pyBregMan: A Python library for Bregman Manifolds

A Bregman manifold is a synonym for a dually flat space in information geometry which admits as a canonical divergence a Bregman divergence. Bregman manifolds are induced by smooth strictly convex functions like the cumulant or partition functions of regular exponential families, the negative entropy of mixture families, or the characteristic functions of regular cones just to list a few such convex Bregman generators. We describe the design of pyBregMan, a library which implements generic operations on Bregman manifolds and instantiate several common Bregman manifolds used in information sciences. At the core of the library is the notion of Legendre-Fenchel duality inducing a canonical pair of dual potential functions and dual Bregman divergences. The library also implements the Fisher-Rao manifolds of categorical/multinomial distributions and multivariate normal distributions. To demonstrate the use of the pyBregMan kernel manipulating those Bregman and Fisher-Rao manifolds, the library also provides several core algorithms for various applications in statistics, machine learning, information fusion, and so on.

Rejection via Learning Density Ratios

Classification with rejection emerges as a learning paradigm which allows models to abstain from making predictions. The predominant approach is to alter the supervised learning pipeline by augmenting typical loss functions, letting model rejection incur a lower loss than an incorrect prediction. Instead, we propose a different distributional perspective, where we seek to find an idealized data distribution which maximizes a pretrained model's performance. This can be formalized via the optimization of a loss's risk with a $ \phi$-divergence regularization term. Through this idealized distribution, a rejection decision can be made by utilizing the density ratio between this distribution and the data distribution. We focus on the setting where our $ \phi $-divergences are specified by the family of $ \alpha $-divergence. Our framework is tested empirically over clean and noisy datasets.

Tradeoffs of Diagonal Fisher Information Matrix Estimators

The Fisher information matrix characterizes the local geometry in the parameter space of neural networks. It elucidates insightful theories and useful tools to understand and optimize neural networks. Given its high computational cost, practitioners often use random estimators and evaluate only the diagonal entries. We examine two such estimators, whose accuracy and sample complexity depend on their associated variances. We derive bounds of the variances and instantiate them in regression and classification networks. We navigate trade-offs of both estimators based on analytical and numerical studies. We find that the variance quantities depend on the non-linearity with respect to different parameter groups and should not be neglected when estimating the Fisher information.

Tempered Calculus for ML: Application to Hyperbolic Model Embedding

Most mathematical distortions used in ML are fundamentally integral in nature: $f$-divergences, Bregman divergences, (regularized) optimal transport distances, integral probability metrics, geodesic distances, etc. In this paper, we unveil a grounded theory and tools which can help improve these distortions to better cope with ML requirements. We start with a generalization of Riemann integration that also encapsulates functions that are not strictly additive but are, more generally, $t$-additive, as in nonextensive statistical mechanics. Notably, this recovers Volterra's product integral as a special case. We then generalize the Fundamental Theorem of calculus using an extension of the (Euclidean) derivative. This, along with a series of more specific Theorems, serves as a basis for results showing how one can specifically design, alter, or change fundamental properties of distortion measures in a simple way, with a special emphasis on geometric- and ML-related properties that are the metricity, hyperbolicity, and encoding. We show how to apply it to a problem that has recently gained traction in ML: hyperbolic embeddings with a "cheap" and accurate encoding along the hyperbolic vs Euclidean scale. We unveil a new application for which the Poincar\'e disk model has very appealing features, and our theory comes in handy: \textit{model} embeddings for boosted combinations of decision trees, trained using the log-loss (trees) and logistic loss (combinations).

Sampled Transformer for Point Sets

The sparse transformer can reduce the computational complexity of the self-attention layers to $O(n)$, whilst still being a universal approximator of continuous sequence-to-sequence functions. However, this permutation variant operation is not appropriate for direct application to sets. In this paper, we proposed an $O(n)$ complexity sampled transformer that can process point set elements directly without any additional inductive bias. Our sampled transformer introduces random element sampling, which randomly splits point sets into subsets, followed by applying a shared Hamiltonian self-attention mechanism to each subset. The overall attention mechanism can be viewed as a Hamiltonian cycle in the complete attention graph, and the permutation of point set elements is equivalent to randomly sampling Hamiltonian cycles. This mechanism implements a Monte Carlo simulation of the $O(n^2)$ dense attention connections. We show that it is a universal approximator for continuous set-to-set functions. Experimental results on point-clouds show comparable or better accuracy with significantly reduced computational complexity compared to the dense transformer or alternative sparse attention schemes.

