论文检索

Probabilistic Methods · Monte Carlo and Sampling Methods

Kejun Tang, Jiayu Zhai, Xiaoliang Wan, Chao Yang

Solving partial differential equations (PDEs) is a central task in scientific computing. Recently, neural network approximation of PDEs has received increasing attention due to its flexible meshless discretization and its potential for high-dimensional problems. One fundamental numerical difficulty is that random samples in the training set introduce statistical errors into the discretization of the loss functional which may become the dominant error in the final approximation, and therefore overshadow the modeling capability of the neural network. In this work, we propose a new minmax formulation to optimize simultaneously the approximate solution, given by a neural network model, and the random samples in the training set, provided by a deep generative model. The key idea is to use a deep generative model to adjust the random samples in the training set such that the residual induced by the neural network model can maintain a smooth profile in the training process. Such an idea is achieved by implicitly embedding the Wasserstein distance between the residual-induced distribution and the uniform distribution into the loss, which is then minimized together with the residual. A nearly uniform residual profile means that its variance is small for any normalized weight function such that the Monte Carlo approximation error of the loss functional is reduced significantly for a certain sample size. The adversarial adaptive sampling (AAS) approach proposed in this work is the first attempt to formulate two essential components, minimizing the residual and seeking the optimal training set, into one minmax objective functional for the neural network approximation of PDEs.

Deep Learning · Large Language Models

Shangbin Feng, Weijia Shi, Yuyang Bai, Vidhisha Balachandran, Tianxing He, Yulia Tsvetkov

By design, large language models (LLMs) are static general-purpose models, expensive to retrain or update frequently. As they are increasingly adopted for knowledge-intensive tasks, it becomes evident that these design choices lead to failures to generate factual, relevant, and up-to-date knowledge. To this end, we propose Knowledge Card, a modular framework to plug in new factual and relevant knowledge into general-purpose LLMs. We first introduce knowledge cards---specialized language models trained on corpora from specific domains and sources. Knowledge cards serve as parametric repositories that are selected at inference time to generate background knowledge for the base LLM. We then propose three content selectors to dynamically select and retain information in documents generated by knowledge cards, specifically controlling for relevance, brevity, and factuality of outputs. Finally, we propose two complementary integration approaches to augment the base LLM with the (relevant, factual) knowledge curated from the specialized LMs. Through extensive experiments, we demonstrate that Knowledge Card achieves state-of-the-art performance on six benchmark datasets. Ultimately, Knowledge Card framework enables dynamic synthesis and updates of knowledge from diverse domains. Its modularity will ensure that relevant knowledge can be continuously updated through the collective efforts of the research community.

Deep Learning

Takuya Furusawa

In this paper, we explore the application of mean field theory, a technique from statistical physics, to deep metric learning and address the high training complexity commonly associated with conventional metric learning loss functions.By adapting mean field theory for deep metric learning, we develop an approach to design classification-based loss functions from pair-based ones, which can be considered complementary to the proxy-based approach.Applying the mean field theory to two pair-based loss functions, we derive two new loss functions, MeanFieldContrastive and MeanFieldClassWiseMultiSimilarity losses, with reduced training complexity.We extensively evaluate these derived loss functions on three image-retrieval datasets and demonstrate that our loss functions outperform baseline methods in two out of the three datasets.

Optimization

Antoine Bambade, Fabian Schramm, Adrien Taylor, Justin Carpentier

Optimization layers within neural network architectures have become increasingly popular for their ability to solve a wide range of machine learning tasks and to model domain-specific knowledge. However, designing optimization layers requires careful consideration as the underlying optimization problems might be infeasible during training. Motivated by applications in learning, control and robotics, this work focuses on convex quadratic programming (QP) layers. The specific structure of this type of optimization layer can be efficiently exploited for faster computations while still allowing rich modeling capabilities. We leverage primal-dual augmented Lagrangian techniques for computing derivatives of both feasible and infeasible QP solutions. More precisely, we propose a unified approach which tackles the differentiability of the closest feasible QP solutions in a classical $\ell_2$ sense. We then harness this approach to enrich the expressive capabilities of existing QP layers. More precisely, we show how differentiating through infeasible QPs during training enables to drive towards feasibility at test time a new range of QP layers. These layers notably demonstrate superior predictive performance in some conventional learning tasks. Additionally, we present alternative formulations that enhance numerical robustness, speed, and accuracy for training such layers. Along with these contributions, we provide an open-source C++ software package called QPLayer for differentiating feasible and infeasible convex QPs and which can be interfaced with modern learning frameworks.

