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We consider the setting of online convex optimization (OCO) with \textit{exp-concave} losses. The best regret bound known for this setting is $O(n\log{}T)$, where $n$ is the dimension and $T$ is the number of prediction rounds (treating all other quantities as constants and assuming $T$ is sufficiently large), and is attainable via the well-known Online Newton Step algorithm (ONS). However, ONS requires on each iteration to compute a projection (according to some matrix-induced norm) onto the feasible convex set, which is often computationally prohibitive in high-dimensional settings and when the feasible set admits a non-trivial structure. In this work we consider projection-free online algorithms for exp-concave and smooth losses, where by projection-free we refer to algorithms that rely only on the availability of a linear optimization oracle (LOO) for the feasible set, which in many applications of interest admits much more efficient implementations than a projection oracle. We present an LOO-based ONS-style algorithm, which using overall $O(T)$ calls to a LOO, guarantees in worst case regret bounded by $\widetilde{O}(n^{2/3}T^{2/3})$ (ignoring all quantities except for $n,T$). However, our algorithm is most interesting in an important and plausible low-dimensional data scenario: if the gradients (approximately) span a subspace of dimension at most $\rho$, $\rho << n$, the regret bound improves to $\widetilde{O}(\rho^{2/3}T^{2/3})$, and by applying standard deterministic sketching techniques, both the space and average additional per-iteration runtime requirements are only $O(\rho{}n)$ (instead of $O(n^2)$). This improves upon recently proposed LOO-based algorithms for OCO which, while having the same state-of-the-art dependence on the horizon $T$, suffer from regret/oracle complexity that scales with $\sqrt{n}$ or worse.

In computer vision, camera pose estimation from correspondences between 3D geometric entities and their projections into the image has been a widely investigated problem. Although most state-of-the-art methods exploit low-level primitives such as points or lines, the emergence of very effective CNN-based object detectors in the recent years has paved the way to the use of higher-level features carrying semantically meaningful information. Pioneering works in that direction have shown that modelling 3D objects by ellipsoids and 2D detections by ellipses offers a convenient manner to link 2D and 3D data. However, the mathematical formalism most often used in the related litterature does not enable to easily distinguish ellipsoids and ellipses from other quadrics and conics, leading to a loss of specificity potentially detrimental in some developments. Moreover, the linearization process of the projection equation creates an over-representation of the camera parameters, also possibly causing an efficiency loss. In this paper, we therefore introduce an ellipsoid-specific theoretical framework and demonstrate its beneficial properties in the context of pose estimation. More precisely, we first show that the proposed formalism enables to reduce the pose estimation problem to a position or orientation-only estimation problem in which the remaining unknowns can be derived in closed-form. Then, we demonstrate that it can be further reduced to a 1 Degree-of-Freedom (1DoF) problem and provide the analytical derivations of the pose as a function of that unique scalar unknown. We illustrate our theoretical considerations by visual examples and include a discussion on the practical aspects. Finally, we release this paper along with the corresponding source code in order to contribute towards more efficient resolutions of ellipsoid-related pose estimation problems.

The identification of choice models is crucial for understanding consumer behavior and informing marketing or operational strategies, policy design, and product development. The identification of parametric choice-based demand models is typically straightforward. However, nonparametric models, which are highly effective and flexible in explaining customer choice, may encounter the challenge of the dimensionality curse, hindering their identification. A prominent example of a nonparametric model is the ranking-based model, which mirrors the random utility maximization (RUM) class and is known to be nonidentifiable from the collection of choice probabilities alone. Our objective in this paper is to develop a new class of nonparametric models that is not subject to the problem of nonidentifiability. Our model assumes bounded rationality of consumers, which results in symmetric demand cannibalization and intriguingly enables full identification. Additionally, our choice model demonstrates competitive prediction accuracy compared to the state-of-the-art benchmarks in a real-world case study, despite incorporating the assumption of bounded rationality which could, in theory, limit the representation power of our model. In addition, we tackle the important problem of finding the optimal assortment under the proposed choice model. We demonstrate the NP-hardness of this problem and provide a fully polynomial-time approximation scheme through dynamic programming. Additionally, we propose an efficient estimation framework using a combination of column generation and expectation-maximization algorithms, which proves to be more tractable than the estimation algorithm of the aforementioned ranking-based model.

