Training Nonlinear Transformers for Efficient In-Context Learning: A Theoretical Learning and Generalization Analysis

Transformer-based large language models have displayed impressive in-context learning capabilities, where a pre-trained model can handle new tasks without fine-tuning by simply augmenting the query with some input-output examples from that task. Despite the empirical success, the mechanics of how to train a Transformer to achieve ICL and the corresponding ICL capacity is mostly elusive due to the technical challenges of analyzing the nonconvex training problems resulting from the nonlinear self-attention and nonlinear activation in Transformers. To the best of our knowledge, this paper provides the first theoretical analysis of the training dynamics of Transformers with nonlinear self-attention and nonlinear MLP, together with the ICL generalization capability of the resulting model. Focusing on a group of binary classification tasks, we train Transformers using data from a subset of these tasks and quantify the impact of various factors on the ICL generalization performance on the remaining unseen tasks with and without data distribution shifts. We also analyze how different components in the learned Transformers contribute to the ICL performance. Furthermore, we provide the first theoretical analysis of how model pruning affects the ICL performance and prove that proper magnitude-based pruning can have a minimal impact on ICL while reducing inference costs. These theoretical findings are justified through numerical experiments.

Decentralized Bilevel Optimization over Graphs: Loopless Algorithmic Update and Transient Iteration Complexity

Stochastic bilevel optimization (SBO) is becoming increasingly essential in machine learning due to its versatility in handling nested structures. To address large-scale SBO, decentralized approaches have emerged as effective paradigms in which nodes communicate with immediate neighbors without a central server, thereby improving communication efficiency and enhancing algorithmic robustness. However, current decentralized SBO algorithms face challenges, including expensive inner-loop updates and unclear understanding of the influence of network topology, data heterogeneity, and the nested bilevel algorithmic structures. In this paper, we introduce a single-loop decentralized SBO (D-SOBA) algorithm and establish its transient iteration complexity, which, for the first time, clarifies the joint influence of network topology and data heterogeneity on decentralized bilevel algorithms. D-SOBA achieves the state-of-the-art asymptotic rate, asymptotic gradient/Hessian complexity, and transient iteration complexity under more relaxed assumptions compared to existing methods. Numerical experiments validate our theoretical findings.

Joint Unsupervised and Supervised Training for Automatic Speech Recognition via Bilevel Optimization

In this paper, we present a novel bilevel optimization-based training approach to training acoustic models for automatic speech recognition (ASR) tasks that we term {bi-level joint unsupervised and supervised training (BL-JUST)}. {BL-JUST employs a lower and upper level optimization with an unsupervised loss and a supervised loss respectively, leveraging recent advances in penalty-based bilevel optimization to solve this challenging ASR problem with affordable complexity and rigorous convergence guarantees.} To evaluate BL-JUST, extensive experiments on the LibriSpeech and TED-LIUM v2 datasets have been conducted. BL-JUST achieves superior performance over the commonly used pre-training followed by fine-tuning strategy.

Soft Random Sampling: A Theoretical and Empirical Analysis

Soft random sampling (SRS) is a simple yet effective approach for efficient training of large-scale deep neural networks when dealing with massive data. SRS selects a subset uniformly at random with replacement from the full data set in each epoch. In this paper, we conduct a theoretical and empirical analysis of SRS. First, we analyze its sampling dynamics including data coverage and occupancy. Next, we investigate its convergence with non-convex objective functions and give the convergence rate. Finally, we provide its generalization performance. We empirically evaluate SRS for image recognition on CIFAR10 and automatic speech recognition on Librispeech and an in-house payload dataset to demonstrate its effectiveness. Compared to existing coreset-based data selection methods, SRS offers a better accuracy-efficiency trade-off. Especially on real-world industrial scale data sets, it is shown to be a powerful training strategy with significant speedup and competitive performance with almost no additional computing cost.

Ontology Revision based on Pre-trained Language Models

Ontology revision aims to seamlessly incorporate new information into an existing ontology and plays a crucial role in tasks such as ontology evolution, ontology maintenance, and ontology alignment. Similar to repair single ontologies, resolving logical incoherence in the task of ontology revision is also important and meaningful since incoherence is a main potential factor to cause inconsistency and reasoning with an inconsistent ontology will obtain meaningless answers. To deal with this problem, various ontology revision methods have been proposed to define revision operators and design ranking strategies for axioms in an ontology. However, they rarely consider axiom semantics which provides important information to differentiate axioms. On the other hand, pre-trained models can be utilized to encode axiom semantics, and have been widely applied in many natural language processing tasks and ontology-related ones in recent years. Therefore, in this paper, we define four scoring functions to rank axioms based on a pre-trained model by considering various information from a rebuttal ontology and its corresponding reliable ontology. Based on such a scoring function, we propose an ontology revision algorithm to deal with unsatisfiable concepts at once. If it is hard to resolve all unsatisfiable concepts in a rebuttal ontology together, an adapted revision algorithm is designed to deal with them group by group. We conduct experiments over 19 ontology pairs and compare our algorithms and scoring functions with existing ones. According to the experiments, it shows that our algorithms could achieve promising performance. The adapted revision algorithm could improve the efficiency largely, and at most 96% time could be saved for some ontology pairs. Some of our scoring functions help a revision algorithm obtain better results in many cases, especially for the challenging pairs.

