We propose Convexity-Driven Projection (CDP), a boundary-free linear method for dimensionality reduction of point clouds that targets preserving detour-induced local non-convexity. CDP builds a $k$-NN graph, identifies admissible pairs whose Euclidean-to-shortest-path ratios are below a threshold, and aggregates their normalized directions to form a positive semidefinite non-convexity structure matrix. The projection uses the top-$k$ eigenvectors of the structure matrix. We give two verifiable guarantees. A pairwise a-posteriori certificate that bounds the post-projection distortion for each admissible pair, and an average-case spectral bound that links expected captured direction energy to the spectrum of the structure matrix, yielding quantile statements for typical distortion. Our evaluation protocol reports fixed- and reselected-pairs detour errors and certificate quantiles, enabling practitioners to check guarantees on their data.

Min-max optimization problems involving nonconvex-nonconcave objectives have found important applications in adversarial training and other multi-agent learning settings. Yet, no known gradient descent-based method is guaranteed to converge to (even local notions of) min-max equilibrium in the nonconvex-nonconcave setting. For all known methods, there exist relatively simple objectives for which they cycle or exhibit other undesirable behavior different from converging to a point, let alone to some game-theoretically meaningful one~\cite{flokas2019poincare,hsieh2021limits}. The only known convergence guarantees hold under the strong assumption that the initialization is very close to a local min-max equilibrium~\cite{wang2019solving}. Moreover, the afore-described challenges are not just theoretical curiosities. All known methods are unstable in practice, even in simple settings. We propose the first method that is guaranteed to converge to a local min-max equilibrium for smooth nonconvex-nonconcave objectives. Our method is second-order and provably escapes limit cycles as long as it is initialized at an easy-to-find initial point. Both the definition of our method and its convergence analysis are motivated by the topological nature of the problem. In particular, our method is not designed to decrease some potential function, such as the distance of its iterate from the set of local min-max equilibria or the projected gradient of the objective, but is designed to satisfy a topological property that guarantees the avoidance of cycles and implies its convergence.

We consider an online binary prediction setting where a forecaster observes a sequence of $T$ bits one by one. Before each bit is revealed, the forecaster predicts the probability that the bit is $1$. The forecaster is called well-calibrated if for each $p \in [0, 1]$, among the $n_p$ bits for which the forecaster predicts probability $p$, the actual number of ones, $m_p$, is indeed equal to $p \cdot n_p$. The calibration error, defined as $\sum_p |m_p - p n_p|$, quantifies the extent to which the forecaster deviates from being well-calibrated. It has long been known that an $O(T^{2/3})$ calibration error is achievable even when the bits are chosen adversarially, and possibly based on the previous predictions. However, little is known on the lower bound side, except an $\Omega(\sqrt{T})$ bound that follows from the trivial example of independent fair coin flips. In this paper, we prove an $\Omega(T^{0.528})$ bound on the calibration error, which is the first super-$\sqrt{T}$ lower bound for this setting to the best of our knowledge. The technical contributions of our work include two lower bound techniques, early stopping and sidestepping, which circumvent the obstacles that have previously hindered strong calibration lower bounds. We also propose an abstraction of the prediction setting, termed the Sign-Preservation game, which may be of independent interest. This game has a much smaller state space than the full prediction setting and allows simpler analyses. The $\Omega(T^{0.528})$ lower bound follows from a general reduction theorem that translates lower bounds on the game value of Sign-Preservation into lower bounds on the calibration error.

In this paper we consider the selection of covariate adjustment variables for off-policy evaluation (Precup et al., 2000) in single time contextual decision making problems. Specifically, we consider the choice of variables that suffice for estimating the value of a point exposure contextual policy by the method of covariate adjustment, when the available data come from a different policy. We assume a causal graphical model with, possibly, hidden variables in which at least one valid adjustment set is fully observable. The value of a policy, also known as the interventional mean, is defined asthe mean ofan outcome (reward)under the policy. In the statistics literature, a policy is referred to as a dynamic treatment regime (Robins, 1993; Murphy et al., 2001; Robins, 2004; Schulte et al., 2014). A practical application of the methods described in this paper is in the design of planned observational studies. Investigators designing such study might use the existing graphical criteria for identifying the class of candidate valid covariate adjustment sets (Pearl, 2000; Kuroki and Miyakawa, 2003; Shpitser et al., 2010), and then apply the methods described in this paper to select an adjustment set that satisfies one of three optimality criteria that we consider here. Each criterion is defined by selecting the observable adjustment set that yields the non-parametrically adjusted estimator with smallest asymptotic variance among those that control for observable adjustment sets in a given class, specifically the class of (i) all adjustment sets, (ii) all minimal adjustment sets, or (iii) all adjustment sets that have minimum cardinality.

Selection and confounding biases are the two most common impediments to the applicability of causal inference methods in large-scale settings. We generalize the notion of backdoor adjustment to account for both biases and leverage external data that may be available without selection bias (e.g., data from census). We introduce the notion of adjustment pair and present complete graphical conditions for identifying causal effects by adjustment. We further design an algorithm for listing all admissible adjustment pairs in polynomial delay, which is useful for researchers interested in evaluating certain properties of some admissible pairs but not all (common properties include cost, variance, and feasibility to measure). Finally, we describe a statistical estimation procedure that can be performed once a set is known to be admissible, which entails different challenges in terms of finite samples.