Weighted empirical risk minimization is a common approach to prediction under distribution drift. This article studies its out-of-sample prediction error under nonstationarity. We provide a general decomposition of the excess risk into a learning term and an error term associated with distribution drift, and prove oracle inequalities for the learning error under mixing conditions. The learning bound holds uniformly over arbitrary weight classes and accounts for the effective sample size induced by the weight vector, the complexity of the weight and hypothesis classes, and potential data dependence. We illustrate the applicability and sharpness of our results in (auto-) regression problems with linear models, basis approximations, and neural networks, recovering minimax-optimal rates (up to logarithmic factors) when specialized to unweighted and stationary settings.

This paper reports on a novel method for LiDAR odometry estimation, which completely parameterizes the system with dual quaternions. To accomplish this, the features derived from the point cloud, including edges, surfaces, and Stable Triangle Descriptor (STD), along with the optimization problem, are expressed in the dual quaternion set. This approach enables the direct combination of translation and orientation errors via dual quaternion operations, greatly enhancing pose estimation, as demonstrated in comparative experiments against other state-of-the-art methods. Our approach reduced drift error compared to other LiDAR-only-odometry methods, especially in scenarios with sharp curves and aggressive movements with large angular displacement. DualQuat-LOAM is benchmarked against several public datasets. In the KITTI dataset it has a translation and rotation error of 0.79% and 0.0039{\deg}/m, with an average run time of 53 ms.

Positioning is a prominent field of study, notably focusing on Visual Inertial Odometry (VIO) and Simultaneous Localization and Mapping (SLAM) methods. Despite their advancements, these methods often encounter dead-reckoning errors that leads to considerable drift in estimated platform motion especially during long traverses. In such cases, the drift error is not negligible and should be rectified. Our proposed approach minimizes the drift error by correcting the estimated motion generated by any SLAM method at each epoch. Our methodology treats positioning measurements rendered by the SLAM solution as random variables formulated jointly in a multivariate distribution. In this setting, The correction of the drift becomes equivalent to finding the mode of this multivariate distribution which jointly maximizes the likelihood of a set of relevant geo-spatial priors about the platform motion and environment. Our method is integrable into any SLAM/VIO method as an correction module. Our experimental results shows the effectiveness of our approach in minimizing the drift error by 10x in long treverses.

We present a new adaptive algorithm for learning discrete distributions under distribution drift. In this setting, we observe a sequence of independent samples from a discrete distribution that is changing over time, and the goal is to estimate the current distribution. Since we have access to only a single sample for each time step, a good estimation requires a careful choice of the number of past samples to use. To use more samples, we must resort to samples further in the past, and we incur a drift error due to the bias introduced by the change in distribution. On the other hand, if we use a small number of past samples, we incur a large statistical error as the estimation has a high variance. We present a novel adaptive algorithm that can solve this trade-off without any prior knowledge of the drift. Unlike previous adaptive results, our algorithm characterizes the statistical error using data-dependent bounds. This technicality enables us to overcome the limitations of the previous work that require a fixed finite support whose size is known in advance and that cannot change over time. Additionally, we can obtain tighter bounds depending on the complexity of the drifting distribution, and also consider distributions with infinite support.

We develop and analyze a general technique for learning with an unknown distribution drift. Given a sequence of independent observations from the last $T$ steps of a drifting distribution, our algorithm agnostically learns a family of functions with respect to the current distribution at time $T$. Unlike previous work, our technique does not require prior knowledge about the magnitude of the drift. Instead, the algorithm adapts to the sample data. Without explicitly estimating the drift, the algorithm learns a family of functions with almost the same error as a learning algorithm that knows the magnitude of the drift in advance. Furthermore, since our algorithm adapts to the data, it can guarantee a better learning error than an algorithm that relies on loose bounds on the drift. We demonstrate the application of our technique in two fundamental learning scenarios: binary classification and linear regression.

We introduce an adaptive method with formal quality guarantees for weak supervision in a non-stationary setting. Our goal is to infer the unknown labels of a sequence of data by using weak supervision sources that provide independent noisy signals of the correct classification for each data point. This setting includes crowdsourcing and programmatic weak supervision. We focus on the non-stationary case, where the accuracy of the weak supervision sources can drift over time, e.g., because of changes in the underlying data distribution. Due to the drift, older data could provide misleading information to infer the label of the current data point. Previous work relied on a priori assumptions on the magnitude of the drift to decide how much data to use from the past. Comparatively, our algorithm does not require any assumptions on the drift, and it adapts based on the input. In particular, at each step, our algorithm guarantees an estimation of the current accuracies of the weak supervision sources over a window of past observations that minimizes a trade-off between the error due to the variance of the estimation and the error due to the drift. Experiments on synthetic and real-world labelers show that our approach indeed adapts to the drift. Unlike fixed-window-size strategies, it dynamically chooses a window size that allows it to consistently maintain good performance.