Yufeng Liu

Yufeng Liu, Hang Yu, Juntu Zhao et al.

Action chunking enables Vision Language Action (VLA) models to run in real time, but naive chunked execution often exhibits discontinuities at chunk boundaries. Real-Time Chunking (RTC) alleviates this issue but is external to the policy, leading to spurious multimodal switching and trajectories that are not intrinsically smooth. We propose Legato, a training-time continuation method for action-chunked flow-based VLA policies. Specifically, Legato initializes denoising from a schedule-shaped mixture of known actions and noise, exposing the model to partial action information. Moreover, Legato reshapes the learned flow dynamics to ensure that the denoising process remains consistent between training and inference under per-step guidance. Legato further uses randomized schedule condition during training to support varying inference delays and achieve controllable smoothness. Empirically, Legato produces smoother trajectories and reduces spurious multimodal switching during execution, leading to less hesitation and shorter task completion time. Extensive real-world experiments show that Legato consistently outperforms RTC across five manipulation tasks, achieving approximately 10% improvements in both trajectory smoothness and task completion time.

Hang Yu, Juntu Zhao, Yufeng Liu et al.

Vision-Language-Action (VLA) models align vision and language with embodied control, but their object referring ability remains limited when relying solely on text prompt, especially in cluttered or out-of-distribution (OOD) scenes. In this study, we introduce the Point-VLA, a plug-and-play policy that augments language instructions with explicit visual cues (e.g., bounding boxes) to resolve referential ambiguity and enable precise object-level grounding. To efficiently scale visually grounded datasets, we further develop an automatic data annotation pipeline requiring minimal human effort. We evaluate Point-VLA on diverse real-world referring tasks and observe consistently stronger performance than text-only instruction VLAs, particularly in cluttered or unseen-object scenarios, with robust generalization. These results demonstrate that Point-VLA effectively resolves object referring ambiguity through pixel-level visual grounding, achieving more generalizable embodied control.

Tianyu Liu, Simeng Han, Xiao Luo et al.

Large language models (LLMs) and large multimodal models (LMMs) promise to accelerate biomedical discovery, yet their reliability remains unclear. We introduce ARIEL (AI Research Assistant for Expert-in-the-Loop Learning), an open-source evaluation and optimization framework that pairs a curated multimodal biomedical corpus with expert-vetted tasks to probe two capabilities: full-length article summarization and fine-grained figure interpretation. Using uniform protocols and blinded PhD-level evaluation, we find that state-of-the-art models generate fluent but incomplete summaries, whereas LMMs struggle with detailed visual reasoning. We later observe that prompt engineering and lightweight fine-tuning substantially improve textual coverage, and a compute-scaled inference strategy enhances visual question answering. We build an ARIEL agent that integrates textual and visual cues, and we show it can propose testable mechanistic hypotheses. ARIEL delineates current strengths and limitations of foundation models, and provides a reproducible platform for advancing trustworthy AI in biomedicine.

Yufeng Liu

Nighttime unmanned aerial vehicle (UAV) tracking has been facilitated with indispensable plug-and-play low-light enhancers. However, the introduction of low-light enhancers increases the extra computational burden for the UAV, significantly hindering the development of real-time UAV applications. Meanwhile, these state-of-the-art (SOTA) enhancers lack tight coupling with the advanced daytime UAV tracking approach. To solve the above issues, this work proposes a novel mutual-learning knowledge distillation framework for nighttime UAV tracking, i.e., MLKD. This framework is constructed to learn a compact and fast nighttime tracker via knowledge transferring from the teacher and knowledge sharing among various students. Specifically, an advanced teacher based on a SOTA enhancer and a superior tracking backbone is adopted for guiding the student based only on the tight coupling-aware tracking backbone to directly extract nighttime object features. To address the biased learning of a single student, diverse lightweight students with different distillation methods are constructed to focus on various aspects of the teacher's knowledge. Moreover, an innovative mutual-learning room is designed to elect the superior student candidate to assist the remaining students frame-by-frame in the training phase. Furthermore, the final best student, i.e., MLKD-Track, is selected through the testing dataset. Extensive experiments demonstrate the effectiveness and superiority of MLKD and MLKD-Track. The practicality of the MLKD-Track is verified in real-world tests with different challenging situations. The code is available at https://github.com/lyfeng001/MLKD.

Xingyu Chen, Yufeng Liu, Yajiao Dong et al.

