On the Impacts of the Random Initialization in the Neural Tangent Kernel Theory

This paper aims to discuss the impact of random initialization of neural networks in the neural tangent kernel (NTK) theory, which is ignored by most recent works in the NTK theory. It is well known that as the network's width tends to infinity, the neural network with random initialization converges to a Gaussian process $f^{\mathrm{GP}}$, which takes values in $L^{2}(\mathcal{X})$, where $\mathcal{X}$ is the domain of the data. In contrast, to adopt the traditional theory of kernel regression, most recent works introduced a special mirrored architecture and a mirrored (random) initialization to ensure the network's output is identically zero at initialization. Therefore, it remains a question whether the conventional setting and mirrored initialization would make wide neural networks exhibit different generalization capabilities. In this paper, we first show that the training dynamics of the gradient flow of neural networks with random initialization converge uniformly to that of the corresponding NTK regression with random initialization $f^{\mathrm{GP}}$. We then show that $\mathbf{P}(f^{\mathrm{GP}} \in [\mathcal{H}^{\mathrm{NT}}]^{s}) = 1$ for any $s < \frac{3}{d+1}$ and $\mathbf{P}(f^{\mathrm{GP}} \in [\mathcal{H}^{\mathrm{NT}}]^{s}) = 0$ for any $s \geq \frac{3}{d+1}$, where $[\mathcal{H}^{\mathrm{NT}}]^{s}$ is the real interpolation space of the RKHS $\mathcal{H}^{\mathrm{NT}}$ associated with the NTK. Consequently, the generalization error of the wide neural network trained by gradient descent is $\Omega(n^{-\frac{3}{d+3}})$, and it still suffers from the curse of dimensionality. On one hand, the result highlights the benefits of mirror initialization. On the other hand, it implies that NTK theory may not fully explain the superior performance of neural networks.

An Offline Adaptation Framework for Constrained Multi-Objective Reinforcement Learning

In recent years, significant progress has been made in multi-objective reinforcement learning (RL) research, which aims to balance multiple objectives by incorporating preferences for each objective. In most existing studies, specific preferences must be provided during deployment to indicate the desired policies explicitly. However, designing these preferences depends heavily on human prior knowledge, which is typically obtained through extensive observation of high-performing demonstrations with expected behaviors. In this work, we propose a simple yet effective offline adaptation framework for multi-objective RL problems without assuming handcrafted target preferences, but only given several demonstrations to implicitly indicate the preferences of expected policies. Additionally, we demonstrate that our framework can naturally be extended to meet constraints on safety-critical objectives by utilizing safe demonstrations, even when the safety thresholds are unknown. Empirical results on offline multi-objective and safe tasks demonstrate the capability of our framework to infer policies that align with real preferences while meeting the constraints implied by the provided demonstrations.

Improving Adaptivity via Over-Parameterization in Sequence Models

It is well known that eigenfunctions of a kernel play a crucial role in kernel regression. Through several examples, we demonstrate that even with the same set of eigenfunctions, the order of these functions significantly impacts regression outcomes. Simplifying the model by diagonalizing the kernel, we introduce an over-parameterized gradient descent in the realm of sequence model to capture the effects of various orders of a fixed set of eigen-functions. This method is designed to explore the impact of varying eigenfunction orders. Our theoretical results show that the over-parameterization gradient flow can adapt to the underlying structure of the signal and significantly outperform the vanilla gradient flow method. Moreover, we also demonstrate that deeper over-parameterization can further enhance the generalization capability of the model. These results not only provide a new perspective on the benefits of over-parameterization and but also offer insights into the adaptivity and generalization potential of neural networks beyond the kernel regime.

On the Pinsker bound of inner product kernel regression in large dimensions

Building on recent studies of large-dimensional kernel regression, particularly those involving inner product kernels on the sphere $\mathbb{S}^{d}$, we investigate the Pinsker bound for inner product kernel regression in such settings. Specifically, we address the scenario where the sample size $n$ is given by $\alpha d^{\gamma}(1+o_{d}(1))$ for some $\alpha, \gamma>0$. We have determined the exact minimax risk for kernel regression in this setting, not only identifying the minimax rate but also the exact constant, known as the Pinsker constant, associated with the excess risk.

On the Saturation Effect of Kernel Ridge Regression

The saturation effect refers to the phenomenon that the kernel ridge regression (KRR) fails to achieve the information theoretical lower bound when the smoothness of the underground truth function exceeds certain level. The saturation effect has been widely observed in practices and a saturation lower bound of KRR has been conjectured for decades. In this paper, we provide a proof of this long-standing conjecture.

