# the MaD Seminar fall 2019

The MaD seminar features leading specialists at the interface of Applied Mathematics, Statistics and Machine Learning. It is partly supported by the Moore-Sloan Data Science Environment at NYU.

MaD seminars are now recorded and streamed live, starting 11/7/19. Links to the videos are available below.

**Room:** Auditorium Hall 150, Center for Data Science, NYU, 60 5th ave.

**Time:** 2:00pm-3:00pm, Reception will follow.

**Subscribe to the Seminar Mailing list here**

### Schedule with Confirmed Speakers

### Abstracts

#### Lenaic Chizat: Analysis of Gradient Methods for the Optimization of wide two layer neural networks

Gradient-based optimization algorithms applied to artificial neural networks with many parameters typically lead to models with good train and test performance. Two lines of theoretical research have recently emerged as attempts to explain this phenomenon: the lazy training and the mean-field analysis. The lazy training analysis studies the situation when a model remains in a small neighborhood of its initialization throughout training and implicitly performs a kernel regression; and the mean-field analysis is a general description of the training dynamics of infinitely wide two-layer neural networks. In this talk, I will present the insights brought by these analyses, their connections, strengths and limitations.

#### Marco Cuturi: Differentiable Ranks and Quantiles using Optimal Transport

We propose a framework to sort values that is algorithmically differentiable. We leverage the fact that sorting can be seen as a particular instance of the optimal transport (OT) problem on R, from input values to a target predefined array of sorted values (e.g. 1,2,…, n if the input array has n elements). Building upon this link, we propose generalized ranks, CDFs and quantile operators by varying the size and introducing weights for the target array. We recover differentiable algorithms by adding to the OT problem an entropic regularization, and approximate it using a few Sinkhorn iterations. We call these operators soft-ranks and soft-sort operators. Using the soft-rank operator, we propose a new classification training loss that is a differentiable proxy of the 0/1 loss. Using the soft-sort operator, we propose a new differentiable quantile regression loss. These losses outperform respectively the cross-entropy and the pinball loss in our experiments.

#### David Blei: The Blessings of Multiple Causes

Causal inference from observational data is a vital problem, but it comes with strong assumptions. Most methods require that we observe all confounders, variables that affect both the causal variables and the outcome variables. But whether we have observed all confounders is a famously untestable assumption. We describe the deconfounder, a way to do causal inference with weaker assumptions than the classical methods require.

How does the deconfounder work? While traditional causal methods measure the effect of a single cause on an outcome, many modern scientific studies involve multiple causes, different variables whose effects are simultaneously of interest. The deconfounder uses the correlation among multiple causes as evidence for unobserved confounders, combining unsupervised machine learning and predictive model checking to perform causal inference. We demonstrate the deconfounder on real-world data and simulation studies, and describe the theoretical requirements for the deconfounder to provide unbiased causal estimates.

This is joint work with Yixin Wang. https://arxiv.org/abs/1805.06826

#### Song Mei: Generalization error of linearized neural networks: staircase and double-descent

Deep learning methods operate in regimes that defy the traditional statistical mindset. Neural network architectures often contain more parameters than training samples, and are so rich that they can interpolate the observed labels, even if the latter are replaced by pure noise. Despite their huge complexity, the same architectures achieve small generalization error on real data. On the other hand, tangent kernel theory provides an interesting perspective on the training dynamics of neural networks. In a proper scaling limit, the gradient flow dynamics of multi-layers neural networks become a linear dynamics associated with a kernel, and converges to a global minimizer of the training loss. The tangent model associated with the neural network can be understood as a linearization of neural network around a random initialization. In this talk, I will discuss two interesting phenomena of the generalization of linearized neural networks: the staircase phenomenon and the double-descent phenomenon. I will use them to discuss the benefits and limitations of the tangent kernel theory.

#### Po-Ling Loh: Estimating location parameters in entangled single-sample distributions

We consider the problem of estimating the common mean of independently sampled data, where samples are drawn in a possibly non-identical manner from symmetric, unimodal distributions with a common mean. This generalizes the setting of Gaussian mixture modeling, since the number of distinct mixture components may diverge with the number of observations. We propose an estimator that adapts to the level of heterogeneity in the data, achieving near-optimality in both the i.i.d. setting and some heterogeneous settings, where the fraction of “low-noise” points is as small as log(n)/n. Our estimator is a hybrid of the modal interval, shorth, and median estimators from classical statistics; however, the key technical contributions rely on novel empirical process theory results that we derive for independent but non-i.i.d. data. In the multivariate setting, we generalize our theory to mean estimation for mixtures of radially symmetric distributions, and derive minimax lower bounds on the expected error of any estimator that is agnostic to the scales of individual data points. Finally, we describe an extension of our estimators applicable to linear regression. In the multivariate mean estimation and regression settings, we present computationally feasible versions of our estimators that run in time polynomial in the number of data points.

