Prove Your Point!: Bringing Proof-Enhancement Principles to Argumentative Essay Generation

Argumentative essay generation (AEG) aims to generate complete texts on specific controversial topics or debates. Although current AEG methods can generate individual opinions, they often overlook the high-level connections between these opinions. This often leads to the generated results being mired in logical confusion, unable to proof their own arguments effectively. The generated essay may present evidence that contradicts the claims or they may fail to assemble the claims into logical flow. In this paper, we present a unified two-stage framework: Proof-Enhancement and Self-Annotation (PESA) for AEG with a focus on logical enhancement. Specifically, we first construct pseudo-labels for logical information,claims and grounds, using a large language model. We then propose a tree planning approach that introduces proof principles and ensures logical consistency. Extensive experimental results show that, benefiting from proof principle guidance, PESA generates argumentative essays with better logical validity and persuasiveness than strong baseline models.

How Transformers Implement Induction Heads: Approximation and Optimization Analysis

Transformers have demonstrated exceptional in-context learning capabilities, yet the theoretical understanding of the underlying mechanisms remain limited. A recent work (Elhage et al., 2021) identified a "rich" in-context mechanism known as induction head, contrasting with "lazy" $n$-gram models that overlook long-range dependencies. In this work, we provide both approximation and optimization analyses of how transformers implement induction heads. In the approximation analysis, we formalize both standard and generalized induction head mechanisms, and examine how transformers can efficiently implement them, with an emphasis on the distinct role of each transformer submodule. For the optimization analysis, we study the training dynamics on a synthetic mixed target, composed of a 4-gram and an in-context 2-gram component. This setting enables us to precisely characterize the entire training process and uncover an {\em abrupt transition} from lazy (4-gram) to rich (induction head) mechanisms as training progresses.

DTactive: A Vision-Based Tactile Sensor with Active Surface

The development of vision-based tactile sensors has significantly enhanced robots' perception and manipulation capabilities, especially for tasks requiring contact-rich interactions with objects. In this work, we present DTactive, a novel vision-based tactile sensor with active surfaces. DTactive inherits and modifies the tactile 3D shape reconstruction method of DTact while integrating a mechanical transmission mechanism that facilitates the mobility of its surface. Thanks to this design, the sensor is capable of simultaneously performing tactile perception and in-hand manipulation with surface movement. Leveraging the high-resolution tactile images from the sensor and the magnetic encoder data from the transmission mechanism, we propose a learning-based method to enable precise angular trajectory control during in-hand manipulation. In our experiments, we successfully achieved accurate rolling manipulation within the range of [ -180{\deg},180{\deg} ] on various objects, with the root mean square error between the desired and actual angular trajectories being less than 12{\deg} on nine trained objects and less than 19{\deg} on three novel objects. The results demonstrate the potential of DTactive for in-hand object manipulation in terms of effectiveness, robustness and precision.

Why Do You Grok? A Theoretical Analysis of Grokking Modular Addition

We present a theoretical explanation of the ``grokking'' phenomenon, where a model generalizes long after overfitting,for the originally-studied problem of modular addition. First, we show that early in gradient descent, when the ``kernel regime'' approximately holds, no permutation-equivariant model can achieve small population error on modular addition unless it sees at least a constant fraction of all possible data points. Eventually, however, models escape the kernel regime. We show that two-layer quadratic networks that achieve zero training loss with bounded $\ell_{\infty}$ norm generalize well with substantially fewer training points, and further show such networks exist and can be found by gradient descent with small $\ell_{\infty}$ regularization. We further provide empirical evidence that these networks as well as simple Transformers, leave the kernel regime only after initially overfitting. Taken together, our results strongly support the case for grokking as a consequence of the transition from kernel-like behavior to limiting behavior of gradient descent on deep networks.

Improving Generalization and Convergence by Enhancing Implicit Regularization

In this work, we propose an Implicit Regularization Enhancement (IRE) framework to accelerate the discovery of flat solutions in deep learning, thereby improving generalization and convergence. Specifically, IRE decouples the dynamics of flat and sharp directions, which boosts the sharpness reduction along flat directions while maintaining the training stability in sharp directions. We show that IRE can be practically incorporated with {\em generic base optimizers} without introducing significant computational overload. Experiments show that IRE consistently improves the generalization performance for image classification tasks across a variety of benchmark datasets (CIFAR-10/100, ImageNet) and models (ResNets and ViTs). Surprisingly, IRE also achieves a $2\times$ {\em speed-up} compared to AdamW in the pre-training of Llama models (of sizes ranging from 60M to 229M) on datasets including Wikitext-103, Minipile, and Openwebtext. Moreover, we provide theoretical guarantees, showing that IRE can substantially accelerate the convergence towards flat minima in Sharpness-aware Minimization (SAM).

