Deep learning · optimization

Neural Collapse (NC) is a recently observed phenomenon in neural networks that characterises the solution space of the final classifier layer when trained until zero training loss. Specifically, NC suggests that the final classifier layer converges to a Simplex Equiangular Tight Frame (ETF), which maximally separates the weights corresponding to each class. By duality, the penultimate layer feature means also converge to the same simplex ETF. Since this simple symmetric structure is optimal, our idea is to utilise this property to improve convergence speed. Specifically, we introduce the notion of \textit{nearest simplex ETF geometry} for the penultimate layer features at any given training iteration, by formulating it as a Riemannian optimisation. Then, at each iteration, the classifier weights are implicitly set to the nearest simplex ETF by solving this inner-optimisation, which is encapsulated within a declarative node to allow backpropagation. Our experiments on synthetic and real-world architectures on classification tasks demonstrate that our approach accelerates convergence and enhances training stability.

Gradient estimation is critical in zeroth-order optimization methods, which aims to obtain the descent direction by sampling update directions and querying function evaluations. Extensive research has been conducted including smoothing and linear interpolation. The former methods smooth the objective function, causing a biased gradient estimation, while the latter often enjoys more accurate estimates, at the cost of large amounts of samples and queries at each iteration to update variables. This paper resorts to the linear interpolation strategy and proposes to reduce the complexity of gradient estimation by reusing queries in the prior iterations while maintaining the sample size unchanged. Specifically, we model the gradient estimation as a quadratically constrained linear program problem and manage to derive the analytical solution. It innovatively decouples the required sample size from the variable dimension without extra conditions required, making it able to leverage the queries in the prior iterations. Moreover, part of the intermediate variables that contribute to the gradient estimation can be directly indexed, significantly reducing the computation complexity. Experiments on both simulation functions and real scenarios (black-box adversarial attacks neural architecture search, and parameter-efficient fine-tuning for large language models), show its efficacy and efficiency. Our code is available at https://github.com/Thinklab-SJTU/ReLIZO.git.

Can we modify the training data distribution to encourage the underlying optimization method toward finding solutions with superior generalization performance on in-distribution data? In this work, we approach this question for the first time by comparing the inductive bias of gradient descent (GD) with that of sharpness-aware minimization (SAM). By studying a two-layer CNN, we rigorously prove that SAM learns different features more uniformly, particularly in early epochs. That is, SAM is less susceptible to simplicity bias compared to GD. We also show that examples constraining features that are learned early are separable from the rest based on the model’s output. Based on this observation, we propose a method that (i) clusters examples based on the network output early in training, (ii) identifies a cluster of examples with similar network output, and (iii) upsamples the rest of examples only once to alleviate the simplicity bias. We show empirically that USEFUL effectively improves the generalization performance on the original data distribution when training with various gradient methods, including (S)GD and SAM. Notably, we demonstrate that our method can be combined with SAM variants and existing data augmentation strategies to achieve, to the best of our knowledge, state-of-the-art performance for training ResNet18 on CIFAR10, STL10, CINIC10, Tiny-ImageNet; ResNet34 on CIFAR100; and VGG19 and DenseNet121 on CIFAR10.

Unsupervised learning (UL)-based solvers for combinatorial optimization (CO) train a neural network that generates a soft solution by directly optimizing the CO objective using a continuous relaxation strategy. These solvers offer several advantages over traditional methods and other learning-based methods, particularly for large-scale CO problems. However, UL-based solvers face two practical issues: (I) an optimization issue, where UL-based solvers are easily trapped at local optima, and (II) a rounding issue, where UL-based solvers require artificial post-learning rounding from the continuous space back to the original discrete space, undermining the robustness of the results. This study proposes a Continuous Relaxation Annealing (CRA) strategy, an effective rounding-free learning method for UL-based solvers. CRA introduces a penalty term that dynamically shifts from prioritizing continuous solutions, effectively smoothing the non-convexity of the objective function, to enforcing discreteness, eliminating artificial rounding. Experimental results demonstrate that CRA significantly enhances the performance of UL-based solvers, outperforming existing UL-based solvers and greedy algorithms in complex CO problems. Additionally, CRA effectively eliminates artificial rounding and accelerates the learning process.

