It seems that, anymore, the perception of machine learning is often reduced to passing a series of arguments to a growing number of libraries and APIs, hoping for magic, and awaiting the results. Maybe you have a very good idea of what's going on under the hood in these libraries -- from data preparation to model building to results interpretation and visualization and beyond -- but you are still relying on these various tools to get the job done. Using well-tested and proven implementations of tools for performing regular tasks makes sense for a whole host of reasons. Reinventing wheels which don't roll efficiently is not best practice... it's limiting, and it takes an unnecessarily long time. Whether you are using open source or proprietary tools to get your work done, these implementations have been honed by teams of individuals ensuring that you get your hands on the best quality instruments with which to accomplish your goals.

Exploring and applying machine learning algorithms to datasets that are too large to fit into memory is pretty common. In this post, I want to offer some common suggestions you may want to consider. Some machine learning tools or libraries may be limited by a default memory configuration. Check if you can re-configure your tool or library to allocate more memory. A good example is Weka, where you can increase the memory as a parameter when starting the application. Are you sure you need to work with all of the data?

We show how the discovery of robust scalable numerical solvers for arbitrary bounded linear operators can be automated as a Game Theory problem by reformulating the process of computing with partial information and limited resources as that of playing underlying hierarchies of adversarial information games. When the solution space is a Banach space $B$ endowed with a quadratic norm $\|\cdot\|$, the optimal measure (mixed strategy) for such games (e.g. the adversarial recovery of $u\in B$, given partial measurements $[\phi_i, u]$ with $\phi_i\in B^*$, using relative error in $\|\cdot\|$-norm as a loss) is a centered Gaussian field $\xi$ solely determined by the norm $\|\cdot\|$, whose conditioning (on measurements) produces optimal bets. When measurements are hierarchical, the process of conditioning this Gaussian field produces a hierarchy of elementary bets (gamblets). These gamblets generalize the notion of Wavelets and Wannier functions in the sense that they are adapted to the norm $\|\cdot\|$ and induce a multi-resolution decomposition of $B$ that is adapted to the eigensubspaces of the operator defining the norm $\|\cdot\|$. When the operator is localized, we show that the resulting gamblets are localized both in space and frequency and introduce the Fast Gamblet Transform (FGT) with rigorous accuracy and (near-linear) complexity estimates. As the FFT can be used to solve and diagonalize arbitrary PDEs with constant coefficients, the FGT can be used to decompose a wide range of continuous linear operators (including arbitrary continuous linear bijections from $H^s_0$ to $H^{-s}$ or to $L^2$) into a sequence of independent linear systems with uniformly bounded condition numbers and leads to $\mathcal{O}(N \operatorname{polylog} N)$ solvers and eigenspace adapted Multiresolution Analysis (resulting in near linear complexity approximation of all eigensubspaces).

This paper considers the problem of minimizing an expectation function over a closed convex set, coupled with an expectation constraint on either decision variables or problem parameters. We first present a new stochastic approximation (SA) type algorithm, namely the cooperative SA (CSA), to handle problems with the expectation constraint on devision variables. We show that this algorithm exhibits the optimal ${\cal O}(1/\sqrt{N})$ rate of convergence, in terms of both optimality gap and constraint violation, when the objective and constraint functions are generally convex, where $N$ denotes the number of iterations. Moreover, we show that this rate of convergence can be improved to ${\cal O}(1/N)$ if the objective and constraint functions are strongly convex. We then present a variant of CSA, namely the cooperative stochastic parameter approximation (CSPA) algorithm, to deal with the situation when the expectation constraint is defined over problem parameters and show that it exhibits similar optimal rate of convergence to CSA. It is worth noting that CSA and CSPA are primal methods which do not require the iterations on the dual space and/or the estimation on the size of the dual variables. To the best of our knowledge, this is the first time that such optimal SA methods for solving expectation constrained stochastic optimization are presented in the literature.

Probabilistic machine learning [Ghahramani, 2015] allows us to model the structure and dependencies of latent variables and observational data as a joint probability distribution. Once a model is defined, we can perform inference to update our prior beliefs about latent variables in light of observed data to obtain the posterior distribution. The posterior can be used to answer any questions we might have about the latent quantities while coherently accounting for our uncertainty about the world. We introduce a method for simultaneous model learning and inference amortization [Gershman and Goodman, 2014], given an unlabeled dataset of observations. The model is specified partially, the rest being specified using a generative network whose weights are to be learned. Inference amortization refers to spending additional time before inference to obtain an amortization artifact which is used to speed up inference during test time.

