Recursive partitioning methods provide computationally efficient surrogates for the Wasserstein distance, yet their statistical behavior and their resolution in the small-discrepancy regime remain insufficiently understood. We study Recursive Rank Matching (RRM) as a representative instance of this class under a population-anchored reference. In this setting, we establish consistency and an explicit convergence rate for the anchored empirical RRM under the quadratic cost. We then identify a dominant mismatch mechanism responsible for the loss of resolution in the small-discrepancy regime. Based on this analysis, we introduce Selective Recursive Rank Matching (SRRM), which suppresses the resulting dominant mismatches and yields a higher-fidelity practical surrogate for the Wasserstein distance at moderate additional computational cost.

Riemannian setting is a highly challenging affair.

Informally, our main result may be stated as follows: Under any stochastic Riemannian Robbins-Monro method, the probability of converging to a strict saddle point (or a submanifold thereof) is zero.

Preference-based Reinforcement Learning (PbRL) provides a way to learn high-performance policies in environments where the reward signal is hard to specify, avoiding heuristic and time-consuming reward design. However, PbRL can suffer from slow convergence speed since it requires training in a reward model. Prior work has proposed learning a reward model from demonstrations and fine-tuning it using preferences. However, when the model is a neural network, using different loss functions for pre-training and fine-tuning can pose challenges to reliable optimization. In this paper, we propose a method to effectively leverage prior knowledge with a Residual Reward Model (RRM). An RRM assumes that the true reward of the environment can be split into a sum of two parts: a prior reward and a learned reward. The prior reward is a term available before training, for example, a user's ``best guess'' reward function, or a reward function learned from inverse reinforcement learning (IRL), and the learned reward is trained with preferences. We introduce state-based and image-based versions of RRM and evaluate them on several tasks in the Meta-World environment suite. Experimental results show that our method substantially improves the performance of a common PbRL method. Our method achieves performance improvements for a variety of different types of prior rewards, including proxy rewards, a reward obtained from IRL, and even a negated version of the proxy reward. We also conduct experiments with a Franka Panda to show that our method leads to superior performance on a real robot. It significantly accelerates policy learning for different tasks, achieving success in fewer steps than the baseline. The videos are presented at https://sunlighted.github.io/RRM-web/.

Reward models (RMs) play a pivotal role in aligning large language models (LLMs) with human preferences. However, traditional RM training, which relies on response pairs tied to specific prompts, struggles to disentangle prompt-driven preferences from prompt-independent artifacts, such as response length and format. In this work, we expose a fundamental limitation of current RM training methods, where RMs fail to effectively distinguish between contextual signals and irrelevant artifacts when determining preferences. To address this, we introduce a causal framework that learns preferences independent of these artifacts and propose a novel data augmentation technique designed to eliminate them. Extensive experiments show that our approach successfully filters out undesirable artifacts, yielding a more robust reward model (RRM). Our RRM improves the performance of a pairwise reward model trained on Gemma-2-9b-it, on Reward-Bench, increasing accuracy from 80.61% to 84.15%. Additionally, we train two DPO policies using both the RM and RRM, demonstrating that the RRM significantly enhances DPO-aligned policies, improving MT-Bench scores from 7.27 to 8.31 and length-controlled win-rates in AlpacaEval-2 from 33.46% to 52.49%. Reinforcement Learning from Human Feedback (RLHF) has become a cornerstone in aligning large language models (LLMs) with human preferences to produce responses that are more helpful, honest, and harmless (Ouyang et al., 2022; Bai et al., 2022a). This approach involves training a reward model (RM) on human feedback, which then guides the LLM to generate high-quality responses through reinforcement learning.

Labeling errors in datasets are common, if not systematic, in practice. They naturally arise in a variety of contexts--human labeling, noisy labeling, and weak labeling (i.e., image classification), for example. This presents a persistent and pervasive stress on machine learning practice. In particular, neural network (NN) architectures can withstand minor amounts of dataset imperfection with traditional countermeasures such as regularization, data augmentation, and batch normalization. However, major dataset imperfections often prove insurmountable. We propose and study the implementation of Rockafellian Relaxation (RR), a new loss reweighting, architecture-independent methodology, for neural network training. Experiments indicate RR can enhance standard neural network methods to achieve robust performance across classification tasks in computer vision and natural language processing (sentiment analysis). We find that RR can mitigate the effects of dataset corruption due to both (heavy) labeling error and/or adversarial perturbation, demonstrating effectiveness across a variety of data domains and machine learning tasks.

Many modern machine learning applications - from online principal component analysis to covariance matrix identification and dictionary learning - can be formulated as minimization problems on Riemannian manifolds, and are typically solved with a Riemannian stochastic gradient method (or some variant thereof). However, in many cases of interest, the resulting minimization problem is not geodesically convex, so the convergence of the chosen solver to a desirable solution - i.e., a local minimizer - is by no means guaranteed. In this paper, we study precisely this question, that is, whether stochastic Riemannian optimization algorithms are guaranteed to avoid saddle points with probability 1. For generality, we study a family of retraction-based methods which, in addition to having a potentially much lower per-iteration cost relative to Riemannian gradient descent, include other widely used algorithms, such as natural policy gradient methods and mirror descent in ordinary convex spaces. In this general setting, we show that, under mild assumptions for the ambient manifold and the oracle providing gradient information, the policies under study avoid strict saddle points / submanifolds with probability 1, from any initial condition. This result provides an important sanity check for the use of gradient methods on manifolds as it shows that, almost always, the limit state of a stochastic Riemannian algorithm can only be a local minimizer.

Time-series forecasts play a critical role in business planning. However, forecasters typically optimize objectives that are agnostic to downstream business goals and thus can produce forecasts misaligned with business preferences. In this work, we demonstrate that optimization of conventional forecasting metrics can often lead to sub-optimal downstream business performance. Focusing on the inventory management setting, we derive an efficient procedure for computing and optimizing proxies of common downstream business metrics in an end-to-end differentiable manner. We explore a wide range of plausible cost trade-off scenarios, and empirically demonstrate that end-to-end optimization often outperforms optimization of standard business-agnostic forecasting metrics (by up to 45.7% for a simple scaling model, and up to 54.0% for an LSTM encoder-decoder model). Finally, we discuss how our findings could benefit other business contexts.