Transformer-based trajectory optimization methods have demonstrated exceptional performance in offline Reinforcement Learning (offline RL). Yet, it poses challenges due to substantial parameter size and limited scalability, which is particularly critical in sequential decision-making scenarios where resources are constrained such as in robots and drones with limited computational power. Mamba, a promising new linear-time sequence model, offers performance on par with transformers while delivering substantially fewer parameters on long sequences. As it remains unclear whether Mamba is compatible with trajectory optimization, this work aims to conduct comprehensive experiments to explore the potential of Decision Mamba (dubbed DeMa) in offline RL from the aspect of data structures and essential components with the following insights: (1) Long sequences impose a significant computational burden without contributing to performance improvements since DeMa's focus on sequences diminishes approximately exponentially. Consequently, we introduce a Transformer-like DeMa as opposed to an RNN-like DeMa. (2) For the components of DeMa, we identify the hidden attention mechanism as a critical factor in its success, which can also work well with other residual structures and does not require position embedding. Extensive evaluations demonstrate that our specially designed DeMa is compatible with trajectory optimization and surpasses previous methods, outperforming Decision Transformer (DT) with higher performance while using 30\% fewer parameters in Atari, and exceeding DT with only a quarter of the parameters in MuJoCo.

Continuously adapting pre-trained models to local data on resource constrained edge devices is the \emph{last mile} for model deployment. However, as models increase in size and depth, backpropagation requires a large amount of memory, which becomes prohibitive for edge devices. In addition, most existing low power neural processing engines (e.g., NPUs, DSPs, MCUs, etc.) are designed as fixed-point inference accelerators, without training capabilities. Forward gradients, solely based on directional derivatives computed from two forward calls, have been recently used for model training, with substantial savings in computation and memory. However, the performance of quantized training with fixed-point forward gradients remains unclear. In this paper, we investigate the feasibility of on-device training using fixed-point forward gradients, by conducting comprehensive experiments across a variety of deep learning benchmark tasks in both vision and audio domains. We propose a series of algorithm enhancements that further reduce the memory footprint, and the accuracy gap compared to backpropagation. An empirical study on how training with forward gradients navigates in the loss landscape is further explored. Our results demonstrate that on the last mile of model customization on edge devices, training with fixed-point forward gradients is a feasible and practical approach.

We initiate an investigation into the optimization properties of next-token prediction (NTP), the dominant training paradigm for modern language models. Specifically, we study the structural properties of the solutions selected by gradient-based optimizers among the many possible minimizers of the NTP objective. By framing NTP as cross-entropy minimization across \emph{distinct} contexts, each tied with a \emph{sparse} conditional probability distribution across a finite vocabulary of tokens, we introduce ``NTP-separability conditions'' that enable reaching the data-entropy lower bound. With this setup, and focusing on linear models with fixed context embeddings, we characterize the optimization bias of gradient descent (GD): Within the data subspace defined by the sparsity patterns of distinct contexts, GD selects parameters that equate the logits' differences of in-support tokens to their log-odds. In the orthogonal subspace, the GD parameters diverge in norm and select the direction that maximizes a margin specific to NTP. These findings extend previous research on implicit bias in one-hot classification to the NTP setting, highlighting key differences and prompting further research into the optimization and generalization properties of NTP, irrespective of the specific architecture used to generate the context embeddings.

Adam has become one of the most favored optimizers in deep learning problems. Despite its success in practice, numerous mysteries persist regarding its theoretical understanding. In this paper, we study the implicit bias of Adam in linear logistic regression. Specifically, we show that when the training data are linearly separable, the iterates of Adam converge towards a linear classifier that achieves the maximum $\ell_\infty$-margin in direction. Notably, for a general class of diminishing learning rates, this convergence occurs within polynomial time. Our result shed light on the difference between Adam and (stochastic) gradient descent from a theoretical perspective.

