Optimizing and certifying the positivity of polynomials are fundamental primitives across mathematics and engineering applications, from dynamical systems to operations research. However, solving these problems in practice requires large semidefinite programs, with poor scaling in dimension and degree. In this work, we demonstrate for the first time that neural networks can effectively solve such problems in a data-driven fashion, achieving tenfold speedups while retaining high accuracy. Moreover, we observe that these polynomial learning problems are equivariant to the non-compact group $SL(2,\mathbb{R})$, which consists of area-preserving linear transformations. We therefore adapt our learning pipelines to accommodate this structure, including data augmentation, a new $SL(2,\mathbb{R})$-equivariant architecture, and an architecture equivariant with respect to its maximal compact subgroup, $SO(2, \mathbb{R})$. Surprisingly, the most successful approaches in practice do not enforce equivariance to the entire group, which we prove arises from an unusual lack of architecture universality for $SL(2,\mathbb{R})$ in particular. A consequence of this result, which is of independent interest, is that there exists an equivariant function for which there is no sequence of equivariant polynomials multiplied by arbitrary invariants that approximates the original function. This is a rare example of a symmetric problem where data augmentation outperforms a fully equivariant architecture, and provides interesting lessons in both theory and practice for other problems with non-compact symmetries.
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Theory · Learning Theory
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.
Deep Learning · Everything Else
Hypernetworks, neural networks that predict the parameters of another neural network, are powerful models that have been successfully used in diverse applications from image generation to multi-task learning. Unfortunately, existing hypernetworks are often challenging to train. Training typically converges far more slowly than for non-hypernetwork models, and the rate of convergence can be very sensitive to hyperparameter choices. In this work, we identify a fundamental and previously unidentified problem that contributes to the challenge of training hypernetworks: a magnitude proportionality between the inputs and outputs of the hypernetwork. We demonstrate both analytically and empirically that this can lead to unstable optimization, thereby slowing down convergence, and sometimes even preventing any learning. We present a simple solution to this problem using a revised hypernetwork formulation that we call Magnitude Invariant Parametrizations (MIP). We demonstrate the proposed solution on several hypernetwork tasks, where it consistently stabilizes training and achieves faster convergence. Furthermore, we perform a comprehensive ablation study including choices of activation function, normalization strategies, input dimensionality, and hypernetwork architecture; and find that MIP improves training in all scenarios. We provide easy-to-use code that can turn existing networks into MIP-based hypernetworks.
Deep Learning · Robustness
One prominent approach toward resolving the adversarial vulnerability of deep neural networks is the two-player zero-sum paradigm of adversarial training, in which predictors are trained against adversarially chosen perturbations of data. Despite the promise of this approach, algorithms based on this paradigm have not engendered sufficient levels of robustness and suffer from pathological behavior like robust overfitting. To understand this shortcoming, we first show that the commonly used surrogate-based relaxation used in adversarial training algorithms voids all guarantees on the robustness of trained classifiers. The identification of this pitfall informs a novel non-zero-sum bilevel formulation of adversarial training, wherein each player optimizes a different objective function. Our formulation yields a simple algorithmic framework that matches and in some cases outperforms state-of-the-art attacks, attains comparable levels of robustness to standard adversarial training algorithms, and does not suffer from robust overfitting.
Deep Learning · Generative Models and Autoencoders
Despite the recent advances in the field of computational Schrödinger Bridges (SB), most existing SB solvers are still heavy-weighted and require complex optimization of several neural networks. It turns out that there is no principal solver which plays the role of simple-yet-effective baseline for SB just like, e.g., $k$-means method in clustering, logistic regression in classification or Sinkhorn algorithm in discrete optimal transport. We address this issue and propose a novel fast and simple SB solver. Our development is a smart combination of two ideas which recently appeared in the field: (a) parameterization of the Schrödinger potentials with sum-exp quadratic functions and (b) viewing the log-Schrödinger potentials as the energy functions. We show that combined together these ideas yield a lightweight, simulation-free and theoretically justified SB solver with a simple straightforward optimization objective. As a result, it allows solving SB in moderate dimensions in a matter of minutes on CPU without a painful hyperparameter selection. Our light solver resembles the Gaussian mixture model which is widely used for density estimation. Inspired by this similarity, we also prove an important theoretical result showing that our light solver is a universal approximator of SBs. Furthemore, we conduct the analysis of the generalization error of our light solver. The code for our solver can be found at https://github.com/ngushchin/LightSB.
