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Deep Learning · Theory

Atsushi Nitanda

Mean-field Langevin dynamics (MFLD) minimizes an entropy-regularized nonlinear convex functional defined over the space of probability distributions. MFLD has gained attention due to its connection with noisy gradient descent for mean-field two-layer neural networks. Unlike standard Langevin dynamics, the nonlinearity of the objective functional induces particle interactions, necessitating multiple particles to approximate the dynamics in a finite-particle setting. Recent works (Chen et al., 2022; Suzuki et al., 2023b) have demonstrated the uniform-in-time propagation of chaos for MFLD, showing that the gap between the particle system and its mean-field limit uniformly shrinks over time as the number of particles increases. In this work, we improve the dependence on logarithmic Sobolev inequality (LSI) constants in their particle approximation errors, which can exponentially deteriorate with the regularization coefficient. Specifically, we establish an LSI-constant-free particle approximation error concerning the objective gap by leveraging the problem structure in risk minimization. As the application, we demonstrate improved convergence of MFLD, sampling guarantee for the mean-field stationary distribution, and uniform-in-time Wasserstein propagation of chaos in terms of particle complexity.

Probabilistic Methods · Bayesian Models and Methods

Tristan Cinquin, Marvin Pförtner, Vincent Fortuin, Philipp Hennig, Robert Bamler

Laplace approximations are popular techniques for endowing deep networks with epistemic uncertainty estimates as they can be applied without altering the predictions of the trained network, and they scale to large models and datasets. While the choice of prior strongly affects the resulting posterior distribution, computational tractability and lack of interpretability of the weight space typically limit the Laplace approximation to isotropic Gaussian priors, which are known to cause pathological behavior as depth increases. As a remedy, we directly place a prior on function space. More precisely, since Lebesgue densities do not exist on infinite-dimensional function spaces, we recast training as finding the so-called weak mode of the posterior measure under a Gaussian process (GP) prior restricted to the space of functions representable by the neural network. Through the GP prior, one can express structured and interpretable inductive biases, such as regularity or periodicity, directly in function space, while still exploiting the implicit inductive biases that allow deep networks to generalize. After model linearization, the training objective induces a negative log-posterior density to which we apply a Laplace approximation, leveraging highly scalable methods from matrix-free linear algebra. Our method provides improved results where prior knowledge is abundant (as is the case in many scientific inference tasks). At the same time, it stays competitive for black-box supervised learning problems, where neural networks typically excel.

Reinforcement Learning · Function Approximation

Weitong Zhang, Zhiyuan Fan, Jiafan He, Quanquan Gu

We study the constant regret guarantees in reinforcement learning (RL). Our objective is to design an algorithm that incurs only finite regret over infinite episodes with high probability. We introduce an algorithm, Cert-LSVI-UCB, for misspecified linear Markov decision processes (MDPs) where both the transition kernel and the reward function can be approximated by some linear function up to misspecification level $\zeta$. At the core of Cert-LSVI-UCB is an innovative certified estimator, which facilitates a fine-grained concentration analysis for multi-phase value-targeted regression, enabling us to establish an instance-dependent regret bound that is constant w.r.t. the number of episodes. Specifically, we demonstrate that for a linear MDP characterized by a minimal suboptimality gap $\Delta$, Cert-LSVI-UCB has a cumulative regret of $\tilde{\mathcal{O}}(d^3H^5/\Delta)$ with high probability, provided that the misspecification level $\zeta$ is below $\tilde{\mathcal{O}}(\Delta / (\sqrt{d}H^2))$. Here $d$ is the dimension of the feature space and $H$ is the horizon. Remarkably, this regret bound is independent of the number of episodes $K$. To the best of our knowledge, Cert-LSVI-UCB is the first algorithm to achieve a constant, instance-dependent, high-probability regret bound in RL with linear function approximation without relying on prior distribution assumptions.