Reinforcement Learning · Deep RL

Juan Rocamonde, Victoriano Montesinos, Elvis Nava, Ethan Perez, David Lindner

Reinforcement learning (RL) requires either manually specifying a reward function, which is often infeasible, or learning a reward model from a large amount of human feedback, which is often very expensive. We study a more sample-efficient alternative: using pretrained vision-language models (VLMs) as zero-shot reward models (RMs) to specify tasks via natural language. We propose a natural and general approach to using VLMs as reward models, which we call VLM-RMs. We use VLM-RMs based on CLIP to train a MuJoCo humanoid to learn complex tasks without a manually specified reward function, such as kneeling, doing the splits, and sitting in a lotus position. For each of these tasks, we only provide _a single sentence text prompt_ describing the desired task with minimal prompt engineering. We provide videos of the trained agents at: https://sites.google.com/view/vlm-rm. We can improve performance by providing a second "baseline" prompt and projecting out parts of the CLIP embedding space irrelevant to distinguish between goal and baseline. Further, we find a strong scaling effect for VLM-RMs: larger VLMs trained with more compute and data are better reward models. The failure modes of VLM-RMs we encountered are all related to known capability limitations of current VLMs, such as limited spatial reasoning ability or visually unrealistic environments that are far off-distribution for the VLM. We find that VLM-RMs are remarkably robust as long as the VLM is large enough. This suggests that future VLMs will become more and more useful reward models for a wide range of RL applications.

Deep Learning · Large Language Models

Yuchen Hu, CHEN CHEN, Chao-Han Huck Yang, Ruizhe Li, Chao Zhang, Pin-Yu Chen, Ensiong Chng

Recent advances in large language models (LLMs) have promoted generative error correction (GER) for automatic speech recognition (ASR), which leverages the rich linguistic knowledge and powerful reasoning ability of LLMs to improve recognition results. The latest work proposes a GER benchmark with "HyPoradise" dataset to learn the mapping from ASR N-best hypotheses to ground-truth transcription by efficient LLM finetuning, which shows great effectiveness but lacks specificity on noise-robust ASR. In this work, we extend the benchmark to noisy conditions and investigate if we can teach LLMs to perform denoising for GER just like what robust ASR do, where one solution is introducing noise information as a conditioner into LLM. However, directly incorporating noise embeddings from audio encoder could harm the LLM tuning due to cross-modality gap. To this end, we propose to extract a language-space noise embedding from the N-best list to represent the noise conditions of source speech, which can promote the denoising process in GER. Furthermore, in order to enhance its representation ability of audio noise, we design a knowledge distillation (KD) approach via mutual information estimation to distill the real noise information in audio embeddings to our language embedding. Experiments on various latest LLMs demonstrate our approach achieves a new breakthrough with up to 53.9% correction improvement in terms of word error rate while with limited training data. Analysis shows that our language-space noise embedding can well represent the noise conditions of source speech, under which off-the-shelf LLMs show strong ability of language-space denoising.

Applications · Neuroscience, Cognitive Science

Max F. Burg, Thomas Zenkel, Michaela Vystrčilová, Jonathan Oesterle, Larissa Höfling, Konstantin F. Willeke, Jan Lause, Sarah Müller, Paul Fahey, Zhiwei Ding 等

Identifying cell types and understanding their functional properties is crucial for unraveling the mechanisms underlying perception and cognition. In the retina, functional types can be identified by carefully selected stimuli, but this requires expert domain knowledge and biases the procedure towards previously known cell types. In the visual cortex, it is still unknown what functional types exist and how to identify them. Thus, for unbiased identification of the functional cell types in retina and visual cortex, new approaches are needed. Here we propose an optimization-based clustering approach using deep predictive models to obtain functional clusters of neurons using Most Discriminative Stimuli (MDS). Our approach alternates between stimulus optimization with cluster reassignment akin to an expectation-maximization algorithm. The algorithm recovers functional clusters in mouse retina, marmoset retina and macaque visual area V4. This demonstrates that our approach can successfully find discriminative stimuli across species, stages of the visual system and recording techniques. The resulting most discriminative stimuli can be used to assign functional cell types fast and on the fly, without the need to train complex predictive models or show a large natural scene dataset, paving the way for experiments that were previously limited by experimental time. Crucially, MDS are interpretable: they visualize the distinctive stimulus patterns that most unambiguously identify a specific type of neuron.

