Stochastic Gradient Langevin Unlearning

``The right to be forgotten'' ensured by laws for user data privacy becomes increasingly important. Machine unlearning aims to efficiently remove the effect of certain data points on the trained model parameters so that it can be approximately the same as if one retrains the model from scratch. This work proposes stochastic gradient Langevin unlearning, the first unlearning framework based on noisy stochastic gradient descent (SGD) with privacy guarantees for approximate unlearning problems under convexity assumption. Our results show that mini-batch gradient updates provide a superior privacy-complexity trade-off compared to the full-batch counterpart. There are numerous algorithmic benefits of our unlearning approach, including complexity saving compared to retraining, and supporting sequential and batch unlearning. To examine the privacy-utility-complexity trade-off of our method, we conduct experiments on benchmark datasets compared against prior works. Our approach achieves a similar utility under the same privacy constraint while using $2\%$ and $10\%$ of the gradient computations compared with the state-of-the-art gradient-based approximate unlearning methods for mini-batch and full-batch settings, respectively.

Mean-Field Analysis for Learning Subspace-Sparse Polynomials with Gaussian Input

In this work, we study the mean-field flow for learning subspace-sparse polynomials using stochastic gradient descent and two-layer neural networks, where the input distribution is standard Gaussian and the output only depends on the projection of the input onto a low-dimensional subspace. We propose a basis-free generalization of the merged-staircase property in Abbe et al. (2022) and establish a necessary condition for the SGD-learnability. In addition, we prove that the condition is almost sufficient, in the sense that a condition slightly stronger than the necessary condition can guarantee the exponential decay of the loss functional to zero.

Rethinking the Capacity of Graph Neural Networks for Branching Strategy

Graph neural networks (GNNs) have been widely used to predict properties and heuristics of mixed-integer linear programs (MILPs) and hence accelerate MILP solvers. This paper investigates the capacity of GNNs to represent strong branching (SB) scores that provide an efficient strategy in the branch-and-bound algorithm. Although message-passing GNN (MP-GNN), as the simplest GNN structure, is frequently employed in the existing literature to learn SB scores, we prove a fundamental limitation in its expressive power -- there exist two MILP instances with different SB scores that cannot be distinguished by any MP-GNN, regardless of the number of parameters. In addition, we establish a universal approximation theorem for another GNN structure called the second-order folklore GNN (2-FGNN). We show that for any data distribution over MILPs, there always exists a 2-FGNN that can approximate the SB score with arbitrarily high accuracy and arbitrarily high probability. A small-scale numerical experiment is conducted to directly validate our theoretical findings.

Langevin Unlearning: A New Perspective of Noisy Gradient Descent for Machine Unlearning

Machine unlearning has raised significant interest with the adoption of laws ensuring the ``right to be forgotten''. Researchers have provided a probabilistic notion of approximate unlearning under a similar definition of Differential Privacy (DP), where privacy is defined as statistical indistinguishability to retraining from scratch. We propose Langevin unlearning, an unlearning framework based on noisy gradient descent with privacy guarantees for approximate unlearning problems. Langevin unlearning unifies the DP learning process and the privacy-certified unlearning process with many algorithmic benefits. These include approximate certified unlearning for non-convex problems, complexity saving compared to retraining, sequential and batch unlearning for multiple unlearning requests. We verify the practicality of Langevin unlearning by studying its privacy-utility-complexity trade-off via experiments on benchmark datasets, and also demonstrate its superiority against gradient-decent-plus-output-perturbation based approximate unlearning.

