This work tackles a critical challenge in AI safety research under limited compute: given a fixed computation budget, how can one maximize the strength of iterative adversarial attacks? Coarsely reducing the number of attack iterations lowers cost but substantially weakens effectiveness. To fulfill the attainable attack efficacy within a constrained budget, we propose a fine-grained control mechanism that selectively recomputes layer activations across both iteration-wise and layer-wise levels. Extensive experiments show that our method consistently outperforms existing baselines at equal cost. Moreover, when integrated into adversarial training, it attains comparable performance with only 30% of the original budget. Adversarial attacks, which craft imperceptible perturbations to input data to degrade the performance of deep learning models, have become a central topic in the safety and robustness of AI systems. From the attack perspective, iterative adversarial methods such as Projected Gradient Descent (PGD) (Madry et al., 2017) are widely adopted as strong oracles to benchmark the robustness of modern neural networks.

This paper proposes Stochastic Geographic Gradient Fusion (SGFusion), a novel training algorithm to leverage the geographic information of mobile users in Federated Learning (FL). SGFusion maps the data collected by mobile devices onto geographical zones and trains one FL model per zone, which adapts well to the data and behaviors of users in that zone. SGFusion models the local data-based correlation among geographical zones as a hierarchical random graph (HRG) optimized by Markov Chain Monte Carlo sampling. At each training step, every zone fuses its local gradient with gradients derived from a small set of other zones sampled from the HRG. This approach enables knowledge fusion and sharing among geographical zones in a probabilistic and stochastic gradient fusion process with self-attention weights, such that "more similar" zones have "higher probabilities" of sharing gradients with "larger attention weights." SGFusion remarkably improves model utility without introducing undue computational cost. Extensive theoretical and empirical results using a heart-rate prediction dataset collected across 6 countries show that models trained with SGFusion converge with upper-bounded expected errors and significantly improve utility in all countries compared to existing approaches without notable cost in system scalability.

Within the current sphere of deep learning research, despite the extensive application of optimization algorithms such as Stochastic Gradient Descent (SGD) and Adaptive Moment Estimation (Adam), there remains a pronounced inadequacy in their capability to address fluctuations in learning efficiency, meet the demands of complex models, and tackle non-convex optimization issues. These challenges primarily arise from the algorithms' limitations in handling complex data structures and models, for instance, difficulties in selecting an appropriate learning rate, avoiding local optima, and navigating through high-dimensional spaces. To address these issues, this paper introduces a novel optimization algorithm named DWMGrad. This algorithm, building on the foundations of traditional methods, incorporates a dynamic guidance mechanism reliant on historical data to dynamically update momentum and learning rates. This allows the optimizer to flexibly adjust its reliance on historical information, adapting to various training scenarios. This strategy not only enables the optimizer to better adapt to changing environments and task complexities but also, as validated through extensive experimentation, demonstrates DWMGrad's ability to achieve faster convergence rates and higher accuracies under a multitude of scenarios.

The heterogeneity of multimodal data leads to inconsistencies and imbalance, allowing a dominant modality to steer gradient updates. Existing solutions mainly focus on optimization- or data-based strategies but rarely exploit the information inherent in multimodal imbalance or conduct its quantitative analysis. To address this gap, we propose a novel quantitative analysis framework for Multimodal Imbalance and design a sample-level adaptive loss function. We define the Modality Gap as the Softmax score difference between modalities for the correct class and model its distribution using a bimodal Gaussian Mixture Model(GMM), representing balanced and imbalanced samples. Using Bayes' theorem, we estimate each sample's posterior probability of belonging to these two groups. Based on this, our adaptive loss (1) minimizes the overall Modality Gap, (2) aligns imbalanced samples with balanced ones, and (3) adaptively penalizes each according to its imbalance degree. A two-stage training strategy-warm-up and adaptive phases,yields state-of-the-art performance on CREMA-D (80.65%), AVE (70.40%), and KineticSound (72.42%). Fine-tuning with high-quality samples identified by the GMM further improves results, highlighting their value for effective multimodal fusion.

We investigate the semi-discrete Optimal Transport (OT) problem, where a continuous source measure $μ$ is transported to a discrete target measure $ν$, with particular attention to the OT map approximation. In this setting, Stochastic Gradient Descent (SGD) based solvers have demonstrated strong empirical performance in recent machine learning applications, yet their theoretical guarantee to approximate the OT map is an open question. In this work, we answer it positively by providing both computational and statistical convergence guarantees of SGD. Specifically, we show that SGD methods can estimate the OT map with a minimax convergence rate of $\mathcal{O}(1/\sqrt{n})$, where $n$ is the number of samples drawn from $μ$. To establish this result, we study the averaged projected SGD algorithm, and identify a suitable projection set that contains a minimizer of the objective, even when the source measure is not compactly supported. Our analysis holds under mild assumptions on the source measure and applies to MTW cost functions,whic include $\|\cdot\|^p$ for $p \in (1, \infty)$. We finally provide numerical evidence for our theoretical results.