Deep Learning · Self-Supervised Learning

Rohan Sharma, Kaiyi Ji, Zhiqiang Xu, Changyou Chen

Self-supervised learning through contrastive representations is an emergent and promising avenue, aiming at alleviating the availability of labeled data. Recent research in the field also demonstrates its viability for several downstream tasks, henceforth leading to works that implement the contrastive principle through innovative loss functions and methods. However, despite achieving impressive progress, most methods depend on prohibitively large batch sizes and compute requirements for good performance. In this work, we propose the $\textbf{AUC}$-$\textbf{C}$ontrastive $\textbf{L}$earning, a new approach to contrastive learning that demonstrates robust and competitive performance in compute-limited regimes. We propose to incorporate the contrastive objective within the AUC-maximization framework, by noting that the AUC metric is maximized upon enhancing the probability of the network's binary prediction difference between positive and negative samples which inspires adequate embedding space arrangements in representation learning. Unlike standard contrastive methods, when performing stochastic optimization, our method maintains unbiased stochastic gradients and thus is more robust to batchsizes as opposed to standard stochastic optimization problems.Remarkably, our method with a batch size of 256, outperforms several state-of-the-art methods that may need much larger batch sizes (e.g., 4096), on ImageNet and other standard datasets. Experiments on transfer learning, few-shot learning, and other downstream tasks also demonstrate the viability of our method.

Miscellaneous Aspects of Machine Learning · Causality

Yong Wu, Yanwei Fu, Shouyan Wang, Xinwei Sun

Proximal causal learning is a powerful framework for identifying the causal effect under the existence of unmeasured confounders. Within this framework, the doubly robust (DR) estimator was derived and has shown its effectiveness in estimation, especially when the model assumption is violated. However, the current form of the DR estimator is restricted to binary treatments, while the treatments can be continuous in many real-world applications. The primary obstacle to continuous treatments resides in the delta function present in the original DR estimator, making it infeasible in causal effect estimation and introducing a heavy computational burden in nuisance function estimation. To address these challenges, we propose a kernel-based DR estimator that can well handle continuous treatments for proximal causal learning. Equipped with its smoothness, we show that its oracle form is a consistent approximation of the influence function. Further, we propose a new approach to efficiently solve the nuisance functions. We then provide a comprehensive convergence analysis in terms of the mean square error. We demonstrate the utility of our estimator on synthetic datasets and real-world applications.

Deep Learning · Self-Supervised Learning

Stylianos Poulakakis-Daktylidis, Hadi Jamali-Rad

Learning quickly from very few labeled samples is a fundamental attribute that separates machines and humans in the era of deep representation learning. Unsupervised few-shot learning (U-FSL) aspires to bridge this gap by discarding the reliance on annotations at training time. Intrigued by the success of contrastive learning approaches in the realm of U-FSL, we structurally approach their shortcomings in both pretraining and downstream inference stages. We propose a novel Dynamic Clustered mEmory (DyCE) module to promote a highly separable latent representation space for enhancing positive sampling at the pretraining phase and infusing implicit class-level insights into unsupervised contrastive learning. We then tackle the, somehow overlooked yet critical, issue of sample bias at the few-shot inference stage. We propose an iterative Optimal Transport-based distribution Alignment (OpTA) strategy and demonstrate that it efficiently addresses the problem, especially in low-shot scenarios where FSL approaches suffer the most from sample bias. We later on discuss that DyCE and OpTA are two intertwined pieces of a novel end-to-end approach (we coin as BECLR), constructively magnifying each other's impact. We then present a suite of extensive quantitative and qualitative experimentation to corroborate that BECLR sets a new state-of-the-art across ALL existing U-FSL benchmarks (to the best of our knowledge), and significantly outperforms the best of the current baselines (codebase available at https://github.com/stypoumic/BECLR).

