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Series or orthogonal basis regression is one of the most popular non-parametric regression techniques in practice, obtained by regressing the response on features generated by evaluating the basis functions at observed covariate values. The most routinely used series estimator is based on ordinary least squares fitting, which is known to be minimax rate optimal in various settings, albeit under stringent restrictions on the basis functions and the distribution of covariates. In this work, inspired by the recently developed Forster-Warmuth (FW) learner, we propose an alternative series regression estimator that can attain the minimax estimation rate under strictly weaker conditions imposed on the basis functions and the joint law of covariates, than existing series estimators in the literature. Moreover, a key contribution of this work generalizes the FW-learner to a so-called counterfactual regression problem, in which the response variable of interest may not be directly observed (hence, the name ``counterfactual'') on all sampled units, and therefore needs to be inferred in order to identify and estimate the regression in view from the observed data. Although counterfactual regression is not entirely a new area of inquiry, we propose the first-ever systematic study of this challenging problem from a unified pseudo-outcome perspective. In fact, we provide what appears to be the first generic and constructive approach for generating the pseudo-outcome (to substitute for the unobserved response) which leads to the estimation of the counterfactual regression curve of interest with small bias, namely bias of second order. Several applications are used to illustrate the resulting FW-learner including many nonparametric regression problems in missing data and causal inference literature, for which we establish high-level conditions for minimax rate optimality of the proposed FW-learner.

Modern ML predictions models are surprisingly accurate in practice and incorporating their power into algorithms has led to a new research direction. Algorithms with predictions have already been used to improve on worst-case optimal bounds for online problems and for static graph problems. With this work, we initiate the study of the complexity of {\em data structures with predictions}, with an emphasis on dynamic graph problems. Unlike the independent work of v.d.~Brand et al.~[arXiv:2307.09961] that aims at upper bounds, our investigation is focused on establishing conditional fine-grained lower bounds for various notions of predictions. Our lower bounds are conditioned on the Online Matrix Vector (OMv) hypothesis. First we show that a prediction-based algorithm for OMv provides a smooth transition between the known bounds, for the offline and the online setting, and then show that this algorithm is essentially optimal under the OMv hypothesis. Further, we introduce and study four different kinds of predictions. (1) For {\em $\varepsilon$-accurate predictions}, where $\varepsilon \in (0,1)$, we show that any lower bound from the non-prediction setting carries over, reduced by a factor of $1-\varepsilon$. (2) For {\em $L$-list accurate predictions}, we show that one can efficiently compute a $(1/L)$-accurate prediction from an $L$-list accurate prediction. (3) For {\em bounded delay predictions} and {\em bounded delay predictions with outliers}, we show that a lower bound from the non-prediction setting carries over, if the reduction fulfills a certain reordering condition (which is fulfilled by many reductions from OMv for dynamic graph problems). This is demonstrated by showing lower and almost tight upper bounds for a concrete, dynamic graph problem, called $\# s \textrm{-} \triangle$, where the number of triangles that contain a fixed vertex $s$ must be reported.

The dynamic ranking, due to its increasing importance in many applications, is becoming crucial, especially with the collection of voluminous time-dependent data. One such application is sports statistics, where dynamic ranking aids in forecasting the performance of competitive teams, drawing on historical and current data. Despite its usefulness, predicting and inferring rankings pose challenges in environments necessitating time-dependent modeling. This paper introduces a spectral ranker called Kernel Rank Centrality, designed to rank items based on pairwise comparisons over time. The ranker operates via kernel smoothing in the Bradley-Terry model, utilizing a Markov chain model. Unlike the maximum likelihood approach, the spectral ranker is nonparametric, demands fewer model assumptions and computations, and allows for real-time ranking. We establish the asymptotic distribution of the ranker by applying an innovative group inverse technique, resulting in a uniform and precise entrywise expansion. This result allows us to devise a new inferential method for predictive inference, previously unavailable in existing approaches. Our numerical examples showcase the ranker's utility in predictive accuracy and constructing an uncertainty measure for prediction, leveraging data from the National Basketball Association (NBA). The results underscore our method's potential compared to the gold standard in sports, the Arpad Elo rating system.

This paper proposes a non-centered parameterization based infinite-dimensional mean-field variational inference (NCP-iMFVI) approach for solving the hierarchical Bayesian inverse problems. This method can generate available estimates from the approximated posterior distribution efficiently. To avoid the mutually singular obstacle that occurred in the infinite-dimensional hierarchical approach, we propose a rigorous theory of the non-centered variational Bayesian approach. Since the non-centered parameterization weakens the connection between the parameter and the hyper-parameter, we can introduce the hyper-parameter to all terms of the eigendecomposition of the prior covariance operator. We also show the relationships between the NCP-iMFVI and infinite-dimensional hierarchical approaches with centered parameterization. The proposed algorithm is applied to three inverse problems governed by the simple smooth equation, the Helmholtz equation, and the steady-state Darcy flow equation. Numerical results confirm our theoretical findings, illustrate the efficiency of solving the iMFVI problem formulated by large-scale linear and nonlinear statistical inverse problems, and verify the mesh-independent property.

