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We consider $L^2$-approximation on weighted reproducing kernel Hilbert spaces of functions depending on infinitely many variables. We focus on unrestricted linear information, admitting evaluations of arbitrary continuous linear functionals. We distinguish between ANOVA and non-ANOVA spaces, where, by ANOVA spaces, we refer to function spaces whose norms are induced by an underlying ANOVA function decomposition. In ANOVA spaces, we provide an optimal algorithm to solve the approximation problem using linear information. We determine the upper and lower error bounds on the polynomial convergence rate of $n$-th minimal worst-case errors, which match if the weights decay regularly. For non-ANOVA spaces, we also establish upper and lower error bounds. Our analysis reveals that for weights with a regular and moderate decay behavior, the convergence rate of $n$-th minimal errors is strictly higher in ANOVA than in non-ANOVA spaces.

Algorithms for online learning typically require one or more boundedness assumptions: that the domain is bounded, that the losses are Lipschitz, or both. In this paper, we develop a new setting for online learning with unbounded domains and non-Lipschitz losses. For this setting we provide an algorithm which guarantees $R_{T}(u)\le \tilde O(G\|u\|\sqrt{T}+L\|u\|^{2}\sqrt{T})$ regret on any problem where the subgradients satisfy $\|g_{t}\|\le G+L\|w_{t}\|$, and show that this bound is unimprovable without further assumptions. We leverage this algorithm to develop new saddle-point optimization algorithms that converge in duality gap in unbounded domains, even in the absence of meaningful curvature. Finally, we provide the first algorithm achieving non-trivial dynamic regret in an unbounded domain for non-Lipschitz losses, as well as a matching lower bound. The regret of our dynamic regret algorithm automatically improves to a novel $L^{*}$ bound when the losses are smooth.

We study the problem of locally private mean estimation of high-dimensional vectors in the Euclidean ball. Existing algorithms for this problem either incur sub-optimal error or have high communication and/or run-time complexity. We propose a new algorithmic framework, ProjUnit, for private mean estimation that yields algorithms that are computationally efficient, have low communication complexity, and incur optimal error up to a $1+o(1)$-factor. Our framework is deceptively simple: each randomizer projects its input to a random low-dimensional subspace, normalizes the result, and then runs an optimal algorithm such as PrivUnitG in the lower-dimensional space. In addition, we show that, by appropriately correlating the random projection matrices across devices, we can achieve fast server run-time. We mathematically analyze the error of the algorithm in terms of properties of the random projections, and study two instantiations. Lastly, our experiments for private mean estimation and private federated learning demonstrate that our algorithms empirically obtain nearly the same utility as optimal ones while having significantly lower communication and computational cost.

Many real-world tasks include some kind of parameter estimation, i.e., determination of a parameter encoded in a probability distribution. Often, such probability distributions arise from stochastic processes. For a stationary stochastic process with temporal correlations, the random variables that constitute it are identically distributed but not independent. This is the case, for instance, for quantum continuous measurements. In this paper we prove two fundamental results concerning the estimation of parameters encoded in a memoryful stochastic process. First, we show that for processes with finite Markov order, the Fisher information is always asymptotically linear in the number of outcomes, and determined by the conditional distribution of the process' Markov order. Second, we prove with suitable examples that correlations do not necessarily enhance the metrological precision. In fact, we show that unlike for entropic information quantities, in general nothing can be said about the sub- or super-additivity of the joint Fisher information, in the presence of correlations. We discuss how the type of correlations in the process affects the scaling. We then apply these results to the case of thermometry on a spin chain.

We consider the problem of continuous-time policy evaluation. This consists in learning through observations the value function associated with an uncontrolled continuous-time stochastic dynamic and a reward function. We propose two original variants of the well-known TD(0) method using vanishing time steps. One is model-free and the other is model-based. For both methods, we prove theoretical convergence rates that we subsequently verify through numerical simulations. Alternatively, those methods can be interpreted as novel reinforcement learning approaches for approximating solutions of linear PDEs (partial differential equations) or linear BSDEs (backward stochastic differential equations).