Theory · Deep Learning
Vision tasks are characterized by the properties of locality and translation invariance. The superior performance of convolutional neural networks (CNNs) on these tasks is widely attributed to the inductive bias of locality and weight sharing baked into their architecture. Existing attempts to quantify the statistical benefits of these biases in CNNs over locally connected convolutional neural networks (LCNs) and fully connected neural networks (FCNs) fall into one of the following categories: either they disregard the optimizer and only provide uniform convergence upper bounds with no separating lower bounds, or they consider simplistic tasks that do not truly mirror the locality and translation invariance as found in real-world vision tasks. To address these deficiencies, we introduce the Dynamic Signal Distribution (DSD) classification task that models an image as consisting of $k$ patches, each of dimension $d$, and the label is determined by a $d$-sparse signal vector that can freely appear in any one of the $k$ patches. On this task, for any orthogonally equivariant algorithm like gradient descent, we prove that CNNs require $\tilde{O}(k+d)$ samples, whereas LCNs require $\Omega(kd)$ samples, establishing the statistical advantages of weight sharing in translation invariant tasks. Furthermore, LCNs need $\tilde{O}(k(k+d))$ samples, compared to $\Omega(k^2d)$ samples for FCNs, showcasing the benefits of locality in local tasks. Additionally, we develop information theoretic tools for analyzing randomized algorithms, which may be of interest for statistical research.
In our era of enormous neural networks, empirical progress has been driven by the philosophy that *more is better.*Recent deep learning practice has found repeatedly that larger model size, more data, and more computation (resulting in lower training loss) optimizing to near-interpolation improves performance. In this paper, we give theoretical backing to these empirical observations by showing that these three properties hold in random feature (RF) regression, a class of models equivalent to shallow networks with only the last layer trained.Concretely, we first show that the test risk of RF regression decreases monotonically with both the number of features and samples, provided the ridge penalty is tuned optimally. In particular, this implies that infinite width RF architectures are preferable to those of any finite width. We then proceed to demonstrate that, for a large class of tasks characterized by powerlaw eigenstructure, training to near-zero training loss is *obligatory:* near-optimal performance can *only* be achieved when the training error is much smaller than the test error. Grounding our theory in real-world data, we find empirically that standard computer vision tasks with convolutional neural kernels clearly fall into this class. Taken together, our results tell a simple, testable story of the benefits of overparameterization and overfitting in random feature models.
Probabilistic Methods · Bayesian Models and Methods
A newly-arising uncertainty estimation method named Evidential Deep Learning (EDL), which can obtain reliable predictive uncertainty in a single forward pass, has garnered increasing interest. Guided by the subjective logic theory, EDL obtains Dirichlet concentration parameters from deep neural networks, thus constructing a Dirichlet probability density function (PDF) to model the distribution of class probabilities. Despite its great success, we argue that EDL keeps nonessential settings in both stages of model construction and optimization.In this work, our analysis indicates that (1) in the construction of the Dirichlet PDF, a commonly ignored parameter termed prior weight governs the balance between leveraging the proportion of evidence and its magnitude in deriving predictive scores, and (2) in model optimization, a variance-minimized regularization term adopted by traditional EDL encourages the Dirichlet PDF to approach a Dirac delta function, potentially exacerbating overconfidence. Therefore, we propose the R-EDL (Relaxed-EDL) method by relaxing these nonessential settings. Specifically, R-EDL treats the prior weight as an adjustable hyper-parameter instead of a fixed scalar, and directly optimizes the expectation of the Dirichlet PDF provided to deprecate the variance-minimized regularization term. Extensive experiments and SOTA performances demonstrate the effectiveness of our method. Source codes are provided in Appendix E.