In this work, we propose a framework for single-view hand mesh reconstruction, which can simultaneously achieve high reconstruction accuracy, fast inference speed, and temporal coherence. Specifically, for 2D encoding, we propose lightweight yet effective stacked structures. Regarding 3D decoding, we provide an efficient graph operator, namely depth-separable spiral convolution. Moreover, we present a novel feature lifting module for bridging the gap between 2D and 3D representations. This module begins with a map-based position regression (MapReg) block to integrate the merits of both heatmap encoding and position regression paradigms for improved 2D accuracy and temporal coherence. Furthermore, MapReg is followed by pose pooling and pose-to-vertex lifting approaches, which transform 2D pose encodings to semantic features of 3D vertices. Overall, our hand reconstruction framework, called MobRecon, comprises affordable computational costs and miniature model size, which reaches a high inference speed of 83FPS on Apple A14 CPU. Extensive experiments on popular datasets such as FreiHAND, RHD, and HO3Dv2 demonstrate that our MobRecon achieves superior performance on reconstruction accuracy and temporal coherence. Our code is publicly available at https://github.com/SeanChenxy/HandMesh.

Xingyu Chen, Yufeng Liu, Chongyang Ma et al.

Recent years have witnessed significant progress in 3D hand mesh recovery. Nevertheless, because of the intrinsic 2D-to-3D ambiguity, recovering camera-space 3D information from a single RGB image remains challenging. To tackle this problem, we divide camera-space mesh recovery into two sub-tasks, i.e., root-relative mesh recovery and root recovery. First, joint landmarks and silhouette are extracted from a single input image to provide 2D cues for the 3D tasks. In the root-relative mesh recovery task, we exploit semantic relations among joints to generate a 3D mesh from the extracted 2D cues. Such generated 3D mesh coordinates are expressed relative to a root position, i.e., wrist of the hand. In the root recovery task, the root position is registered to the camera space by aligning the generated 3D mesh back to 2D cues, thereby completing cameraspace 3D mesh recovery. Our pipeline is novel in that (1) it explicitly makes use of known semantic relations among joints and (2) it exploits 1D projections of the silhouette and mesh to achieve robust registration. Extensive experiments on popular datasets such as FreiHAND, RHD, and Human3.6M demonstrate that our approach achieves stateof-the-art performance on both root-relative mesh recovery and root recovery. Our code is publicly available at https://github.com/SeanChenxy/HandMesh.

Daiqi Gao, Yufeng Liu, Donglin Zeng

Dynamic treatment regimes or policies are a sequence of decision functions over multiple stages that are tailored to individual features. One important class of treatment policies in practice, namely multi-stage stationary treatment policies, prescribes treatment assignment probabilities using the same decision function across stages, where the decision is based on the same set of features consisting of time-evolving variables (e.g., routinely collected disease biomarkers). Although there has been extensive literature on constructing valid inference for the value function associated with dynamic treatment policies, little work has focused on the policies themselves, especially in the presence of high-dimensional feature variables. We aim to fill the gap in this work. Specifically, we first estimate the multi-stage stationary treatment policy using an augmented inverse probability weighted estimator for the value function to increase asymptotic efficiency, and further apply a penalty to select important feature variables. We then construct one-step improvements of the policy parameter estimators for valid inference. Theoretically, we show that the improved estimators are asymptotically normal, even if nuisance parameters are estimated at a slow convergence rate and the dimension of the feature variables increases with the sample size. Our numerical studies demonstrate that the proposed method estimates a sparse policy with a near-optimal value function and conducts valid inference for the policy parameters.

Seong Jin Lee, Will Wei Sun, Yufeng Liu

Reinforcement learning from human feedback (RLHF) has become a cornerstone for aligning large language models with human preferences. However, the heterogeneity of human feedback, driven by diverse individual contexts and preferences, poses significant challenges for reward learning. To address this, we propose a Low-rank Contextual RLHF (LoCo-RLHF) framework that integrates contextual information to better model heterogeneous feedback while maintaining computational efficiency. Our approach builds on a contextual preference model, leveraging the intrinsic low-rank structure of the interaction between user contexts and query-answer pairs to mitigate the high dimensionality of feature representations. Furthermore, we address the challenge of distributional shifts in feedback through our Pessimism in Reduced Subspace (PRS) policy, inspired by pessimistic offline reinforcement learning techniques. We theoretically demonstrate that our policy achieves a tighter sub-optimality gap compared to existing methods. Extensive experiments validate the effectiveness of LoCo-RLHF, showcasing its superior performance in personalized RLHF settings and its robustness to distribution shifts.