Estimation of heterogeneous causal effects - i.e., how effects of policies and treatments vary across subjects - is a fundamental task in causal inference, playing a crucial role in optimal treatment allocation, generalizability, subgroup effects, and more. Many flexible methods for estimating conditional average treatment effects (CATEs) have been proposed in recent years, but questions surrounding optimality have remained largely unanswered. In particular, a minimax theory of optimality has yet to be developed, with the minimax rate of convergence and construction of rate-optimal estimators remaining open problems. In this paper we derive the minimax rate for CATE estimation, in a nonparametric model where distributional components are Holder-smooth, and present a new local polynomial estimator, giving high-level conditions under which it is minimax optimal. More specifically, our minimax lower bound is derived via a localized version of the method of fuzzy hypotheses, combining lower bound constructions for nonparametric regression and functional estimation. Our proposed estimator can be viewed as a local polynomial R-Learner, based on a localized modification of higher-order influence function methods; it is shown to be minimax optimal under a condition on how accurately the covariate distribution is estimated. The minimax rate we find exhibits several interesting features, including a non-standard elbow phenomenon and an unusual interpolation between nonparametric regression and functional estimation rates. The latter quantifies how the CATE, as an estimand, can be viewed as a regression/functional hybrid. We conclude with some discussion of a few remaining open problems.

Participant noncompliance, in which participants do not follow their assigned treatment protocol, often obscures the causal relationship between treatment and treatment effect in randomized trials. In the longitudinal setting, the G-computation algorithm can adjust for confounding to estimate causal effects. Typically, G-computation assumes that both 1) compliance is observed; and 2) the densities of the confounders can be correctly specified. We aim to develop a G-computation estimator in the setting where both assumptions are violated. For 1), in place of unobserved compliance, we substitute in probability weights derived from modeling a biomarker associated with compliance. For 2), we fit semiparametric models using predictive mean matching. Specifically, we parametrically specify only the conditional mean of the confounders, and then use predictive mean matching to randomly generate confounder data for G-computation. In both the simulation and application, we compare multiple causal estimators already established in the literature with those derived from our method. For the simulation, we generated data across different sample sizes and levels of confounding. For the application, we apply our method to a trial that sought to evaluate the effect of cigarettes with low nicotine on cigarette consumption (Center for the Evaluation of Nicotine in Cigarettes Project 2 - CENIC-P2).

This paper studies covariate adjusted estimation of the average treatment effect (ATE) in stratified experiments. We work in the stratified randomization framework of Cytrynbaum (2021), which includes matched tuples designs (e.g. matched pairs), coarse stratification, and complete randomization as special cases. Interestingly, we show that the Lin (2013) interacted regression is generically asymptotically inefficient, with efficiency only in the edge case of complete randomization. Motivated by this finding, we derive the optimal linear covariate adjustment for a given stratified design, constructing several new estimators that achieve the minimal variance. Conceptually, we show that optimal linear adjustment of a stratified design is equivalent in large samples to doubly-robust semiparametric adjustment of an independent design. We also develop novel asymptotically exact inference for the ATE over a general family of adjusted estimators, showing in simulations that the usual Eicker-Huber-White confidence intervals can significantly overcover. Our inference methods produce shorter confidence intervals by fully accounting for the precision gains from both covariate adjustment and stratified randomization. Simulation experiments and an empirical application to the Oregon Health Insurance Experiment data (Finkelstein et al. (2012)) demonstrate the value of our proposed methods.

