In this talk, we explore some of the findings of NSF funded “Studying Successful Doctoral Students in Mathematics from Underrepresented Groups.” In this study, 75 people of color at various stages of graduate school in the mathematical sciences, ranging from just beginning, to just finishing, were interviewed about their experiences. We will reflect on their experiences with select quotes and glean some important lessons from the results including the need for proper mentoring and policies that clarify steps to success. We will provide general practices on recruitment and retention

Recruitment and Retention of Underrepresented Doctoral Students in STEM

Approximate number perception is noisy, but it is unclear whether that noise reflects an error-prone attempt to report one's best guess of the value, or an inherently stochastic process such as sampling. Here we demonstrate that approximate number estimation should be thought of as a sampling procedure, as the average of two estimates of the same stimulus tends to outperform either estimate alone. Additionally, the average difference between the two estimates of a given number linearly increases as a function of number, consistent with Weber's law. Finally, we provide evidence that the classic psychophysical curves associated with number representation are represented within the mind after a single brief viewing of a stimulus, and not only useful for describing a summary of behavior over many trials.

Sampling in Approximate Number Perception

When an agent votes, she typically ranks the set of available alternatives. Occasionally, she may also wish to report the intensity of her preferences by indicating adjacent pairs of alternatives in her ranking between which her preference is acutely decisive; for instance, she may suggest that she likes alternative $a$ more than $b$, but $b$ \emph{much more} than $c$. We design near-optimal voting rules which aggregate such \emph{preference rankings with intensities} using the recently-popular distortion framework. We also show that traditional voting rules, which aggregate preference rankings while ignoring (or not eliciting) intensities, can incur significant welfare loss.

Voting with Preference Intensities

The ability to measure the satisfaction of (groups of) voters is a crucial prerequisite for formulating proportionality axioms in approval-based participatory budgeting elections. Two common -- but very different -- ways to measure the satisfaction of a voter consider (i) the number of approved projects and (ii) the total cost of approved projects, respectively. In general, it is difficult to decide which measure of satisfaction best reflects the voters' true utilities. In this paper, we study proportionality axioms with respect to large classes of approval-based satisfaction functions. We establish logical implications among our axioms and related notions from the literature, and we ask whether outcomes can be achieved that are proportional with respect to more than one satisfaction function. We show that this is impossible for the two commonly used satisfaction functions when considering proportionality notions based on extended justified representation, but achievable for a notion based on proportional justified representation. For the latter result, we introduce a strengthening of priceability and show that it is satisfied by several polynomial-time computable rules, including the Method of Equal Shares and Phragmén's sequential rule. 

Proportionality in Approval-Based Participatory Budgeting

Cake-cutting is a fundamental model of dividing a heterogeneous resource, such as land, broadcast time, and advertisement space. In this study, we consider the problem of dividing indivisible goods fairly under the connectivity constraints of a path. We prove that a connected division of indivisible items satisfying a discrete counterpart of envy-freeness, called {\em envy-freeness up to one good} (EF1), always exists for any number of agents $n$ with monotone valuations. Our result settles an open question raised by Bil\`o et al. (2019), who proved that an EF1 connected division always exists for the number of agents $n \le 4$. Moreover, the proof can be extended to show the following (1) ``secretive" and (2) ``extra" versions: (1) for $n$ agents with monotone valuations, the path can be divided into $n$ connected bundles such that an EF1 assignment of the remaining bundles can be made to the other agents for any selection made by the “secretive agent”; (2) for $n+1$ agents with monotone valuations, the path can be divided into $n$ connected bundles such that when any ``extra agent” leaves, an EF1 assignment of the bundles can be made to the remaining agents.  

How to cut a discrete cake fairly

We consider the fair division problem of indivisible items. It is well-known that an envy-free allocation may not exist, and a relaxed version of envy-freeness, envy-freeness up to one item (EF1), has been widely considered. In an EF1 allocation, an agent may envy others' allocated shares, but only up to one item. In many applications, we may wish to specify a subset of prioritized agents where strict envy-freeness needs to be guaranteed from these agents to the remaining agents, while ensuring the whole allocation is still EF1. Prioritized agents may be those agents who are envious in a previous EF1 allocation, those agents who belong to underrepresented groups, etc. Motivated by this, we propose a new fairness notion named envy-freeness with prioritized agents EFprior, and study the existence and the algorithmic aspects for the problem of computing an EFprior allocation. With additive valuations, the simple round-robin algorithm is able to compute an EFprior allocation. In this paper, we mainly focus on general valuations. In particular, we present a polynomial-time algorithm that outputs an EFprior allocation with most of the items allocated. When all the items need to be allocated, we also present polynomial-time algorithms for some well-motivated special cases.