Seong Jin Lee, Will Wei Sun, Yufeng Liu

As e-commerce expands, delivering real-time personalized recommendations from vast catalogs poses a critical challenge for retail platforms. Maximizing revenue requires careful consideration of both individual customer characteristics and available item features to optimize assortments over time. In this paper, we consider the dynamic assortment problem with dual contexts -- user and item features. In high-dimensional scenarios, the quadratic growth of dimensions complicates computation and estimation. To tackle this challenge, we introduce a new low-rank dynamic assortment model to transform this problem into a manageable scale. Then we propose an efficient algorithm that estimates the intrinsic subspaces and utilizes the upper confidence bound approach to address the exploration-exploitation trade-off in online decision making. Theoretically, we establish a regret bound of $\tilde{O}((d_1+d_2)r\sqrt{T})$, where $d_1, d_2$ represent the dimensions of the user and item features respectively, $r$ is the rank of the parameter matrix, and $T$ denotes the time horizon. This bound represents a substantial improvement over prior literature, made possible by leveraging the low-rank structure. Extensive simulations and an application to the Expedia hotel recommendation dataset further demonstrate the advantages of our proposed method.

Juntu Zhao, Wenbo Lu, Di Zhang et al.

Imitation-learning-based visuomotor policies have been widely used in robot manipulation, where both visual observations and proprioceptive states are typically adopted together for precise control. However, in this study, we find that this common practice makes the policy overly reliant on the proprioceptive state input, which causes overfitting to the training trajectories and results in poor spatial generalization. On the contrary, we propose the State-free Policy, removing the proprioceptive state input and predicting actions only conditioned on visual observations. The State-free Policy is built in the relative end-effector action space, and should ensure the full task-relevant visual observations, here provided by dual wide-angle wrist cameras. Empirical results demonstrate that the State-free policy achieves significantly stronger spatial generalization than the state-based policy: in real-world tasks such as pick-and-place, challenging shirt-folding, and complex whole-body manipulation, spanning multiple robot embodiments, the average success rate improves from 0% to 85% in height generalization and from 6% to 64% in horizontal generalization. Furthermore, they also show advantages in data efficiency and cross-embodiment adaptation, enhancing their practicality for real-world deployment. Discover more by visiting: https://statefreepolicy.github.io.

Dhruv Patel, Hui Shen, Shankar Bhamidi et al.

In the context of unsupervised learning, Lloyd's algorithm is one of the most widely used clustering algorithms. It has inspired a plethora of work investigating the correctness of the algorithm under various settings with ground truth clusters. In particular, in 2016, Lu and Zhou have shown that the mis-clustering rate of Lloyd's algorithm on $n$ independent samples from a sub-Gaussian mixture is exponentially bounded after $O(\log(n))$ iterations, assuming proper initialization of the algorithm. However, in many applications, the true samples are unobserved and need to be learned from the data via pre-processing pipelines such as spectral methods on appropriate data matrices. We show that the mis-clustering rate of Lloyd's algorithm on perturbed samples from a sub-Gaussian mixture is also exponentially bounded after $O(\log(n))$ iterations under the assumptions of proper initialization and that the perturbation is small relative to the sub-Gaussian noise. In canonical settings with ground truth clusters, we derive bounds for algorithms such as $k$-means$++$ to find good initializations and thus leading to the correctness of clustering via the main result. We show the implications of the results for pipelines measuring the statistical significance of derived clusters from data such as SigClust. We use these general results to derive implications in providing theoretical guarantees on the misclustering rate for Lloyd's algorithm in a host of applications, including high-dimensional time series, multi-dimensional scaling, and community detection for sparse networks via spectral clustering.

Weibin Mo, Zhengling Qi, Yufeng Liu

We thank the opportunity offered by editors for this discussion and the discussants for their insightful comments and thoughtful contributions. We also want to congratulate Kallus (2020) for his inspiring work in improving the efficiency of policy learning by retargeting. Motivated from the discussion in Dukes and Vansteelandt (2020), we first point out interesting connections and distinctions between our work and Kallus (2020) in Section 1. In particular, the assumptions and sources of variation for consideration in these two papers lead to different research problems with different scopes and focuses. In Section 2, following the discussions in Li et al. (2020); Liang and Zhao (2020), we also consider the efficient policy evaluation problem when we have some data from the testing distribution available at the training stage. We show that under the assumption that the sample sizes from training and testing are growing in the same order, efficient value function estimates can deliver competitive performance. We further show some connections of these estimates with existing literature. However, when the growth of testing sample size available for training is in a slower order, efficient value function estimates may not perform well anymore. In contrast, the requirement of the testing sample size for DRITR is not as strong as that of efficient policy evaluation using the combined data. Finally, we highlight the general applicability and usefulness of DRITR in Section 3.