Fair Wrapping for Black-box Predictions

We introduce a new family of techniques to post-process ("wrap") a black-box classifier in order to reduce its bias. Our technique builds on the recent analysis of improper loss functions whose optimisation can correct any twist in prediction, unfairness being treated as a twist. In the post-processing, we learn a wrapper function which we define as an {\alpha}-tree, which modifies the prediction. We provide two generic boosting algorithms to learn {\alpha}-trees. We show that our modification has appealing properties in terms of composition of{\alpha}-trees, generalization, interpretability, and KL divergence between modified and original predictions. We exemplify the use of our technique in three fairness notions: conditional value at risk, equality of opportunity, and statistical parity; and provide experiments on several readily available datasets.

Linking Across Data Granularity: Fitting Multivariate Hawkes Processes to Partially Interval-Censored Data

This work introduces a novel multivariate temporal point process, the Partial Mean Behavior Poisson (PMBP) process, which can be leveraged to fit the multivariate Hawkes process to partially interval-censored data consisting of a mix of event timestamps on a subset of dimensions and interval-censored event counts on the complementary dimensions. First, we define the PMBP process via its conditional intensity and derive the regularity conditions for subcriticality. We show that both the Hawkes process and the MBP process (Rizoiu et al. (2021)) are special cases of the PMBP process. Second, we provide numerical schemes that enable calculating the conditional intensity and sampling event histories of the PMBP process. Third, we demonstrate the applicability of the PMBP process by empirical testing using synthetic and real-world datasets: We test the capability of the PMBP process to recover multivariate Hawkes parameters given sample event histories of the Hawkes process. Next, we evaluate the PMBP process on the Youtube popularity prediction task and show that it outperforms the current state-of-the-art Hawkes Intensity process (Rizoiu et al. (2017b)). Lastly, on a curated dataset of COVID19 daily case counts and COVID19-related news articles for a sample of countries, we show that clustering on the PMBP-fitted parameters enables a categorization of countries with respect to the country-level interaction of cases and news reporting.

On the Variance of the Fisher Information for Deep Learning

The Fisher information matrix (FIM) has been applied to the realm of deep learning. It is closely related to the loss landscape, the variance of the parameters, second order optimization, and deep learning theory. The exact FIM is either unavailable in closed form or too expensive to compute. In practice, it is almost always estimated based on empirical samples. We investigate two such estimators based on two equivalent representations of the FIM. They are both unbiased and consistent with respect to the underlying "true" FIM. Their estimation quality is characterized by their variance given in closed form. We bound their variances and analyze how the parametric structure of a deep neural network can impact the variance. We discuss the meaning of this variance measure and our bounds in the context of deep learning.

Interval-censored Hawkes processes

Hawkes processes are a popular means of modeling the event times of self-exciting phenomena, such as earthquake strikes or tweets on a topical subject. Classically, these models are fit to historical event time data via likelihood maximization. However, in many scenarios, the exact times of historical events are not recorded for either privacy (e.g., patient admittance to hospitals) or technical limitations (e.g., most transport data records the volume of vehicles passing loop detectors but not the individual times). The interval-censored setting denotes when only the aggregate counts of events at specific time intervals are observed. Fitting the parameters of interval-censored Hawkes processes requires designing new training objectives that do not rely on the exact event times. In this paper, we propose a model to estimate the parameters of a Hawkes process in interval-censored settings. Our model builds upon the existing Hawkes Intensity Process (HIP) of in several important directions. First, we observe that while HIP is formulated in terms of expected intensities, it is more natural to work instead with expected counts; further, one can express the latter as the solution to an integral equation closely related to the defining equation of HIP. Second, we show how a non-homogeneous Poisson approximation to the Hawkes process admits a tractable likelihood in the interval-censored setting; this approximation recovers the original HIP objective as a special case, and allows for the use of a broader class of Bregman divergences as loss function. Third, we explicate how to compute a tighter approximation to the ground truth in the likelihood. Finally, we show how our model can incorporate information about varying interval lengths. Experiments on synthetic and real-world data confirm our HIPPer model outperforms HIP and several other baselines on the task of interval-censored inference.

Data Preprocessing to Mitigate Bias with Boosted Fair Mollifiers

In a recent paper, Celis et al. (2020) introduced a new approach to fairness that corrects the data distribution itself. The approach is computationally appealing, but its approximation guarantees with respect to the target distribution can be quite loose as they need to rely on a (typically limited) number of constraints on data-based aggregated statistics; also resulting on a fairness guarantee which can be data dependent. Our paper makes use of a mathematical object recently introduced in privacy -- mollifiers of distributions -- and a popular approach to machine learning -- boosting -- to get an approach in the same lineage as Celis et al. but without those impediments, including in particular, better guarantees in terms of accuracy and finer guarantees in terms of fairness. The approach involves learning the sufficient statistics of an exponential family. When training data is tabular, it is defined by decision trees whose interpretability can provide clues on the source of (un)fairness. Experiments display the quality of the results obtained for simulated and real-world data.