Reinforcement Learning · Everything Else

Finn Rietz, Erik Schaffernicht, Stefan Heinrich, Johannes A. Stork

Reinforcement learning (RL) for complex tasks remains a challenge, primarily due to the difficulties of engineering scalar reward functions and the inherent inefficiency of training models from scratch. Instead, it would be better to specify complex tasks in terms of elementary subtasks and to reuse subtask solutions whenever possible. In this work, we address continuous space lexicographic multi-objective RL problems, consisting of prioritized subtasks, which are notoriously difficult to solve. We show that these can be scalarized with a subtask transformation and then solved incrementally using value decomposition. Exploiting this insight, we propose prioritized soft Q-decomposition (PSQD), a novel algorithm for learning and adapting subtask solutions under lexicographic priorities in continuous state-action spaces. PSQD offers the ability to reuse previously learned subtask solutions in a zero-shot composition, followed by an adaptation step. Its ability to use retained subtask training data for offline learning eliminates the need for new environment interaction during adaptation. We demonstrate the efficacy of our approach by presenting successful learning, reuse, and adaptation results for both low- and high-dimensional simulated robot control tasks, as well as offline learning results. In contrast to baseline approaches, PSQD does not trade off between conflicting subtasks or priority constraints and satisfies subtask priorities during learning. PSQD provides an intuitive framework for tackling complex RL problems, offering insights into the inner workings of the subtask composition.

Social Aspects · Trustworthy Machine Learning

Xianjun Yang, Wei Cheng, Yue Wu, Linda Petzold, William Wang, Haifeng Chen

Large language models (LLMs) have notably enhanced the fluency and diversity of machine-generated text. However, this progress also presents a significant challenge in detecting the origin of a given text, and current research on detection methods lags behind the rapid evolution of LLMs. Conventional training-based methods have limitations in flexibility, particularly when adapting to new domains, and they often lack explanatory power. To address this gap, we propose a novel training-free detection strategy called Divergent N-Gram Analysis (DNA-GPT). Given a text, we first truncate it in the middle and then use only the preceding portion as input to the LLMs to regenerate the new remaining parts. By analyzing the differences between the original and new remaining parts through N-gram analysis in black-box or probability divergence in white-box, we can clearly illustrate significant discrepancies between machine-generated and human-written text. We conducted extensive experiments on the most advanced LLMs from OpenAI, including text-davinci-003, GPT-3.5-turbo, and GPT-4, as well as open-source models such as GPT-NeoX-20B and LLaMa-13B. Results show that our zero-shot approach exhibits state-of-the-art performance in distinguishing between human and GPT-generated text on four English and one German dataset, outperforming OpenAI's own classifier, which is trained on millions of text. Additionally, our methods provide reasonable explanations and evidence to support our claim, which is a unique feature of explainable detection. Our method is also robust under the revised text attack and can additionally solve model sourcing.

Applications · Computer Vision

Rujie Wu, Xiaojian Ma, Zhenliang Zhang, Wei Wang, Qing Li, Song-Chun Zhu, Yizhou Wang

We introduce Bongard-OpenWorld, a new benchmark for evaluating real-world few-shot reasoning for machine vision. It originates from the classical Bongard Problems (BPs): Given two sets of images (positive and negative), the model needs to identify the set that query images belong to by inducing the visual concepts, which is exclusively depicted by images from the positive set. Our benchmark inherits the few-shot concept induction of the original BPs while adding the two novel layers of challenge: 1) open-world free-form concepts, as the visual concepts in Bongard-OpenWorld are unique compositions of terms from an open vocabulary, ranging from object categories to abstract visual attributes and commonsense factual knowledge; 2) real-world images, as opposed to the synthetic diagrams used by many counterparts. In our exploration, Bongard-OpenWorld already imposes a significant challenge to current few-shot reasoning algorithms. We further investigate to which extent the recently introduced Large Language Models (LLMs) and Vision-Language Models (VLMs) can solve our task, by directly probing VLMs, and combining VLMs and LLMs in an interactive reasoning scheme. We even conceived a neuro-symbolic reasoning approach that reconciles LLMs & VLMs with logical reasoning to emulate the human problem-solving process for Bongard Problems. However, none of these approaches manage to close the human-machine gap, as the best learner achieves 64% accuracy while human participants easily reach 91%. We hope Bongard-OpenWorld can help us better understand the limitations of current visual intelligence and facilitate future research on visual agents with stronger few-shot visual reasoning capabilities.

