The constant evolution of technology has meant data and information is being generated at a rate unlike ever before, and it's only on the rise. Furthermore, the demand for people skilled in analyzing, interpreting and using this data is already high and is set to grow exponentially over the coming years. These new roles cover all aspect from strategy, operations to governance. Hence, the current and future demand will require more data scientists, data engineers, data strategists, and Chief Data Officers. In this blog, we will be looking at different set of interview questions that can certainly help if you are planning to give a shift to your career towards data science.

Gradient descent is the most commonly used optimization method deployed in machine learning and deep learning algorithms. It's used to train a machine learning model and is based on a convex function. It does this to minimize a given cost function to its local minimum. Gradient descent was invented by French mathematician Louis Augustin Cauchy in 1847. Most machine learning and deep learning algorithms involve some sort of optimization. Optimization refers to the process of either minimizing or maximizing some function by altering its parameters.

Multichannel blind deconvolution is the problem of recovering an unknown signal $f$ and multiple unknown channels $x_i$ from convolutional measurements $y_i=x_i \circledast f$ ($i=1,2,\dots,N$). We consider the case where the $x_i$'s are sparse, and convolution with $f$ is invertible. Our nonconvex optimization formulation solves for a filter $h$ on the unit sphere that produces sparse output $y_i\circledast h$. Under some technical assumptions, we show that all local minima of the objective function correspond to the inverse filter of $f$ up to an inherent sign and shift ambiguity, and all saddle points have strictly negative curvatures. This geometric structure allows successful recovery of $f$ and $x_i$ using a simple manifold gradient descent algorithm with random initialization. Our theoretical findings are complemented by numerical experiments, which demonstrate superior performance of the proposed approach over the previous methods.

Asynchronous momentum stochastic gradient descent algorithms (Async-MSGD) have been widely used in distributed machine learning, e.g., training large collaborative filtering systems and deep neural networks. Due to current technical limit, however, establishing convergence properties of Async-MSGD for these highly complicated nonoconvex problems is generally infeasible. Therefore, we propose to analyze the algorithm through a simpler but nontrivial nonconvex problems --- streaming PCA. This allows us to make progress toward understanding Aync-MSGD and gaining new insights for more general problems. Specifically, by exploiting the diffusion approximation of stochastic optimization, we establish the asymptotic rate of convergence of Async-MSGD for streaming PCA. Our results indicate a fundamental tradeoff between asynchrony and momentum: To ensure convergence and acceleration through asynchrony, we have to reduce the momentum (compared with Sync-MSGD). To the best of our knowledge, this is the first theoretical attempt on understanding Async-MSGD for distributed nonconvex stochastic optimization. Numerical experiments on both streaming PCA and training deep neural networks are provided to support our findings for Async-MSGD.

Multichannel blind deconvolution is the problem of recovering an unknown signal $f$ and multiple unknown channels $x_i$ from convolutional measurements $y_i=x_i \circledast f$ ($i=1,2,\dots,N$). We consider the case where the $x_i$'s are sparse, and convolution with $f$ is invertible. Our nonconvex optimization formulation solves for a filter $h$ on the unit sphere that produces sparse output $y_i\circledast h$. Under some technical assumptions, we show that all local minima of the objective function correspond to the inverse filter of $f$ up to an inherent sign and shift ambiguity, and all saddle points have strictly negative curvatures. This geometric structure allows successful recovery of $f$ and $x_i$ using a simple manifold gradient descent algorithm with random initialization. Our theoretical findings are complemented by numerical experiments, which demonstrate superior performance of the proposed approach over the previous methods.

Asynchronous momentum stochastic gradient descent algorithms (Async-MSGD) have been widely used in distributed machine learning, e.g., training large collaborative filtering systems and deep neural networks. Due to current technical limit, however, establishing convergence properties of Async-MSGD for these highly complicated nonoconvex problems is generally infeasible. Therefore, we propose to analyze the algorithm through a simpler but nontrivial nonconvex problems --- streaming PCA. This allows us to make progress toward understanding Aync-MSGD and gaining new insights for more general problems. Specifically, by exploiting the diffusion approximation of stochastic optimization, we establish the asymptotic rate of convergence of Async-MSGD for streaming PCA. Our results indicate a fundamental tradeoff between asynchrony and momentum: To ensure convergence and acceleration through asynchrony, we have to reduce the momentum (compared with Sync-MSGD). To the best of our knowledge, this is the first theoretical attempt on understanding Async-MSGD for distributed nonconvex stochastic optimization. Numerical experiments on both streaming PCA and training deep neural networks are provided to support our findings for Async-MSGD.

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.

The performance of neural networks on high-dimensional data distributions suggests that it may be possible to parameterize a representation of a given high-dimensional function with controllably small errors, potentially outperforming standard interpolation methods. We demonstrate, both theoretically and numerically, that this is indeed the case. We map the parameters of a neural network to a system of particles relaxing with an interaction potential determined by the loss function. We show that in the limit that the number of parameters $n$ is large, the landscape of the mean-squared error becomes convex and the representation error in the function scales as $O(n^{-1})$. In this limit, we prove a dynamical variant of the universal approximation theorem showing that the optimal representation can be attained by stochastic gradient descent, the algorithm ubiquitously used for parameter optimization in machine learning. In the asymptotic regime, we study the fluctuations around the optimal representation and show that they arise at a scale $O(n^{-1})$. These fluctuations in the landscape identify the natural scale for the noise in stochastic gradient descent. Our results apply to both single and multi-layer neural networks, as well as standard kernel methods like radial basis functions.

We consider an inexact variant of the popular Riemannian trust-region algorithm for structured big-data minimization problems. The proposed algorithm approximates the gradient and the Hessian in addition to the solution of a trust-region sub-problem. Addressing large-scale finite-sum problems, we specifically propose sub-sampled algorithms with a fixed bound on sub-sampled Hessian and gradient sizes, where the gradient and Hessian are computed by a random sampling technique. Numerical evaluations demonstrate that the proposed algorithms outperform state-of-the-art Riemannian deterministic and stochastic gradient algorithms across different applications.

Modern large-scale machine learning applications require stochastic optimization algorithms to be implemented on distributed computational architectures. A key bottleneck is the communication overhead for exchanging information such as stochastic gradients among different workers. In this paper, to reduce the communication cost, we propose a convex optimization formulation to minimize the coding length of stochastic gradients. The key idea is to randomly drop out coordinates of the stochastic gradient vectors and amplify the remaining coordinates appropriately to ensure the sparsified gradient to be unbiased. To solve the optimal sparsification efficiently, several simple and fast algorithms are proposed for an approximate solution, with a theoretical guarantee for sparseness. Experiments on $\ell_2$ regularized logistic regression, support vector machines, and convolutional neural networks validate our sparsification approaches.