Optimization · Learning for Optimization

Yao Shu, Jiongfeng Fang, Ying He, Fei Yu

First-order optimization (FOO) algorithms are pivotal in numerous computational domains, such as reinforcement learning and deep learning. However, their application to complex tasks often entails significant optimization inefficiency due to their need of many sequential iterations for convergence. In response, we introduce first-order optimization expedited with approximately parallelized iterations (OptEx), the first general framework that enhances the time efficiency of FOO by leveraging parallel computing to directly mitigate its requirement of many sequential iterations for convergence. To achieve this, OptEx utilizes a kernelized gradient estimation that is based on the history of evaluated gradients to predict the gradients required by the next few sequential iterations in FOO, which helps to break the inherent iterative dependency and hence enables the approximate parallelization of iterations in FOO. We further establish theoretical guarantees for the estimation error of our kernelized gradient estimation and the iteration complexity of SGD-based OptEx, confirming that the estimation error diminishes to zero as the history of gradients accumulates and that our SGD-based OptEx enjoys an effective acceleration rate of Θ(√N ) over standard SGD given parallelism of N, in terms of the sequential iterations required for convergence. Finally, we provide extensive empirical studies, including synthetic functions, reinforcement learning tasks, and neural network training on various datasets, to underscore the substantial efficiency improvements achieved by our OptEx in practice.

Miscellaneous Aspects of Machine Learning · Unsupervised and Semi-supervised Learning

Zhibin Gu, Songhe Feng

While Tensor-based Multi-view Subspace Clustering (TMSC) has garnered significant attention for its capacity to effectively capture high-order correlations among multiple views, three notable limitations in current TMSC methods necessitate consideration: 1) high computational complexity and reliance on dictionary completeness resulting from using observed data as the dictionary, 2) inaccurate subspace representation stemming from the oversight of local geometric information and 3) under-penalization of noise-related singular values within tensor data caused by treating all singular values equally. To address these limitations, this paper presents a \textbf{S}calable TMSC framework with \textbf{T}riple inf\textbf{O}rmatio\textbf{N} \textbf{E}nhancement (\textbf{STONE}). Notably, an enhanced anchor dictionary learning mechanism has been utilized to recover the low-rank anchor structure, resulting in reduced computational complexity and increased resilience, especially in scenarios with inadequate dictionaries. Additionally, we introduce an anchor hypergraph Laplacian regularizer to preserve the inherent geometry of the data within the subspace representation. Simultaneously, an improved hyperbolic tangent function has been employed as a precise approximation for tensor rank, effectively capturing the significant variations in singular values. Extensive experimentation on a variety of datasets demonstrates that our approach surpasses SOTA methods in both effectiveness and efficiency.

Probabilistic Methods · Variational Inference

Jen Ning Lim, Adam Johansen

Semi-implicit variational inference (SIVI) enriches the expressiveness of variationalfamilies by utilizing a kernel and a mixing distribution to hierarchically define thevariational distribution. Existing SIVI methods parameterize the mixing distributionusing implicit distributions, leading to intractable variational densities. As a result,directly maximizing the evidence lower bound (ELBO) is not possible, so theyresort to one of the following: optimizing bounds on the ELBO, employing costlyinner-loop Markov chain Monte Carlo runs, or solving minimax objectives. In thispaper, we propose a novel method for SIVI called Particle Variational Inference(PVI) which employs empirical measures to approximate the optimal mixingdistributions characterized as the minimizer of a free energy functional. PVI arisesnaturally as a particle approximation of a Euclidean–Wasserstein gradient flow and,unlike prior works, it directly optimizes the ELBO whilst making no parametricassumption about the mixing distribution. Our empirical results demonstrate thatPVI performs favourably compared to other SIVI methods across various tasks.Moreover, we provide a theoretical analysis of the behaviour of the gradient flowof a related free energy functional: establishing the existence and uniqueness ofsolutions as well as propagation of chaos results.

Privacy

Hilal Asi, Daogao Liu, Kevin Tian

We study the problem of differentially private stochastic convex optimization (DP-SCO) with heavy-tailed gradients, where we assume a $k^{\text{th}}$-moment bound on the Lipschitz constants of sample functions, rather than a uniform bound. We propose a new reduction-based approach that enables us to obtain the first optimal rates (up to logarithmic factors) in the heavy-tailed setting, achieving error $G_2 \cdot \frac 1 {\sqrt n} + G_k \cdot (\frac{\sqrt d}{n\epsilon})^{1 - \frac 1 k}$ under $(\epsilon, \delta)$-approximate differential privacy, up to a mild $\textup{polylog}(\frac{1}{\delta})$ factor, where $G_2^2$ and $G_k^k$ are the $2^{\text{nd}}$ and $k^{\text{th}}$ moment bounds on sample Lipschitz constants, nearly-matching a lower bound of [LR23].We then give a suite of private algorithms for DP-SCO with heavy-tailed gradients improving our basic result under additional assumptions, including an optimal algorithm under a known-Lipschitz constant assumption, a near-linear time algorithm for smooth functions, and an optimal linear time algorithm for smooth generalized linear models.