Exploring Neural Network Landscapes: Star-Shaped and Geodesic Connectivity

One of the most intriguing findings in the structure of neural network landscape is the phenomenon of mode connectivity: For two typical global minima, there exists a path connecting them without barrier. This concept of mode connectivity has played a crucial role in understanding important phenomena in deep learning. In this paper, we conduct a fine-grained analysis of this connectivity phenomenon. First, we demonstrate that in the overparameterized case, the connecting path can be as simple as a two-piece linear path, and the path length can be nearly equal to the Euclidean distance. This finding suggests that the landscape should be nearly convex in a certain sense. Second, we uncover a surprising star-shaped connectivity: For a finite number of typical minima, there exists a center on minima manifold that connects all of them simultaneously via linear paths. These results are provably valid for linear networks and two-layer ReLU networks under a teacher-student setup, and are empirically supported by models trained on MNIST and CIFAR-10.

A Duality Analysis of Kernel Ridge Regression in the Noiseless Regime

In this paper, we conduct a comprehensive analysis of generalization properties of Kernel Ridge Regression (KRR) in the noiseless regime, a scenario crucial to scientific computing, where data are often generated via computer simulations. We prove that KRR can attain the minimax optimal rate, which depends on both the eigenvalue decay of the associated kernel and the relative smoothness of target functions. Particularly, when the eigenvalue decays exponentially fast, KRR achieves the spectral accuracy, i.e., a convergence rate faster than any polynomial. Moreover, the numerical experiments well corroborate our theoretical findings. Our proof leverages a novel extension of the duality framework introduced by Chen et al. (2023), which could be useful in analyzing kernel-based methods beyond the scope of this work.

The Implicit Bias of Gradient Noise: A Symmetry Perspective

We characterize the learning dynamics of stochastic gradient descent (SGD) when continuous symmetry exists in the loss function, where the divergence between SGD and gradient descent is dramatic. We show that depending on how the symmetry affects the learning dynamics, we can divide a family of symmetry into two classes. For one class of symmetry, SGD naturally converges to solutions that have a balanced and aligned gradient noise. For the other class of symmetry, SGD will almost always diverge. Then, we show that our result remains applicable and can help us understand the training dynamics even when the symmetry is not present in the loss function. Our main result is universal in the sense that it only depends on the existence of the symmetry and is independent of the details of the loss function. We demonstrate that the proposed theory offers an explanation of progressive sharpening and flattening and can be applied to common practical problems such as representation normalization, matrix factorization, and the use of warmup.

Achieving Margin Maximization Exponentially Fast via Progressive Norm Rescaling

In this work, we investigate the margin-maximization bias exhibited by gradient-based algorithms in classifying linearly separable data. We present an in-depth analysis of the specific properties of the velocity field associated with (normalized) gradients, focusing on their role in margin maximization. Inspired by this analysis, we propose a novel algorithm called Progressive Rescaling Gradient Descent (PRGD) and show that PRGD can maximize the margin at an {\em exponential rate}. This stands in stark contrast to all existing algorithms, which maximize the margin at a slow {\em polynomial rate}. Specifically, we identify mild conditions on data distribution under which existing algorithms such as gradient descent (GD) and normalized gradient descent (NGD) {\em provably fail} in maximizing the margin efficiently. To validate our theoretical findings, we present both synthetic and real-world experiments. Notably, PRGD also shows promise in enhancing the generalization performance when applied to linearly non-separable datasets and deep neural networks.

The Local Landscape of Phase Retrieval Under Limited Samples

In this paper, we provide a fine-grained analysis of the local landscape of phase retrieval under the regime with limited samples. Our aim is to ascertain the minimal sample size necessary to guarantee a benign local landscape surrounding global minima in high dimensions. Let $n$ and $d$ denote the sample size and input dimension, respectively. We first explore the local convexity and establish that when $n=o(d\log d)$, for almost every fixed point in the local ball, the Hessian matrix must have negative eigenvalues as long as $d$ is sufficiently large. Consequently, the local landscape is highly non-convex. We next consider the one-point strong convexity and show that as long as $n=\omega(d)$, with high probability, the landscape is one-point strongly convex in the local annulus: $\{w\in\mathbb{R}^d: o_d(1)\leqslant \|w-w^*\|\leqslant c\}$, where $w^*$ is the ground truth and $c$ is an absolute constant. This implies that gradient descent initialized from any point in this domain can converge to an $o_d(1)$-loss solution exponentially fast. Furthermore, we show that when $n=o(d\log d)$, there is a radius of $\widetilde\Theta\left(\sqrt{1/d}\right)$ such that one-point convexity breaks in the corresponding smaller local ball. This indicates an impossibility to establish a convergence to exact $w^*$ for gradient descent under limited samples by relying solely on one-point convexity.