This is joint work with Ankit Pensia and Varun Jog.

#### Ryan Tibshirani: Surprises in High-Dimensional Ridgeless Least Squares Interpolation (or: What Deep Learning Taught me About Linear Models)

Interpolators—estimators that achieve zero training error—have attracted growing attention in machine learning, mainly because state-of-the art neural networks appear to be models of this type. We study minimum L2 norm (“ridgeless”) interpolation in high-dimensional least squares regression. We consider two different models for the feature distribution: a linear model, where the feature vectors are obtained by applying a linear transform to a vector of i.i.d. entries, and a nonlinear model, where the feature vectors are obtained by passing the input through a random one-layer neural network. We recover—-in a precise quantitative way—-several phenomena that have been observed in large-scale neural networks and kernel machines, including the “double descent” behavior of the prediction risk, and the potential benefits of overparametrization.

This represents work with Trevor Hastie, Andrea Montanari, and Saharon Rosset.

#### Peyman Milanfar: Denoising as a Fundamental Building Block: Form, function, and regularization of inverse problems

Denoising of images has reached impressive levels of quality – almost as good as we can ever hope. There are thousands of papers on this topic, and their scope is so vast and approaches so diverse that putting them in some order is useful and challenging. I will speak about why we should still care deeply about this topic, what we can say about this general class of operators on images, and what makes them so special. Of particular interest is how we can use denoisers as building blocks for broader image processing tasks, including as regularizers for general inverse problems. I’ll also show examples of applications including high dynamic range enhancement, deblurring, and super-resolution.

#### Rahul Mazumder: Learning Structured Sparse Problems at Scale: Continuous and Mixed Integer Programming Perspectives

Structured sparsity plays an important role in high dimensional statistics and machine learning. They are naturally cast as solutions to nonconvex optimization problems. A major focus in this area has been on convex relaxations and/or greedy algorithms. Mixed Integer Programming (MIP) presents a flexible and effective framework for modeling and computation of these problems (to optimality). Despite promising recent research in this area, there is a considerable gap between the problem-sizes that can be handled via efficient MIP solvers versus fast algorithms to solve the convex relaxations. Compared to first order methods in convex optimization used in sparse learning, current efficient MIP solvers (e.g., commercial solvers) are less transparent, do not effectively exploit (statistical) problem-structure and can be computationally expensive. Convex optimization methods for sparse learning may provide insights into solving the corresponding MIP problems at scale.

To this end, we will discuss our recent work on sparse learning (e.g., best subset selection, hierarchical sparsity, sparse PCA). Our framework allows us to obtain near-optimal solutions to the discrete sparse learning problems at scales much larger than current state-of-the-art commercial solvers. This enables us to algorithmically understand statistical properties of high-dimensional sparse estimators. This sheds interesting insights into the behavior of sparse learning estimators (e.g., the curious behavior of best-subsets across different SNR regimes) — properties that seem to be less known due to computational limitations.

#### Madeleine Udell: Optimal storage SDP

This talk develops new storage-optimal algorithms that provably solve generic semidefinite programs (SDPs) in standard form. The methods are particularly effective for weakly constrained SDPs.

The key idea is to formulate an approximate complementarity principle: Given an approximate solution to the dual SDP, the primal SDP has an approximate solution whose range is contained in the null space of the dual slack matrix. For weakly constrained SDPs, this null space has very low dimension, so this observation significantly reduces the search space for the primal solution.

This result suggests an algorithmic strategy that can be implemented with minimal storage: (1) Solve the dual SDP approximately; (2) compress the primal SDP to the null space of the dual slack matrix; (3) solve the compressed primal SDP.

#### Dustin Mixon:SqueezeFit: Label aware dimensionality reduction via semidefinite programming

Given labeled points in a high-dimensional vector space, we seek a low-dimensional subspace such that projecting onto this subspace maintains some prescribed distance between points of differing labels. Intended applications include compressive classification. This talk will introduce a semidefinite relaxation of this problem, along with various performance guarantees. (Joint work with Culver McWhirter (OSU) and Soledad Villar (NYU).)

#### Facundo Memoli: Gromov-Wasserstein distances and distributional invariants of datasets

The Gromov-Wasserstein (GW) distance is a generalization of the standard Wasserstein distance between two probability measures on a given ambient metric space. The GW distance assumes that these two probability measures might live on different ambient spaces and therefore implements an actual comparison of pairs of metric measure spaces. Metric-measure spaces are triples (X,dX,muX) where (X,dX) is a metric space and muX is a Borel probability measure over X and serve as a model for datasets.

In practical applications, this distance is estimated either directly via gradient based optimization approaches, or through the computation of lower bounds which arise from distributional invariants of metric-measure spaces. One particular such invariant is the so called ‘global distance distribution’ of pairwise distances.

This talk will overview the construction of the GW distance, the stability of distribution based invariants, and will discuss some results regarding the injectivity of the global distribution of distances for smooth planar curves.