We provide several convergence theorems for SGD for two large classes of structured non-convex functions: (i) the Quasar (Strongly) Convex functions and (ii) the functions satisfying the Polyak-Lojasiewicz condition. Our analysis relies on the Expected Residual condition which we show is a strictly weaker assumption as compared to previously used growth conditions, expected smoothness or bounded variance assumptions. We provide theoretical guarantees for the convergence of SGD for different step size selections including constant, decreasing and the recently proposed stochastic Polyak step size. In addition, all of our analysis holds for the arbitrary sampling paradigm, and as such, we are able to give insights into the complexity of minibatching and determine an optimal minibatch size. In particular we recover the best known convergence rates of full gradient descent and single element sampling SGD as a special case. Finally, we show that for models that interpolate the training data, we can dispense of our Expected Residual condition and give state-of-the-art results in this setting.

This article provides an overview of the Collective Knowledge technology (CK or cKnowledge). CK attempts to make it easier to reproduce ML&systems research, deploy ML models in production, and adapt them to continuously changing data sets, models, research techniques, software, and hardware. The CK concept is to decompose complex systems and ad-hoc research projects into reusable sub-components with unified APIs, CLI, and JSON meta description. Such components can be connected into portable workflows using DevOps principles combined with reusable automation actions, software detection plugins, meta packages, and exposed optimization parameters. CK workflows can automatically plug in different models, data and tools from different vendors while building, running and benchmarking research code in a unified way across diverse platforms and environments. Such workflows also help to perform whole system optimization, reproduce results, and compare them using public or private scoreboards on the CK platform (https://cKnowledge.io). For example, the modular CK approach was successfully validated with industrial partners to automatically co-design and optimize software, hardware, and machine learning models for reproducible and efficient object detection in terms of speed, accuracy, energy, size, and other characteristics. The long-term goal is to simplify and accelerate the development and deployment of ML models and systems by helping researchers and practitioners to share and reuse their knowledge, experience, best practices, artifacts, and techniques using open CK APIs.

Coreset is usually a small weighted subset of $n$ input points in $\mathbb{R}^d$, that provably approximates their loss function for a given set of queries (models, classifiers, etc.). Coresets become increasingly common in machine learning since existing heuristics or inefficient algorithms may be improved by running them possibly many times on the small coreset that can be maintained for streaming distributed data. Coresets can be obtained by sensitivity (importance) sampling, where its size is proportional to the total sum of sensitivities. Unfortunately, computing the sensitivity of each point is problem dependent and may be harder to compute than the original optimization problem at hand. We suggest a generic framework for computing sensitivities (and thus coresets) for wide family of loss functions which we call near-convex functions. This is by suggesting the $f$-SVD factorization that generalizes the SVD factorization of matrices to functions. Example applications include coresets that are either new or significantly improves previous results, such as SVM, Logistic regression, M-estimators, and $\ell_z$-regression. Experimental results and open source are also provided.

A convex optimization model predicts an output from an input by solving a convex optimization problem. The class of convex optimization models is large, and includes as special cases many well-known models like linear and logistic regression. We propose a heuristic for learning the parameters in a convex optimization model given a dataset of input-output pairs, using recently developed methods for differentiating the solution of a convex optimization problem with respect to its parameters. We describe three general classes of convex optimization models, maximum a posteriori (MAP) models, utility maximization models, and agent models, and present a numerical experiment for each.

The causal relationships among a set of random variables are commonly represented by a Directed Acyclic Graph (DAG), where there is a directed edge from variable $X$ to variable $Y$ if $X$ is a direct cause of $Y$. From the purely observational data, the true causal graph can be identified up to a Markov Equivalence Class (MEC), which is a set of DAGs with the same conditional independencies between the variables. The size of an MEC is a measure of complexity for recovering the true causal graph by performing interventions. We propose a method for efficient iteration over possible MECs given intervention results. We utilize the proposed method for computing MEC sizes and experiment design in active and passive learning settings. Compared to previous work for computing the size of MEC, our proposed algorithm reduces the time complexity by a factor of $O(n)$ for sparse graphs where $n$ is the number of variables in the system. Additionally, integrating our approach with dynamic programming, we design an optimal algorithm for passive experiment design. Experimental results show that our proposed algorithms for both computing the size of MEC and experiment design outperform the state of the art.