Federated learning faces challenges due to the heterogeneity in data volumes and distributions at different clients, which can compromise model generalization ability to various distributions. Existing approaches to address this issue based on group distributionally robust optimization (GDRO) often lead to high communication and sample complexity. To this end, this work introduces algorithms tailored for communication-efficient Federated Group Distributionally Robust Optimization (FGDRO). Our contributions are threefold: Firstly, we introduce the FGDRO-CVaR algorithm, which optimizes the average top-K losses while reducing communication complexity to $O(1/\epsilon^4)$, where $\epsilon$ denotes the desired precision level. Secondly, our FGDRO-KL algorithm is crafted to optimize KL regularized FGDRO, cutting communication complexity to $O(1/\epsilon^3)$. Lastly, we propose FGDRO-KL-Adam to utilize Adam-type local updates in FGDRO-KL, which not only maintains a communication cost of $O(1/\epsilon^3)$ but also shows potential to surpass SGD-type local steps in practical applications. The effectiveness of our algorithms has been demonstrated on a variety of real-world tasks, including natural language processing and computer vision.

We propose a Regularized Adaptive Momentum Dual Averaging (RAMDA) algorithm for training structured neural networks. Similar to existing regularized adaptive methods, the subproblem for computing the update direction of RAMDA involves a nonsmooth regularizer and a diagonal preconditioner, and therefore does not possess a closed-form solution in general. We thus also carefully devise an implementable inexactness condition that retains convergence guarantees similar to the exact versions, and propose a companion efficient solver for the subproblems of both RAMDA and existing methods to make them practically feasible. We leverage the theory of manifold identification in variational analysis to show that, even in the presence of such inexactness, the iterates of RAMDA attain the ideal structure induced by the regularizer at the stationary point of asymptotic convergence. This structure is locally optimal near the point of convergence, so RAMDA is guaranteed to obtain the best structure possible among all methods converging to the same point, making it the first regularized adaptive method outputting models that possess outstanding predictive performance while being (locally) optimally structured. Extensive numerical experiments in large-scale modern computer vision, language modeling, and speech tasks show that the proposed RAMDA is efficient and consistently outperforms state of the art for training structured neural network. Implementation of our algorithm is available at https://www.github.com/ismoptgroup/RAMDA.

In this paper, we study Adam in non-convex smooth scenarios with potential unbounded gradients and affine variance noise. We consider a general noise model which governs affine variance noise, bounded noise, and sub-Gaussian noise. We show that Adam with a specific hyper-parameter setup can find a stationary point with a $\mathcal{O}(\text{poly}(\log T)/\sqrt{T})$ rate in high probability under this general noise model where $T$ denotes total number iterations, matching the lower rate of stochastic first-order algorithms up to logarithm factors. We also provide a probabilistic convergence result for Adam under a generalized smooth condition which allows unbounded smoothness parameters and has been illustrated empirically to capture the smooth property of many practical objective functions more accurately.

Deep learning succeeds by doing hierarchical feature learning, yet tuning hyper-parameters (HP) such as initialization scales, learning rates etc., only give indirect control over this behavior. In this paper, we introduce a key notion to predict and control feature learning: the angle $\theta_\ell$ between the feature updates and the backward pass (at layer index $\ell$). We show that the magnitude of feature updates after one GD step, at any training time, can be expressed via a simple and general *feature speed formula* in terms of this angle $\theta_\ell$, the loss decay, and the magnitude of the backward pass. This angle $\theta_\ell$ is controlled by the conditioning of the layer-to-layer Jacobians and at random initialization, it is determined by the spectrum of a certain kernel, which coincides with the Neural Tangent Kernel when $\ell=\text{depth}$. Given $\theta_\ell$, the feature speed formula provides us with rules to adjust HPs (scales and learning rates) so as to satisfy certain dynamical properties, such as feature learning and loss decay. We investigate the implications of our approach for ReLU MLPs and ResNets in the large width-then-depth limit. Relying on prior work, we show that in ReLU MLPs with iid initialization, the angle degenerates with depth as $\cos(\theta_\ell)=\Theta(1/\sqrt{\ell})$. In contrast, ResNets with branch scale $O(1/\sqrt{\text{depth}})$ maintain a non-degenerate angle $\cos(\theta_\ell)=\Theta(1)$. We use these insights to recover key properties of known HP scalings (such as $\mu$P), and also introduce a new HP scaling for large depth ReLU MLPs with favorable theoretical properties.

