We propose a tensor-based criterion for benign landscape in phase retrieval and establish boundedness of gradient trajectories. This implies that gradient descent will converge to a global minimum for almost every initial point.

Black-box optimization algorithms have been widely used in various machine learning problems, including reinforcement learning and prompt fine-tuning. However, directly optimizing the training loss value, as commonly done in existing black-box optimization methods, could lead to suboptimal model quality and generalization performance. To address those problems in black-box optimization, we propose a novel Sharpness-Aware Black-box Optimization (SABO) algorithm, which applies a sharpness-aware minimization strategy to improve the model generalization. Specifically, the proposed SABO method first reparameterizes the objective function by its expectation over a Gaussian distribution. Then it iteratively updates the parameterized distribution by approximated stochastic gradients of the maximum objective value within a small neighborhood around the current solution in the Gaussian distribution space. Theoretically, we prove the convergence rate and generalization bound of the proposed SABO algorithm. Empirically, extensive experiments on the black-box prompt fine-tuning tasks demonstrate the effectiveness of the proposed SABO method in improving model generalization performance.

Reinforcement Learning (RL), when combined with function approximators such as Artificial Neural Networks (ANNs), has shown success in learning policies that outperform humans in complex games by leveraging extensive datasets (see, e.g., 33, 19, 39, 40). While ANNs were previously used as value function approximators [29], the introduction of Deep Q-Networks (DQN) by [24, 25] marked a significant breakthrough by improving learning stability through two mechanisms: the target network and experience replay. The experience replay (see 22) stores the agent's interactions within the environment, allowing sampling of past interactions in a random way that disrupts their correlation. The target network further stabilizes the learning process by periodically copying the parameters of the learning network. This strategy is crucial because the Bellman update --using estimations to update other estimations-- would otherwise occur using the same network, potentially causing divergence. By leveraging the target network, gradient steps are directed towards a periodically fixed target, ensuring more stability in the learning process. Additionally, the learning rate hyperparameter controls the magnitude of these gradient steps in optimizers such as the stochastic gradient descent algorithm, affecting the training convergence. The learning rate is one of the most important hyperparameters, with previous work demonstrating that decreasing its value during policy finetuning can enhance performance by up to 25% in vanilla DQN [3].

Adversarial training can be used to learn models that are robust against perturbations. For linear models, it can be formulated as a convex optimization problem. Compared to methods proposed in the context of deep learning, leveraging the optimization structure allows significantly faster convergence rates. Still, the use of generic convex solvers can be inefficient for large-scale problems. Here, we propose tailored optimization algorithms for the adversarial training of linear models, which render large-scale regression and classification problems more tractable. For regression problems, we propose a family of solvers based on iterative ridge regression and, for classification, a family of solvers based on projected gradient descent. The methods are based on extended variable reformulations of the original problem. We illustrate their efficiency in numerical examples.

Adversarial attacks, characterized by subtle data perturbations that destabilize neural network predictions, have been a topic of significant interest for over a decade [48, 16, 32, 5]. These attacks have evolved into various forms, depending on the knowledge of the model's architecture (white-box, gray-box, black-box) [49], the type of data being targeted (graphs, images, text, etc.) [12, 47, 16, 57], and the specific adversarial objectives (targeted, untargeted, defense-oriented) [55, 29]. While numerous defense strategies aim to broadly stabilize models against adversarial attacks, independent of the specific attack mechanism [7, 14, 15, 41], the most effective and widely-used defenses focus on adversarial training, where the model is trained to withstand particular attacks [29, 50]. Adversarial training is known for producing robust models efficiently, but its effectiveness hinges on the availability of adversarial attacks that are both potent in degrading model accuracy and efficient in terms of computational resources. However, the most aggressive attacks often require significant computational resources, making them less practical for adversarial training. The projected gradient descent (PGD) attack [29] is popular in adversarial training due to its balance between aggressiveness and computational efficiency. In this work, we observe that in many cases the perturbations generated by PGD predominantly affect the lower part of the singular value spectrum of input images, indicating that these perturbations are approximately low-rank. Additionally, we find that the size of PGD-generated attacks differs significantly between standard and adversarially trained models when measured by their nuclear norm, which sums the singular values of the attack. This metric provides insight into the frequency profile of the attack when analyzed using the singular value decomposition (SVD) transform, aligning with known frequency profiles observed under discrete Fourier transform (DFT) and discrete cosine transform (DCT) analyses of PGD attacks [54, 31].