Weibin Mo, Yufeng Liu

Recent development in data-driven decision science has seen great advances in individualized decision making. Given data with individual covariates, treatment assignments and outcomes, researchers can search for the optimal individualized treatment rule (ITR) that maximizes the expected outcome. Existing methods typically require initial estimation of some nuisance models. The double robustness property that can protect from misspecification of either the treatment-free effect or the propensity score has been widely advocated. However, when model misspecification exists, a doubly robust estimate can be consistent but may suffer from downgraded efficiency. Other than potential misspecified nuisance models, most existing methods do not account for the potential problem when the variance of outcome is heterogeneous among covariates and treatment. We observe that such heteroscedasticity can greatly affect the estimation efficiency of the optimal ITR. In this paper, we demonstrate that the consequences of misspecified treatment-free effect and heteroscedasticity can be unified as a covariate-treatment dependent variance of residuals. To improve efficiency of the estimated ITR, we propose an Efficient Learning (E-Learning) framework for finding an optimal ITR in the multi-armed treatment setting. We show that the proposed E-Learning is optimal among a regular class of semiparametric estimates that can allow treatment-free effect misspecification. In our simulation study, E-Learning demonstrates its effectiveness if one of or both misspecified treatment-free effect and heteroscedasticity exist. Our analysis of a Type 2 Diabetes Mellitus (T2DM) observational study also suggests the improved efficiency of E-Learning.

Yumeng Zhang, Li Chen, Yufeng Liu et al.

Deep learning methods have achieved excellent performance in pose estimation, but the lack of robustness causes the keypoints to change drastically between similar images. In view of this problem, a stable heatmap regression method is proposed to alleviate network vulnerability to small perturbations. We utilize the correlation between different rows and columns in a heatmap to alleviate the multi-peaks problem, and design a highly differentiated heatmap regression to make a keypoint discriminative from surrounding points. A maximum stability training loss is used to simplify the optimization difficulty when minimizing the prediction gap of two similar images. The proposed method achieves a significant advance in robustness over state-of-the-art approaches on two benchmark datasets and maintains high performance.

Junguang Jiang, Yifei Ji, Ximei Wang et al.

Domain adaptation (DA) aims at transferring knowledge from a labeled source domain to an unlabeled target domain. Though many DA theories and algorithms have been proposed, most of them are tailored into classification settings and may fail in regression tasks, especially in the practical keypoint detection task. To tackle this difficult but significant task, we present a method of regressive domain adaptation (RegDA) for unsupervised keypoint detection. Inspired by the latest theoretical work, we first utilize an adversarial regressor to maximize the disparity on the target domain and train a feature generator to minimize this disparity. However, due to the high dimension of the output space, this regressor fails to detect samples that deviate from the support of the source. To overcome this problem, we propose two important ideas. First, based on our observation that the probability density of the output space is sparse, we introduce a spatial probability distribution to describe this sparsity and then use it to guide the learning of the adversarial regressor. Second, to alleviate the optimization difficulty in the high-dimensional space, we innovatively convert the minimax game in the adversarial training to the minimization of two opposite goals. Extensive experiments show that our method brings large improvement by 8% to 11% in terms of PCK on different datasets.

Weibin Mo, Zhengling Qi, Yufeng Liu

Recent development in the data-driven decision science has seen great advances in individualized decision making. Given data with individual covariates, treatment assignments and outcomes, policy makers best individualized treatment rule (ITR) that maximizes the expected outcome, known as the value function. Many existing methods assume that the training and testing distributions are the same. However, the estimated optimal ITR may have poor generalizability when the training and testing distributions are not identical. In this paper, we consider the problem of finding an optimal ITR from a restricted ITR class where there is some unknown covariate changes between the training and testing distributions. We propose a novel distributionally robust ITR (DR-ITR) framework that maximizes the worst-case value function across the values under a set of underlying distributions that are "close" to the training distribution. The resulting DR-ITR can guarantee the performance among all such distributions reasonably well. We further propose a calibrating procedure that tunes the DR-ITR adaptively to a small amount of calibration data from a target population. In this way, the calibrated DR-ITR can be shown to enjoy better generalizability than the standard ITR based on our numerical studies.