Trustworthy Machine Learning

Hongchao Zhang, Zhizhen Qin, Sicun Gao, Andrew Clark

Neural Control Barrier Functions (NCBFs) have shown significant promise in enforcing safety constraints on nonlinear autonomous systems. State-of-the-art exact approaches to verifying safety of NCBF-based controllers exploit the piecewise-linear structure of ReLU neural networks, however, such approaches still rely on enumerating all of the activation regions of the network near the safety boundary, thus incurring high computation cost. In this paper, we propose a framework for Synthesis with Efficient Exact Verification (SEEV). Our framework consists of two components, namely (i) an NCBF synthesis algorithm that introduces a novel regularizer to reduce the number of activation regions at the safety boundary, and (ii) a verification algorithm that exploits tight over-approximations of the safety conditions to reduce the cost of verifying each piecewise-linear segment. Our simulations show that SEEV significantly improves verification efficiency while maintaining the CBF quality across various benchmark systems and neural network structures. Our code is available at https://github.com/HongchaoZhang-HZ/SEEV.

Optimization · Convex

Qiujiang Jin, Ruichen Jiang, Aryan Mokhtari

In this paper, we present the first explicit and non-asymptotic global convergence rates of the BFGS method when implemented with an inexact line search scheme satisfying the Armijo-Wolfe conditions. We show that BFGS achieves a global linear convergence rate of $(1 - \frac{1}{\kappa})^t$ for $\mu$-strongly convex functions with $L$-Lipschitz gradients, where $\kappa = \frac{L}{\mu}$ represents the condition number. Additionally, if the objective function's Hessian is Lipschitz, BFGS with the Armijo-Wolfe line search achieves a linear convergence rate that depends solely on the line search parameters, independent of the condition number. We also establish a global superlinear convergence rate of $\mathcal{O}((\frac{1}{t})^t)$. These global bounds are all valid for any starting point $x_0$ and any symmetric positive definite initial Hessian approximation matrix $B_0$, though the choice of $B_0$ impacts the number of iterations needed to achieve these rates. By synthesizing these results, we outline the first global complexity characterization of BFGS with the Armijo-Wolfe line search. Additionally, we clearly define a mechanism for selecting the step size to satisfy the Armijo-Wolfe conditions and characterize its overall complexity.

Reinforcement Learning · Deep RL

Hongming Zhang, Chenjun Xiao, Chao Gao, Han Wang, bo xu, Martin Müller

Reinforcement learning (RL) algorithms are typically based on optimizing a Markov Decision Process (MDP) using the optimal Bellman equation. Recent studies have revealed that focusing the optimization of Bellman equations solely on in-sample actions tends to result in more stable optimization, especially in the presence of function approximation. Upon on these findings, in this paper, we propose an Empirical MDP Iteration (EMIT) framework. EMIT constructs a sequence of empirical MDPs using data from the growing replay memory. For each of these empirical MDPs, it learns an estimated Q-function denoted as $\widehat{Q}$. The key strength is that by restricting the Bellman update to in-sample bootstrapping, each empirical MDP converges to a unique optimal $\widehat{Q}$ function. Furthermore, gradually expanding from the empirical MDPs to the original MDP induces a monotonic policy improvement. Instead of creating entirely new algorithms, we demonstrate that EMIT can be seamlessly integrated with existing online RL algorithms, effectively acting as a regularizer for contemporary Q-learning methods. We show this by implementing EMIT for two representative RL algorithms, DQN and TD3. Experimental results on Atari and MuJoCo benchmarks show that EMIT significantly reduces estimation errors and substantially improves the performance of both algorithms.