In literature on imprecise probability little attention is paid to the fact that imprecise probabilities are precise on some events. We call these sets system of precision. We show that, under mild assumptions, the system of precision of a lower and upper probability form a so-called (pre-)Dynkin-system. Interestingly, there are several settings, ranging from machine learning on partial data over frequential probability theory to quantum probability theory and decision making under uncertainty, in which a priori the probabilities are only desired to be precise on a specific underlying set system. At the core of all of these settings lies the observation that precise beliefs, probabilities or frequencies on two events do not necessarily imply this precision to hold for the intersection of those events. Here, (pre-)Dynkin-systems have been adopted as systems of precision, too. We show that, under extendability conditions, those pre-Dynkin-systems equipped with probabilities can be embedded into algebras of sets. Surprisingly, the extendability conditions elaborated in a strand of work in quantum physics are equivalent to coherence in the sense of Walley (1991, Statistical reasoning with imprecise probabilities, p. 84). Thus, literature on probabilities on pre-Dynkin-systems gets linked to the literature on imprecise probability. Finally, we spell out a lattice duality which rigorously relates the system of precision to credal sets of probabilities. In particular, we provide a hitherto undescribed, parametrized family of coherent imprecise probabilities.

This article tackles the old problem of prediction via a nonparametric transformation model (NTM) in a new Bayesian way. Estimation of NTMs is known challenging due to model unidentifiability though appealing because of its robust prediction capability in survival analysis. Inspired by the uniqueness of the posterior predictive distribution, we achieve efficient prediction via the NTM aforementioned under the Bayesian paradigm. Our strategy is to assign weakly informative priors to nonparametric components rather than identify the model by adding complicated constraints in the existing literature. The Bayesian success pays tribute to i) a subtle cast of NTMs by an exponential transformation for the purpose of compressing spaces of infinite-dimensional parameters to positive quadrants considering non-negativity of the failure time; ii) a newly constructed weakly informative quantile-knots I-splines prior for the recast transformation function together with the Dirichlet process mixture model assigned to the error distribution. In addition, we provide a convenient and precise estimator for the identified parameter component subject to the general unit-norm restriction through posterior modification, enabling effective relative risks. Simulations and applications on real datasets reveal that our method is robust and outperforms the competing methods. An R package BuLTM is available to predict survival curves, estimate relative risks, and facilitate posterior checking.

Federated learning methods, that is, methods that perform model training using data situated across different sources, whilst simultaneously not having the data leave their original source, are of increasing interest in a number of fields. However, despite this interest, the classes of models for which easily-applicable and sufficiently general approaches are available is limited, excluding many structured probabilistic models. We present a general yet elegant resolution to the aforementioned issue. The approach is based on adopting structured variational inference, an approach widely used in Bayesian machine learning, to the federated setting. Additionally, a communication-efficient variant analogous to the canonical FedAvg algorithm is explored. The effectiveness of the proposed algorithms are demonstrated, and their performance is compared on Bayesian multinomial regression, topic modelling, and mixed model examples.

Recent contrastive representation learning methods rely on estimating mutual information (MI) between multiple views of an underlying context. E.g., we can derive multiple views of a given image by applying data augmentation, or we can split a sequence into views comprising the past and future of some step in the sequence. Contrastive lower bounds on MI are easy to optimize, but have a strong underestimation bias when estimating large amounts of MI. We propose decomposing the full MI estimation problem into a sum of smaller estimation problems by splitting one of the views into progressively more informed subviews and by applying the chain rule on MI between the decomposed views. This expression contains a sum of unconditional and conditional MI terms, each measuring modest chunks of the total MI, which facilitates approximation via contrastive bounds. To maximize the sum, we formulate a contrastive lower bound on the conditional MI which can be approximated efficiently. We refer to our general approach as Decomposed Estimation of Mutual Information (DEMI). We show that DEMI can capture a larger amount of MI than standard non-decomposed contrastive bounds in a synthetic setting, and learns better representations in a vision domain and for dialogue generation.