On the Convergence of Adam-Type Algorithm for Bilevel Optimization under Unbounded Smoothness

Adam has become one of the most popular optimizers for training modern deep neural networks, such as transformers. However, its applicability is largely restricted to single-level optimization problems. In this paper, we aim to extend vanilla Adam to tackle bilevel optimization problems, which have important applications in machine learning, such as meta-learning. In particular, we study stochastic bilevel optimization problems where the lower-level function is strongly convex and the upper-level objective is nonconvex with potentially unbounded smoothness. This unbounded smooth objective function covers a broad class of neural networks, including transformers, which may exhibit non-Lipschitz gradients. In this work, we introduce AdamBO, a single-loop Adam-type method that achieves $\widetilde{O}(\epsilon^{-4})$ oracle complexity to find $\epsilon$-stationary points, where the oracle calls involve stochastic gradient or Hessian/Jacobian-vector product evaluations. The key to our analysis is a novel randomness decoupling lemma that provides refined control over the lower-level variable. We conduct extensive experiments on various machine learning tasks involving bilevel formulations with recurrent neural networks (RNNs) and transformers, demonstrating the effectiveness of our proposed Adam-type algorithm.

Personalized Federated Fine-tuning for Heterogeneous Data: An Automatic Rank Learning Approach via Two-Level LoRA

We study the task of personalized federated fine-tuning with heterogeneous data in the context of language models, where clients collaboratively fine-tune a language model (e.g., BERT, GPT) without sharing their local data, achieving personalization simultaneously. While recent efforts have applied parameter-efficient fine-tuning techniques like low-rank adaptation (LoRA) in federated settings, they typically use single or multiple independent low-rank adapters with predefined maximal and minimal ranks, which may not be optimal for diverse data sources over clients. To address this issue, we propose PF2LoRA, a new personalized federated fine-tuning algorithm built on a novel \emph{automatic rank learning approach via two-level LoRA}. Given the pretrained language model whose weight is frozen, our algorithm aims to learn two levels of adaptation simultaneously: the first level aims to learn a common adapter for all clients, while the second level fosters individual client personalization. A key advantage of PF2LoRA is its ability to adaptively determine a suitable rank based on an individual client's data, rather than relying on a predefined rank that is agnostic to data heterogeneity. We present a synthetic example that highlights how PF2LoRA automatically learns the ground-truth rank for each client, tailoring the adaptation to match the properties of their individual data. Notably, this approach introduces minimal additional memory overhead, as the second-level adaptation comprises a small number of parameters compared to the first level. Our experiments on natural language understanding and generation tasks demonstrate that PF2LoRA significantly outperforms existing federated fine-tuning methods.

Local Steps Speed Up Local GD for Heterogeneous Distributed Logistic Regression

We analyze two variants of Local Gradient Descent applied to distributed logistic regression with heterogeneous, separable data and show convergence at the rate $O(1/KR)$ for $K$ local steps and sufficiently large $R$ communication rounds. In contrast, all existing convergence guarantees for Local GD applied to any problem are at least $\Omega(1/R)$, meaning they fail to show the benefit of local updates. The key to our improved guarantee is showing progress on the logistic regression objective when using a large stepsize $\eta \gg 1/K$, whereas prior analysis depends on $\eta \leq 1/K$.

A Nearly Optimal Single Loop Algorithm for Stochastic Bilevel Optimization under Unbounded Smoothness

This paper studies the problem of stochastic bilevel optimization where the upper-level function is nonconvex with potentially unbounded smoothness and the lower-level function is strongly convex. This problem is motivated by meta-learning applied to sequential data, such as text classification using recurrent neural networks, where the smoothness constant of the upper-level loss function scales linearly with the gradient norm and can be potentially unbounded. Existing algorithm crucially relies on the nested loop design, which requires significant tuning efforts and is not practical. In this paper, we address this issue by proposing a Single Loop bIlevel oPtimizer (SLIP). The proposed algorithm first updates the lower-level variable by a few steps of stochastic gradient descent, and then simultaneously updates the upper-level variable by normalized stochastic gradient descent with momentum and the lower-level variable by stochastic gradient descent. Under standard assumptions, we show that our algorithm finds an $\epsilon$-stationary point within $\widetilde{O}(1/\epsilon^4)$\footnote{Here $\widetilde{O}(\cdot)$ compresses logarithmic factors of $1/\epsilon$ and $1/\delta$, where $\delta\in(0,1)$ denotes the failure probability.} oracle calls of stochastic gradient or Hessian-vector product, both in expectation and with high probability. This complexity result is nearly optimal up to logarithmic factors without mean-square smoothness of the stochastic gradient oracle. Our proof relies on (i) a refined characterization and control of the lower-level variable and (ii) establishing a novel connection between bilevel optimization and stochastic optimization under distributional drift. Our experiments on various tasks show that our algorithm significantly outperforms strong baselines in bilevel optimization.