Naim U. Rashid, Daniel J. Luckett, Jingxiang Chen et al.

The complexity of human cancer often results in significant heterogeneity in response to treatment. Precision medicine offers potential to improve patient outcomes by leveraging this heterogeneity. Individualized treatment rules (ITRs) formalize precision medicine as maps from the patient covariate space into the space of allowable treatments. The optimal ITR is that which maximizes the mean of a clinical outcome in a population of interest. Patient-derived xenograft (PDX) studies permit the evaluation of multiple treatments within a single tumor and thus are ideally suited for estimating optimal ITRs. PDX data are characterized by correlated outcomes, a high-dimensional feature space, and a large number of treatments. Existing methods for estimating optimal ITRs do not take advantage of the unique structure of PDX data or handle the associated challenges well. In this paper, we explore machine learning methods for estimating optimal ITRs from PDX data. We analyze data from a large PDX study to identify biomarkers that are informative for developing personalized treatment recommendations in multiple cancers. We estimate optimal ITRs using regression-based approaches such as Q-learning and direct search methods such as outcome weighted learning. Finally, we implement a superlearner approach to combine a set of estimated ITRs and show that the resulting ITR performs better than any of the input ITRs, mitigating uncertainty regarding user choice of any particular ITR estimation methodology. Our results indicate that PDX data are a valuable resource for developing individualized treatment strategies in oncology.

Zhengling Qi, Ying Cui, Yufeng Liu et al.

This paper has two main goals: (a) establish several statistical properties---consistency, asymptotic distributions, and convergence rates---of stationary solutions and values of a class of coupled nonconvex and nonsmoothempirical risk minimization problems, and (b) validate these properties by a noisy amplitude-based phase retrieval problem, the latter being of much topical interest.Derived from available data via sampling, these empirical risk minimization problems are the computational workhorse of a population risk model which involves the minimization of an expected value of a random functional. When these minimization problems are nonconvex, the computation of their globally optimal solutions is elusive. Together with the fact that the expectation operator cannot be evaluated for general probability distributions, it becomes necessary to justify whether the stationary solutions of the empirical problems are practical approximations of the stationary solution of the population problem. When these two features, general distribution and nonconvexity, are coupled with nondifferentiability that often renders the problems "non-Clarke regular", the task of the justification becomes challenging. Our work aims to address such a challenge within an algorithm-free setting. The resulting analysis is therefore different from the much of the analysis in the recent literature that is based on local search algorithms. Furthermore, supplementing the classical minimizer-centric analysis, our results offer a first step to close the gap between computational optimization and asymptotic analysis of coupled nonconvex nonsmooth statistical estimation problems, expanding the former with statistical properties of the practically obtained solution and providing the latter with a more practical focus pertaining to computational tractability.

Yumeng Zhang, Li Chen, Yufeng Liu et al.

Insufficient labeled training datasets is one of the bottlenecks of 3D hand pose estimation from monocular RGB images. Synthetic datasets have a large number of images with precise annotations, but the obvious difference with real-world datasets impacts the generalization. Little work has been done to bridge the gap between two domains over their wide difference. In this paper, we propose a domain adaptation method called Adaptive Wasserstein Hourglass (AW Hourglass) for weakly-supervised 3D hand pose estimation, which aims to distinguish the difference and explore the common characteristics (e.g. hand structure) of synthetic and real-world datasets. Learning the common characteristics helps the network focus on pose-related information. The similarity of the characteristics makes it easier to enforce domain-invariant constraints. During training, based on the relation between these common characteristics and 3D pose learned from fully-annotated synthetic datasets, it is beneficial for the network to restore the 3D pose of weakly labeled real-world datasets with the aid of 2D annotations and depth images. While in testing, the network predicts the 3D pose with the input of RGB.

Zhengling Qi, Ying Cui, Yufeng Liu et al.

