Recently, gradient-based discrete sampling has emerged as a highly efficient, general-purpose solver for various combinatorial optimization (CO) problems, achieving performance comparable to or surpassing the popular data-driven approaches. However, we identify a critical issue in these methods, which we term ''wandering in contours''. This behavior refers to sampling new different solutions that share very similar objective values for a long time, leading to computational inefficiency and suboptimal exploration of potential solutions. In this paper, we introduce a novel reheating mechanism inspired by the concept of critical temperature and specific heat in physics, aimed at overcoming this limitation. Empirically, our method demonstrates superiority over existing sampling-based and data-driven algorithms across a diverse array of CO problems.

Recent advances in deep learning optimization have unveiled two intriguing phenomena under large learning rates: Edge of Stability (EoS) and Progressive Sharpening (PS), challenging classical Gradient Descent (GD) analyses. Current research approaches, using either generalist frameworks or minimalist examples, face significant limitations in explaining these phenomena. This paper advances the minimalist approach by introducing a two-layer network with a two-dimensional input, where one dimension is relevant to the response and the other is irrelevant. Through this model, we rigorously prove the existence of progressive sharpening and self-stabilization under large learning rates, and establish non-asymptotic analysis of the training dynamics and sharpness along the entire GD trajectory. Besides, we connect our minimalist example to existing works by reconciling the existence of a well-behaved ``stable set" between minimalist and generalist analyses, and extending the analysis of Gradient Flow Solution sharpness to our two-dimensional input scenario. These findings provide new insights into the EoS phenomenon from both parameter and input data distribution perspectives, potentially informing more effective optimization strategies in deep learning practice.

The Unlearning Sensitive Content from Large Language Models task aims to remove targeted datapoints from trained models while minimally affecting their general knowledge. In our work, we leverage parameter-efficient, gradient-based unlearning using low-rank (LoRA) adaptation and layer-focused fine-tuning. To further enhance unlearning effectiveness, we employ data chunking, splitting forget data into disjoint partitions and merging them with cyclically sampled retain samples at a pre-defined ratio. Our task-agnostic method achieves an outstanding forget-retain balance, ranking first on leaderboards and significantly outperforming baselines and competing systems.

Machine unlearning aims to remove the influence of problematic training data after a model has been trained. The primary challenge in machine unlearning is ensuring that the process effectively removes specified data without compromising the model's overall performance on the remaining dataset. Many existing machine unlearning methods address this challenge by carefully balancing gradient ascent on the unlearn data with the gradient descent on a retain set representing the training data. Here, we propose OrthoGrad, a novel approach that mitigates interference between the unlearn set and the retain set rather than competing ascent and descent processes. Our method projects the gradient of the unlearn set onto the subspace orthogonal to all gradients in the retain batch, effectively avoiding any gradient interference. We demonstrate the effectiveness of OrthoGrad on multiple machine unlearning benchmarks, including automatic speech recognition, outperforming competing methods.

This study reviews popular stochastic gradient-based schemes based on large least-square problems. These schemes, often called optimizers in machine learning, play a crucial role in finding better model parameters. Hence, this study focuses on viewing such optimizers with different hyper-parameters and analyzing them based on least square problems. Codes that produced results in this work are available on https://github.com/q-viper/gradients-based-methods-on-large-least-square.

We consider nonconvex optimization problem over simplex, and more generally, a product of simplices. We provide an algorithm, Langevin Multiplicative Weights Update (LMWU) for solving global optimization problems by adding a noise scaling with the non-Euclidean geometry in the simplex. Non-convex optimization has been extensively studied by machine learning community due to its application in various scenarios such as neural network approximation and finding Nash equilibrium. Despite recent progresses on provable guarantee of escaping and avoiding saddle point (convergence to local minima) and global convergence of Langevin gradient based method without constraints, the global optimization with constraints is less studied. We show that LMWU algorithm is provably convergent to interior global minima with a non-asymptotic convergence analysis. We verify the efficiency of the proposed algorithm in real data set from polynomial portfolio management, where optimization of a highly non-linear objective function plays a crucial role.

