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We study the implicit bias of flatness / low (loss) curvature and its effects on generalization in two-layer overparameterized ReLU networks with multivariate inputs---a problem well motivated by the minima stability and edge-of-stability phenomena in gradient-descent training. Existing work either requires interpolation or focuses only on univariate inputs. This paper presents new and somewhat surprising theoretical results for multivariate inputs. On two natural settings (1) generalization gap for flat solutions, and (2) mean-squared error (MSE) in nonparametric function estimation by stable minima, we prove upper and lower bounds, which establish that while flatness does imply generalization, the resulting rates of convergence necessarily deteriorate exponentially as the input dimension grows. This gives an exponential separation between the flat solutions compared to low-norm solutions (i.e., weight decay), which are known not to suffer from the curse of dimensionality. In particular, our minimax lower bound construction, based on a novel packing argument with boundary-localized ReLU neurons, reveals how flat solutions can exploit a kind of "neural shattering" where neurons rarely activate, but with high weight magnitudes. This leads to poor performance in high dimensions. We corroborate these theoretical findings with extensive numerical simulations. To the best of our knowledge, our analysis provides the first systematic explanation for why flat minima may fail to generalize in high dimensions. 

Deep neural networks are well-known for their generalization capabilities, largely attributed to optimizers’ ability to find "good" solutions in high-dimensional loss landscapes. This work aims to deepen the understanding of optimization specifically through the lens of loss landscapes. We propose a generalized framework for adaptive optimization that favors convergence to these "good" solutions. Our approach shifts the optimization paradigm from merely finding solutions quickly to discovering solutions that generalize well, establishing a careful balance between optimization efficiency and model generalization. We empirically validate our claims using two-layer, fully connected neural network with ReLU activation and demonstrate practical applicability through binary quantization of ResNets. Our numerical results demonstrate that these adaptive optimizers facilitate exploration leading to faster convergence speeds and narrow the generalization gap between stochastic gradient descent and other adaptive methods. 

Mobile devices such as smartphones, laptops, and tablets can often connect to multiple access networks (e.g., Wi-Fi, LTE, and 5G) simultaneously. Recent advancements facilitate seamless integration of these connections below the transport layer, enhancing the experience for apps that lack inherent multi-path support. This optimization hinges on dynamically determining the traffic distribution across networks for each device, a process referred to as \textit{multi-access traffic splitting}. This paper introduces \textit{NetworkGym}, a high-fidelity network environment simulator that facilitates generating multiple network traffic flows and multi-access traffic splitting. This simulator facilitates training and evaluating different RL-based solutions for the multi-access traffic splitting problem. Our initial explorations demonstrate that the majority of existing state-of-the-art offline RL algorithms (e.g. CQL) fail to outperform certain hand-crafted heuristic policies on average. This illustrates the urgent need to evaluate offline RL algorithms against a broader range of benchmarks, rather than relying solely on popular ones such as D4RL. We also propose an extension to the TD3+BC algorithm, named Pessimistic TD3 (PTD3), and demonstrate that it outperforms many state-of-the-art offline RL algorithms. PTD3's behavioral constraint mechanism, which relies on value-function pessimism, is theoretically motivated and relatively simple to implement. 

We study the generalization of two-layer ReLU neural networks in a univariate nonparametric regression problem with noisy labels. This is a problem where kernels (\emph{e.g.} NTK) are provably sub-optimal and benign overfitting does not happen, thus disqualifying existing theory for interpolating (0-loss, global optimal) solutions. We present a new theory of generalization for local minima that gradient descent with a constant learning rate can \emph{stably} converge to. We show that gradient descent with a fixed learning rate η can only find local minima that represent smooth functions with a certain weighted \emph{first order total variation} bounded by 1/η−1/2+O˜(σ+MSE‾‾‾‾‾√) where σ is the label noise level, MSE is short for mean squared error against the ground truth, and O˜(⋅) hides a logarithmic factor. Under mild assumptions, we also prove a nearly-optimal MSE bound of O˜(n−4/5) within the strict interior of the support of the n data points. Our theoretical results are validated by extensive simulation that demonstrates large learning rate training induces sparse linear spline fits. To the best of our knowledge, we are the first to obtain generalization bound via minima stability in the non-interpolation case and the first to show ReLU NNs without regularization can achieve near-optimal rates in nonparametric regression. 