Facet-Aware Multi-Head Mixture-of-Experts Model for Sequential Recommendation

Sequential recommendation (SR) systems excel at capturing users' dynamic preferences by leveraging their interaction histories. Most existing SR systems assign a single embedding vector to each item to represent its features, and various types of models are adopted to combine these item embeddings into a sequence representation vector to capture the user intent. However, we argue that this representation alone is insufficient to capture an item's multi-faceted nature (e.g., movie genres, starring actors). Besides, users often exhibit complex and varied preferences within these facets (e.g., liking both action and musical films in the facet of genre), which are challenging to fully represent. To address the issues above, we propose a novel structure called Facet-Aware Multi-Head Mixture-of-Experts Model for Sequential Recommendation (FAME). We leverage sub-embeddings from each head in the last multi-head attention layer to predict the next item separately. This approach captures the potential multi-faceted nature of items without increasing model complexity. A gating mechanism integrates recommendations from each head and dynamically determines their importance. Furthermore, we introduce a Mixture-of-Experts (MoE) network in each attention head to disentangle various user preferences within each facet. Each expert within the MoE focuses on a specific preference. A learnable router network is adopted to compute the importance weight for each expert and aggregate them. We conduct extensive experiments on four public sequential recommendation datasets and the results demonstrate the effectiveness of our method over existing baseline models.

Federated Learning under Periodic Client Participation and Heterogeneous Data: A New Communication-Efficient Algorithm and Analysis

In federated learning, it is common to assume that clients are always available to participate in training, which may not be feasible with user devices in practice. Recent works analyze federated learning under more realistic participation patterns, such as cyclic client availability or arbitrary participation. However, all such works either require strong assumptions (e.g., all clients participate almost surely within a bounded window), do not achieve linear speedup and reduced communication rounds, or are not applicable in the general non-convex setting. In this work, we focus on nonconvex optimization and consider participation patterns in which the chance of participation over a fixed window of rounds is equal among all clients, which includes cyclic client availability as a special case. Under this setting, we propose a new algorithm, named Amplified SCAFFOLD, and prove that it achieves linear speedup, reduced communication, and resilience to data heterogeneity simultaneously. In particular, for cyclic participation, our algorithm is proved to enjoy $\mathcal{O}(\epsilon^{-2})$ communication rounds to find an $\epsilon$-stationary point in the non-convex stochastic setting. In contrast, the prior work under the same setting requires $\mathcal{O}(\kappa^2 \epsilon^{-4})$ communication rounds, where $\kappa$ denotes the data heterogeneity. Therefore, our algorithm significantly reduces communication rounds due to better dependency in terms of $\epsilon$ and $\kappa$. Our analysis relies on a fine-grained treatment of the nested dependence between client participation and errors in the control variates, which results in tighter guarantees than previous work. We also provide experimental results with (1) synthetic data and (2) real-world data with a large number of clients $(N = 250)$, demonstrating the effectiveness of our algorithm under periodic client participation.

Mask-based Membership Inference Attacks for Retrieval-Augmented Generation

Retrieval-Augmented Generation (RAG) has been an effective approach to mitigate hallucinations in large language models (LLMs) by incorporating up-to-date and domain-specific knowledge. Recently, there has been a trend of storing up-to-date or copyrighted data in RAG knowledge databases instead of using it for LLM training. This practice has raised concerns about Membership Inference Attacks (MIAs), which aim to detect if a specific target document is stored in the RAG system's knowledge database so as to protect the rights of data producers. While research has focused on enhancing the trustworthiness of RAG systems, existing MIAs for RAG systems remain largely insufficient. Previous work either relies solely on the RAG system's judgment or is easily influenced by other documents or the LLM's internal knowledge, which is unreliable and lacks explainability. To address these limitations, we propose a Mask-Based Membership Inference Attacks (MBA) framework. Our framework first employs a masking algorithm that effectively masks a certain number of words in the target document. The masked text is then used to prompt the RAG system, and the RAG system is required to predict the mask values. If the target document appears in the knowledge database, the masked text will retrieve the complete target document as context, allowing for accurate mask prediction. Finally, we adopt a simple yet effective threshold-based method to infer the membership of target document by analyzing the accuracy of mask prediction. Our mask-based approach is more document-specific, making the RAG system's generation less susceptible to distractions from other documents or the LLM's internal knowledge. Extensive experiments demonstrate the effectiveness of our approach compared to existing baseline models.

