The nadir objective vector plays a key role in solving multi-objective optimization problems (MOPs), where it is often used to normalize the objective space and guide the search. The current methods for estimating the nadir objective vector perform effectively only on specific MOPs. This paper reveals the limitations of these methods: exact methods can only work on discrete MOPs, while heuristic methods cannot deal with the MOP with a complicated feasible objective region. To fill this gap, we propose a general and rigorous method, namely boundary decomposition for nadir objective vector estimation (BDNE).

Abstract--Class-incremental learning requires a learning system to continually learn knowledge of new classes and meanwhile try to preserve previously learned knowledge of old classes. As current state-of-the-art methods based on Vision-Language Models (VLMs) still suffer from the issue of differentiating classes across learning tasks. Here a novel VLM-based continual learning framework for image classification is proposed. In this framework, task-specific adapters are added to the pre-trained and frozen image encoder to learn new knowledge, and a novel cross-task representation calibration strategy based on a mixture of light-weight projectors is used to help better separate all learned classes in a unified feature space, alleviating class confusion across tasks. In addition, a novel inference strategy guided by prediction uncertainty is developed to more accurately select the most appropriate image feature for class prediction. Extensive experiments on multiple datasets under various settings demonstrate the superior performance of our method compared to existing ones.

Sparse Mixture-of-Experts (SMoE) architectures have enabled a new frontier in scaling Large Language Models (LLMs), offering superior performance by activating only a fraction of their total parameters during inference. However, their practical deployment is severely hampered by substantial static memory overhead, as all experts must be loaded into memory. Existing post-training pruning methods, while reducing model size, often derive their pruning criteria from a single, general-purpose corpus. This leads to a critical limitation: a catastrophic performance degradation when the pruned model is applied to other domains, necessitating a costly re-pruning for each new domain. To address this generalization gap, we introduce Mosaic Pruning (MoP). The core idea of MoP is to construct a functionally comprehensive set of experts through a structured ``cluster-then-select" process. This process leverages a similarity metric that captures expert performance across different task domains to functionally cluster the experts, and subsequently selects the most representative expert from each cluster based on our proposed Activation Variability Score. Unlike methods that optimize for a single corpus, our proposed Mosaic Pruning ensures that the pruned model retains a functionally complementary set of experts, much like the tiles of a mosaic that together form a complete picture of the original model's capabilities, enabling it to handle diverse downstream tasks.Extensive experiments on various MoE models demonstrate the superiority of our approach. MoP significantly outperforms prior work, achieving a 7.24\% gain on general tasks and 8.92\% on specialized tasks like math reasoning and code generation.

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The nadir objective vector plays a key role in solving multi-objective optimization problems (MOPs), where it is often used to normalize the objective space and guide the search. The current methods for estimating the nadir objective vector perform effectively only on specific MOPs. This paper reveals the limitations of these methods: exact methods can only work on discrete MOPs, while heuristic methods cannot deal with the MOP with a complicated feasible objective region. To fill this gap, we propose a general and rigorous method, namely boundary decomposition for nadir objective vector estimation (BDNE). By utilizing bilevel optimization, boundary subproblems are optimized and adjusted alternately, thereby refining their optimal solutions to align with the nadir objective vector.

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Specifying all desirable properties of a language model is challenging, but certain requirements seem essential. Given samples from an unknown language, the trained model should produce valid strings not seen in training and be expressive enough to capture the language's full richness. Otherwise, outputting invalid strings constitutes "hallucination," and failing to capture the full range leads to "mode collapse." We ask if a language model can meet both requirements. We investigate this within a statistical language generation setting building on Gold and Angluin. Here, the model receives random samples from a distribution over an unknown language K, which belongs to a possibly infinite collection of languages. The goal is to generate unseen strings from K. We say the model generates from K with consistency and breadth if, as training size increases, its output converges to all unseen strings in K. Kleinberg and Mullainathan [KM24] asked if consistency and breadth in language generation are possible. We answer this negatively: for a large class of language models, including next-token prediction models, this is impossible for most collections of candidate languages. This contrasts with [KM24]'s result, showing consistent generation without breadth is possible for any countable collection of languages. Our finding highlights that generation with breadth fundamentally differs from generation without breadth. As a byproduct, we establish near-tight bounds on the number of samples needed for generation with or without breadth. Finally, our results offer hope: consistent generation with breadth is achievable for any countable collection of languages when negative examples (strings outside K) are available alongside positive ones. This suggests that post-training feedback, which encodes negative examples, can be crucial in reducing hallucinations while limiting mode collapse.

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Large Language Models (LLMs) exhibit strong generalization capabilities to novel tasks when prompted with language instructions and in-context demos. Since this ability sensitively depends on the quality of prompts, various methods have been explored to automate the instruction design. While these methods demonstrated promising results, they also restricted the searched prompt to one instruction. Such simplification significantly limits their capacity, as a single demo-free instruction might not be able to cover the entire complex problem space of the targeted task. To alleviate this issue, we adopt the Mixture-of-Expert paradigm and divide the problem space into a set of sub-regions; Each sub-region is governed by a specialized expert, equipped with both an instruction and a set of demos. A two-phase process is developed to construct the specialized expert for each region: (1) demo assignment: Inspired by the theoretical connection between in-context learning and kernel regression, we group demos into experts based on their semantic similarity; (2) instruction assignment: A region-based joint search of an instruction per expert complements the demos assigned to it, yielding a synergistic effect. The resulting method, codenamed Mixture-of-Prompts (MoP), achieves an average win rate of 81% against prior arts across several major benchmarks.