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Clans are representations of generalized algebraic theories that contain more information than the finite-limit categories associated to the locally finitely presentable categories of models via Gabriel-Ulmer duality. Extending Gabriel-Ulmer duality to account for this additional information, we present a duality theory between clans and locally finitely presentable categories equipped with a weak factorization system of a certain kind.

Fr\'echet regression has received considerable attention to model metric-space valued responses that are complex and non-Euclidean data, such as probability distributions and vectors on the unit sphere. However, existing Fr\'echet regression literature focuses on the classical setting where the predictor dimension is fixed, and the sample size goes to infinity. This paper proposes sparse Fr\'echet sufficient dimension reduction with graphical structure among high-dimensional Euclidean predictors. In particular, we propose a convex optimization problem that leverages the graphical information among predictors and avoids inverting the high-dimensional covariance matrix. We also provide the Alternating Direction Method of Multipliers (ADMM) algorithm to solve the optimization problem. Theoretically, the proposed method achieves subspace estimation and variable selection consistency under suitable conditions. Extensive simulations and a real data analysis are carried out to illustrate the finite-sample performance of the proposed method.

In the theory of lossy compression, the rate-distortion (R-D) function $R(D)$ describes how much a data source can be compressed (in bit-rate) at any given level of fidelity (distortion). Obtaining $R(D)$ for a given data source establishes the fundamental performance limit for all compression algorithms. We propose a new method to estimate $R(D)$ from the perspective of optimal transport. Unlike the classic Blahut--Arimoto algorithm which fixes the support of the reproduction distribution in advance, our Wasserstein gradient descent algorithm learns the support of the optimal reproduction distribution by moving particles. We prove its local convergence and analyze the sample complexity of our R-D estimator based on a connection to entropic optimal transport. Experimentally, we obtain comparable or tighter bounds than state-of-the-art neural network methods on low-rate sources while requiring considerably less tuning and computation effort. We also highlight a connection to maximum-likelihood deconvolution and introduce a new class of sources that can be used as test cases with known solutions to the R-D problem.

We consider maximizing a monotonic, submodular set function $f: 2^{[n]} \rightarrow [0,1]$ under stochastic bandit feedback. Specifically, $f$ is unknown to the learner but at each time $t=1,\dots,T$ the learner chooses a set $S_t \subset [n]$ with $|S_t| \leq k$ and receives reward $f(S_t) + \eta_t$ where $\eta_t$ is mean-zero sub-Gaussian noise. The objective is to minimize the learner's regret over $T$ times with respect to ($1-e^{-1}$)-approximation of maximum $f(S_*)$ with $|S_*| = k$, obtained through greedy maximization of $f$. To date, the best regret bound in the literature scales as $k n^{1/3} T^{2/3}$. And by trivially treating every set as a unique arm one deduces that $\sqrt{ {n \choose k} T }$ is also achievable. In this work, we establish the first minimax lower bound for this setting that scales like $\mathcal{O}(\min_{i \le k}(in^{1/3}T^{2/3} + \sqrt{n^{k-i}T}))$. Moreover, we propose an algorithm that is capable of matching the lower bound regret.

We present an approach to the verification of systems for whose description some elements - constants or functions - are underspecified and can be regarded as parameters, and, in particular, describe a method for automatically generating constraints on such parameters under which certain safety conditions are guaranteed to hold. We present an implementation and illustrate its use on several examples.

Sharpness-Aware Minimization (SAM) is an optimizer that takes a descent step based on the gradient at a perturbation $y_t = x_t + \rho \frac{\nabla f(x_t)}{\lVert \nabla f(x_t) \rVert}$ of the current point $x_t$. Existing studies prove convergence of SAM for smooth functions, but they do so by assuming decaying perturbation size $\rho$ and/or no gradient normalization in $y_t$, which is detached from practice. To address this gap, we study deterministic/stochastic versions of SAM with practical configurations (i.e., constant $\rho$ and gradient normalization in $y_t$) and explore their convergence properties on smooth functions with (non)convexity assumptions. Perhaps surprisingly, in many scenarios, we find out that SAM has limited capability to converge to global minima or stationary points. For smooth strongly convex functions, we show that while deterministic SAM enjoys tight global convergence rates of $\tilde \Theta(\frac{1}{T^2})$, the convergence bound of stochastic SAM suffers an inevitable additive term $O(\rho^2)$, indicating convergence only up to neighborhoods of optima. In fact, such $O(\rho^2)$ factors arise for stochastic SAM in all the settings we consider, and also for deterministic SAM in nonconvex cases; importantly, we prove by examples that such terms are unavoidable. Our results highlight vastly different characteristics of SAM with vs. without decaying perturbation size or gradient normalization, and suggest that the intuitions gained from one version may not apply to the other.