If $X$ is a subset of vertices of a graph $G$, then vertices $u$ and $v$ are $X$-visible if there exists a shortest $u,v$-path $P$ such that $V(P)\cap X \subseteq \{u,v\}$. If each two vertices from $X$ are $X$-visible, then $X$ is a mutual-visibility set. The mutual-visibility number of $G$ is the cardinality of a largest mutual-visibility set of $G$ and has been already investigated. In this paper a variety of mutual-visibility problems is introduced based on which natural pairs of vertices are required to be $X$-visible. This yields the total, the dual, and the outer mutual-visibility numbers. We first show that these graph invariants are related to each other and to the classical mutual-visibility number, and then we prove that the three newly introduced mutual-visibility problems are computationally difficult. According to this result, we compute or bound their values for several graphs classes that include for instance grid graphs and tori. We conclude the study by presenting some inter-comparison between the values of such parameters, which is based on the computations we made for some specific families.

Price discrimination, which refers to the strategy of setting different prices for different customer groups, has been widely used in online retailing. Although it helps boost the collected revenue for online retailers, it might create serious concerns about fairness, which even violates the regulation and laws. This paper studies the problem of dynamic discriminatory pricing under fairness constraints. In particular, we consider a finite selling horizon of length $T$ for a single product with two groups of customers. Each group of customers has its unknown demand function that needs to be learned. For each selling period, the seller determines the price for each group and observes their purchase behavior. While existing literature mainly focuses on maximizing revenue, ensuring fairness among different customers has not been fully explored in the dynamic pricing literature. This work adopts the fairness notion from Cohen et al. (2022). For price fairness, we propose an optimal dynamic pricing policy regarding regret, which enforces the strict price fairness constraint. In contrast to the standard $\sqrt{T}$-type regret in online learning, we show that the optimal regret in our case is $\tilde{O}(T^{4/5})$. We further extend our algorithm to a more general notion of fairness, which includes demand fairness as a special case. To handle this general class, we propose a soft fairness constraint and develop a dynamic pricing policy that achieves $\tilde{O}(T^{4/5})$ regret. We also demonstrate that our algorithmic techniques can be adapted to more general scenarios such as fairness among multiple groups of customers.

A nonlinear optimization method is proposed for the solution of inverse medium problems with spatially varying properties. To avoid the prohibitively large number of unknown control variables resulting from standard grid-based representations, the misfit is instead minimized in a small subspace spanned by the first few eigenfunctions of a judicious elliptic operator, which itself depends on the previous iteration. By repeatedly adapting both the dimension and the basis of the search space, regularization is inherently incorporated at each iteration without the need for extra Tikhonov penalization. Convergence is proved under an angle condition, which is included into the resulting \emph{Adaptive Spectral Inversion} (ASI) algorithm. The ASI approach compares favorably to standard grid-based inversion using $L^2$-Tikhonov regularization when applied to an elliptic inverse problem. The improved accuracy resulting from the newly included angle condition is further demonstrated via numerical experiments from time-dependent inverse scattering problems.

Modelling in biology must adapt to increasingly complex and massive data. The efficiency of the inference algorithms used to estimate model parameters is therefore questioned. Many of these are based on stochastic optimization processes which waste a significant part of the computation time due to their rejection sampling approaches. We introduce the Fixed Landscape Inference MethOd (flimo), a new likelihood-free inference method for continuous state-space stochastic models. It applies deterministic gradient-based optimization algorithms to obtain a point estimate of the parameters, minimizing the difference between the data and some simulations according to some prescribed summary statistics. In this sense, it is analogous to Approximate Bayesian Computation (ABC). Like ABC, it can also provide an approximation of the distribution of the parameters. Three applications are proposed: a usual theoretical example, namely the inference of the parameters of g-and-k distributions; a population genetics problem, not so simple as it seems, namely the inference of a selective value from time series in a Wright-Fisher model; and simulations from a Ricker model, representing chaotic population dynamics. In the two first applications, the results show a drastic reduction of the computational time needed for the inference phase compared to the other methods, despite an equivalent accuracy. Even when likelihood-based methods are applicable, the simplicity and efficiency of flimo make it a compelling alternative. Implementations in Julia and in R are available on //metabarcoding.org/flimo. To run flimo, the user must simply be able to simulate data according to the chosen model.

Classic algorithms and machine learning systems like neural networks are both abundant in everyday life. While classic computer science algorithms are suitable for precise execution of exactly defined tasks such as finding the shortest path in a large graph, neural networks allow learning from data to predict the most likely answer in more complex tasks such as image classification, which cannot be reduced to an exact algorithm. To get the best of both worlds, this thesis explores combining both concepts leading to more robust, better performing, more interpretable, more computationally efficient, and more data efficient architectures. The thesis formalizes the idea of algorithmic supervision, which allows a neural network to learn from or in conjunction with an algorithm. When integrating an algorithm into a neural architecture, it is important that the algorithm is differentiable such that the architecture can be trained end-to-end and gradients can be propagated back through the algorithm in a meaningful way. To make algorithms differentiable, this thesis proposes a general method for continuously relaxing algorithms by perturbing variables and approximating the expectation value in closed form, i.e., without sampling. In addition, this thesis proposes differentiable algorithms, such as differentiable sorting networks, differentiable renderers, and differentiable logic gate networks. Finally, this thesis presents alternative training strategies for learning with algorithms.

The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.