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Online Learning

Abhishek Sinha, Rahul Vaze

A well-studied generalization of the standard online convex optimization (OCO) framework is constrained online convex optimization (COCO). In COCO, on every round, a convex cost function and a convex constraint function are revealed to the learner after it chooses the action for that round. The objective is to design an online learning policy that simultaneously achieves a small regret while ensuring a small cumulative constraint violation (CCV) against an adaptive adversary interacting over a horizon of length $T$. A long-standing open question in COCO is whether an online policy can simultaneously achieve $O(\sqrt{T})$ regret and $\tilde{O}(\sqrt{T})$ CCV without any restrictive assumptions. For the first time, we answer this in the affirmative and show that a simple first-order policy can simultaneously achieve these bounds. Furthermore, in the case of strongly convex cost and convex constraint functions, the regret guarantee can be improved to $O(\log T)$ while keeping the CCV bound the same as above. We establish these results by effectively combining adaptive OCO policies as a blackbox with Lyapunov optimization - a classic tool from control theory. Surprisingly, the analysis is short and elegant.

Generative Models · New Approaches

Ching-An Cheng, Allen Nie, Adith Swaminathan

We study a class of optimization problems motivated by automating the design and update of AI systems like coding assistants, robots, and copilots. AutoDiff frameworks, like PyTorch, enable efficient end-to-end optimization of differentiable systems. However, general computational workflows can be non-differentiable and involve rich feedback (e.g. console output or user’s responses), heterogeneous parameters (e.g. prompts, codes), and intricate objectives (beyond maximizing a score). We investigate end-to-end generative optimization – using generative models such as LLMs within the optimizer for automatic updating of general computational workflows. We discover that workflow execution traces are akin to back-propagated gradients in AutoDiff and can provide key information to interpret feedback for efficient optimization. Formally, we frame a new mathematical setup, Optimization with Trace Oracle (OPTO). In OPTO, an optimizer receives an execution trace along with feedback on the computed output and updates parameters iteratively. We provide a Python library, Trace, that efficiently converts a workflow optimization problem into an OPTO instance using PyTorch-like syntax. Using Trace, we develop a general LLM-based generative optimizer called OptoPrime. In empirical studies, we find that OptoPrime is capable of first-order numerical optimization, prompt optimization, hyper-parameter tuning, robot controller design, code debugging, etc., and is often competitive with specialized optimizers for each domain. We envision Trace as an open research platform for devising novel generative optimizers and developing the next generation of interactive learning agents. Website: https://microsoft.github.io/Trace/.

Online Learning

Julian Zimmert, Teodor Vanislavov Marinov

Prod is a seminal algorithm in full-information online learning, which has been conjectured to be fundamentally sub-optimal for multi-armed bandits.By leveraging the interpretation of Prod as a first-order OMD approximation, we present the following surprising results:1. Variants of Prod can obtain optimal regret for adversarial multi-armed bandits. 2. There exists a simple and (arguably) importance-weighting free variant with optimal rate. 3. One can even achieve best-both-worlds guarantees with logarithmic regret in the stochastic regime.The bandit algorithms in this work use simple arithmetic update rules without the need of solving optimization problems typical in prior work. Finally, the results directly improve the state of the art of incentive-compatible bandits.

Yuwei Xia, Ding Wang, Qiang Liu, Liang Wang, Shu Wu, Xiao-Yu Zhang

Temporal Knowledge Graph (TKG) forecasting aims to predict future facts based on given histories. Most recent graph-based models excel at capturing structural information within TKGs but lack semantic comprehension abilities. Nowadays, with the surge of LLMs, the LLM-based TKG prediction model has emerged. However, the existing LLM-based model exhibits three shortcomings: (1) It only focuses on the first-order history for prediction while ignoring high-order historical information, resulting in the provided information for LLMs being extremely limited. (2) LLMs struggle with optimal reasoning performance under heavy historical information loads. (3) For TKG prediction, the temporal reasoning capability of LLM alone is limited. To address the first two challenges, we propose Chain-of-History (CoH) reasoning which explores high-order histories step-by-step, achieving effective utilization of high-order historical information for LLMs on TKG prediction. To address the third issue, we design CoH as a plug-and-play module to enhance the performance of graph-based models for TKG prediction. Extensive experiments on three datasets and backbones demonstrate the effectiveness of CoH.