Multi-task learning has gained popularity due to the advantages it provides with respect to resource usage and performance. Nonetheless, the joint optimization of parameters with respect to multiple tasks remains an active research topic. Sub-partitioning the parameters between different tasks has proven to be an efficient way to relax the optimization constraints over the shared weights, may the partitions be disjoint or overlapping. However, one drawback of this approach is that it can weaken the inductive bias generally set up by the joint task optimization. In this work, we present a novel way to partition the parameter space without weakening the inductive bias. Specifically, we propose Maximum Roaming, a method inspired by dropout that randomly varies the parameter partitioning, while forcing them to visit as many tasks as possible at a regulated frequency, so that the network fully adapts to each update. We study the properties of our method through experiments on a variety of visual multi-task data sets. Experimental results suggest that the regularization brought by roaming has more impact on performance than usual partitioning optimization strategies. The overall method is flexible, easily applicable, provides superior regularization and consistently achieves improved performances compared to recent multi-task learning formulations.

Robust Markov decision processes (MDPs) allow to compute reliable solutions for dynamic decision problems whose evolution is modeled by rewards and partially-known transition probabilities. Unfortunately, accounting for uncertainty in the transition probabilities significantly increases the computational complexity of solving robust MDPs, which severely limits their scalability. This paper describes new efficient algorithms for solving the common class of robust MDPs with s- and sa-rectangular ambiguity sets defined by weighted $L_1$ norms. We propose partial policy iteration, a new, efficient, flexible, and general policy iteration scheme for robust MDPs. We also propose fast methods for computing the robust Bellman operator in quasi-linear time, nearly matching the linear complexity the non-robust Bellman operator. Our experimental results indicate that the proposed methods are many orders of magnitude faster than the state-of-the-art approach which uses linear programming solvers combined with a robust value iteration.

In this paper, we consider large-scale finite-sum nonconvex problems arising from machine learning. Since the Hessian is often a summation of a relative cheap and accessible part and an expensive or even inaccessible part, a stochastic quasi-Newton matrix is constructed using partial Hessian information as much as possible. By further exploiting the low-rank structures based on the Nystr\"om approximation, the computation of the quasi-Newton direction is affordable. To make full use of the gradient estimation, we also develop an extra-step strategy for this framework. Global convergence to stationary point in expectation and local suplinear convergence rate are established under some mild assumptions. Numerical experiments on logistic regression, deep autoencoder networks and deep learning problems show that the efficiency of our proposed method is at least comparable with the state-of-the-art methods.

Many real-world vehicle routing problems involve rich sets of constraints with respect to the capacities of the vehicles, time windows for customers etc. While in recent years first machine learning models have been developed to solve basic vehicle routing problems faster than optimization heuristics, complex constraints rarely are taken into consideration. Due to their general procedure to construct solutions sequentially route by route, these methods generalize unfavorably to such problems. In this paper, we develop a policy model that is able to start and extend multiple routes concurrently by using attention on the joint action space of several tours. In that way the model is able to select routes and customers and thus learns to make difficult trade-offs between routes. In comprehensive experiments on three variants of the vehicle routing problem with time windows we show that our model called JAMPR works well for different problem sizes and outperforms the existing state-of-the-art constructive model. For two of the three variants it also creates significantly better solutions than a comparable meta-heuristic solver.

Compared to minimization problems, the min-max landscape in machine learning applications is considerably more convoluted because of the existence of cycles and similar phenomena. Such oscillatory behaviors are well-understood in the convex-concave regime, and many algorithms are known to overcome them. In this paper, we go beyond the convex-concave setting and we characterize the convergence properties of a wide class of zeroth-, first-, and (scalable) second-order methods in non-convex/non-concave problems. In particular, we show that these state-of-the-art min-max optimization algorithms may converge with arbitrarily high probability to attractors that are in no way min-max optimal or even stationary. Spurious convergence phenomena of this type can arise even in two-dimensional problems, a fact which corroborates the empirical evidence surrounding the formidable difficulty of training GANs.