Fair Division with Prioritized Agents

Satisfaction of the strict saddle property has become a standard assumption in non-convex optimization, and it ensures that many first-order optimization
algorithms will almost always escape saddle points. However, functions exist in machine learning that do not satisfy this property, such as the loss function of a neural network with at least two hidden layers. First-order methods such as gradient descent may converge to non-strict saddle points of such functions, and there do not currently exist any first-order methods that reliably escape non-strict saddle points. To address this need, we demonstrate that regularizing a function with a linear term enforces the strict saddle property, and we provide justification for only regularizing locally, i.e., when the norm of the gradient falls below a certain threshold. We analyze bifurcations that may result from this form of regularization, and then we provide a selection rule for regularizers that depends only on the gradient of an objective function. This rule is shown to guarantee that gradient descent will escape the neighborhoods around a broad class of non-strict saddle points, and this behavior is demonstrated on numerical examples of non-strict saddle points common in the optimization literature.

Linear Regularizers Enforce the Strict Saddle Property

Time or money? That is a question! In this paper, we consider this dilemma in the pricing regime, in which we try to find the optimal pricing scheme for identical items with heterogenous time-sensitive buyers. We characterize the revenue-optimal solution and propose an efficient algorithm to find it in a Bayesian setting. Our results also demonstrate the tight ratio between the value of wasted time and the seller's revenue, as well as that of two common-used pricing schemes, the $k$-step function and the fixed pricing. To explore the nature of the optimal scheme in the general setting, we present the closed forms over the product distribution and show by examples that positive correlation between the valuation of the item and the cost per unit time could help increase revenue. To the best of our knowledge, it is the first step towards understanding the impact of the time factor as a part of the buyer cost in pricing problems, in the computational view.

Optimal Pricing Schemes for Identical Items with Time-Sensitive Buyers

Difference-of-Convex (DC) minimization, referring to the problem of minimizing the difference of two convex functions, has been found rich applications in statistical learning and studied extensively for decades. However, existing methods are primarily based on multi-stage convex relaxation, only leading to weak optimality of critical points. This paper proposes a coordinate descent method for minimizing a class of DC functions based on sequential nonconvex approximation. Our approach iteratively solves a nonconvex one-dimensional subproblem globally, and it is guaranteed to converge to a coordinate-wise stationary point. We prove that this new optimality condition is always stronger than the standard critical point condition and directional point condition under a mild \textit{locally bounded nonconvexity assumption}. For comparisons, we also include a naive variant of coordinate descent methods based on sequential convex approximation in our study. When the objective function satisfies a \textit{globally bounded nonconvexity assumption} and \textit{Luo-Tseng error bound assumption}, coordinate descent methods achieve \textit{Q-linear} convergence rate. Also, for many applications of interest, we show that the nonconvex one-dimensional subproblem can be computed exactly and efficiently using a breakpoint searching method. Finally, we have conducted extensive experiments on several statistical learning tasks to show the superiority of our approach.

Coordinate Descent Methods for DC Minimization: Optimality Conditions and Global Convergence

This paper presents SVAM (Sequential Variance-Altered MLE), a unified framework for learning generalized linear models under adversarial label corruption in training data. SVAM extends to tasks such as least squares regression, logistic regression, and gamma regression, whereas many existing works on learning with label corruptions focus only on least squares regression. SVAM is based on a novel variance reduction technique that may be of independent interest and works by iteratively solving weighted MLEs over variance-altered versions of the GLM objective. SVAM offers provable model recovery guarantees superior to the state-of-the-art for robust regression even when a constant fraction of training labels are adversarially corrupted. SVAM also empirically outperforms several existing problem-specific techniques for robust regression and classification. Code for SVAM is available at https://github.com/purushottamkar/svam/

Corruption-tolerant Algorithms for Generalized Linear Models

We consider a voting scenario in which the resource to be voted upon may consist of both indivisible and divisible goods. This generalizes both the well-studied model of multiwinner voting and the recently introduced model of cake sharing. Under approval votes, we propose two variants of the extended justified representation (EJR) notion from multiwinner voting, a stronger one called EJR for mixed goods (EJR-M) and a weaker one called EJR up to 1 (EJR-1). We extend three multiwinner voting rules to our setting—GreedyEJR, the method of equal shares (MES), and proportional approval voting (PAV)—and show that while all three generalizations satisfy EJR-1, only the first one provides EJR-M. In addition, we derive tight bounds on the proportionality degree implied by EJR-M and EJR-1, and investigate the proportionality degree of our proposed rules.

Approval-Based Voting with Mixed Goods

Adversarial robustness studies the worst-case performance
of a machine learning model to ensure safety and reliability. With the proliferation of deep-learning-based technology, the potential risks associated with model development and deployment can be amplified and become dreadful vulnerabilities. This paper provides a comprehensive overview of research topics and foundational principles of research methods for adversarial robustness of deep learning models, including attacks, defenses, verification, and novel applications.

Underline digital video library

Recruitment and Retention of Underrepresented Doctoral Students in STEM

Sampling in Approximate Number Perception

Voting with Preference Intensities

Proportionality in Approval-Based Participatory Budgeting

How to cut a discrete cake fairly

Fair Division with Prioritized Agents

Linear Regularizers Enforce the Strict Saddle Property

Optimal Pricing Schemes for Identical Items with Time-Sensitive Buyers

Coordinate Descent Methods for DC Minimization: Optimality Conditions and Global Convergence

Corruption-tolerant Algorithms for Generalized Linear Models

Approval-Based Voting with Mixed Goods

Holistic Adversarial Robustness of Deep Learning Models

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