An accurate model of patient-specific kidney graft survival distributions can help to improve shared-decision making in the treatment and care of patients. In this paper, we propose a deep learning method that directly models the survival function instead of estimating the hazard function to predict survival times for graft patients based on the principle of multi-task learning. By learning to jointly predict the time of the event, and its rank in the cox partial log likelihood framework, our deep learning approach outperforms, in terms of survival time prediction quality and concordance index, other common methods for survival analysis, including the Cox Proportional Hazards model and a network trained on the cox partial log-likelihood.

Coresets are small sets of points that approximate the properties of a larger point-set. For example, given a compact set $\mathcal{S} \subseteq \mathbb{R}^d$, a coreset could be defined as a (weighted) subset of $\mathcal{S}$ that approximates the sum of squared distances from $\mathcal{S}$ to every linear subspace of $\mathbb{R}^d$. As such, coresets can be used as a proxy to the full dataset and provide an important technique to speed up algorithms for solving problems including principal component analysis, latent semantic indexing, etc. In this paper, we provide a structural result that connects the construction of such coresets to approximating matrix products. This structural result implies a simple, randomized algorithm that constructs coresets whose sizes are independent of the number and dimensionality of the input points. The expected size of the resulting coresets yields an improvement over the state-of-the-art deterministic approach. Finally, we evaluate the proposed randomized algorithm on synthetic and real data, and demonstrate its effective performance relative to its deterministic counterpart.

Consistency of the kernel density estimator requires that the kernel bandwidth tends to zero as the sample size grows. In this paper we investigate the question of whether consistency is possible when the bandwidth is fixed, if we consider a more general class of weighted KDEs. To answer this question in the affirmative, we introduce the fixed-bandwidth KDE (fbKDE), obtained by solving a quadratic program, and prove that it consistently estimates any continuous square-integrable density. We also establish rates of convergence for the fbKDE with radial kernels and the box kernel under appropriate smoothness assumptions. Furthermore, in an experimental study we demonstrate that the fbKDE compares favorably to the standard KDE and the previously proposed variable bandwidth KDE.

Many applications of machine learning, for example in health care, would benefit from methods that can guarantee privacy of data subjects. Differential privacy (DP) has become established as a standard for protecting learning results. The standard DP algorithms require a single trusted party to have access to the entire data, which is a clear weakness. We consider DP Bayesian learning in a distributed setting, where each party only holds a single sample or a few samples of the data. We propose a learning strategy based on a secure multi-party sum function for aggregating summaries from data holders and the Gaussian mechanism for DP. Our method builds on an asymptotically optimal and practically efficient DP Bayesian inference with rapidly diminishing extra cost.

This paper deals with finding an $n$-dimensional solution $x$ to a system of quadratic equations of the form $y_i=|\langle{a}_i,x\rangle|^2$ for $1\le i \le m$, which is also known as phase retrieval and is NP-hard in general. We put forth a novel procedure for minimizing the amplitude-based least-squares empirical loss, that starts with a weighted maximal correlation initialization obtainable with a few power or Lanczos iterations, followed by successive refinements based upon a sequence of iteratively reweighted (generalized) gradient iterations. The two (both the initialization and gradient flow) stages distinguish themselves from prior contributions by the inclusion of a fresh (re)weighting regularization technique. The overall algorithm is conceptually simple, numerically scalable, and easy-to-implement. For certain random measurement models, the novel procedure is shown capable of finding the true solution $x$ in time proportional to reading the data $\{(a_i;y_i)\}_{1\le i \le m}$. This holds with high probability and without extra assumption on the signal $x$ to be recovered, provided that the number $m$ of equations is some constant $c>0$ times the number $n$ of unknowns in the signal vector, namely, $m>cn$. Empirically, the upshots of this contribution are: i) (almost) $100\%$ perfect signal recovery in the high-dimensional (say e.g., $n\ge 2,000$) regime given only an information-theoretic limit number of noiseless equations, namely, $m=2n-1$ in the real-valued Gaussian case; and, ii) (nearly) optimal statistical accuracy in the presence of additive noise of bounded support. Finally, substantial numerical tests using both synthetic data and real images corroborate markedly improved signal recovery performance and computational efficiency of our novel procedure relative to state-of-the-art approaches.