Di Fang

Run He, Kai Tong, Di Fang et al.

In this paper, we introduce analytic federated learning (AFL), a new training paradigm that brings analytical (i.e., closed-form) solutions to the federated learning (FL) with pre-trained models. Our AFL draws inspiration from analytic learning -- a gradient-free technique that trains neural networks with analytical solutions in one epoch. In the local client training stage, the AFL facilitates a one-epoch training, eliminating the necessity for multi-epoch updates. In the aggregation stage, we derive an absolute aggregation (AA) law. This AA law allows a single-round aggregation, reducing heavy communication overhead and achieving fast convergence by removing the need for multiple aggregation rounds. More importantly, the AFL exhibits a property that \textit{invariance to data partitioning}, meaning that regardless of how the full dataset is distributed among clients, the aggregated result remains identical. This could spawn various potentials, such as data heterogeneity invariance and client-number invariance. We conduct experiments across various FL settings including extremely non-IID ones, and scenarios with a large number of clients (e.g., $\ge 1000$). In all these settings, our AFL constantly performs competitively while existing FL techniques encounter various obstacles. Our codes are available at https://github.com/ZHUANGHP/Analytic-federated-learning.

Hsin-Yuan Huang, Yu Tong, Di Fang et al.

Learning a many-body Hamiltonian from its dynamics is a fundamental problem in physics. In this work, we propose the first algorithm to achieve the Heisenberg limit for learning an interacting $N$-qubit local Hamiltonian. After a total evolution time of $\mathcal{O}(ε^{-1})$, the proposed algorithm can efficiently estimate any parameter in the $N$-qubit Hamiltonian to $ε$-error with high probability. The proposed algorithm is robust against state preparation and measurement error, does not require eigenstates or thermal states, and only uses $\mathrm{polylog}(ε^{-1})$ experiments. In contrast, the best previous algorithms, such as recent works using gradient-based optimization or polynomial interpolation, require a total evolution time of $\mathcal{O}(ε^{-2})$ and $\mathcal{O}(ε^{-2})$ experiments. Our algorithm uses ideas from quantum simulation to decouple the unknown $N$-qubit Hamiltonian $H$ into noninteracting patches, and learns $H$ using a quantum-enhanced divide-and-conquer approach. We prove a matching lower bound to establish the asymptotic optimality of our algorithm.

Di Fang, Yinan Zhu, Runze Fang et al.

Continual learning enables AI models to learn new data sequentially without retraining in real-world scenarios. Most existing methods assume the training data are balanced, aiming to reduce the catastrophic forgetting problem that models tend to forget previously generated data. However, data imbalance and the mixture of new and old data in real-world scenarios lead the model to ignore categories with fewer training samples. To solve this problem, we propose an analytic imbalance rectifier algorithm (AIR), a novel online exemplar-free continual learning method with an analytic (i.e., closed-form) solution for data-imbalanced class-incremental learning (CIL) and generalized CIL scenarios in real-world continual learning. AIR introduces an analytic re-weighting module (ARM) that calculates a re-weighting factor for each class for the loss function to balance the contribution of each category to the overall loss and solve the problem of imbalanced training data. AIR uses the least squares technique to give a non-discriminatory optimal classifier and its iterative update method in continual learning. Experimental results on multiple datasets show that AIR significantly outperforms existing methods in long-tailed and generalized CIL scenarios. The source code is available at https://github.com/fang-d/AIR.

Huiping Zhuang, Yizhu Chen, Di Fang et al.

Class incremental learning (CIL) trains a network on sequential tasks with separated categories in each task but suffers from catastrophic forgetting, where models quickly lose previously learned knowledge when acquiring new tasks. The generalized CIL (GCIL) aims to address the CIL problem in a more real-world scenario, where incoming data have mixed data categories and unknown sample size distribution. Existing attempts for the GCIL either have poor performance or invade data privacy by saving exemplars. In this paper, we propose a new exemplar-free GCIL technique named generalized analytic continual learning (GACL). The GACL adopts analytic learning (a gradient-free training technique) and delivers an analytical (i.e., closed-form) solution to the GCIL scenario. This solution is derived via decomposing the incoming data into exposed and unexposed classes, thereby attaining a weight-invariant property, a rare yet valuable property supporting an equivalence between incremental learning and its joint training. Such an equivalence is crucial in GCIL settings as data distributions among different tasks no longer pose challenges to adopting our GACL. Theoretically, this equivalence property is validated through matrix analysis tools. Empirically, we conduct extensive experiments where, compared with existing GCIL methods, our GACL exhibits a consistently leading performance across various datasets and GCIL settings. Source code is available at https://github.com/CHEN-YIZHU/GACL.