We propose a novel and practical privacy notion called $f$-Membership Inference Privacy ($f$-MIP), which explicitly considers the capabilities of realistic adversaries under the membership inference attack threat model. Consequently, $f$-MIP offers interpretable privacy guarantees and improved utility (e.g., better classification accuracy). In particular, we derive a parametric family of $f$-MIP guarantees that we refer to as $\mu$-Gaussian Membership Inference Privacy ($\mu$-GMIP) by theoretically analyzing likelihood ratio-based membership inference attacks on stochastic gradient descent (SGD). Our analysis highlights that models trained with standard SGD already offer an elementary level of MIP. Additionally, we show how $f$-MIP can be amplified by adding noise to gradient updates. Our analysis further yields an analytical membership inference attack that offers two distinct advantages over previous approaches. First, unlike existing state-of-the-art attacks that require training hundreds of shadow models, our attack does not require any shadow model. Second, our analytical attack enables straightforward auditing of our privacy notion $f$-MIP. Finally, we quantify how various hyperparameters (e.g., batch size, number of model parameters) and specific data characteristics determine an attacker's ability to accurately infer a point's membership in the training set. We demonstrate the effectiveness of our method on models trained on vision and tabular datasets.

The persistent homology transform (PHT) represents a shape with a multiset of persistence diagrams parameterized by the sphere of directions in the ambient space. In this work, we describe a finite set of diagrams that discretize the PHT such that it faithfully represents the underlying shape. We provide a discretization that is exponential in the dimension of the shape. Moreover, we show that this discretization is stable with respect to various perturbations. Furthermore, we provide an algorithm for computing the discretization. Our approach relies only on knowing the heights and dimensions of topological events, which means that it can be adapted to provide discretizations of other dimension-returning topological transforms, including the Betti curve transform. With mild alterations, we also adapt our methods to faithfully discretize the Euler Characteristic curve transform.

Object detection typically assumes that training and test data are drawn from an identical distribution, which, however, does not always hold in practice. Such a distribution mismatch will lead to a significant performance drop. In this work, we aim to improve the cross-domain robustness of object detection. We tackle the domain shift on two levels: 1) the image-level shift, such as image style, illumination, etc, and 2) the instance-level shift, such as object appearance, size, etc. We build our approach based on the recent state-of-the-art Faster R-CNN model, and design two domain adaptation components, on image level and instance level, to reduce the domain discrepancy. The two domain adaptation components are based on H-divergence theory, and are implemented by learning a domain classifier in adversarial training manner. The domain classifiers on different levels are further reinforced with a consistency regularization to learn a domain-invariant region proposal network (RPN) in the Faster R-CNN model. We evaluate our newly proposed approach using multiple datasets including Cityscapes, KITTI, SIM10K, etc. The results demonstrate the effectiveness of our proposed approach for robust object detection in various domain shift scenarios.

Dynamic programming (DP) solves a variety of structured combinatorial problems by iteratively breaking them down into smaller subproblems. In spite of their versatility, DP algorithms are usually non-differentiable, which hampers their use as a layer in neural networks trained by backpropagation. To address this issue, we propose to smooth the max operator in the dynamic programming recursion, using a strongly convex regularizer. This allows to relax both the optimal value and solution of the original combinatorial problem, and turns a broad class of DP algorithms into differentiable operators. Theoretically, we provide a new probabilistic perspective on backpropagating through these DP operators, and relate them to inference in graphical models. We derive two particular instantiations of our framework, a smoothed Viterbi algorithm for sequence prediction and a smoothed DTW algorithm for time-series alignment. We showcase these instantiations on two structured prediction tasks and on structured and sparse attention for neural machine translation.