trained on historical data from drilling accidents that can detect different types of accidents using real-time signals. The results show that using the time-series comparison, based on aggregated statistics and gradient boosting classification, it is possible to detect an anomaly and identify its type by comparing current measurements while drilling with the stored ones from the database of accidents.

Practical Deep Learning with Bayesian Principles Machine Learning

Bayesian methods promise to fix many shortcomings of deep learning, but they are impractical and rarely match the performance of standard methods, let alone improve them. In this paper, we demonstrate practical training of deep networks with natural-gradient variational inference. By applying techniques such as batch normalisation, data augmentation, and distributed training, we achieve similar performance in about the same number of epochs as the Adam optimiser, even on large datasets such as ImageNet. Importantly, the benefits of Bayesian principles are preserved: predictive probabilities are well-calibrated and uncertainties on out-of-distribution data are improved. This work enables practical deep learning while preserving benefits of Bayesian principles. A PyTorch implementation will be available as a plug-and-play optimiser.

Primal-Dual Block Frank-Wolfe Machine Learning

Particularly, we are interested in problems whose solution has special "simple" structure like low-rank or sparsity. The sparsity constraint applies to large-scale multiclass/multi-label classification, low-degree polynomial data mapping [5], random feature kernel machines [32], and Elastic Net [39]. Motivated by recent applications in low-rank multi-class SVM, phase retrieval, matrix completion, affine rank minimization and other problems (e.g., [9, 31, 2, 3]), we also consider settings where the constraint x C (e.g., trace norm ball) while convex, may be difficult to project onto. A wish-list for this class of problems would include an algorithm that (1) exploits the function finite-sum form and the simple structure of the solution, (2) achieves linear convergence for smooth and strongly convex problems, (3) does not pay a heavy price for the projection step. We propose a Frank-Wolfe (FW) type method that attains these three goals. This does not come without challenges: Although it is currently well-appreciated that FW type algorithms avoid the cost of projection [14, 1], the benefits are limited to constraints that are hard to project onto, like the trace norm ball. For problems like phase retrieval and ERM for multi-label multi-class classification, the gradient computation requires large matrix multiplications. This dominates the per-iteration cost, and the existing FW type methods do not asymptotically reduce time complexity per iteration, even without paying the expensive projection step.

ODE Analysis of Stochastic Gradient Methods with Optimism and Anchoring for Minimax Problems and GANs Machine Learning

Despite remarkable empirical success, the training dynamics of generative adversarial networks (GAN), which involves solving a minimax game using stochastic gradients, is still poorly understood. In this work, we analyze last-iterate convergence of simultaneous gradient descent (simGD) and its variants under the assumption of convex-concavity, guided by a continuous-time analysis with differential equations. First, we show that simGD, as is, converges with stochastic sub-gradients under strict convexity in the primal variable. Second, we generalize optimistic simGD to accommodate an optimism rate separate from the learning rate and show its convergence with full gradients. Finally, we present anchored simGD, a new method, and show convergence with stochastic subgradients.

Data Sketching for Faster Training of Machine Learning Models Artificial Intelligence

Many machine learning problems reduce to the problem of minimizing an expected risk, defined as the sum of a large number of, often convex, component functions. Iterative gradient methods are popular techniques for the above problems. However, they are in general slow to converge, in particular for large data sets. In this work, we develop analysis for selecting a subset (or sketch) of training data points with their corresponding learning rates in order to provide faster convergence to a close neighbordhood of the optimal solution. We show that subsets that minimize the upper-bound on the estimation error of the full gradient, maximize a submodular facility location function. As a result, by greedily maximizing the facility location function we obtain subsets that yield faster convergence to a close neighborhood of the optimum solution. We demonstrate the real-world effectiveness of our algorithm, SIG, confirming our analysis, through an extensive set of experiments on several applications, including logistic regression and training neural networks. We also include a method that provides a deliberate deterministic ordering of the data subset that is quite effective in practice. We observe that our method, while achieving practically the same loss, speeds up gradient methods by up to 10x for convex and 3x for non-convex (deep) functions.

Machine Learning Estimation of Heterogeneous Treatment Effects with Instruments Machine Learning

We consider the estimation of heterogeneous treatment effects with arbitrary machine learning methods in the presence of unobserved confounders with the aid of a valid instrument. Such settings arise in A/B tests with an intent-to-treat structure, where the experimenter randomizes over which user will receive a recommendation to take an action, and we are interested in the effect of the downstream action. We develop a statistical learning approach to the estimation of heterogeneous effects, reducing the problem to the minimization of an appropriate loss function that depends on a set of auxiliary models (each corresponding to a separate prediction task). The reduction enables the use of all recent algorithmic advances (e.g. neural nets, forests). We show that the estimated effect model is robust to estimation errors in the auxiliary models, by showing that the loss satisfies a Neyman orthogonality criterion. Our approach can be used to estimate projections of the true effect model on simpler hypothesis spaces. When these spaces are parametric, then the parameter estimates are asymptotically normal, which enables construction of confidence sets. We applied our method to estimate the effect of membership on downstream webpage engagement on TripAdvisor, using as an instrument an intent-to-treat A/B test among 4 million TripAdvisor users, where some users received an easier membership sign-up process. We also validate our method on synthetic data and on public datasets for the effects of schooling on income.