The esoteric nuances of machine learning algorithms and terminology can easily overwhelm the machine learning novice. As I was reading the Deep Learning book by Yoshua Bengio, Aaron Courville, and Ian Goodfellow, I was ecstatic when I reached the section that explained the common "recipe" that almost all machine learning algorithms share -- a dataset, a cost function, an optimization procedure, and a model. In this article, I summarize each universal'ingredient' of machine learning algorithms by dissecting them into their simplest components. With these'ingredients' in mind, you no longer have to view each new machine learning algorithm you encounter as an entity isolated from the others, but rather a unique combination of the four common elements described below. There are many types of machine learning algorithms.

We study online optimization in a setting where an online learner seeks to optimize a per-round hitting cost, which may be non-convex, while incurring a movement cost when changing actions between rounds. We ask: \textit{under what general conditions is it possible for an online learner to leverage predictions of future cost functions in order to achieve near-optimal costs?} Prior work has provided near-optimal online algorithms for specific combinations of assumptions about hitting and switching costs, but no general results are known. In this work, we give two general sufficient conditions that specify a relationship between the hitting and movement costs which guarantees that a new algorithm, Synchronized Fixed Horizon Control (SFHC), provides a $1+O(1/w)$ competitive ratio, where $w$ is the number of predictions available to the learner. Our conditions do not require the cost functions to be convex, and we also derive competitive ratio results for non-convex hitting and movement costs. Our results provide the first constant, dimension-free competitive ratio for online non-convex optimization with movement costs. Further, we give an example of a natural instance, Convex Body Chasing (CBC), where the sufficient conditions are not satisfied and we can prove that no online algorithm can have a competitive ratio that converges to 1.

Since this phenomenon was first observed, researchers have attempted to develop methods which produce models that are robust to adversarial perturbations under specific attack models (Wong and Kolter (2018); Sinha et al. (2018); Raghunathan et al. (2018); Mirman et al. (2018); Madry et al. (2018); Zhang et al. (2019)). As machine learning proliferates into society, including security-critical settings like health care (Esteva et al. (2017)) or autonomous vehicles (Codevilla et al. (2018)), it is crucial to develop methods that allow us to understand the vulnerability of our models and design appropriate countermeasures. Additionally there is a growing literature on the theory of adversarial examples. Many of these results attempt to understand adversarial examples by constructing examples of learning problems for which it is difficult to construct a classifier that is robust to adversarial perturbations. This difficultly may arise due to sample complexity (Schmidt et al. (2018)), computational constraints (Bubeck et al. (2019); Degwekar et al. (2019)), or the high-dimensional geometry of the initial feature space (Shafahi et al. (2019); Khoury and Hadfield-Menell (2018)).

Lipschitz constraints under L2 norm on deep neural networks are useful for provable adversarial robustness bounds, stable training, and Wasserstein distance estimation. While heuristic approaches such as the gradient penalty have seen much practical success, it is challenging to achieve similar practical performance while provably enforcing a Lipschitz constraint. In principle, one can design Lipschitz constrained architectures using the composition property of Lipschitz functions, but Anil et al. recently identified a key obstacle to this approach: gradient norm attenuation. They showed how to circumvent this problem in the case of fully connected networks by designing each layer to be gradient norm preserving. We extend their approach to train scalable, expressive, provably Lipschitz convolutional networks. In particular, we present the Block Convolution Orthogonal Parameterization (BCOP), an expressive parameterization of orthogonal convolution operations. We show that even though the space of orthogonal convolutions is disconnected, the largest connected component of BCOP with 2n channels can represent arbitrary BCOP convolutions over n channels. Our BCOP parameterization allows us to train large convolutional networks with provable Lipschitz bounds. Empirically, we find that it is competitive with existing approaches to provable adversarial robustness and Wasserstein distance estimation.

We present a global optimization approach for solving the classical maximum a-posteriori (MAP) estimation problem for the Gaussian mixture model. Our approach formulates the MAP estimation problem as a mixed-integer nonlinear optimization problem (MINLP). Our method provides a certificate of global optimality, can accommodate side constraints, and is extendable to other finite mixture models. We propose an approximation to the MINLP hat transforms it into a mixed integer quadratic program (MIQP) which preserves global optimality within desired accuracy and improves computational aspects. Numerical experiments compare our method to standard estimation approaches and show that our method finds the globally optimal MAP for some standard data sets, providing a benchmark for comparing estimation methods.

