Learning in High Dimensional Spaces

We propose a data-driven approach to solve multiscale elliptic PDEs with random coefficients based on the intrinsic low dimension structure of the underlying elliptic differential operators. Our method consists of offline and online stages. At the offline stage, a low dimension space and its basis are extracted from the data to achieve significant dimension reduction in the solution space. At the online stage, the extracted basis will be used to solve a new multiscale elliptic PDE efficiently. The existence of low dimension structure is established by showing the high separability of the underlying Green's functions. Different online construction methods are proposed depending on the problem setup. We provide error analysis based on the sampling error and the truncation threshold in building the data-driven basis. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method.

Modern machine learning is distributed and the work of several machines is typically aggregated by \emph{averaging} which is the optimal rule in terms of speed, offering a speedup of $n$ (with respect to using a single machine) when $n$ processes are learning together. Distributing data and models poses however fundamental vulnerabilities, be they to software bugs, asynchrony, or worse, to malicious attackers controlling some machines or injecting misleading data in the network. Such behavior is best modeled as Byzantine failures, and averaging does not tolerate a single one from a worker. Krum, the first provably Byzantine resilient aggregation rule for SGD only uses one worker per step, which hampers its speed of convergence, especially in best case conditions when none of the workers is actually Byzantine. An idea, coined multi-Krum, of using $m$ different workers per step was mentioned, without however any proof neither on its Byzantine resilience nor on its slowdown. More recently, it was shown that in high dimensional machine learning, guaranteeing convergence is not a sufficient condition for \emph{strong} Byzantine resilience. A improvement on Krum, coined Bulyan, was proposed and proved to guarantee stronger resilience. However, Bulyan suffers from the same weakness of Krum: using only one worker per step. This adds up to the aforementioned open problem and leaves the crucial need for both fast and strong Byzantine resilience unfulfilled. The present paper proposes using Bulyan over Multi-Krum (we call it Multi-Bulyan), a combination for which we provide proofs of strong Byzantine resilience, as well as an ${\frac{m}{n}}$ slowdown, compared to averaging, the fastest (but non Byzantine resilient) rule for distributed machine learning, finally we prove that Multi-Bulyan inherits the $O(d)$ merits of both multi-Krum and Bulyan.

Classical dimension reduction problem can be loosely formulated as a problem of finding a $k$-dimensional affine subspace of ${\mathbb R}^n$ onto which data points ${\mathbf x}_1,\cdots, {\mathbf x}_N$ can be projected without loss of valuable information. We reformulate this problem in the language of tempered distributions, i.e. as a problem of approximating an empirical probability density function $p_{\rm{emp}}({\mathbf x}) = \frac{1}{N} \sum_{i=1}^N \delta^n (\bold{x} - \bold{x}_i)$, where $\delta^n$ is an $n$-dimensional Dirac delta function, by another tempered distribution $q({\mathbf x})$ whose density is supported in some $k$-dimensional subspace. Thus, our problem is reduced to the minimization of a certain loss function $I(q)$ measuring the distance from $q$ to $p_{\rm{emp}}$ over a pertinent set of generalized functions, denoted $\mathcal{G}_k$. Another classical problem of data analysis is the sufficient dimension reduction problem. We show that it can be reduced to the following problem: given a function $f: {\mathbb R}^n\rightarrow {\mathbb R}$ and a probability density function $p({\mathbf x})$, find a function of the form $g({\mathbf w}^T_1{\mathbf x}, \cdots, {\mathbf w}^T_k{\mathbf x})$ that minimizes the loss ${\mathbb E}_{{\mathbf x}\sim p} |f({\mathbf x})-g({\mathbf w}^T_1{\mathbf x}, \cdots, {\mathbf w}^T_k{\mathbf x})|^2$. We first show that search spaces of the latter two problems are in one-to-one correspondence which is defined by the Fourier transform. We introduce a nonnegative penalty function $R(f)$ and a set of ordinary functions $\Omega_\epsilon = \{f| R(f)\leq \epsilon\}$ in such a way that $\Omega_\epsilon$ `approximates' the space $\mathcal{G}_k$ when $\epsilon \rightarrow 0$. Then we present an algorithm for minimization of $I(f)+\lambda R(f)$, based on the idea of two-step iterative computation.

I continue exploring the curse of dimensionality. Following the analysis form Part I., I want to discuss another consequence of sparse sampling in high dimensions: sample points are close to an edge of the sample. This post is based on The Elements of Statistical Learning, Section 2.5, which I encourage to read! Consider \(N\) data points uniformly distributed in a \(p\)-dimensional unit ball centered at the origin. Suppose we consider a nearest-neighbor estimate at the origin.

We will now investigate this curse. Let us prepare the notebook. Let \(\lambda: 0.1\) represent the locality input parameter. Now let us write a function which verifies if a point \(x \in [0,1]\) belongs to a given interval. Now we write a simulation function.