Deep Learning · Graph Neural Networks
Equivariance is an important structural property that is captured by architectures such as graph neural networks (GNNs). However, equivariant graph functions cannot produce different outputs for similar nodes, which may be undesirable when the function is trying to optimize some global graph property. In this paper, we define orbit-equivariance, a relaxation of equivariance which allows for such functions whilst retaining important structural inductive biases. We situate the property in the hierarchy of graph functions, define a taxonomy of orbit-equivariant functions, and provide four different ways to achieve non-equivariant GNNs. For each, we analyze their expressivity with respect to orbit-equivariance and evaluate them on two novel datasets, one of which stems from a real-world use-case of designing optimal bioisosteres.
Social Aspects · Trustworthy Machine Learning
Adversarial patch attacks, where a small patch is placed in the scene to fool neural networks, have been studied for numerous applications. Focusing on image classification, we consider the setting of a black-box transfer attack where an attacker does not know the target model. Instead of forcing corrupted image representations to cross the nearest decision boundaries or converge to a particular point, we propose a distribution-oriented approach. We rely on optimal transport to push the feature distribution of attacked images towards an already modeled distribution. We show that this new distribution-oriented approach leads to better transferable patches. Through digital experiments conducted on ImageNet-1K, we provide evidence that our new patches are the only ones that can simultaneously influence multiple Transformer models and Convolutional Neural Networks. Physical world experiments demonstrate that our patch can affect systems in deployment without explicit knowledge.
Optimization · Learning for Optimization
Modern solvers for solving mixed integer programming (MIP) often rely on the branch-and-bound (B&B) algorithm which could be of high time complexity, and presolving techniques are well designed to simplify the instance as pre-processing before B&B. However, such presolvers in existing literature or open-source solvers are mostly set by default agnostic to specific input instances, and few studies have been reported on tailoring presolving settings. In this paper, we aim to dive into this open question and show that the MIP solver can be indeed largely improved when switching the default instance-agnostic presolving into instance-specific presolving. Specifically, we propose a combination of supervised learning and classic heuristics to achieve efficient presolving adjusting, avoiding tedious reinforcement learning. Notably, our approach is orthogonal from many recent efforts in incorporating learning modules into the B&B framework after the presolving stage, and to our best knowledge, this is the first work for introducing learning to presolve in MIP solvers. Experiments on multiple real-world datasets show that well-trained neural networks can infer proper presolving for arbitrary incoming MIP instances in less than 0.5s, which is neglectable compared with the solving time often hours or days.
Deep Learning · Theory
Neural networks trained by gradient descent (GD) have exhibited a number of surprising generalization behaviors. First, they can achieve a perfect fit to noisy training data and still generalize near-optimally, showing that overfitting can sometimes be benign. Second, they can undergo a period of classical, harmful overfitting---achieving a perfect fit to training data with near-random performance on test data---before transitioning (''grokking'') to near-optimal generalization later in training. In this work, we show that both of these phenomena provably occur in two-layer ReLU networks trained by GD on XOR cluster data where a constant fraction of the training labels are flipped. In this setting, we show that after the first step of GD, the network achieves 100\% training accuracy, perfectly fitting the noisy labels in the training data, but achieves near-random test accuracy. At a later training step, the network achieves near-optimal test accuracy while still fitting the random labels in the training data, exhibiting a ''grokking'' phenomenon. This provides the first theoretical result of benign overfitting in neural network classification when the data distribution is not linearly separable. Our proofs rely on analyzing the feature learning process under GD, which reveals that the network implements a non-generalizable linear classifier after one step and gradually learns generalizable features in later steps.