Differentially private stochastic gradient descent (DP-SGD) is the most widely used method for training machine learning models with provable privacy guarantees. A key challenge in DP-SGD is setting the per-sample gradient clipping threshold, which significantly affects the trade-off between privacy and utility. While recent adaptive methods improve performance by adjusting this threshold during training, they operate in the standard coordinate system and fail to account for correlations across the coordinates of the gradient. We propose GeoClip, a geometry-aware framework that clips and perturbs gradients in a transformed basis aligned with the geometry of the gradient distribution. GeoClip adaptively estimates this transformation using only previously released noisy gradients, incurring no additional privacy cost. We provide convergence guarantees for GeoClip and derive a closed-form solution for the optimal transformation that minimizes the amount of noise added while keeping the probability of gradient clipping under control. Experiments on both tabular and image datasets demonstrate that GeoClip consistently outperforms existing adaptive clipping methods under the same privacy budget.

The theory of training deep networks has become a central question of modern machine learning and has inspired many practical advancements. In particular, the gradient descent (GD) optimization algorithm has been extensively studied in recent years. A key assumption about GD has appeared in several recent works: the \emph{GD map is non-singular} -- it preserves sets of measure zero under preimages. Crucially, this assumption has been used to prove that GD avoids saddle points and maxima, and to establish the existence of a computable quantity that determines the convergence to global minima (both for GD and stochastic GD). However, the current literature either assumes the non-singularity of the GD map or imposes restrictive assumptions, such as Lipschitz smoothness of the loss (for example, Lipschitzness does not hold for deep ReLU networks with the cross-entropy loss) and restricts the analysis to GD with small step-sizes. In this paper, we investigate the neural network map as a function on the space of weights and biases. We also prove, for the first time, the non-singularity of the gradient descent (GD) map on the loss landscape of realistic neural network architectures (with fully connected, convolutional, or softmax attention layers) and piecewise analytic activations (which includes sigmoid, ReLU, leaky ReLU, etc.) for almost all step-sizes. Our work significantly extends the existing results on the convergence of GD and SGD by guaranteeing that they apply to practical neural network settings and has the potential to unlock further exploration of learning dynamics.

We study the problem of training neural networks with quantized parameters. Learning low-precision quantized parameters by enabling computation of gradients via the Straight-Through Estimator (STE) can be challenging. While the STE enables back-propagation, which is a first-order method, recent works have explored the use of zeroth-order (ZO) gradient descent for fine-tuning. We note that the STE provides high-quality biased gradients, and ZO gradients are unbiased but can be expensive. We thus propose First-Order-Guided Zeroth-Order Gradient Descent (FOGZO) that reduces STE bias while reducing computations relative to ZO methods. Empirically, we show FOGZO improves the tradeoff between quality and training time in Quantization-Aware Pre-Training. Specifically, versus STE at the same number of iterations, we show a 1-8\% accuracy improvement for DeiT Tiny/Small, 1-2\% accuracy improvement on ResNet 18/50, and 1-22 perplexity point improvement for LLaMA models with up to 0.3 billion parameters. For the same loss, FOGZO yields a 796$\times$ reduction in computation versus n-SPSA for a 2-layer MLP on MNIST. Code is available at https://github.com/1733116199/fogzo.

Adaptive gradient methods such as Adam and Adagrad are widely used in machine learning, yet their effect on the generalization of learned models -- relative to methods like gradient descent -- remains poorly understood. Prior work on binary classification suggests that Adam exhibits a ``richness bias,'' which can help it learn nonlinear decision boundaries closer to the Bayes-optimal decision boundary relative to gradient descent. However, the coordinate-wise preconditioning scheme employed by Adam renders the overall method sensitive to orthogonal transformations of feature space. We show that this sensitivity can manifest as a reversal of Adam's competitive advantage: even small rotations of the underlying data distribution can make Adam forfeit its richness bias and converge to a linear decision boundary that is farther from the Bayes-optimal decision boundary than the one learned by gradient descent. To alleviate this issue, we show that a recently proposed reparameterization method -- which applies an orthogonal transformation to the optimization objective -- endows any first-order method with equivariance to data rotations, and we empirically demonstrate its ability to restore Adam's bias towards rich decision boundaries.

Protein design using structure prediction models such as AlphaFold2 has shown remarkable success, but existing approaches like relaxed sequence optimization (RSO) rely on single-path gradient descent and ignore sequence-space constraints, limiting diversity and designability. We introduce Relaxed Sequence Sampling (RSS), a Markov chain Monte Carlo (MCMC) framework that integrates structural and evolutionary information for protein design. RSS operates in continuous logit space, combining gradient-guided exploration with protein language model-informed jumps. Its energy function couples AlphaFold2-derived structural objectives with ESM2-derived sequence priors, balancing accuracy and biological plausibility. In an in silico protein binder design task, RSS produces 5$\times$ more designable structures and 2-3$\times$ greater structural diversity than RSO baselines, at equal computational cost. These results highlight RSS as a principled approach for efficiently exploring the protein design landscape.