Ahmed Abdulaal, Adamos Hadjivasiliou, Nina Montaña-Brown, Tiantian He, Ayodeji Ijishakin, Ivana Drobnjak, Daniel Castro, Daniel Alexander

Scientific discovery hinges on the effective integration of metadata, which refers to a set of 'cognitive' operations such as determining what information is relevant for inquiry, and data, which encompasses physical operations such as observation and experimentation. This paper introduces the Causal Modelling Agent (CMA), a novel framework that synergizes the metadata-based reasoning capabilities of Large Language Models (LLMs) with the data-driven modelling of Deep Structural Causal Models (DSCMs) for the task of causal discovery. We evaluate the CMA's performance on a number of benchmarks, as well as on the real-world task of modelling the clinical and radiological phenotype of Alzheimer's Disease (AD). Our experimental results indicate that the CMA can outperform previous data-driven or metadata-driven approaches to causal discovery. In our real-world application, we use the CMA to derive new insights into the causal relationships among biomarkers of AD.

Deep Learning · Large Language Models

Athul Jacob, Yikang Shen, Gabriele Farina, Jacob Andreas

When applied to question answering and other text generation tasks, language models (LMs) may be queried generatively (by sampling answers from their output distribution) or discriminatively (by using them to score or rank a set of candidate answers). These procedures sometimes yield very different predictions. How do we reconcile mutually incompatible scoring procedures to obtain coherent LM predictions? We introduce a new, a training-free, game-theoretic procedure for language model decoding. Our approach casts language model decoding as a regularized imperfect-information sequential signaling game—which we term the concensus game—in which a generator seeks to communicate an abstract correctness parameter using natural language sentences to a discriminator. We develop computational procedures for finding approximate equilibria of this game, resulting in a decoding algorithm we call equilibrium-ranking. Applied to a large number of tasks (including reading comprehension, commonsense reasoning, mathematical problem-solving, and assistive dialog), equilibrium-ranking consistently improves performance over existing LM decoding procedures. These improvements are sometimes substantial—on multiple benchmarks, we observe that applying equilibrium-ranking to LLaMA-7B outperforms the much larger LLaMA-65B and PaLM-540B models.

Renjie Pi, Lewei Yao, Jianhua Han, Xiaodan Liang, Wei Zhang, Hang Xu

This paper introduces Instruction-oriented Object Detection (IOD), a new task that enhances human-computer interaction by enabling object detectors to understand user instructions and locate relevant objects. Unlike traditional open-vocabulary object detection tasks that rely on users providing a list of required category names, IOD requires models to comprehend natural-language instructions, contextual reasoning, and output the name and location of the desired categories. This poses fresh challenges for modern object detection systems. To develop an IOD system, we create a dataset called IOD-Bench, which consists of instruction-guided detections, along with specialized evaluation metrics. We leverage large-scale language models (LLMs) to generate a diverse set of instructions (8k+) based on existing public object detection datasets, covering a wide range of real-world scenarios. As an initial approach to the IOD task, we propose a model called Ins-DetCLIP. It harnesses the extensive knowledge within LLMs to empower the detector with instruction-following capabilities. Specifically, our Ins-DetCLIP employs a visual encoder (i.e., DetCLIP, an open-vocabulary detector) to extract object-level features. These features are then aligned with the input instructions using a cross-modal fusion module integrated into a pre-trained LLM. Experimental results conducted on IOD-Bench demonstrate that our model consistently outperforms baseline methods that directly combine LLMs with detection models. This research aims to pave the way for a more adaptable and versatile interaction paradigm in modern object detection systems, making a significant contribution to the field.

Reinforcement Learning · Inverse

Matteo Pirotta, Andrea Tirinzoni, Ahmed Touati, Alessandro Lazaric, Yann Ollivier

Imitation learning (IL) aims at producing agents that can imitate any behavior given a few expert demonstrations. Yet existing approaches require many demonstrations and/or running (online or offline) reinforcement learning (RL) algorithms for each new imitation task. Here we show that recent RL foundation models based on successor measures can imitate any expert behavior almost instantly with just a few demonstrations and no need for RL or fine-tuning, while accommodating several IL principles (behavioral cloning, feature matching, reward-based, and goal-based reductions). In our experiments, imitation via RL foundation models matches, and often surpasses, the performance of SOTA offline IL algorithms, and produces imitation policies from new demonstrations within seconds instead of hours.