Applications · Physics
Advancements in deep learning have led to the development of physics-informed neural networks (PINNs) for solving partial differential equations (PDEs) without being supervised by PDE solutions. While vanilla PINNs require training one network per PDE configuration, recent works have showed the potential to meta-learn PINNs across a range of PDE configurations. It is however known that PINN training is associated with different levels of difficulty, depending on the underlying PDE configurations or the number of residual sampling points available. Existing meta-learning approaches, however, treat all PINN tasks equally. We address this gap by introducing a novel difficulty-aware task sampler (DATS) for meta-learning of PINNs. We derive an optimal analytical solution to optimize the probability for sampling individual PINN tasks in order to minimize their validation loss across tasks. We further present two alternative strategies to utilize this sampling probability to either adaptively weigh PINN tasks, or dynamically allocate optimal residual points across tasks. We evaluated DATS against uniform and self-paced task-sampling baselines on two representative meta-PINN models, across four benchmark PDEs as well as three different residual point sampling strategies. The results demonstrated that DATS was able to improve the accuracy of meta-learned PINN solutions when reducing performance disparity across PDE configurations, at only a fraction of residual sampling budgets required by its baselines.
Deep Learning · Graph Neural Networks
Recently, hypergraph neural networks (HGNNs) exhibit the potential to tackle tasks with high-order correlations and have achieved success in many tasks. However, existing evolution on the hypergraph has poor controllability and lacks sufficient theoretical support (like dynamic systems), thus yielding sub-optimal performance. One typical scenario is that only one or two layers of HGNNs can achieve good results and more layers lead to degeneration of performance. Under such circumstances, it is important to increase the controllability of HGNNs. In this paper, we first introduce hypergraph dynamic systems (HDS), which bridge hypergraphs and dynamic systems and characterize the continuous dynamics of representations. We then propose a control-diffusion hypergraph dynamic system by an ordinary differential equation (ODE). We design a multi-layer HDS$^{ode}$ as a neural implementation, which contains control steps and diffusion steps. HDS$^{ode}$ has the properties of controllability and stabilization and is allowed to capture long-range correlations among vertices. Experiments on $9$ datasets demonstrate HDS$^{ode}$ beat all compared methods. HDS$^{ode}$ achieves stable performance with increased layers and solves the poor controllability of HGNNs. We also provide the feature visualization of the evolutionary process to demonstrate the controllability and stabilization of HDS$^{ode}$.
Optimization
Stochastic Gradient Descent (SGD) stands as a cornerstone optimization algorithm with proven real-world empirical successes but relatively limited theoretical understanding. Recent research has illuminated a key factor contributing to its practical efficacy: the implicit regularization it instigates. Several studies have investigated the linear stability property of SGD in the vicinity of a stationary point as a predictive proxy for sharpness and generalization error in overparameterized neural networks (Wu et al., 2022; Jastrzebski et al., 2019; Cohen et al., 2021). In this paper, we delve deeper into the relationship between linear stability and sharpness. More specifically, we meticulously delineate the necessary and sufficient conditions for linear stability, contingent on hyperparameters of SGD and the sharpness at the optimum. Towards this end, we introduce a novel coherence measure of the loss Hessian that encapsulates pertinent geometric properties of the loss function that are relevant to the linear stability of SGD. It enables us to provide a simplified sufficient condition for identifying linear instability at an optimum. Notably, compared to previous works, our analysis relies on significantly milder assumptions and is applicable for a broader class of loss functions than known before, encompassing not only mean-squared error but also cross-entropy loss.
Reinforcement Learning · Deep RL
Partially Observable Markov Decision Processes (POMDPs) are used to model environments where the state cannot be perceived, necessitating reasoning based on past observations and actions. However, remembering the full history is generally intractable due to the exponential growth in the history space. Maintaining a probability distribution that models the belief over the current state can be used as a sufficient statistic of the history, but its computation requires access to the model of the environment and is often intractable. While SOTA algorithms use Recurrent Neural Networks to compress the observation-action history aiming to learn a sufficient statistic, they lack guarantees of success and can lead to sub-optimal policies. To overcome this, we propose the Wasserstein Belief Updater, an RL algorithm that learns a latent model of the POMDP and an approximation of the belief update under the assumption that the state is observable during training. Our approach comes with theoretical guarantees on the quality of our approximation ensuring that our latent beliefs allow for learning the optimal value function.
