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This paper provides the first sample complexity lower bounds for the estimation of simple diffusion models, including the Bass model (used in modeling consumer adoption) and the SIR model (used in modeling epidemics). We show that one cannot hope to learn such models until quite late in the diffusion. Specifically, we show that the time required to collect a number of observations that exceeds our sample complexity lower bounds is large. For Bass models with low innovation rates, our results imply that one cannot hope to predict the eventual number of adopting customers until one is at least two-thirds of the way to the time at which the rate of new adopters is at its peak. In a similar vein, our results imply that in the case of an SIR model, one cannot hope to predict the eventual number of infections until one is approximately two-thirds of the way to the time at which the infection rate has peaked. This lower bound in estimation further translates into a lower bound in regret for decision-making in epidemic interventions. Our results formalize the challenge of accurate forecasting and highlight the importance of incorporating additional data sources. To this end, we analyze the benefit of a seroprevalence study in an epidemic, where we characterize the size of the study needed to improve SIR model estimation. Extensive empirical analyses on product adoption and epidemic data support our theoretical findings.

This study develops an asymptotic theory for estimating the time-varying characteristics of locally stationary functional time series (LSFTS). We investigate a kernel-based method to estimate the time-varying covariance operator and the time-varying mean function of an LSFTS. In particular, we derive the convergence rate of the kernel estimator of the covariance operator and associated eigenvalue and eigenfunctions and establish a central limit theorem for the kernel-based locally weighted sample mean. As applications of our results, we discuss methods for testing the equality of time-varying mean functions in two functional samples.

Probabilistic logical rule learning has shown great strength in logical rule mining and knowledge graph completion. It learns logical rules to predict missing edges by reasoning on existing edges in the knowledge graph. However, previous efforts have largely been limited to only modeling chain-like Horn clauses such as $R_1(x,z)\land R_2(z,y)\Rightarrow H(x,y)$. This formulation overlooks additional contextual information from neighboring sub-graphs of entity variables $x$, $y$ and $z$. Intuitively, there is a large gap here, as local sub-graphs have been found to provide important information for knowledge graph completion. Inspired by these observations, we propose Logical Entity RePresentation (LERP) to encode contextual information of entities in the knowledge graph. A LERP is designed as a vector of probabilistic logical functions on the entity's neighboring sub-graph. It is an interpretable representation while allowing for differentiable optimization. We can then incorporate LERP into probabilistic logical rule learning to learn more expressive rules. Empirical results demonstrate that with LERP, our model outperforms other rule learning methods in knowledge graph completion and is comparable or even superior to state-of-the-art black-box methods. Moreover, we find that our model can discover a more expressive family of logical rules. LERP can also be further combined with embedding learning methods like TransE to make it more interpretable.

Training a robust policy is critical for policy deployment in real-world systems or dealing with unknown dynamics mismatch in different dynamic systems. Domain Randomization~(DR) is a simple and elegant approach that trains a conservative policy to counter different dynamic systems without expert knowledge about the target system parameters. However, existing works reveal that the policy trained through DR tends to be over-conservative and performs poorly in target domains. Our key insight is that dynamic systems with different parameters provide different levels of difficulty for the policy, and the difficulty of behaving well in a system is constantly changing due to the evolution of the policy. If we can actively sample the systems with proper difficulty for the policy on the fly, it will stabilize the training process and prevent the policy from becoming over-conservative or over-optimistic. To operationalize this idea, we introduce Active Dynamics Preference~(ADP), which quantifies the informativeness and density of sampled system parameters. ADP actively selects system parameters with high informativeness and low density. We validate our approach in four robotic locomotion tasks with various discrepancies between the training and testing environments. Extensive results demonstrate that our approach has superior robustness for system inconsistency compared to several baselines.

Model selection is a strategy aimed at creating accurate and robust models. A key challenge in designing these algorithms is identifying the optimal model for classifying any particular input sample. This paper addresses this challenge and proposes a novel framework for differentiable model selection integrating machine learning and combinatorial optimization. The framework is tailored for ensemble learning, a strategy that combines the outputs of individually pre-trained models, and learns to select appropriate ensemble members for a particular input sample by transforming the ensemble learning task into a differentiable selection program trained end-to-end within the ensemble learning model. Tested on various tasks, the proposed framework demonstrates its versatility and effectiveness, outperforming conventional and advanced consensus rules across a variety of settings and learning tasks.