An Accelerated Algorithm for Stochastic Bilevel Optimization under Unbounded Smoothness

This paper investigates a class of stochastic bilevel optimization problems where the upper-level function is nonconvex with potentially unbounded smoothness and the lower-level problem is strongly convex. These problems have significant applications in sequential data learning, such as text classification using recurrent neural networks. The unbounded smoothness is characterized by the smoothness constant of the upper-level function scaling linearly with the gradient norm, lacking a uniform upper bound. Existing state-of-the-art algorithms require $\widetilde{O}(1/\epsilon^4)$ oracle calls of stochastic gradient or Hessian/Jacobian-vector product to find an $\epsilon$-stationary point. However, it remains unclear if we can further improve the convergence rate when the assumptions for the function in the population level also hold for each random realization almost surely (e.g., Lipschitzness of each realization of the stochastic gradient). To address this issue, we propose a new Accelerated Bilevel Optimization algorithm named AccBO. The algorithm updates the upper-level variable by normalized stochastic gradient descent with recursive momentum and the lower-level variable by the stochastic Nesterov accelerated gradient descent algorithm with averaging. We prove that our algorithm achieves an oracle complexity of $\widetilde{O}(1/\epsilon^3)$ to find an $\epsilon$-stationary point. Our proof relies on a novel lemma characterizing the dynamics of stochastic Nesterov accelerated gradient descent algorithm under distribution drift with high probability for the lower-level variable, which is of independent interest and also plays a crucial role in analyzing the hypergradient estimation error over time. Experimental results on various tasks confirm that our proposed algorithm achieves the predicted theoretical acceleration and significantly outperforms baselines in bilevel optimization.

Beyond Single-Model Views for Deep Learning: Optimization versus Generalizability of Stochastic Optimization Algorithms

Despite an extensive body of literature on deep learning optimization, our current understanding of what makes an optimization algorithm effective is fragmented. In particular, we do not understand well whether enhanced optimization translates to improved generalizability. Current research overlooks the inherent stochastic nature of stochastic gradient descent (SGD) and its variants, resulting in a lack of comprehensive benchmarking and insight into their statistical performance. This paper aims to address this gap by adopting a novel approach. Rather than solely evaluating the endpoint of individual optimization trajectories, we draw from an ensemble of trajectories to estimate the stationary distribution of stochastic optimizers. Our investigation encompasses a wide array of techniques, including SGD and its variants, flat-minima optimizers, and new algorithms we propose under the Basin Hopping framework. Through our evaluation, which encompasses synthetic functions with known minima and real-world problems in computer vision and natural language processing, we emphasize fair benchmarking under a statistical framework, comparing stationary distributions and establishing statistical significance. Our study uncovers several key findings regarding the relationship between training loss and hold-out accuracy, as well as the comparable performance of SGD, noise-enabled variants, and novel optimizers utilizing the BH framework. Notably, these algorithms demonstrate performance on par with flat-minima optimizers like SAM, albeit with half the gradient evaluations. We anticipate that our work will catalyze further exploration in deep learning optimization, encouraging a shift away from single-model approaches towards methodologies that acknowledge and leverage the stochastic nature of optimizers.

Bilevel Optimization under Unbounded Smoothness: A New Algorithm and Convergence Analysis

Bilevel optimization is an important formulation for many machine learning problems. Current bilevel optimization algorithms assume that the gradient of the upper-level function is Lipschitz. However, recent studies reveal that certain neural networks such as recurrent neural networks (RNNs) and long-short-term memory networks (LSTMs) exhibit potential unbounded smoothness, rendering conventional bilevel optimization algorithms unsuitable. In this paper, we design a new bilevel optimization algorithm, namely BO-REP, to address this challenge. This algorithm updates the upper-level variable using normalized momentum and incorporates two novel techniques for updating the lower-level variable: \textit{initialization refinement} and \textit{periodic updates}. Specifically, once the upper-level variable is initialized, a subroutine is invoked to obtain a refined estimate of the corresponding optimal lower-level variable, and the lower-level variable is updated only after every specific period instead of each iteration. When the upper-level problem is nonconvex and unbounded smooth, and the lower-level problem is strongly convex, we prove that our algorithm requires $\widetilde{\mathcal{O}}(1/\epsilon^4)$ iterations to find an $\epsilon$-stationary point in the stochastic setting, where each iteration involves calling a stochastic gradient or Hessian-vector product oracle. Notably, this result matches the state-of-the-art complexity results under the bounded smoothness setting and without mean-squared smoothness of the stochastic gradient, up to logarithmic factors. Our proof relies on novel technical lemmas for the periodically updated lower-level variable, which are of independent interest. Our experiments on hyper-representation learning, hyperparameter optimization, and data hyper-cleaning for text classification tasks demonstrate the effectiveness of our proposed algorithm.