Jeonghoon Kim, Heesoo Jung, Hyeju Jang, Hogun Park

Multi-hop logical reasoning on knowledge graphs is a pivotal task in natural language processing, with numerous approaches aiming to answer First-Order Logic (FOL) queries. Recent geometry (e.g., box, cone) and probability (e.g., beta distribution)-based methodologies have effectively addressed complex FOL queries. However, a common challenge across these methods lies in determining accurate geometric bounds or probability parameters for these queries. The challenge arises because existing methods rely on linear sequential operations within their computation graphs, overlooking the logical structure of the query and the relation-induced information that can be gleaned from the relations of the query, which we call the context of the query. To address the problem, we propose a model-agnostic methodology that enhances the effectiveness of existing multi-hop logical reasoning approaches by fully integrating the context of the FOL query graph. Our approach distinctively discerns (1) the structural context inherent to the query structure and (2) the relation-induced context unique to each node in the query graph as delineated in the corresponding knowledge graph. This dual-context paradigm helps nodes within a query graph attain refined internal representations throughout the multi-hop reasoning steps. Through experiments on two datasets, our method consistently enhances the three multi-hop reasoning foundation models, achieving performance improvements of up to 19.5%. Our codes are available at https://github.com/kjh9503/caqr.

Guiyang Hou, Wenqi Zhang, Yongliang Shen, Linjuan Wu, Weiming Lu

Theory of Mind (ToM)—the cognitive ability to reason about mental states of ourselves and others, is the foundation of social interaction. Although ToM comes naturally to humans, it poses a significant challenge to even the most advanced Large Language Models (LLMs). Due to the complex logical chains in ToM reasoning, especially in higher-order ToM questions, simply utilizing reasoning methods like Chain of Thought (CoT) will not improve the ToM capabilities of LLMs. We present TimeToM, which constructs a temporal space and uses it as the foundation to improve the ToM capabilities of LLMs in multiple scenarios. Specifically, within the temporal space, we construct Temporal Belief State Chain (TBSC) for each character and inspired by the cognition perspective of the social world model, we divide TBSC into self-world beliefs and social world beliefs, aligning with first-order ToM (first-order beliefs) and higher-order ToM (higher-order beliefs) questions, respectively. Moreover, we design a novel tool-belief solver that, by considering belief communication between characters in temporal space, can transform a character’s higher-order beliefs into another character’s first-order beliefs under belief communication period.

Shenzhi Wang, Chang Liu, Zilong Zheng, Siyuan Qi, Shuo Chen, Qisen Yang, Andrew Zhao, Chaofei Wang, Shiji Song, Gao Huang

Recent advances in large language models (LLMs) have led to significant success in using LLMs as agents. Nevertheless, a common assumption that LLMs always process honest information neglects the widespread deceptive or misleading content in human and AI-generated material. This oversight might expose LLMs to malicious manipulations. To enhance LLMs’ ability to identify and counteract deceptive information, in this paper, inspired by humans’ recursive thinking and perspective-taking, we introduce a novel cognitive framework, Recursive Contemplation (ReCon). ReCon combines formulation and refinement contemplation processes; formulation contemplation produces initial thoughts and speech, while refinement contemplation further polishes them. Additionally, we incorporate first-order and second-order perspective transitions into these processes respectively. Specifically, the first-order allows an LLM agent to infer others’ mental states, and the second-order involves understanding how others perceive the agent’s mental state. After integrating ReCon with various LLMs, extensive experiment results from the Avalon game and BigTom benchmark indicate ReCon’s efficacy in aiding LLMs to discern and maneuver around deceptive information without extra fine-tuning and data. Finally, we demonstrate ReCon’s scaling trend with model parameters, and explore the current limitations of LLMs in terms of safety and reasoning, potentially furnishing insights for subsequent research. Our project page can be found at https://shenzhi-wang.github.io/avalon_recon.

Xiaojuan Tang, Song-Chun Zhu, Yitao Liang, Muhan Zhang

Knowledge graph reasoning is an important problem for knowledge graphs. In this paper, we propose a novel and principled framework called RulE (stands for Rule Embedding) to effectively leverage logical rules to enhance KG reasoning. Unlike knowledge graph embedding methods, RulE learns rule embeddings from existing triplets and first-order rules by jointly representing entities, relations and logical rules in a unified embedding space. Based on the learned rule embeddings, a confidence score can be calculated for each rule, reflecting its consistency with the observed triplets. This allows us to perform logical rule inference in a soft way, thus alleviating the brittleness of logic. On the other hand, RulE injects prior logical rule information into the embedding space, enriching and regularizing the entity/relation embeddings. This makes KGE alone perform better too. RulE is conceptually simple and empirically effective. We conduct extensive experiments to verify each component of RulE.Results on multiple benchmarks reveal that our model outperforms the majority of existing embedding-based and rule-based approaches.