Off-policy evaluation (OPE) is the problem of estimating the value of a target policy using historical data collected under a different logging policy. OPE methods typically assume overlap between the target and logging policy, enabling solutions based on importance weighting and/or imputation. In this work, we approach OPE without assuming either overlap or a well-specified model by considering a strategy based on partial identification under non-parametric assumptions on the conditional mean function, focusing especially on Lipschitz smoothness. Under such smoothness assumptions, we formulate a pair of linear programs whose optimal values upper and lower bound the contributions of the no-overlap region to the off-policy value. We show that these linear programs have a concise closed form solution that can be computed efficiently and that their solutions converge, under the Lipschitz assumption, to the sharp partial identification bounds on the off-policy value. Furthermore, we show that the rate of convergence is minimax optimal, up to log factors. We deploy our methods on two semi-synthetic examples, and obtain informative and valid bounds that are tighter than those possible without smoothness assumptions.

For a given elliptic curve $E$ over a finite local ring, we denote by $E^{\infty}$ its subgroup at infinity. Every point $P \in E^{\infty}$ can be described solely in terms of its $x$-coordinate $P_x$, which can be therefore used to parameterize all its multiples $nP$. We refer to the coefficient of $(P_x)^i$ in the parameterization of $(nP)_x$ as the $i$-th multiplication polynomial. We show that this coefficient is a degree-$i$ rational polynomial without a constant term in $n$. We also prove that no primes greater than $i$ may appear in the denominators of its terms. As a consequence, for every finite field $\mathbb{F}_q$ and any $k\in\mathbb{N}^*$, we prescribe the group structure of a generic elliptic curve defined over $\mathbb{F}_q[X]/(X^k)$, and we show that their ECDLP on $E^{\infty}$ may be efficiently solved.

Estimating the rigid transformation between two LiDAR scans through putative 3D correspondences is a typical point cloud registration paradigm. Current 3D feature matching approaches commonly lead to numerous outlier correspondences, making outlier-robust registration techniques indispensable. Many recent studies have adopted the branch and bound (BnB) optimization framework to solve the correspondence-based point cloud registration problem globally and deterministically. Nonetheless, BnB-based methods are time-consuming to search the entire 6-dimensional parameter space, since their computational complexity is exponential to the dimension of the solution domain. In order to enhance algorithm efficiency, existing works attempt to decouple the 6 degrees of freedom (DOF) original problem into two 3-DOF sub-problems, thereby reducing the dimension of the parameter space. In contrast, our proposed approach introduces a novel pose decoupling strategy based on residual projections, effectively decomposing the raw problem into three 2-DOF rotation search sub-problems. Subsequently, we employ a novel BnB-based search method to solve these sub-problems, achieving efficient and deterministic registration. Furthermore, our method can be adapted to address the challenging problem of simultaneous pose and correspondence registration (SPCR). Through extensive experiments conducted on synthetic and real-world datasets, we demonstrate that our proposed method outperforms state-of-the-art methods in terms of efficiency, while simultaneously ensuring robustness.

We give query complexity lower bounds for convex optimization and the related feasibility problem. We show that quadratic memory is necessary to achieve the optimal oracle complexity for first-order convex optimization. In particular, this shows that center-of-mass cutting-planes algorithms in dimension $d$ which use $\tilde O(d^2)$ memory and $\tilde O(d)$ queries are Pareto-optimal for both convex optimization and the feasibility problem, up to logarithmic factors. Precisely, we prove that to minimize $1$-Lipschitz convex functions over the unit ball to $1/d^4$ accuracy, any deterministic first-order algorithms using at most $d^{2-\delta}$ bits of memory must make $\tilde\Omega(d^{1+\delta/3})$ queries, for any $\delta\in[0,1]$. For the feasibility problem, in which an algorithm only has access to a separation oracle, we show a stronger trade-off: for at most $d^{2-\delta}$ memory, the number of queries required is $\tilde\Omega(d^{1+\delta})$. This resolves a COLT 2019 open problem of Woodworth and Srebro.

With the explosive growth of information technology, multi-view graph data have become increasingly prevalent and valuable. Most existing multi-view clustering techniques either focus on the scenario of multiple graphs or multi-view attributes. In this paper, we propose a generic framework to cluster multi-view attributed graph data. Specifically, inspired by the success of contrastive learning, we propose multi-view contrastive graph clustering (MCGC) method to learn a consensus graph since the original graph could be noisy or incomplete and is not directly applicable. Our method composes of two key steps: we first filter out the undesirable high-frequency noise while preserving the graph geometric features via graph filtering and obtain a smooth representation of nodes; we then learn a consensus graph regularized by graph contrastive loss. Results on several benchmark datasets show the superiority of our method with respect to state-of-the-art approaches. In particular, our simple approach outperforms existing deep learning-based methods.