Recent exploration of optimal individualized decision rules (IDRs) for patients in precision medicine has attracted a lot of attention due to the heterogeneous responses of patients to different treatments. In the existing literature of precision medicine, an optimal IDR is defined as a decision function mapping from the patients' covariate space into the treatment space that maximizes the expected outcome of each individual. Motivated by the concept of Optimized Certainty Equivalent (OCE) introduced originally in \cite{ben1986expected} that includes the popular conditional-value-of risk (CVaR) \cite{rockafellar2000optimization}, we propose a decision-rule based optimized covariates dependent equivalent (CDE) for individualized decision making problems. Our proposed IDR-CDE broadens the existing expected-mean outcome framework in precision medicine and enriches the previous concept of the OCE. Numerical experiments demonstrate that our overall approach outperforms existing methods in estimating optimal IDRs under heavy-tail distributions of the data.

Will Wei Sun, Guang Cheng, Yufeng Liu

Stability is an important aspect of a classification procedure because unstable predictions can potentially reduce users' trust in a classification system and also harm the reproducibility of scientific conclusions. The major goal of our work is to introduce a novel concept of classification instability, i.e., decision boundary instability (DBI), and incorporate it with the generalization error (GE) as a standard for selecting the most accurate and stable classifier. Specifically, we implement a two-stage algorithm: (i) initially select a subset of classifiers whose estimated GEs are not significantly different from the minimal estimated GE among all the candidate classifiers; (ii) the optimal classifier is chosen as the one achieving the minimal DBI among the subset selected in stage (i). This general selection principle applies to both linear and nonlinear classifiers. Large-margin classifiers are used as a prototypical example to illustrate the above idea. Our selection method is shown to be consistent in the sense that the optimal classifier simultaneously achieves the minimal GE and the minimal DBI. Various simulations and real examples further demonstrate the advantage of our method over several alternative approaches.

Botao Hao, Will Wei Sun, Yufeng Liu et al.

We consider joint estimation of multiple graphical models arising from heterogeneous and high-dimensional observations. Unlike most previous approaches which assume that the cluster structure is given in advance, an appealing feature of our method is to learn cluster structure while estimating heterogeneous graphical models. This is achieved via a high dimensional version of Expectation Conditional Maximization (ECM) algorithm (Meng and Rubin, 1993). A joint graphical lasso penalty is imposed on the conditional maximization step to extract both homogeneity and heterogeneity components across all clusters. Our algorithm is computationally efficient due to fast sparse learning routines and can be implemented without unsupervised learning knowledge. The superior performance of our method is demonstrated by extensive experiments and its application to a Glioblastoma cancer dataset reveals some new insights in understanding the Glioblastoma cancer. In theory, a non-asymptotic error bound is established for the output directly from our high dimensional ECM algorithm, and it consists of two quantities: statistical error (statistical accuracy) and optimization error (computational complexity). Such a result gives a theoretical guideline in terminating our ECM iterations.

Yuying Xie, Yufeng Liu, William Valdar

Gaussian graphical models are widely used to represent conditional dependence among random variables. In this paper, we propose a novel estimator for data arising from a group of Gaussian graphical models that are themselves dependent. A motivating example is that of modeling gene expression collected on multiple tissues from the same individual: here the multivariate outcome is affected by dependencies acting not only at the level of the specific tissues, but also at the level of the whole body; existing methods that assume independence among graphs are not applicable in this case. To estimate multiple dependent graphs, we decompose the problem into two graphical layers: the systemic layer, which affects all outcomes and thereby induces cross- graph dependence, and the category-specific layer, which represents graph-specific variation. We propose a graphical EM technique that estimates both layers jointly, establish estimation consistency and selection sparsistency of the proposed estimator, and confirm by simulation that the EM method is superior to a simple one-step method. We apply our technique to mouse genomics data and obtain biologically plausible results.

Patrick K. Kimes, D. Neil Hayes, J. S. Marron et al.

Binary classification is a common statistical learning problem in which a model is estimated on a set of covariates for some outcome indicating the membership of one of two classes. In the literature, there exists a distinction between hard and soft classification. In soft classification, the conditional class probability is modeled as a function of the covariates. In contrast, hard classification methods only target the optimal prediction boundary. While hard and soft classification methods have been studied extensively, not much work has been done to compare the actual tasks of hard and soft classification. In this paper we propose a spectrum of statistical learning problems which span the hard and soft classification tasks based on fitting multiple decision rules to the data. By doing so, we reveal a novel collection of learning tasks of increasing complexity. We study the problems using the framework of large-margin classifiers and a class of piecewise linear convex surrogates, for which we derive statistical properties and a corresponding sub-gradient descent algorithm. We conclude by applying our approach to simulation settings and a magnetic resonance imaging (MRI) dataset from the Alzheimer's Disease Neuroimaging Initiative (ADNI) study.