In this work, we consider the optimization process of minibatch stochastic gradient descent (SGD) on a 2-layer neural network with data separated by a quadratic ground truth function. We prove that with data drawn from the $d$-dimensional Boolean hypercube labeled by the quadratic ``XOR'' function $y = -x_ix_j$, it is possible to train to a population error $o(1)$ with $d \:\text{polylog}(d)$ samples. Our result considers simultaneously training both layers of the two-layer-neural network with ReLU activations via standard minibatch SGD on the logistic loss. To our knowledge, this work is the first to give a sample complexity of $\tilde{O}(d)$ for efficiently learning the XOR function on isotropic data on a standard neural network with standard training. Our main technique is showing that the network evolves in two phases: a $\textit{signal-finding}$ phase where the network is small and many of the neurons evolve independently to find features, and a $\textit{signal-heavy}$ phase, where SGD maintains and balances the features. We leverage the simultaneous training of the layers to show that it is sufficient for only a small fraction of the neurons to learn features, since those neurons will be amplified by the simultaneous growth of their second layer weights.

Batch reinforcement learning (RL) defines the task of learning from a fixed batch of data lacking exhaustive exploration. Worst-case optimality algorithms, which calibrate a value-function model class from logged experience and perform some type of pessimistic evaluation under the learned model, have emerged as a promising paradigm for batch RL. However, contemporary works on this stream have commonly overlooked the hierarchical decision-making structure hidden in the optimization landscape. In this paper, we adopt a game-theoretical viewpoint and model the policy learning diagram as a two-player general-sum game with a leader-follower structure. We propose a novel stochastic gradient-based learning algorithm: StackelbergLearner, in which the leader player updates according to the total derivative of its objective instead of the usual individual gradient, and the follower player makes individual updates and ensures transition-consistent pessimistic reasoning. The derived learning dynamic naturally lends StackelbergLearner to a game-theoretic interpretation and provides a convergence guarantee to differentiable Stackelberg equilibria. From a theoretical standpoint, we provide instance-dependent regret bounds with general function approximation, which shows that our algorithm can learn a best-effort policy that is able to compete against any comparator policy that is covered by batch data. Notably, our theoretical regret guarantees only require realizability without any data coverage and strong function approximation conditions, e.g., Bellman closedness, which is in contrast to prior works lacking such guarantees. Through comprehensive experiments, we find that our algorithm consistently performs as well or better as compared to state-of-the-art methods in batch RL benchmark and real-world datasets.

It is widely recognized that the generalization ability of neural networks can be greatly enhanced through carefully tuning the training procedure. The current state-of-the-art training approach involves utilizing stochastic gradient descent (SGD) or Adam optimization algorithms along with a combination of additional regularization techniques such as weight decay, dropout, or noise injection. Optimal generalization can only be achieved by tuning a multitude of hyper-parameters extensively, which can be time-consuming and necessitates the additional validation dataset. To address this issue, we present a nearly tuning-free PAC-Bayes training framework that requires no extra regularization. This framework achieves test performance comparable to that of SGD/Adam, even when the latter are optimized through a complete grid search and supplemented with additional regularization terms. To understand the underlying benefits of these strategies, numerous studies have focused on studying individual strategies. For instance, it has been shown that larger learning rates (Cohen et al., 2021; Barrett & Dherin, 2020), momentum (Ghosh et al., 2022), smaller batch sizes (Lee & Jang, 2022) and batch normalization (Luo et al., 2018) individually induce higher degrees of implicit regularization on the sharpness of the loss function, yielding better generalization. Additionally, the intensity of explicit regularization techniques such as weight decay (Loshchilov & Hutter, 2017), dropout (Wei et al., 2020), parameter noise injection (Neelakantan et al., 2015; Orvieto et al., 2022), label noise (Damian et al., 2021) can significantly affect generalization. Despite these observations and explanations, it's unclear why seeking optimal combinations of these regularizations is still crucial in practice. Adjusting the intensity of each regularization based on different scenarios can be a tedious job, especially when previous research has indicated that some techniques can conflict with each other (Li et al., 2019). We summarize this challenge for conventional training in (Q1). Alternatively, PAC-Bayes generalization bounds provide foundational insights into generalization in the absence of validation and testing data (Shawe-Taylor & Williamson, 1997). Jiang et al. (2019) further suggests that PAC-Bayes bounds are among the best for evaluating generalization capabilities. Although PAC-Bayes bounds were traditionally used only in the post-training stage for quality control (Vapnik, 1998; McAllester, 1999), the recent work (Dziugaite & Roy, 2017b) has opened the door to using these bounds during the training phase. They showed that one can directly train a network via optimizing the PAC-Bayes bound, a strategy we refer to as PAC-Bayes training, and obtain reasonable performances.