Run He, Di Fang, Yicheng Xu et al.

Exemplar-Free Class-Incremental Learning (EFCIL) aims to sequentially learn from distinct categories without retaining exemplars but easily suffers from catastrophic forgetting of learned knowledge. While existing EFCIL methods leverage knowledge distillation to alleviate forgetting, they still face two critical challenges: semantic shift and decision bias. Specifically, the embeddings of old tasks shift in the embedding space after learning new tasks, and the classifier becomes biased towards new tasks due to training solely with new data, hindering the balance between old and new knowledge. To address these issues, we propose the Dual-Projection Shift Estimation and Classifier Reconstruction (DPCR) approach for EFCIL. DPCR effectively estimates semantic shift through a dual-projection, which combines a learnable transformation with a row-space projection to capture both task-wise and category-wise shifts. Furthermore, to mitigate decision bias, DPCR employs ridge regression to reformulate a classifier reconstruction process. This reconstruction exploits previous in covariance and prototype of each class after calibration with estimated shift, thereby reducing decision bias. Extensive experiments demonstrate that, on various datasets, DPCR effectively balances old and new tasks, outperforming state-of-the-art EFCIL methods. Our codes are available at https://github.com/RHe502/ICML25-DPCR.

Xiang Zhang, Run He, Jiao Chen et al.

Class-incremental learning (CIL) enables models to learn new classes continually without forgetting previously acquired knowledge. Multi-label CIL (MLCIL) extends CIL to a real-world scenario where each sample may belong to multiple classes, introducing several challenges: label absence, which leads to incomplete historical information due to missing labels, and class imbalance, which results in the model bias toward majority classes. To address these challenges, we propose Label-Augmented Analytic Adaptation (L3A), an exemplar-free approach without storing past samples. L3A integrates two key modules. The pseudo-label (PL) module implements label augmentation by generating pseudo-labels for current phase samples, addressing the label absence problem. The weighted analytic classifier (WAC) derives a closed-form solution for neural networks. It introduces sample-specific weights to adaptively balance the class contribution and mitigate class imbalance. Experiments on MS-COCO and PASCAL VOC datasets demonstrate that L3A outperforms existing methods in MLCIL tasks. Our code is available at https://github.com/scut-zx/L3A.

Suoxin Zhang, Run He, Di Fang et al.

Low-rank adaptation (LoRA) is a widely used parameter-efficient fine-tuning method that places trainable low-rank adapters into frozen pre-trained models. Recent studies show that using fewer LoRA adapters may still maintain or even improve performance, but existing methods still distribute adapters broadly, leaving where to place a limited number of adapters to maximize performance largely open. To investigate this, we introduce PAGE (Projected Adapter Gradient Energy), a gradient-based sensitivity probe that estimates the initial trainable gradient energy available to each candidate LoRA adapter. Surprisingly, we find that PAGE is highly concentrated on a single shallow FFN down-projection across two model families and four downstream tasks. We term this module the dominant adaptation module and show that its layer index is architecture-dependent but task-stable. Motivated by this finding, we propose DomLoRA, a placement method that places a single adapter at the dominant adaptation module. With only ~0.7% of vanilla LoRA's trainable parameters, DomLoRA outperforms it on average across various downstream tasks, including instruction following, mathematical reasoning, code generation, and multi-turn conversation. This method also improves other LoRA variants, supporting the dominant adaptation module perspective as a practical placement guideline.

Di Fang, Jianfeng Lu, Yu Tong et al.