Mihir Parmar, Nisarg Patel, Neeraj Varshney, Mutsumi Nakamura, Man Luo, Santosh Mashetty, Arindam Mitra, Chitta Baral

Recently developed large language models (LLMs) have been shown to perform remarkably well on a wide range of language understanding tasks. But, can they really “reason” over the natural language? This question has been receiving significant research attention and many reasoning skills such as commonsense, numerical, and qualitative have been studied. However, the crucial skill pertaining to ‘logical reasoning’ has remained underexplored. Existing work investigating this reasoning ability of LLMs has focused only on a couple of inference rules (such as modus ponens and modus tollens) of propositional and first-order logic. Addressing the above limitation, we comprehensively evaluate the logical reasoning ability of LLMs on 25 different reasoning patterns spanning over propositional, first-order, and non-monotonic logics. To enable systematic evaluation, we introduce LogicBench, a natural language question-answering dataset focusing on the use of a single inference rule. We conduct detailed analysis with a range of LLMs such as GPT-4, ChatGPT, Gemini, Llama-2, and Mistral using chain-of-thought prompting. Experimental results show that existing LLMs do not fare well on LogicBench; especially, they struggle with instances involving complex reasoning and negations. Furthermore, they sometimes tend to prioritize parametric knowledge over contextual information and overlook the correct reasoning chain. We believe that our work and findings facilitate future research for evaluating and enhancing the logical reasoning ability of LLMs.

Jundong Xu, Hao Fei, Liangming Pan, Qian Liu, Mong-Li Lee, Wynne Hsu

While the recent Chain-of-Thought (CoT) technique enhances the reasoning ability of large language models (LLMs) with the theory of mind, it might still struggle in handling logical reasoning that relies much on symbolic expressions and rigid deducing rules. To strengthen the logical reasoning capability of LLMs, we propose a novel Symbolic Chain-of-Thought, namely SymbCoT, a fully LLM-based framework that integrates symbolic expressions and logic rules with CoT prompting. Technically, building upon an LLM, SymbCoT 1) first translates the natural language context into the symbolic format, and then 2) derives a step-by-step plan to solve the problem with symbolic logical rules, 3) followed by a verifier to check the translation and reasoning chain. Via thorough evaluations on 5 standard datasets with both First-Order Logic and Constraint Optimization symbolic expressions, SymbCoT shows striking improvements over the CoT method consistently, meanwhile refreshing the current state-of-the-art performances. We further demonstrate that our system advances in more faithful, flexible, and explainable logical reasoning. To our knowledge, this is the first attempt at combining symbolic expressions and rules into CoT for logical reasoning with LLMs. Code is open at https://github.com/Aiden0526/SymbCoT.

Yuan Yang, Siheng Xiong, Ali Payani, Ehsan Shareghi, Faramarz Fekri

Advancements in logical reasoning, utilizing LLMs to convert natural language into logical symbolism, combined with the use of external theorem provers, have repositioned the symbolic approach as a central point of interest. The main challenge within this paradigm lies in the LLMs’ capability to accurately translate natural language (NL) statements into first-order-logic (FOL) expressions. Although LLMs have shown notable success, there remains a gap in understanding the limitations and challenges they encounter in NL-FOL translation. This is primarily due to the absence of datasets and evaluation test beds at the required fine-grained level. We present MALLS, a dataset of 28K diverse and verified sentence-level NL-FOL pairs collected from GPT4. We utilize a combined strategy of FOL rule parsing, human annotation, and automatic filtering to ensure quality. We also present LogicLLaMA, a LLaMA2-7B/13B fine-tuned on MALLS for NL-FOL translation, which can be used standalone or to correct previously generated rules by GPT3.5 after being further fine-tuned via a novel reinforcement learning with human feedback (RLHF) framework. We benchmark a wide range of LLMs on MALLS and previous datasets, highlighting weaknesses in them in NL-FOL translation and demonstrating the advantages of MALLS. We also show that LogicLLaMA achieves GPT4-level performance and can generalize to other datasets. Project repo is available at https://github.com/gblackout/LogicLLaMA