Classical machine learning models such as deep neural networks are usually trained by using Stochastic Gradient Descent-based (SGD) algorithms. The classical SGD can be interpreted as a discretization of the stochastic gradient flow. In this paper we propose a novel, robust and accelerated stochastic optimizer that relies on two key elements: (1) an accelerated Nesterov-like Stochastic Differential Equation (SDE) and (2) its semi-implicit Gauss-Seidel type discretization. The convergence and stability of the obtained method, referred to as NAG-GS, are first studied extensively in the case of the minimization of a quadratic function. This analysis allows us to come up with an optimal learning rate in terms of the convergence rate while ensuring the stability of NAG-GS. This is achieved by the careful analysis of the spectral radius of the iteration matrix and the covariance matrix at stationarity with respect to all hyperparameters of our method. Further, we show that NAG- GS is competitive with state-of-the-art methods such as momentum SGD with weight decay and AdamW for the training of machine learning models such as the logistic regression model, the residual networks models on standard computer vision datasets, Transformers in the frame of the GLUE benchmark and the recent Vision Transformers.

We introduce a clipping strategy for Stochastic Gradient Descent (SGD) which uses quantiles of the gradient norm as clipping thresholds. We prove that this new strategy provides a robust and efficient optimization algorithm for smooth objectives (convex or non-convex), that tolerates heavy-tailed samples (including infinite variance) and a fraction of outliers in the data stream akin to Huber contamination. Our mathematical analysis leverages the connection between constant step size SGD and Markov chains and handles the bias introduced by clipping in an original way. For strongly convex objectives, we prove that the iteration converges to a concentrated distribution and derive high probability bounds on the final estimation error. In the non-convex case, we prove that the limit distribution is localized on a neighborhood with low gradient. We propose an implementation of this algorithm using rolling quantiles which leads to a highly efficient optimization procedure with strong robustness properties, as confirmed by our numerical experiments.

Multi-Task Learning (MTL) is a widely-used and powerful learning paradigm for training deep neural networks that allows learning more than one objective by a single backbone. Compared to training tasks separately, MTL significantly reduces computational costs, improves data efficiency, and potentially enhances model performance by leveraging knowledge across tasks. Hence, it has been adopted in a variety of applications, ranging from computer vision to natural language processing and speech recognition. Among them, there is an emerging line of work in MTL that focuses on manipulating the task gradient to derive an ultimate gradient descent direction to benefit all tasks. Despite achieving impressive results on many benchmarks, directly applying these approaches without using appropriate regularization techniques might lead to suboptimal solutions on real-world problems. In particular, standard training that minimizes the empirical loss on the training data can easily suffer from overfitting to low-resource tasks or be spoiled by noisy-labeled ones, which can cause negative transfer between tasks and overall performance drop. To alleviate such problems, we propose to leverage a recently introduced training method, named Sharpness-aware Minimization, which can enhance model generalization ability on single-task learning. Accordingly, we present a novel MTL training methodology, encouraging the model to find task-based flat minima for coherently improving its generalization capability on all tasks. Finally, we conduct comprehensive experiments on a variety of applications to demonstrate the merit of our proposed approach to existing gradient-based MTL methods, as suggested by our developed theory.