This paper considers the problem for finding the $(\delta,\epsilon)$-Goldstein stationary point of Lipschitz continuous objective, which is a rich function class to cover a great number of important applications. We construct a novel zeroth-order quantum estimator for the gradient of the smoothed surrogate. Based on such estimator, we propose a novel quantum algorithm that achieves a query complexity of $\tilde{\mathcal{O}}(d^{3/2}\delta^{-1}\epsilon^{-3})$ on the stochastic function value oracle, where $d$ is the dimension of the problem. We also enhance the query complexity to $\tilde{\mathcal{O}}(d^{3/2}\delta^{-1}\epsilon^{-7/3})$ by introducing a variance reduction variant. Our findings demonstrate the clear advantages of utilizing quantum techniques for non-convex non-smooth optimization, as they outperform the optimal classical methods on the dependency of $\epsilon$ by a factor of $\epsilon^{-2/3}$.

We introduce camera ray matching (CRAYM) into the joint optimization of camera poses and neural fields from multi-view images. The optimized field, referred to as a feature volume, can be “probed” by the camera rays for novel view synthesis (NVS) and 3D geometry reconstruction. One key reason for matching camera rays, instead of pixels as in prior works, is that the camera rays can be parameterized by the feature volume to carry both geometric and photometric information. Multi-view consistencies involving the camera rays and scene rendering can be naturally integrated into the joint optimization and network training, to impose physically meaningful constraints to improve the final quality of both the geometric reconstruction and photorealistic rendering. We formulate our per-ray optimization and matched ray coherence by focusing on camera rays passing through keypoints in the input images to elevate both the efficiency and accuracy of scene correspondences. Accumulated ray features along the feature volume provide a means to discount the coherence constraint amid erroneous ray matching. We demonstrate the effectiveness of CRAYM for both NVS and geometry reconstruction, over dense- or sparse-view settings, with qualitative and quantitative comparisons to state-of-the-art alternatives.

Recent studies on deep ensembles have identified the sharpness of the local minima of individual learners and the diversity of the ensemble members as key factors in improving test-time performance. Building on this, our study investigates the interplay between sharpness and diversity within deep ensembles, illustrating their crucial role in robust generalization to both in-distribution (ID) and out-of-distribution (OOD) data. We discover a trade-off between sharpness and diversity: minimizing the sharpness in the loss landscape tends to diminish the diversity of individual members within the ensemble, adversely affecting the ensemble's improvement. The trade-off is justified through our rigorous theoretical analysis and verified empirically through extensive experiments. To address the issue of reduced diversity, we introduce SharpBalance, a novel training approach that balances sharpness and diversity within ensembles. Theoretically, we show that our training strategy achieves a better sharpness-diversity trade-off. Empirically, we conducted comprehensive evaluations in various data sets (CIFAR-10, CIFAR-100, TinyImageNet) and showed that SharpBalance not only effectively improves the sharpness-diversity trade-off but also significantly improves ensemble performance in ID and OOD scenarios.

In deep learning, model performance often deteriorates when trained on highly imbalanced datasets, especially when evaluation metrics require robust generalization across underrepresented classes. To address the challenges posed by imbalanced data distributions, this study introduces a novel method utilizing density ratio estimation for dynamic class weight adjustment, termed as Re-weighting with Density Ratio (RDR). Our method adaptively adjusts the importance of each class during training, mitigates overfitting on dominant classes and enhances model adaptability across diverse datasets. Extensive experiments conducted on various large scale benchmark datasets validate the effectiveness of our method. Results demonstrate substantial improvements in generalization capabilities, particularly under severely imbalanced conditions.

Federated Learning (FL) is a prevalent machine learning paradigm designed to address challenges posed by heterogeneous client data while preserving data privacy. Unlike distributed training, it typically orchestrates resource-constrained edge devices to communicate via a low-bandwidth communication network with a central server. This urges the development of more computation and communication efficient training algorithms. In this paper, we propose an efficient FL paradigm, where the local models in the clients are trained with low-precision operations and communicated with the server in low precision format, while only the model aggregation in the server is performed with high-precision computation. We surprisingly find that high precision models can be recovered from the low precision local models with proper aggregation in the server. In this way, both the workload in the client-side and the communication cost can be significantly reduced. We theoretically show that our proposed paradigm can converge to the optimal solution as the training goes on, which demonstrates that low precision local training is enough for FL. Our paradigm can be integrated with existing FL algorithms flexibly. Experiments across extensive benchmarks are conducted to showcase the effectiveness of our proposed method. Notably, the models trained by our method with the precision as low as 8 bits are comparable to those from the full precision training. As a by-product, we show that low precision local training can relieve the over-fitting issue in local training, which under heterogeneous client data can cause the client models drift further away from each other and lead to the failure in model aggregation. Code is released at https://github.com/digbangbang/LPT-FL.