On the Convergence and Sample Complexity Analysis of Deep Q-Networks with $ε$-Greedy Exploration

This paper provides a theoretical understanding of Deep Q-Network (DQN) with the $\varepsilon$-greedy exploration in deep reinforcement learning. Despite the tremendous empirical achievement of the DQN, its theoretical characterization remains underexplored. First, the exploration strategy is either impractical or ignored in the existing analysis. Second, in contrast to conventional Q-learning algorithms, the DQN employs the target network and experience replay to acquire an unbiased estimation of the mean-square Bellman error (MSBE) utilized in training the Q-network. However, the existing theoretical analysis of DQNs lacks convergence analysis or bypasses the technical challenges by deploying a significantly overparameterized neural network, which is not computationally efficient. This paper provides the first theoretical convergence and sample complexity analysis of the practical setting of DQNs with $\epsilon$-greedy policy. We prove an iterative procedure with decaying $\epsilon$ converges to the optimal Q-value function geometrically. Moreover, a higher level of $\epsilon$ values enlarges the region of convergence but slows down the convergence, while the opposite holds for a lower level of $\epsilon$ values. Experiments justify our established theoretical insights on DQNs.

FedLogic: Interpretable Federated Multi-Domain Chain-of-Thought Prompt Selection for Large Language Models

Leveraging ``chain-of-thought (CoT)'' reasoning to elicit rapid and precise responses from large language models (LLMs) is rapidly attracting research interest. A notable challenge here is how to design or select optimal prompts. The process of prompt selection relies on trial and error, involving continuous adjustments and combinations of input prompts by users based on the corresponding new responses generated from LLMs. Furthermore, minimal research has been conducted to explore how LLMs employ the mathematical problem-solving capabilities learned from user interactions to address issues in narrative writing. To improve interpretability and explore the balance principle between generality and personalization under a multi-domain CoT prompt selection scenario, we propose the Federated Logic rule learning approach (FedLogic). We introduce a theoretical formalization and interactive emulation of the multi-domain CoT prompt selection dilemma in the context of federated LLMs. We cast the problem of joint probability modeling as a bilevel program, where the CoT prompt selection intricacy can be likened to a fuzzy score-based rule selection with the LLMs function as rule generators. FedLogic solves this problem through variational expectation maximization (V-EM). In addition, we incorporate two KL-divergence constraints within this probabilistic modeling framework to surmount the intricacies of managing extensive search spaces and accomplishing cross-domain personalization of CoTs. To the best of our knowledge, FedLogic is the first interpretable and principled federated multi-domain CoT prompt selection approach for LLMs.

How Can Context Help? Exploring Joint Retrieval of Passage and Personalized Context

The integration of external personalized context information into document-grounded conversational systems has significant potential business value, but has not been well-studied. Motivated by the concept of personalized context-aware document-grounded conversational systems, we introduce the task of context-aware passage retrieval. We also construct a dataset specifically curated for this purpose. We describe multiple baseline systems to address this task, and propose a novel approach, Personalized Context-Aware Search (PCAS), that effectively harnesses contextual information during passage retrieval. Experimental evaluations conducted on multiple popular dense retrieval systems demonstrate that our proposed approach not only outperforms the baselines in retrieving the most relevant passage but also excels at identifying the pertinent context among all the available contexts. We envision that our contributions will serve as a catalyst for inspiring future research endeavors in this promising direction.

A Generalized Alternating Method for Bilevel Learning under the Polyak-Łojasiewicz Condition

Bilevel optimization has recently regained interest owing to its applications in emerging machine learning fields such as hyperparameter optimization, meta-learning, and reinforcement learning. Recent results have shown that simple alternating (implicit) gradient-based algorithms can achieve the same convergence rate of single-level gradient descent (GD) for bilevel problems with a strongly convex lower-level objective. However, it remains unclear whether this result can be generalized to bilevel problems beyond this basic setting. In this paper, we propose a Generalized ALternating mEthod for bilevel opTimization (GALET) with a nonconvex lower-level objective that satisfies the Polyak-{\L}ojasiewicz (PL) condition. We first introduce a stationary metric for the considered bilevel problems, which generalizes the existing metric. We then establish that GALET achieves an $\epsilon$-stationary metric for the considered problem within $\tilde{\cal O}(\epsilon^{-1})$ iterations, which matches the iteration complexity of GD for smooth nonconvex problems.