Theory · Learning Theory

Na Li, Yuchen Jiao, Hangguan Shan, Shefeng Yan

The thriving field of multi-agent reinforcement learning (MARL) studies how a group of interacting agents make decisions autonomously in a shared dynamic environment. Existing theoretical studies in this area suffer from at least two of the following obstacles: memory inefficiency, the heavy dependence of sample complexity on the long horizon and the large state space, the high computational complexity, non-Markov policy, non-Nash policy, and high burn-in cost. In this work, we take a step towards settling this problem by designing a model-free self-play algorithm \emph{Memory-Efficient Nash Q-Learning (ME-Nash-QL)} for two-player zero-sum Markov games, which is a specific setting of MARL. We prove that ME-Nash-QL can output an $\varepsilon$-approximate Nash policy with remarkable space complexity $O(SABH)$, sample complexity $\widetilde{O}(H^4SAB/\varepsilon^2)$, and computational complexity $O(T\mathrm{poly}(AB))$, where $S$ is the number of states, $\{A, B\}$ is the number of actions for the two players, $H$ is the horizon length, and $T$ is the number of samples. Notably, our approach outperforms in terms of space complexity compared to existing algorithms for tabular cases. It achieves the lowest computational complexity while preserving Markov policies, setting a new standard. Furthermore, our algorithm outputs a Nash policy and achieves the best sample complexity compared with the existing guarantee for long horizons, i.e. when $\min \\{ A, B \\} \ll H^2$. Our algorithm also achieves the best burn-in cost $O(SAB\,\mathrm{poly}(H))$, whereas previous algorithms need at least $O(S^3 AB\,\mathrm{poly}(H))$ to attain the same level of sample complexity with ours.

Deep Learning · Other Representation Learning

Hao Xiong, Yehui Tang, Yunlin He, Wei Tan, Junchi Yan

Network embedding (NE) is a prominent technique for network analysis where the nodes are represented as vectorized embeddings in a continuous space. Existing works tend to resort to the low-dimensional embedding space for efficiency and less risk of over-fitting. In this paper, we explore a new NE paradigm whose embedding dimension goes exponentially high w.r.t. the number of parameters, yet being very efficient and effective. Specifically, the node embeddings are represented as product states that lie in a super high-dimensional (e.g. $2^{32}$-dim) quantum Hilbert space, with a carefully designed optimization approach to guarantee the robustness to work in different scenarios. In the experiments, we show diverse virtues of our methods, including but not limited to: the overwhelming performance on downstream tasks against conventional low-dimensional NE baselines with the similar amount of computing resources, the super high efficiency for a fixed low embedding dimension (e.g. 512) with less than 1/200 memory usage, the robustness when equipped with different objectives and sampling strategies as a fundamental tool for future NE research. As a relatively unexplored topic in literature, the high-dimensional NE paradigm is demonstrated effective both experimentally and theoretically.

Optimization

Lizhang Chen, Bo Liu, Kaizhao Liang, Qiang Liu

Lion (Evolved Sign Momentum), a new optimizer discovered through program search, has shown promising results in training large AI models. It achieves results comparable to AdamW but with greater memory efficiency. As what we can expect from the result of the random search, Lion blends a number of elements from existing algorithms, including signed momentum, decoupled weight decay, Polayk and Nesterov momentum, but doesn't fit into any existing category of theoretically grounded optimizers. Thus, even though Lion appears to perform well as a general-purpose optimizer for a wide range of tasks, its theoretical basis remains uncertain. This absence of theoretical clarity limits opportunities to further enhance and expand Lion's efficacy. This work aims to demystify Lion. Using both continuous-time and discrete-time analysis, we demonstrate that Lion is a novel and theoretically grounded approach for minimizing a general loss function $f(x)$ while enforcing a bound constraint $||x||_\infty \leq 1/\lambda$. Lion achieves this through the incorporation of decoupled weight decay, where $\lambda$ represents the weight decay coefficient. Our analysis is facilitated by the development of a new Lyapunov function for the Lion updates. It applies to a wide range of Lion-$\phi$ algorithms, where the $sign(\cdot)$ operator in Lion is replaced by the subgradient of a convex function $\phi$, leading to the solution of the general composite optimization problem $\min_x f(x) + \phi^*(x)$. Our findings provide valuable insights into the dynamics of Lion and pave the way for further enhancements and extensions of Lion-related algorithms.