Optimization · Optimization and Learning under Uncertainty

Germain Vivier-Ardisson, Alexandre Forel, Axel Parmentier, Thibaut Vidal

Optimization layers in deep neural networks have enjoyed a growing popularity in structured learning, improving the state of the art on a variety of applications. Yet, these pipelines lack interpretability since they are made of two opaque layers: a highly non-linear prediction model, such as a deep neural network, and an optimization layer, which is typically a complex black-box solver. Our goal is to improve the transparency of such methods by providing counterfactual explanations. We build upon variational autoencoders a principled way of obtaining counterfactuals: working in the latent space leads to a natural notion of plausibility of explanations. We finally introduce a variant of the classic loss for VAE training that improves their performance in our specific structured context. These provide the foundations of CF-OPT, a first-order optimization algorithm that can find counterfactual explanations for a broad class of structured learning architectures. Our numerical results show that both close and plausible explanations can be obtained for problems from the recent literature.

Optimization · Learning for Optimization

Bingheng Li, Linxin Yang, Yupeng Chen, Senmiao Wang, Haitao Mao, Qian Chen, Yao Ma, Akang Wang, Tian Ding, Jiliang Tang 等

Solving large-scale linear programming (LP) problems is an important task in various areas such as communication networks, power systems, finance and logistics. Recently, two distinct approaches have emerged to expedite LP solving: (i) First-order methods (FOMs); (ii) Learning to optimize (L2O). In this work, we propose an FOM-unrolled neural network (NN) called PDHG-Net, and propose a two-stage L2O method to solve large-scale LP problems. The new architecture PDHG-Net is designed by unrolling the recently emerged PDHG method into a neural network, combined with channel-expansion techniques borrowed from graph neural networks. We prove that the proposed PDHG-Net can recover PDHG algorithm, thus can approximate optimal solutions of LP instances with a polynomial number of neurons. We propose a two-stage inference approach: first use PDHG-Net to generate an approximate solution, and then apply PDHG algorithm to further improve the solution. Experiments show that our approach can significantly accelerate LP solving, achieving up to a 3$\times$ speedup compared to FOMs for large-scale LP problems.

Peng Wang, Huikang Liu, Druv Pai, Yaodong Yu, Zhihui Zhu, Qing Qu, Yi Ma

The maximal coding rate reduction (MCR$^2$) objective for learning structured and compact deep representations is drawing increasing attention, especially after its recent usage in the derivation of fully explainable and highly effective deep network architectures. However, it lacks a complete theoretical justification: only the properties of its global optima are known, and its global landscape has not been studied. In this work, we give a complete characterization of the properties of all its local and global optima as well as other types of critical points. Specifically, we show that each (local or global) maximizer of the MCR$^2$ problem corresponds to a low-dimensional, discriminative, and diverse representation, and furthermore, each critical point of the objective is either a local maximizer or a strict saddle point. Such a favorable landscape makes MCR$^2$ a natural choice of objective for learning diverse and discriminative representations via first-order optimization. To further verify our theoretical findings, we illustrate these properties with extensive experiments on both synthetic and real data sets.

Optimization · Stochastic

Gergely Neu, Nneka Okolo

We study the performance of stochastic first-order methods for finding saddle points of convex-concave functions. A notorious challenge faced by such methods is that the gradients can grow arbitrarily large during optimization, which may result in instability and divergence. In this paper, we propose a simple and effective regularization technique that stabilizes the iterates and yields meaningful performance guarantees even if the domain and the gradient noise scales linearly with the size of the iterates (and is thus potentially unbounded). Besides providing a set of general results, we also apply our algorithm to a specific problem in reinforcement learning, where it leads to performance guarantees for finding near-optimal policies in an average-reward MDP without prior knowledge of the bias span.

Deep Learning · Attention Mechanisms

Michael Sander, Raja Giryes, Taiji Suzuki, Mathieu Blondel, Gabriel Peyré

Transformers have achieved state-of-the-art performance in language modeling tasks. However, the reasons behind their tremendous success are still unclear. In this paper, towards a better understanding, we train a Transformer model on a simple next token prediction task, where sequences are generated as a first-order autoregressive process $s_{t+1} = W s_t$. We show how a trained Transformer predicts the next token by first learning $W$ in-context, then applying a prediction mapping. We call the resulting procedure *in-context autoregressive learning*. More precisely, focusing on commuting orthogonal matrices $W$, we first show that a trained one-layer linear Transformer implements one step of gradient descent for the minimization of an inner objective function, when considering augmented tokens. When the tokens are not augmented, we characterize the global minima of a one-layer diagonal linear multi-head Transformer. Importantly, we exhibit orthogonality between heads and show that positional encoding captures trigonometric relations in the data. On the experimental side, we consider the general case of non-commuting orthogonal matrices and generalize our theoretical findings.