We study the problem of multi-agent reinforcement learning (MARL) with adaptivity constraints -- a new problem motivated by real-world applications where deployments of new policies are costly and the number of policy updates must be minimized. For two-player zero-sum Markov Games, we design a (policy) elimination based algorithm that achieves a regret of O˜(H3S2ABK‾‾‾‾‾‾‾‾‾‾√), while the batch complexity is only O(H+loglogK). In the above, S denotes the number of states, A,B are the number of actions for the two players respectively, H is the horizon and K is the number of episodes. Furthermore, we prove a batch complexity lower bound Ω(HlogAK+loglogK) for all algorithms with O˜(K‾‾√) regret bound, which matches our upper bound up to logarithmic factors. As a byproduct, our techniques naturally extend to learning bandit games and reward-free MARL within near optimal batch complexity. To the best of our knowledge, these are the first line of results towards understanding MARL with low adaptivity. 

Convolutional residual neural networks (ConvResNets), though overparameterized, can achieve remarkable prediction performance in practice, which cannot be well explained by conventional wisdom. To bridge this gap, we study the performance of ConvResNeXts, which cover ConvResNets as a special case, trained with weight decay from the perspective of nonparametric classification. Our analysis allows for infinitely many building blocks in ConvResNeXts, and shows that weight decay implicitly enforces sparsity on these blocks. Specifically, we consider a smooth target function supported on a low-dimensional manifold, then prove that ConvResNeXts can adapt to the function smoothness and low-dimensional structures and efficiently learn the function without suffering from the curse of dimensionality. Our findings partially justify the advantage of overparameterized ConvResNeXts over conventional machine learning models. 

Private selection mechanisms (e.g., Report Noisy Max, Sparse Vector) are fundamental primitives of differentially private (DP) data analysis with wide applications to private query release, voting, and hyperparameter tuning. Recent work (Liu and Talwar, 2019; Papernot and Steinke, 2022) has made significant progress in both generalizing private selection mechanisms and tightening their privacy analysis using modern numerical privacy accounting tools, e.g., Rényi DP. But Rényi DP is known to be lossy when (ϵ,δ)-DP is ultimately needed, and there is a trend to close the gap by directly handling privacy profiles, i.e., δ as a function of ϵ or its equivalent dual form known as f-DPs. In this paper, we work out an easy-to-use recipe that bounds the privacy profiles of ReportNoisyMax and PrivateTuning using the privacy profiles of the base algorithms they corral. Numerically, our approach improves over the RDP-based accounting in all regimes of interest and leads to substantial benefits in end-to-end private learning experiments. Our analysis also suggests new distributions, e.g., binomial distribution for randomizing the number of rounds that leads to more substantial improvements in certain regimes. 

A recent study by De et al. (2022) has reported that large-scale representation learning through pre-training on a public dataset significantly enhances differentially private (DP) learning in downstream tasks, despite the high dimensionality of the feature space. To theoretically explain this phenomenon, we consider the setting of a layer-peeled model in representation learning, which results in interesting phenomena related to learned features in deep learning and transfer learning, known as Neural Collapse (NC). Within the framework of NC, we establish an error bound indicating that the misclassification error is independent of dimension when the distance between actual features and the ideal ones is smaller than a threshold. Additionally, the quality of the features in the last layer is empirically evaluated under different pre-trained models within the framework of NC, showing that a more powerful transformer leads to a better feature representation. Furthermore, we reveal that DP fine-tuning is less robust compared to fine-tuning without DP, particularly in the presence of perturbations. These observations are supported by both theoretical analyses and experimental evaluation. Moreover, to enhance the robustness of DP fine-tuning, we suggest several strategies, such as feature normalization or employing dimension reduction methods like Principal Component Analysis (PCA). Empirically, we demonstrate a significant improvement in testing accuracy by conducting PCA on the last-layer features.