In this paper, we introduce the problem of Matroid-Constrained Vertex Cover: given a graph with weights on the edges and a matroid imposed on the vertices, our problem is to choose a subset of vertices that is independent in the matroid, with the objective of maximizing the total weight of covered edges. This problem is a generalization of the much studied max $k$-vertex cover problem, in which the matroid is the simple uniform matroid, and it is also a special case of the problem of maximizing a monotone submodular function under a matroid constraint. First, we give a Fixed-Parameter Tractable Approximation Scheme (FPT-AS) when the given matroid is a partition matroid, a laminar matroid, or a transversal matroid. Precisely, if $k$ is the rank of the matroid, we obtain $(1 - \varepsilon)$ approximation using $(1/\varepsilon)^{O(k)}n^{O(1)}$ time for partition and laminar matroids and using $(1/\varepsilon+k)^{O(k)}n^{O(1)}$ time for transversal matroids. This extends a result of Manurangsi for uniform matroids [Manurangsi, 2018]. We also show that these ideas can be applied in the context of (single-pass) streaming algorithms. Besides, our FPT-AS introduces a new technique based on matroid union, which may be of independent interest in extremal combinatorics. In the second part, we consider general matroids. We propose a simple local search algorithm that guarantees $2/3 \approx 0.66$ approximation. For the more general problem where two matroids are imposed on the vertices and a feasible solution must be a common independent set, we show that a local search algorithm gives a $2/3 \cdot (1 - 1/(p+1))$ approximation in $n^{O(p)}$ time, for any integer $p$. We also provide some evidence to show that with the constraint of one or two matroids, the approximation ratio of $2/3$ is likely the best possible, using the currently known techniques of local search.

We introduce MESSY estimation, a Maximum-Entropy based Stochastic and Symbolic densitY estimation method. The proposed approach recovers probability density functions symbolically from samples using moments of a Gradient flow in which the ansatz serves as the driving force. In particular, we construct a gradient-based drift-diffusion process that connects samples of the unknown distribution function to a guess symbolic expression. We then show that when the guess distribution has the maximum entropy form, the parameters of this distribution can be found efficiently by solving a linear system of equations constructed using the moments of the provided samples. Furthermore, we use Symbolic regression to explore the space of smooth functions and find optimal basis functions for the exponent of the maximum entropy functional leading to good conditioning. The cost of the proposed method in each iteration of the random search is linear with the number of samples and quadratic with the number of basis functions. We validate the proposed MESSY estimation method against other benchmark methods for the case of a bi-modal and a discontinuous density, as well as a density at the limit of physical realizability. We find that the addition of a symbolic search for basis functions improves the accuracy of the estimation at a reasonable additional computational cost. Our results suggest that the proposed method outperforms existing density recovery methods in the limit of a small to moderate number of samples by providing a low-bias and tractable symbolic description of the unknown density at a reasonable computational cost.

Humans performing tasks that involve taking a series of multiple dependent actions over time often learn from experience by reflecting on specific cases and points in time, where different actions could have led to significantly better outcomes. While recent machine learning methods to retrospectively analyze sequential decision making processes promise to aid decision makers in identifying such cases, they have focused on environments with finitely many discrete states. However, in many practical applications, the state of the environment is inherently continuous in nature. In this paper, we aim to fill this gap. We start by formally characterizing a sequence of discrete actions and continuous states using finite horizon Markov decision processes and a broad class of bijective structural causal models. Building upon this characterization, we formalize the problem of finding counterfactually optimal action sequences and show that, in general, we cannot expect to solve it in polynomial time. Then, we develop a search method based on the $A^*$ algorithm that, under a natural form of Lipschitz continuity of the environment's dynamics, is guaranteed to return the optimal solution to the problem. Experiments on real clinical data show that our method is very efficient in practice, and it has the potential to offer interesting insights for sequential decision making tasks.

Stochastic optimization is one of the central problems in Machine Learning and Theoretical Computer Science. In the standard model, the algorithm is given a fixed distribution known in advance. In practice though, one may acquire at a cost extra information to make better decisions. In this paper, we study how to buy information for stochastic optimization and formulate this question as an online learning problem. Assuming the learner has an oracle for the original optimization problem, we design a $2$-competitive deterministic algorithm and a $e/(e-1)$-competitive randomized algorithm for buying information. We show that this ratio is tight as the problem is equivalent to a robust generalization of the ski-rental problem, which we call super-martingale stopping. We also consider an adaptive setting where the learner can choose to buy information after taking some actions for the underlying optimization problem. We focus on the classic optimization problem, Min-Sum Set Cover, where the goal is to quickly find an action that covers a given request drawn from a known distribution. We provide an $8$-competitive algorithm running in polynomial time that chooses actions and decides when to buy information about the underlying request.

Personalized treatment effect estimates are often of interest in high-stakes applications -- thus, before deploying a model estimating such effects in practice, one needs to be sure that the best candidate from the ever-growing machine learning toolbox for this task was chosen. Unfortunately, due to the absence of counterfactual information in practice, it is usually not possible to rely on standard validation metrics for doing so, leading to a well-known model selection dilemma in the treatment effect estimation literature. While some solutions have recently been investigated, systematic understanding of the strengths and weaknesses of different model selection criteria is still lacking. In this paper, instead of attempting to declare a global `winner', we therefore empirically investigate success- and failure modes of different selection criteria. We highlight that there is a complex interplay between selection strategies, candidate estimators and the data used for comparing them, and provide interesting insights into the relative (dis)advantages of different criteria alongside desiderata for the design of further illuminating empirical studies in this context.