Optimization

Charles Guille-Escuret, Hiroki Naganuma, Kilian Fatras, Ioannis Mitliagkas

Understanding the optimization dynamics of neural networks is necessary for closing the gap between theory and practice. Stochastic first-order optimization algorithms are known to efficiently locate favorable minima in deep neural networks. This efficiency, however, contrasts with the non-convex and seemingly complex structure of neural loss landscapes. In this study, we delve into the fundamental geometric properties of sampled gradients along optimization paths. We focus on two key quantities, the restricted secant inequality and error bound, as well as their ratio γ, which hold high significance for first-order optimization. Our analysis reveals that these quantities exhibit predictable, consistent behavior throughout training, despite the stochasticity induced by sampling minibatches. Our findings suggest that not only do optimization trajectories never encounter significant obstacles, but they also maintain stable dynamics during the majority of training. These observed properties are sufficiently expressive to theoretically guarantee linear convergence and prescribe learning rate schedules mirroring empirical practices. We conduct our experiments on image classification, semantic segmentation and language modeling across different batch sizes, network architectures, datasets, optimizers, and initialization seeds. We discuss the impact of each factor. Our work provides novel insights into the properties of neural network loss functions, and opens the door to theoretical frameworks more relevant to prevalent practice.

Deep Learning · Graph Neural Networks

Christopher Morris, Fabrizio Frasca, Nadav Dym, Haggai Maron, Ismail Ceylan, Ron Levie, Derek Lim, Michael Bronstein, Martin Grohe, Stefanie Jegelka

Machine learning on graphs, especially using graph neural networks (GNNs), has seen a surge in interest due to the wide availability of graph data across a broad spectrum of disciplines, from life to social and engineering sciences. Despite their practical success, our theoretical understanding of the properties of GNNs remains highly incomplete. Recent theoretical advancements primarily focus on elucidating the coarse-grained expressive power of GNNs, predominantly employing combinatorial techniques. However, these studies do not perfectly align with practice, particularly in understanding the generalization behavior of GNNs when trained with stochastic first-order optimization techniques. In this position paper, we argue that the graph machine learning community needs to shift its attention to developing a balanced theory of graph machine learning, focusing on a more thorough understanding of the interplay of expressive power, generalization, and optimization.

Optimization · Large Scale, Parallel and Distributed

Yuxing Liu, Lesi Chen, Luo Luo

This paper studies decentralized optimization problem, where the local objective on each node is an average of a finite set of convex functions and the global function is strongly convex. We propose an efficient stochastic variance reduced first-order method that allows the different nodes to establish their stochastic local gradient estimator with different mini-batch sizes per iteration. We prove the upper bound on the computation time of the proposed method contains the dependence on the global condition number, which is sharper than the previous results that only depend on the local condition numbers. Compared with the state-of-the-art methods, we also show that our method requires less local incremental first-order oracle calls and comparable communication cost. We further perform numerical experiments to validate the advantage of our method.

Miscellaneous Aspects of Machine Learning · General Machine Learning Techniques

Yang Zhou, Zijie Zhang, Zeru Zhang, Lingjuan Lyu, Wei-Shinn Ku

Graph matching in the setting of federated learning is still an open problem. This paper proposes an unsupervised federated graph matching algorithm, UFGM, for inferring matched node pairs on different graphs across clients while maintaining privacy requirement, by leveraging graphlet theory and trust region optimization. First, the nodes' graphlet features are captured to generate pseudo matched node pairs on different graphs across clients as pseudo training data for tackling the dilemma of unsupervised graph matching in federated setting and leveraging the strength of supervised graph matching. An approximate graphlet enumeration method is proposed to sample a small number of graphlets and capture nodes' graphlet features. Theoretical analysis is conducted to demonstrate that the approximate method is able to maintain the quality of graphlet estimation while reducing its expensive cost. Second, we propose a separate trust region algorithm for pseudo supervised federated graph matching while maintaining the privacy constraints. In order to avoid expensive cost of the second-order Hessian computation in the trust region algorithm, we propose two weak quasi-Newton conditions to construct a positive definite scalar matrix as the Hessian approximation with only first-order gradients. We theoretically derive the error introduced by the separate trust region due to the Hessian approximation and conduct the convergence analysis of the approximation method.