We formalize a new concept for LLMs, context-enhanced learning. It involves standard gradient-based learning on text except that the context is enhanced with additional data on which no auto-regressive gradients are computed. This setting is a gradient-based analog of usual in-context learning (ICL) and appears in some recent works. Using a multi-step reasoning task, we prove in a simplified setting that context-enhanced learning can be exponentially more sample-efficient than standard learning when the model is capable of ICL. At a mechanistic level, we find that the benefit of context-enhancement arises from a more accurate gradient learning signal. We also experimentally demonstrate that it appears hard to detect or recover learning materials that were used in the context during training. This may have implications for data security as well as copyright.

We introduce a novel gradient-based approach for solving sequential tasks by dynamically adjusting the underlying myopic potential field in response to feedback and the world's regularities. This adjustment implicitly considers subgoals encoded in these regularities, enabling the solution of long sequential tasks, as demonstrated by solving the traditional planning domain of Blocks World - without any planning. Unlike conventional planning methods, our feedback-driven approach adapts to uncertain and dynamic environments, as demonstrated by one hundred real-world trials involving drawer manipulation. These experiments highlight the robustness of our method compared to planning and show how interactive perception and error recovery naturally emerge from gradient descent without explicitly implementing them. This offers a computationally efficient alternative to planning for a variety of sequential tasks, while aligning with observations on biological problem-solving strategies.

Stochastic gradient descent (SGD) optimization methods are the method of choice to train deep artificial neural networks (ANNs) in data-driven learning problems (see, for instance, [1, 7, 33, 36, 38, 41] and the references therein) as well as scientific computing problems (see, for example, [3, 13, 19, 22, 26, 40] and the references therein). However, often not the plain vanilla standard SGD optimization method is the employed optimizer but instead more sophisticated accelerated and adaptive variants of the standard SGD method such as the adaptive moment estimation SGD (Adam) optimizer (see [32]) are used in practically relevant deep ANN training problems. We also refer, for instance, to [2, 20, 23, 29, 35, 39] for monographs and surveys treating SGD optimization methods for the training of ANNs. The considered SGD optimization method is used with the aim to minimize the true risk function (the objective function) of the considered ANN learning problem so that, roughly speaking, the realization function of the deep ANN minimizing the true risk function approximates as best as possible the output data given the input data. Despite the omnipresent use of SGD optimization methods in the training of ANNs, it remains, in basically all practically relevant scenarios, a fundamental open problem to provide a rigorous theoretical description and explanation for the convergence (and non-convergence) properties of SGD optimization methods in deep learning. In particular, it remains an open question to prove or disprove convergence of the true risk of SGD optimization methods to the optimal/infimal true risk value in the training of deep ANNs (cf., for example, [11, 18, 31, 34] and the literature review in Subsection 1.4 below). In this work we contribute to this open problem of research in two aspects.

We study the problem of estimating Dynamic Discrete Choice (DDC) models, also known as offline Maximum Entropy-Regularized Inverse Reinforcement Learning (offline MaxEnt-IRL) in machine learning. The objective is to recover reward or $Q^*$ functions that govern agent behavior from offline behavior data. In this paper, we propose a globally convergent gradient-based method for solving these problems without the restrictive assumption of linearly parameterized rewards. The novelty of our approach lies in introducing the Empirical Risk Minimization (ERM) based IRL/DDC framework, which circumvents the need for explicit state transition probability estimation in the Bellman equation. Furthermore, our method is compatible with non-parametric estimation techniques such as neural networks. Therefore, the proposed method has the potential to be scaled to high-dimensional, infinite state spaces. A key theoretical insight underlying our approach is that the Bellman residual satisfies the Polyak-Lojasiewicz (PL) condition -- a property that, while weaker than strong convexity, is sufficient to ensure fast global convergence guarantees. Through a series of synthetic experiments, we demonstrate that our approach consistently outperforms benchmark methods and state-of-the-art alternatives.