Neural networks and neural ODEs tend to be vulnerable to adversarial attacks, rendering robust optimizers critical to curb the success of such attacks. In this regard, the key insight of this work is to interpret Neural ODE optimization as a min-max optimal control problem. More particularly, we present Game Theoretic Second-Order Neural Optimizer (GTSONO), a robust game theoretic optimizer based on the principles of min-max Differential Dynamic Programming.The proposed method exhibits significant computational benefits due to efficient matrix decompositions and provides convergence guarantees to local saddle points.Empirically, the robustness of the proposed optimizer is demonstrated through greater robust accuracy compared to benchmark optimizers when trained on clean images. Additionally, its ability to provide a performance increase when adapted to an already existing adversarial defense technique is also illustrated.Finally, the superiority of the proposed update law over its gradient based counterpart highlights the potential benefits of incorporating robust optimal control paradigms into adversarial training methods.
Deep Learning · Graph Neural Networks
Neural networks that process the parameters of other neural networks find applications in domains as diverse as classifying implicit neural representations, generating neural network weights, and predicting generalization errors. However, existing approaches either overlook the inherent permutation symmetry in the neural network or rely on intricate weight-sharing patterns to achieve equivariance, while ignoring the impact of the network architecture itself. In this work, we propose to represent neural networks as computational graphs of parameters, which allows us to harness powerful graph neural networks and transformers that preserve permutation symmetry. Consequently, our approach enables a single model to encode neural computational graphs with diverse architectures. We showcase the effectiveness of our method on a wide range of tasks, including classification and editing of implicit neural representations, predicting generalization performance, and learning to optimize, while consistently outperforming state-of-the-art methods. The source code is open-sourced at https://github.com/mkofinas/neural-graphs.
Deep Learning · Theory
We study the problem of learning hierarchical polynomials over the standard Gaussian distribution with three-layer neural networks. We specifically consider target functions of the form $h = g \circ p$ where $p : \mathbb{R}^d \rightarrow \mathbb{R}$ is a degree $k$ polynomial and $g: \mathbb{R} \rightarrow \mathbb{R}$ is a degree $q$ polynomial. This function class generalizes the single-index model, which corresponds to $k=1$, and is a natural class of functions possessing an underlying hierarchical structure. Our main result shows that for a large subclass of degree $k$ polynomials $p$, a three-layer neural network trained via layerwise gradient descent on the square loss learns the target $h$ up to vanishing test error in $\widetilde O(d^k)$ samples and polynomial time. This is a strict improvement over kernel methods, which require $\widetilde \Theta(d^{kq})$ samples, as well as existing guarantees for two-layer networks, which require the target function to be low-rank. Our result also generalizes prior works on three-layer neural networks, which were restricted to the case of $p$ being a quadratic. When $p$ is indeed a quadratic, we achieve the information-theoretically optimal sample complexity $\widetilde O(d^2)$, which is an improvement over prior work (Nichani et al., 2023) requiring a sample size of $\widetilde\Theta(d^4)$. Our proof proceeds by showing that during the initial stage of training the network performs feature learning to recover the feature $p$ with $\widetilde O(d^k)$ samples. This work demonstrates the ability of three-layer neural networks to learn complex features and as a result, learn a broad class of hierarchical functions.
Miscellaneous Aspects of Machine Learning · Representation Learning
We identify a new phenomenon in neural network optimization which arises from the interaction of depth and a particular heavy-tailed structure in natural data. Our result offers intuitive explanations for several previously reported observations about network training dynamics, including a conceptually new cause for progressive sharpening and the edge of stability. We further draw connections to related phenomena including grokking and simplicity bias.Experimentally, we demonstrate the significant influence of paired groups of outliers in the training data with strong *Opposing Signals*: consistent, large magnitude features which dominate the network output and provide gradients which point in opposite directions. Due to these outliers, early optimization enters a narrow valley which carefully balances the opposing groups; subsequent sharpening causes their loss to rise rapidly, oscillating between high on one group and then the other, until the overall loss spikes. We carefully study these groups' effect on the network's optimization and behavior, and we complement this with a theoretical analysis of a two-layer linear network under a simplified model.Our finding enables new qualitative predictions of training behavior which we confirm experimentally. It also provides a new lens through which to study and improve modern training practices for stochastic optimization, which we highlight via a case study of Adam versus SGD.