We introduce a distributionally robust minimium mean square error estimation model with a Wasserstein ambiguity set to recover an unknown signal from a noisy observation. The proposed model can be viewed as a zero-sum game between a statistician choosing an estimator---that is, a measurable function of the observation---and a fictitious adversary choosing a prior---that is, a pair of signal and noise distributions ranging over independent Wasserstein balls---with the goal to minimize and maximize the expected squared estimation error, respectively. We show that if the Wasserstein balls are centered at normal distributions, then the zero-sum game admits a Nash equilibrium, where the players' optimal strategies are given by an {\em affine} estimator and a {\em normal} prior, respectively. We further prove that this Nash equilibrium can be computed by solving a tractable convex program. Finally, we develop a Frank-Wolfe algorithm that can solve this convex program orders of magnitude faster than state-of-the-art general purpose solvers. We show that this algorithm enjoys a linear convergence rate and that its direction-finding subproblems can be solved in quasi-closed form.

Bilevel optimizations are at the center of several important machine learning problems such as hyperparameter tuning, data denoising, few-shot learning, data poisoning. Different from simultaneous or multi-objective optimization, obtaining the exact descent direction for continuous bilevel optimization requires computing the inverse of the hessian of the lower-level cost function, even for first order methods. In this paper, we propose a new method for solving bilevel optimization, using the penalty function, which avoids computing the inverse of the hessian. We prove convergence of the method under mild conditions and show that it computes the exact hypergradient asymptotically. Small space and time complexity of our method allows us to solve large-scale bilevel optimization problems involving deep neural networks with up to 3.8M upper-level and 1.4M lower-level variables. We present results of our method for data denoising on MNIST/CIFAR10/SVHN datasets, for few-shot learning on Omniglot/Mini-Imagenet datasets and for training-data poisoning on MNIST/Imagenet datasets. In all experiments, our method outperforms or is comparable to previously proposed methods both in terms of accuracy and run-time.

Deep generative priors offer powerful models for complex-structured data, such as images, audio, and text. Using these priors in inverse problems typically requires estimating the input and/or hidden signals in a multi-layer deep neural network from observation of its output. While these approaches have been successful in practice, rigorous performance analysis is complicated by the non-convex nature of the underlying optimization problems. This paper presents a novel algorithm, Multi-Layer Vector Approximate Message Passing (ML-VAMP), for inference in multi-layer stochastic neural networks. ML-VAMP can be configured to compute maximum a priori (MAP) or approximate minimum mean-squared error (MMSE) estimates for these networks. We show that the performance of ML-VAMP can be exactly predicted in a certain high-dimensional random limit. Furthermore, under certain conditions, ML-VAMP yields estimates that achieve the minimum (i.e., Bayes-optimal) MSE as predicted by the replica method. In this way, ML-VAMP provides a computationally efficient method for multi-layer inference with an exact performance characterization and testable conditions for optimality in the large-system limit.

With IT environments growing more and more complex and demanding, traditional IT systems management is moving into a new era of intelligence-driven automation and optimization. New machine learning and artificial intelligence capabilities, combined with a common management framework, are changing the face of how IT operations are bringing all the pieces together―with automation being key. "The foundation of it begins with automation, because if you can automate, you become repeatable, consistent, and reliable, and those are all good in your data center," says Hewlett Packard Enterprise's Doug de Werd. Automation also supports greater innovation and creativity, he says, because it frees organizations to focus on the business, enabling quicker releases and time to market, among other competitive advantages. Looking forward, organizations will move toward autonomous computing, he adds, where new technologies will adjust compute, storage, and network capacity on the fly in a continuous optimization model. Where is your company on its systems management journey?

Many problems in statistical learning, imaging, and computer vision involve the optimization of a non-convex objective function with singularities at the boundary of the feasible set. For such challenging instances, we develop a new interior-point technique building on the Hessian-barrier algorithm recently introduced in Bomze, Mertikopoulos, Schachinger and Staudigl, [SIAM J. Opt. 2019 29(3), pp. 2100-2127], where the Riemannian metric is induced by a generalized self-concordant function. This class of functions is sufficiently general to include most of the commonly used barrier functions in the literature of interior point methods. We prove global convergence to an approximate stationary point of the method, and in cases where the feasible set admits an easily computable self-concordant barrier, we verify worst-case optimal iteration complexity of the method. Applications in non-convex statistical estimation and $L^{p}$-minimization are discussed to given the efficiency of the method.