Stable concurrent learning and control of dynamical systems is the subject of adaptive control. Despite being an established field with many practical applications and a rich theory, much of the development in adaptive control for nonlinear systems revolves around a few key algorithms. By exploiting strong connections between classical adaptive nonlinear control techniques and recent progress in optimization and machine learning, we show that there exists considerable untapped potential in algorithm development for both adaptive nonlinear control and adaptive dynamics prediction. We begin by introducing first-order adaptation laws inspired by natural gradient descent and mirror descent. We prove that when there are multiple dynamics consistent with the data, these non-Euclidean adaptation laws implicitly regularize the learned model. Local geometry imposed during learning thus may be used to select parameter vectors -- out of the many that will achieve perfect tracking or prediction -- for desired properties such as sparsity. We apply this result to regularized dynamics predictor and observer design, and as concrete examples, we consider Hamiltonian systems, Lagrangian systems, and recurrent neural networks. We subsequently develop a variational formalism based on the Bregman Lagrangian. We show that its Euler Lagrange equations lead to natural gradient and mirror descent-like adaptation laws with momentum, and we recover their first-order analogues in the infinite friction limit. We illustrate our analyses with simulations demonstrating our theoretical results.

Recently, significant progress has been made in understanding the generalization of neural networks (NNs) trained by gradient descent (GD) using the algorithmic stability approach. However, most of the existing research has focused on one-hidden-layer NNs and has not addressed the impact of different network scaling parameters. In this paper, we greatly extend the previous work \cite{lei2022stability,richards2021stability} by conducting a comprehensive stability and generalization analysis of GD for multi-layer NNs. For two-layer NNs, our results are established under general network scaling parameters, relaxing previous conditions. In the case of three-layer NNs, our technical contribution lies in demonstrating its nearly co-coercive property by utilizing a novel induction strategy that thoroughly explores the effects of over-parameterization. As a direct application of our general findings, we derive the excess risk rate of $O(1/\sqrt{n})$ for GD algorithms in both two-layer and three-layer NNs. This sheds light on sufficient or necessary conditions for under-parameterized and over-parameterized NNs trained by GD to attain the desired risk rate of $O(1/\sqrt{n})$. Moreover, we demonstrate that as the scaling parameter increases or the network complexity decreases, less over-parameterization is required for GD to achieve the desired error rates. Additionally, under a low-noise condition, we obtain a fast risk rate of $O(1/n)$ for GD in both two-layer and three-layer NNs.

Active learning (AL) aims to select the most useful data samples from an unlabeled data pool and annotate them to expand the labeled dataset under a limited budget. Especially, uncertainty-based methods choose the most uncertain samples, which are known to be effective in improving model performance. However, AL literature often overlooks training dynamics (TD), defined as the ever-changing model behavior during optimization via stochastic gradient descent, even though other areas of literature have empirically shown that TD provides important clues for measuring the sample uncertainty. In this paper, we propose a novel AL method, Training Dynamics for Active Learning (TiDAL), which leverages the TD to quantify uncertainties of unlabeled data. Since tracking the TD of all the large-scale unlabeled data is impractical, TiDAL utilizes an additional prediction module that learns the TD of labeled data. To further justify the design of TiDAL, we provide theoretical and empirical evidence to argue the usefulness of leveraging TD for AL. Experimental results show that our TiDAL achieves better or comparable performance on both balanced and imbalanced benchmark datasets compared to state-of-the-art AL methods, which estimate data uncertainty using only static information after model training.

Large Language Models (LLMs) have proven their exceptional capabilities in performing language-related tasks. However, their deployment poses significant challenges due to their considerable memory and storage requirements. In response to this issue, weight-only quantization, particularly 3 and 4-bit weight-only quantization, has emerged as one of the most viable solutions. As the number of bits decreases, the quantization grid broadens, thus emphasizing the importance of up and down rounding. While previous studies have demonstrated that fine-tuning up and down rounding with the addition of perturbations can enhance accuracy in some scenarios, our study is driven by the precise and limited boundary of these perturbations, where only the threshold for altering the rounding value is of significance. Consequently, we propose a concise and highly effective approach for optimizing the weight rounding task. Our method, named SignRound, involves lightweight block-wise tuning using signed gradient descent, enabling us to achieve outstanding results within 400 steps. SignRound competes impressively against recent methods without introducing additional inference overhead. The source code will be publicly available at \url{https://github.com/intel/neural-compressor} soon.