Gibbs state preparation is an important subroutine in quantum computing. In this work we use the detectability lemma to improve Gibbs state preparation. Specifically, we design new Gibbs state preparation methods that do not rely on simulating Lindbladian evolution, thus avoiding the overhead from it. For local Lindbladians consisting of $M$ terms, this approach reduces the cost by a factor of $O(M)$. We also combine the detectability lemma operator and quantum singular value transformation to implement ground state projection operators of frustration-free Hamiltonians, resulting in a quadratic speedup in the spectral gap dependence. Applying this method to Lindbladians for the Gibbs state of local commuting Hamiltonians, we achieve quadratically better dependence on the Lindbladian spectral gap.

Run He, Di Fang, Yizhu Chen et al.

Exemplar-free class-incremental learning (EFCIL) aims to mitigate catastrophic forgetting in class-incremental learning (CIL) without available historical training samples as exemplars. Compared with its exemplar-based CIL counterpart that stores exemplars, EFCIL suffers more from forgetting issues. Recently, a new EFCIL branch named Analytic Continual Learning (ACL) introduces a gradient-free paradigm via Recursive Least-Square, achieving a forgetting-resistant classifier training with a frozen backbone during CIL. However, existing ACL suffers from ineffective representations and insufficient utilization of backbone knowledge. In this paper, we propose a representation-enhanced analytic learning (REAL) to address these problems. To enhance the representation, REAL constructs a dual-stream base pretraining followed by representation enhancing distillation process. The dual-stream base pretraining combines self-supervised contrastive learning for general features and supervised learning for class-specific knowledge, followed by the representation enhancing distillation to merge both streams, enhancing representations for subsequent CIL paradigm. To utilize more knowledge from the backbone, REAL presents a feature fusion buffer to multi-layer backbone features, providing informative features for the subsequent classifier training. Our method can be incorporated into existing ACL techniques and provides more competitive performance. Empirical results demonstrate that, REAL achieves state-of-the-art performance on CIFAR-100, ImageNet-100 and ImageNet-1k benchmarks, outperforming exemplar-free methods and rivaling exemplar-based approaches.

Jianheng Tang, Huiping Zhuang, Di Fang et al.

The development of artificial intelligence demands that models incrementally update knowledge by Continual Learning (CL) to adapt to open-world environments. To meet privacy and security requirements, Continual Unlearning (CU) emerges as an important problem, aiming to sequentially forget particular knowledge acquired during the CL phase. However, existing unlearning methods primarily focus on single-shot joint forgetting and face significant limitations when applied to CU. First, most existing methods require access to the retained dataset for re-training or fine-tuning, violating the inherent constraint in CL that historical data cannot be revisited. Second, these methods often suffer from a poor trade-off between system efficiency and model fidelity, making them vulnerable to being overwhelmed or degraded by adversaries through deliberately frequent requests. In this paper, we identify that the limitations of existing unlearning methods stem fundamentally from their reliance on gradient-based updates. To bridge the research gap at its root, we propose a novel gradient-free method for CU, named Analytic Continual Unlearning (ACU), for efficient and exact forgetting with historical data privacy preservation. In response to each unlearning request, our ACU recursively derives an analytical (i.e., closed-form) solution in an interpretable manner using the least squares method. Theoretical and experimental evaluations validate the superiority of our ACU on unlearning effectiveness, model fidelity, and system efficiency.

Di Fang, Lei Li

The Generalized Langevin Equation, in history, arises as a natural fix for the rather traditional Langevin equation when the random force is no longer memoryless. It has been proved that with fractional Gaussian noise (fGn) mostly considered by biologists, the overdamped Generalized Langevin equation satisfying fluctuation-dissipation theorem can be written as a fractional stochastic differential equation (FSDE). While the ergodicity is clear for linear forces, it remains less transparent for nonlinear forces. In this work, we present both a direct and a fast algorithm respectively to this FSDE model. The strong orders of convergence are proved for both schemes, where the role of the memory effects can be clearly observed. We verify the convergence theorems using linear forces, and then present the ergodicity study of the double well potentials in both 1D and 2D setups.