A theory of sufficient dimension reduction (SDR) is developed from an optimizational perspective. In our formulation of the problem, instead of dealing with raw data, we assume that our ground truth includes a mapping ${\mathbf f}: {\mathbb R}^n\rightarrow {\mathbb R}^m$ and a probability distribution function $p$ over ${\mathbb R}^n$, both given analytically. We formulate SDR as a problem of finding a function ${\mathbf g}: {\mathbb R}^k\rightarrow {\mathbb R}^m$ and a matrix $P\in {\mathbb R}^{k\times n}$ such that ${\mathbb E}_{{\mathbf x}\sim p({\mathbf x})} \left|{\mathbf f}({\mathbf x}) - {\mathbf g}(P{\mathbf x})\right|^2$ is minimal. It turns out that the latter problem allows a reformulation in the dual space, i.e. instead of searching for ${\mathbf g}(P{\mathbf x})$ we suggest searching for its Fourier transform. First, we characterize all tempered distributions that can serve as the Fourier transform of such functions. The reformulation in the dual space can be interpreted as a problem of finding a $k$-dimensional linear subspace $S$ and a tempered distribution ${\mathbf t}$ supported in $S$ such that ${\mathbf t}$ is "close" in a certain sense to the Fourier transform of ${\mathbf f}$. Instead of optimizing over generalized functions with a $k$-dimensional support, we suggest minimizing over ordinary functions but with an additional term $R$ that penalizes a strong distortion of the support from any $k$-dimensional linear subspace. For a specific case of $R$, we develop an algorithm that can be formulated for functions given in the initial form as well as for their Fourier transforms. Eventually, we report results of numerical experiments with a discretized version of the latter algorithm.

Estimation is the computational task of recovering a hidden parameter $x$ associated with a distribution $D_x$, given a measurement $y$ sampled from the distribution. High dimensional estimation problems arise naturally in statistics, machine learning, and complexity theory. Many high dimensional estimation problems can be formulated as systems of polynomial equations and inequalities, and thus give rise to natural probability distributions over polynomial systems. Sum-of-squares proofs provide a powerful framework to reason about polynomial systems, and further there exist efficient algorithms to search for low-degree sum-of-squares proofs. Understanding and characterizing the power of sum-of-squares proofs for estimation problems has been a subject of intense study in recent years. On one hand, there is a growing body of work utilizing sum-of-squares proofs for recovering solutions to polynomial systems when the system is feasible. On the other hand, a general technique referred to as pseudocalibration has been developed towards showing lower bounds on the degree of sum-of-squares proofs. Finally, the existence of sum-of-squares refutations of a polynomial system has been shown to be intimately connected to the existence of spectral algorithms. In this article we survey these developments.

We design nearly optimal differentially private algorithms for learning two fundamental families of high-dimensional distributions in total variation distance: multivariate Gaussians in $\mathbb{R}^{d}$ and product distributions on the hypercube. The sample complexity of both our algorithms approaches the sample complexity of non-private learners up to a small multiplicative factor and an additional additive term that is lower order for a wide range of parameters, showing that privacy comes essentially for free for these problems. Our algorithms use a novel technical approach to reducing the sensitivity of the estimation procedure that we call recursive private preconditioning and may find additional applications.

An important task for many if not all the scientific domains is efficient knowledge integration, testing and codification. It is often solved with model construction in a controllable computational environment. In spite of that, the throughput of in-silico simulation-based observations become similarly intractable for thorough analysis. This is especially the case in molecular biology, which served as a subject for this study. In this project, we aimed to test some approaches developed to deal with the curse of dimensionality. Among these we found dimension reduction techniques especially appealing. They can be used to identify irrelevant variability and help to understand critical processes underlying high-dimensional datasets. Additionally, we subjected our data sets to nonlinear time series analysis, as those are well established methods for results comparison. To investigate the usefulness of dimension reduction methods, we decided to base our study on a concrete sample set. The example was taken from the domain of systems biology concerning dynamic evolution of sub-cellular signaling. Particularly, the dataset relates to the yeast pheromone pathway and is studied in-silico with a stochastic model. The model reconstructs signal propagation stimulated by a mating pheromone. In the paper, we elaborate on the reason of multidimensional analysis problem in the context of molecular signaling, and next, we introduce the model of choice, simulation details and obtained time series dynamics. A description of used methods followed by a discussion of results and their biological interpretation finalize the paper.

In the covariate shift learning scenario, the training and test covariate distributions differ, so that a predictor's average loss over the training and test distributions also differ. In this work, we explore the potential of extreme dimension reduction, i.e. to very low dimensions, in improving the performance of importance weighting methods for handling covariate shift, which fail in high dimensions due to potentially high train/test covariate divergence and the inability to accurately estimate the requisite density ratios. We first formulate and solve a problem optimizing over linear subspaces a combination of their predictive utility and train/test divergence within. Applying it to simulated and real data, we show extreme dimension reduction helps sometimes but not always, due to a bias introduced by dimension reduction.