Noisy labels pose a common challenge for training accurate deep neural networks. To mitigate label noise, prior studies have proposed various robust loss functions to achieve noise tolerance in the presence of label noise, particularly symmetric losses. However, they usually suffer from the underfitting issue due to the overly strict symmetric condition. In this work, we propose a simple yet effective approach for relaxing the symmetric condition, namely **$\epsilon$-softmax**, which simply modifies the outputs of the softmax layer to approximate one-hot vectors with a controllable error $\epsilon$. Essentially, ***$\epsilon$-softmax** not only acts as an alternative for the softmax layer, but also implicitly plays the crucial role in modifying the loss function.* We prove theoretically that **$\epsilon$-softmax** can achieve noise-tolerant learning with controllable excess risk bound for almost any loss function. Recognizing that **$\epsilon$-softmax**-enhanced losses may slightly reduce fitting ability on clean datasets, we further incorporate them with one symmetric loss, thereby achieving a better trade-off between robustness and effective learning. Extensive experiments demonstrate the superiority of our method in mitigating synthetic and real-world label noise.

When training neural networks with custom objectives, such as ranking losses and shortest-path losses, a common problem is that they are, per se, non-differentiable. A popular approach is to continuously relax the objectives to provide gradients, enabling learning. However, such differentiable relaxations are often non-convex and can exhibit vanishing and exploding gradients, making them (already in isolation) hard to optimize. Here, the loss function poses the bottleneck when training a deep neural network. We present Newton Losses, a method for improving the performance of existing hard to optimize losses by exploiting their second-order information via their empirical Fisher and Hessian matrices. Instead of training the neural network with second-order techniques, we only utilize the loss function's second-order information to replace it by a Newton Loss, while training the network with gradient descent. This makes our method computationally efficient. We apply Newton Losses to eight differentiable algorithms for sorting and shortest-paths, achieving significant improvements for less-optimized differentiable algorithms, and consistent improvements, even for well-optimized differentiable algorithms.

The immense parameter scale of large language models underscores the necessity for parameter-efficient fine-tuning methods. Methods based on Low-Rank Adaptation (LoRA) assume the low-rank characteristics of the incremental matrix and optimize the matrix obtained from low-rank decomposition. Although effective, these methods are constrained by a fixed and unalterable intrinsic rank, neglecting the variable importance of matrices. Consequently, methods for adaptive rank allocation are proposed, among which AdaLoRA demonstrates excellent fine-tuning performance. AdaLoRA conducts adaptation based on singular value decomposition (SVD), dynamically allocating intrinsic ranks according to importance. However, it still struggles to achieve a balance between fine-tuning effectiveness and efficiency, leading to limited rank allocation space. Additionally, the importance measurement focuses only on parameters with minimal impact on the loss, neglecting the dominant role of singular values in SVD-based matrices and the fluctuations during training. To address these issues, we propose SalientLoRA, which adaptively optimizes intrinsic ranks of LoRA via salience measurement. Firstly, during rank allocation, the salience measurement analyses the variation of singular value magnitudes across multiple time steps and establishes their inter-dependency relationships to assess the matrix importance. This measurement mitigates instability and randomness that may arise during importance assessment. Secondly, to achieve a balance between fine-tuning performance and efficiency, we propose an adaptive adjustment of time-series window, which adaptively controls the size of time-series for significance measurement and rank reduction during training, allowing for rapid rank allocation while maintaining training stability. This mechanism enables matrics to set a higher initial rank, thus expanding the allocation space for ranks. To evaluate the generality of our method across various tasks, we conduct experiments on natural language understanding (NLU), natural language generation (NLG), and large model instruction tuning tasks. Experimental results demonstrate the superiority of SalientLoRA, which outperforms state-of-the-art methods by 0.96\%-3.56\% on multiple datasets. Furthermore, as the rank allocation space expands, our method ensures fine-tuning efficiency, achieving a speed improvement of 94.5\% compared to AdaLoRA. The code is publicly available at https://github.com/Heyest/SalientLoRA.