Modeling complex systems that evolve toward equilibrium distributions is important in various physical applications, including molecular dynamics and robotic control. These systems often follow the stochastic gradient descent of an underlying energy function, converging to stationary distributions around energy minima. The local covariance of these distributions is shaped by the energy landscape's curvature, often resulting in anisotropic characteristics. While flow-based generative models have gained traction in generating samples from equilibrium distributions in such applications, they predominately employ isotropic conditional probability paths, limiting their ability to capture such covariance structures. In this paper, we introduce Hessian-Informed Flow Matching (HI-FM), a novel approach that integrates the Hessian of an energy function into conditional flows within the flow matching framework. This integration allows HI-FM to account for local curvature and anisotropic covariance structures. Our approach leverages the linearization theorem from dynamical systems and incorporates additional considerations such as time transformations and equivariance. Empirical evaluations on the MNIST and Lennard-Jones particles datasets demonstrate that HI-FM improves the likelihood of test samples.

This paper studies the problem of federated training of a deep neural network (DNN) over a random access channel (RACH) such as in computer networks, wireless networks, and cellular systems. More precisely, a set of remote users participate in training a centralized DNN model using SGD under the coordination of a parameter server (PS). The local model updates are transmitted from the remote users to the PS over a RACH using a slotted ALOHA protocol. The PS collects the updates from the remote users, accumulates them, and sends central model updates to the users at regular time intervals. We refer to this setting as the RACH-FL setting. The RACH-FL setting crucially addresses the problem of jointly designing a (i) client selection and (ii) gradient compression strategy which addresses the communication constraints between the remote users and the PS when transmission occurs over a RACH. For the RACH-FL setting, we propose a policy, which we term the ''age-of-gradient'' (AoG) policy in which (i) gradient sparsification is performed using top-K sparsification, (ii) the error correction is performed using memory accumulation, and (iii) the slot transmission probability is obtained by comparing the current local memory magnitude minus the magnitude of the gradient update to a threshold. Intuitively, the AoG measure of ''freshness'' of the memory state is reminiscent of the concept of age-of-information (AoI) in the context of communication theory and provides a rather natural interpretation of this policy. Numerical simulations show the superior performance of the AoG policy as compared to other RACH-FL policies.

In-context learning has been recognized as a key factor in the success of Large Language Models (LLMs). It refers to the model's ability to learn patterns on the fly from provided in-context examples in the prompt during inference. Previous studies have demonstrated that the Transformer architecture used in LLMs can implement a single-step gradient descent update by processing in-context examples in a single forward pass. Recent work has further shown that, during in-context learning, a looped Transformer can implement multi-step gradient descent updates in forward passes. However, their theoretical results require an exponential number of in-context examples, $n = \exp(\Omega(T))$, where $T$ is the number of loops or passes, to achieve a reasonably low error. In this paper, we study linear looped Transformers in-context learning on linear vector generation tasks. We show that linear looped Transformers can implement multi-step gradient descent efficiently for in-context learning. Our results demonstrate that as long as the input data has a constant condition number, e.g., $n = O(d)$, the linear looped Transformers can achieve a small error by multi-step gradient descent during in-context learning. Furthermore, our preliminary experiments validate our theoretical analysis. Our findings reveal that the Transformer architecture possesses a stronger in-context learning capability than previously understood, offering new insights into the mechanisms behind LLMs and potentially guiding the better design of efficient inference algorithms for LLMs.

Motivated by recent progress in the problem of numerical K\"ahler metrics, we survey machine learning techniques in this area, discussing both advantages and drawbacks. We then revisit the algebraic ansatz pioneered by Donaldson. Inspired by his work, we present a novel approach to obtaining Ricci-flat approximations to K\"ahler metrics, applying machine learning within a `principled' framework. In particular, we use gradient descent on the Grassmannian manifold to identify an efficient subspace of sections for calculation of the metric. We combine this approach with both Donaldson's algorithm and learning on the $h$-matrix itself (the latter method being equivalent to gradient descent on the fibre bundle of Hermitian metrics on the tautological bundle over the Grassmannian). We implement our methods on the Dwork family of threefolds, commenting on the behaviour at different points in moduli space. In particular, we observe the emergence of nontrivial local minima as the moduli parameter is increased.

In reinforcement learning, off-policy actor-critic approaches like DDPG and TD3 are based on the deterministic policy gradient. Herein, the Q-function is trained from off-policy environment data and the actor (policy) is trained to maximize the Q-function via gradient ascent. We observe that in complex tasks like dexterous manipulation and restricted locomotion, the Q-value is a complex function of action, having several local optima or discontinuities. This poses a challenge for gradient ascent to traverse and makes the actor prone to get stuck at local optima. To address this, we introduce a new actor architecture that combines two simple insights: (i) use multiple actors and evaluate the Q-value maximizing action, and (ii) learn surrogates to the Q-function that are simpler to optimize with gradient-based methods. We evaluate tasks such as restricted locomotion, dexterous manipulation, and large discrete-action space recommender systems and show that our actor finds optimal actions more frequently and outperforms alternate actor architectures.