The phase diagram of kernel interpolation in large dimensions

The generalization ability of kernel interpolation in large dimensions (i.e., $n \asymp d^{\gamma}$ for some $\gamma>0$) might be one of the most interesting problems in the recent renaissance of kernel regression, since it may help us understand the 'benign overfitting phenomenon' reported in the neural networks literature. Focusing on the inner product kernel on the sphere, we fully characterized the exact order of both the variance and bias of large-dimensional kernel interpolation under various source conditions $s\geq 0$. Consequently, we obtained the $(s,\gamma)$-phase diagram of large-dimensional kernel interpolation, i.e., we determined the regions in $(s,\gamma)$-plane where the kernel interpolation is minimax optimal, sub-optimal and inconsistent.

The Optimality of Kernel Classifiers in Sobolev Space

Kernel methods are widely used in machine learning, especially for classification problems. However, the theoretical analysis of kernel classification is still limited. This paper investigates the statistical performances of kernel classifiers. With some mild assumptions on the conditional probability $\eta(x)=\mathbb{P}(Y=1\mid X=x)$, we derive an upper bound on the classification excess risk of a kernel classifier using recent advances in the theory of kernel regression. We also obtain a minimax lower bound for Sobolev spaces, which shows the optimality of the proposed classifier. Our theoretical results can be extended to the generalization error of overparameterized neural network classifiers. To make our theoretical results more applicable in realistic settings, we also propose a simple method to estimate the interpolation smoothness of $2\eta(x)-1$ and apply the method to real datasets.

Off-Policy Primal-Dual Safe Reinforcement Learning

Primal-dual safe RL methods commonly perform iterations between the primal update of the policy and the dual update of the Lagrange Multiplier. Such a training paradigm is highly susceptible to the error in cumulative cost estimation since this estimation serves as the key bond connecting the primal and dual update processes. We show that this problem causes significant underestimation of cost when using off-policy methods, leading to the failure to satisfy the safety constraint. To address this issue, we propose \textit{conservative policy optimization}, which learns a policy in a constraint-satisfying area by considering the uncertainty in cost estimation. This improves constraint satisfaction but also potentially hinders reward maximization. We then introduce \textit{local policy convexification} to help eliminate such suboptimality by gradually reducing the estimation uncertainty. We provide theoretical interpretations of the joint coupling effect of these two ingredients and further verify them by extensive experiments. Results on benchmark tasks show that our method not only achieves an asymptotic performance comparable to state-of-the-art on-policy methods while using much fewer samples, but also significantly reduces constraint violation during training. Our code is available at https://github.com/ZifanWu/CAL.

Policy-regularized Offline Multi-objective Reinforcement Learning

In this paper, we aim to utilize only offline trajectory data to train a policy for multi-objective RL. We extend the offline policy-regularized method, a widely-adopted approach for single-objective offline RL problems, into the multi-objective setting in order to achieve the above goal. However, such methods face a new challenge in offline MORL settings, namely the preference-inconsistent demonstration problem. We propose two solutions to this problem: 1) filtering out preference-inconsistent demonstrations via approximating behavior preferences, and 2) adopting regularization techniques with high policy expressiveness. Moreover, we integrate the preference-conditioned scalarized update method into policy-regularized offline RL, in order to simultaneously learn a set of policies using a single policy network, thus reducing the computational cost induced by the training of a large number of individual policies for various preferences. Finally, we introduce Regularization Weight Adaptation to dynamically determine appropriate regularization weights for arbitrary target preferences during deployment. Empirical results on various multi-objective datasets demonstrate the capability of our approach in solving offline MORL problems.

Generalization Error Curves for Analytic Spectral Algorithms under Power-law Decay

The generalization error curve of certain kernel regression method aims at determining the exact order of generalization error with various source condition, noise level and choice of the regularization parameter rather than the minimax rate. In this work, under mild assumptions, we rigorously provide a full characterization of the generalization error curves of the kernel gradient descent method (and a large class of analytic spectral algorithms) in kernel regression. Consequently, we could sharpen the near inconsistency of kernel interpolation and clarify the saturation effects of kernel regression algorithms with higher qualification, etc. Thanks to the neural tangent kernel theory, these results greatly improve our understanding of the generalization behavior of training the wide neural networks. A novel technical contribution, the analytic functional argument, might be of independent interest.