On Representing Mixed-Integer Linear Programs by Graph Neural Networks

While Mixed-integer linear programming (MILP) is NP-hard in general, practical MILP has received roughly 100--fold speedup in the past twenty years. Still, many classes of MILPs quickly become unsolvable as their sizes increase, motivating researchers to seek new acceleration techniques for MILPs. With deep learning, they have obtained strong empirical results, and many results were obtained by applying graph neural networks (GNNs) to making decisions in various stages of MILP solution processes. This work discovers a fundamental limitation: there exist feasible and infeasible MILPs that all GNNs will, however, treat equally, indicating GNN's lacking power to express general MILPs. Then, we show that, by restricting the MILPs to unfoldable ones or by adding random features, there exist GNNs that can reliably predict MILP feasibility, optimal objective values, and optimal solutions up to prescribed precision. We conducted small-scale numerical experiments to validate our theoretical findings.

On Representing Linear Programs by Graph Neural Networks

Learning to optimize is a rapidly growing area that aims to solve optimization problems or improve existing optimization algorithms using machine learning (ML). In particular, the graph neural network (GNN) is considered a suitable ML model for optimization problems whose variables and constraints are permutation--invariant, for example, the linear program (LP). While the literature has reported encouraging numerical results, this paper establishes the theoretical foundation of applying GNNs to solving LPs. Given any size limit of LPs, we construct a GNN that maps different LPs to different outputs. We show that properly built GNNs can reliably predict feasibility, boundedness, and an optimal solution for each LP in a broad class. Our proofs are based upon the recently--discovered connections between the Weisfeiler--Lehman isomorphism test and the GNN. To validate our results, we train a simple GNN and present its accuracy in mapping LPs to their feasibilities and solutions.

Dual reparametrized Variational Generative Model for Time-Series Forecasting

This paper propose DualVDT, a generative model for Time-series forecasting. Introduced dual reparametrized variational mechanisms on variational autoencoder (VAE) to tighter the evidence lower bound (ELBO) of the model, prove the advance performance analytically. This mechanism leverage the latent score based generative model (SGM), explicitly denoising the perturbation accumulated on latent vector through reverse time stochastic differential equation and variational ancestral sampling. The posterior of denoised latent distribution fused with dual reparametrized variational density. The KL divergence in ELBO will reduce to reach the better results of the model. This paper also proposed a latent attention mechanisms to extract multivariate dependency explicitly. Build the local temporal dependency simultaneously in factor wised through constructed local topology and temporal wised. The proven and experiment on multiple datasets illustrate, DualVDT, with a novel dual reparametrized structure, which denoise the latent perturbation through the reverse dynamics combining local-temporal inference, has the advanced performance both analytically and experimentally.

A Regularity Theory for Static Schrödinger Equations on $\mathbb{R}^d$ in Spectral Barron Spaces

Spectral Barron spaces have received considerable interest recently as it is the natural function space for approximation theory of two-layer neural networks with a dimension-free convergence rate. In this paper we study the regularity of solutions to the whole-space static Schr\"odinger equation in spectral Barron spaces. We prove that if the source of the equation lies in the spectral Barron space $\mathcal{B}^s(\mathbb{R}^d)$ and the potential function admitting a non-negative lower bound decomposes as a positive constant plus a function in $\mathcal{B}^s(\mathbb{R}^d)$, then the solution lies in the spectral Barron space $\mathcal{B}^{s+2}(\mathbb{R}^d)$.

On the Representation of Solutions to Elliptic PDEs in Barron Spaces

Numerical solutions to high-dimensional partial differential equations (PDEs) based on neural networks have seen exciting developments. This paper derives complexity estimates of the solutions of $d$-dimensional second-order elliptic PDEs in the Barron space, that is a set of functions admitting the integral of certain parametric ridge function against a probability measure on the parameters. We prove under some appropriate assumptions that if the coefficients and the source term of the elliptic PDE lie in Barron spaces, then the solution of the PDE is $\epsilon$-close with respect to the $H^1$ norm to a Barron function. Moreover, we prove dimension-explicit bounds for the Barron norm of this approximate solution, depending at most polynomially on the dimension $d$ of the PDE. As a direct consequence of the complexity estimates, the solution of the PDE can be approximated on any bounded domain by a two-layer neural network with respect to the $H^1$ norm with a dimension-explicit convergence rate.