We provide finite sample upper and lower bounds on the Binomial tail probability which are a direct application of Sanov's theorem. We then use these to obtain high probability upper and lower bounds on the minimum of i.i.d. Both bounds are finite sample, asymptotically tight, and expressed in terms of the KL-divergence. The purpose of this note is to provide, in a self-contained and concise way, both upper and lower bounds on the Binomial tail, and through that, on the minimum of i.i.d. The upper bound on the minimum of i.i.d.

In medical settings, it is critical that all who are in need of care are correctly heard and understood. When this is not the case due to prejudices a listener has, the speaker is experiencing \emph{testimonial injustice}, which, building upon recent work, we quantify by the presence of several categories of unjust vocabulary in medical notes. In this paper, we use FCI, a causal discovery method, to study the degree to which certain demographic features could lead to marginalization (e.g., age, gender, and race) by way of contributing to testimonial injustice. To achieve this, we review physicians' notes for each patient, where we identify occurrences of unjust vocabulary, along with the demographic features present, and use causal discovery to build a Structural Causal Model (SCM) relating those demographic features to testimonial injustice. We analyze and discuss the resulting SCMs to show the interaction of these factors and how they influence the experience of injustice. Despite the potential presence of some confounding variables, we observe how one contributing feature can make a person more prone to experiencing another contributor of testimonial injustice. There is no single root of injustice and thus intersectionality cannot be ignored. These results call for considering more than singular or equalized attributes of who a person is when analyzing and improving their experiences of bias and injustice. This work is thus a first foray at using causal discovery to understand the nuanced experiences of patients in medical settings, and its insights could be used to guide design principles throughout healthcare, to build trust and promote better patient care.

We consider scenarios where a very accurate predictive model using restricted features is available at the time of training of a larger, full-featured, model. This restricted model may be thought of as "side-information", derived either from an auxiliary exhaustive dataset or on the same dataset, by forcing the restriction. How can the restricted model be useful to the full model? We propose an approach for transferring the knowledge of the restricted model to the full model, by aligning the full model's context-restricted performance with that of the restricted model's. We call this methodology Induced Model Matching (IMM) and first illustrate its general applicability by using logistic regression as a toy example. We then explore IMM's use in language modeling, the application that initially inspired it, and where it offers an explicit foundation in contrast to the implicit use of restricted models in techniques such as noising. We demonstrate the methodology on both LSTM and transformer full models, using $N$-grams as restricted models. To further illustrate the potential of the principle whenever it is much cheaper to collect restricted rather than full information, we conclude with a simple RL example where POMDP policies can improve learned MDP policies via IMM.

Categorical models are a natural fit for many problems. When learning the distribution of categories from samples, high-dimensionality may dilute the data. Minimax optimality is too pessimistic to remedy this issue. A serendipitously discovered estimator, absolute discounting, corrects empirical frequencies by subtracting a constant from observed categories, which it then redistributes among the unobserved. It outperforms classical estimators empirically, and has been used extensively in natural language modeling.

The study of fairness in intelligent decision systems has mostly ignored long-term influence on the underlying population. Yet fairness considerations (e.g. affirmative action) have often the implicit goal of achieving balance among groups within the population. The most basic notion of balance is eventual equality between the qualifications of the groups. How can we incorporate influence dynamics in decision making? How well do dynamics-oblivious fairness policies fare in terms of reaching equality? In this paper, we propose a simple yet revealing model that encompasses (1) a selection process where an institution chooses from multiple groups according to their qualifications so as to maximize an institutional utility and (2) dynamics that govern the evolution of the groups' qualifications according to the imposed policies. We focus on demographic parity as the formalism of affirmative action. We then give conditions under which an unconstrained policy reaches equality on its own. In this case, surprisingly, imposing demographic parity may break equality. When it doesn't, one would expect the additional constraint to reduce utility, however, we show that utility may in fact increase. In more realistic scenarios, unconstrained policies do not lead to equality. In such cases, we show that although imposing demographic parity may remedy it, there is a danger that groups settle at a worse set of qualifications. As a silver lining, we also identify when the constraint not only leads to equality, but also improves all groups. This gives quantifiable insight into both sides of the mismatch hypothesis. These cases and trade-offs are instrumental in determining when and how imposing demographic parity can be beneficial in selection processes, both for the institution and for society on the long run.

Categorical models are a natural fit for many problems. When learning the distribution ofcategories from samples, high-dimensionality may dilute the data. Minimax optimality is too pessimistic to remedy this issue. A serendipitously discovered estimator, absolute discounting, corrects empirical frequencies by subtracting aconstant from observed categories, which it then redistributes among the unobserved. It outperforms classical estimators empirically, and has been used extensively innatural language modeling. In this paper, we rigorously explain the prowess of this estimator using less pessimistic notions. We show that (1) absolute discountingrecovers classical minimax KL-risk rates, (2) it is adaptive to an effective dimension rather than the true dimension, (3) it is strongly related to the Good-Turing estimator and inherits its competitive properties. We use powerlaw distributionsas the cornerstone of these results.

Utilizing the structure of a probabilistic model can significantly increase its learning speed. Motivated by several recent applications, in particular bigram models in language processing, we consider learning low-rank conditional probability matrices under expected KL-risk. This choice makes smoothing, that is the careful handling of low-probability elements, paramount. We derive an iterative algorithm that extends classical non-negative matrix factorization to naturally incorporate additive smoothing and prove that it converges to the stationary points of a penalized empirical risk. We then derive sample-complexity bounds for the global minimizer of the penalized risk and show that it is within a small factor of the optimal sample complexity. This framework generalizes to more sophisticated smoothing techniques, including absolute-discounting.

Faced with massive data, is it possible to trade off (statistical) risk, and (computational) space and time? This challenge lies at the heart of large-scale machine learning. Using k-means clustering as a prototypical unsupervised learning problem, we show how we can strategically summarize the data (control space) in order to trade off risk and time when data is generated by a probabilistic model. Our summarization is based on coreset constructions from computational geometry. We also develop an algorithm, TRAM, to navigate the space/time/data/risk tradeoff in practice. In particular, we show that for a fixed risk (or data size), as the data size increases (resp. risk increases) the running time of TRAM decreases. Our extensive experiments on real data sets demonstrate the existence and practical utility of such tradeoffs, not only for k-means but also for Gaussian Mixture Models.

This paper shows that one cannot learn the probability of rare events without imposing further structural assumptions. The event of interest is that of obtaining an outcome outside the coverage of an i.i.d. sample from a discrete distribution. The probability of this event is referred to as the "missing mass". The impossibility result can then be stated as: the missing mass is not distribution-free PAC-learnable in relative error. The proof is semi-constructive and relies on a coupling argument using a dithered geometric distribution. This result formalizes the folklore that in order to predict rare events, one necessarily needs distributions with "heavy tails".