Computer Vision

Zhaoliang Zhang, Tianchen Song, Yongjae Lee, Li Yang, Cheng Peng, Rama Chellappa, Deliang Fan

Recently, 3D Gaussian Splatting (3DGS) has become one of the mainstream methodologies for novel view synthesis (NVS) due to its high quality and fast rendering speed. However, as a point-based scene representation, 3DGS potentially generates a large number of Gaussians to fit the scene, leading to high memory usage. Improvements that have been proposed require either an empirical pre-set pruning ratio or importance score threshold to prune the point cloud. Such hyperparameters require multiple rounds of training to optimize and achieve the maximum pruning ratio while maintaining the rendering quality for each scene. In this work, we propose learning-to-prune 3DGS (LP-3DGS), where a trainable binary mask is applied to the importance score to automatically find a favorable pruning ratio. Instead of using the traditional straight-through estimator (STE) method to approximate the binary mask gradient, we redesign the masking function to leverage the Gumbel-Sigmoid method, making it differentiable and compatible with the existing training process of 3DGS. Extensive experiments have shown that LP-3DGS consistently achieves a good balance between efficiency and high quality.

Probabilistic Methods · Variational Inference

Shiyue Zhang, Longlin Yu, Ziheng Cheng, Cheng Zhang

Recently, through a unified gradient flow perspective of Markov chain Monte Carlo (MCMC) and variational inference (VI), particle-based variational inference methods (ParVIs) have been proposed that tend to combine the best of both worlds. While typical ParVIs such as Stein Variational Gradient Descent (SVGD) approximate the gradient flow within a reproducing kernel Hilbert space (RKHS), many attempts have been made recently to replace RKHS with more expressive function spaces, such as neural networks. While successful, these methods are mainly designed for sampling from unconstrained domains. In this paper, we offer a general solution to constrained sampling by introducing a boundary condition for the gradient flow which would confine the particles within the specific domain. This allows us to propose a new functional gradient ParVI method for constrained sampling, called *constrained functional gradient flow* (CFG), with provable continuous-time convergence in total variation (TV). We also present novel numerical strategies to handle the boundary integral term arising from the domain constraints. Our theory and experiments demonstrate the effectiveness of the proposed framework.

Probabilistic Methods

Rémi Bardenet, Subhroshekhar Ghosh, Hugo Simon-Onfroy, Hoang Son Tran

Determinantal point processes (DPPs) are random configurations of points with tunable negative dependence. Because sampling is tractable, DPPs are natural candidates for subsampling tasks, such as minibatch selection or coreset construction. A \emph{coreset} is a subset of a (large) training set, such that minimizing an empirical loss averaged over the coreset is a controlled replacement for the intractable minimization of the original empirical loss.Typically, the control takes the form of a guarantee that the average loss over the coreset approximates the total loss uniformly across the parameter space.Recent work has provided significant empirical support in favor of using DPPs to build randomized coresets, coupled with interesting theoretical results that are suggestive but leave some key questions unanswered.In particular, the central question of whether the cardinality of a DPP-based coreset is fundamentally smaller than one based on independent sampling remained open.In this paper, we answer this question in the affirmative, demonstrating that \emph{DPPs can provably outperform independently drawn coresets}. In this vein, we contribute a conceptual understanding of coreset loss as a \emph{linear statistic} of the (random) coreset. We leverage this structural observation to connect the coresets problem to a more general problem of concentration phenomena for linear statistics of DPPs, wherein we obtain \emph{effective concentration inequalities that extend well-beyond the state-of-the-art}, encompassing general non-projection, even non-symmetric kernels. The latter have been recently shown to be of interest in machine learning beyond coresets, but come with a limited theoretical toolbox, to the extension of which our result contributes. Finally, we are also able to address the coresets problem for vector-valued objective functions, a novelty in the coresets literature.

Miscellaneous Aspects of Machine Learning · Kernel methods

Hideaki Kim

Kernel methods are widely utilized in machine learning field to learn, from training data, a latent function in a reproducing kernel Hilbert space. It is well known that the approximator thus obtained usually achieves a linear representation, which brings various computational benefits, while maintaining great representation power (i.e., universal approximation). However, when non-negativity constraints are imposed on the function's outputs, the literature usually takes the kernel method-based approximators as offering linear representations at the expense of limited model flexibility or good representation power by allowing for their nonlinear forms. The main contribution of this paper is to derive a sufficient condition for a positive definite kernel so that it may construct flexible and linear approximators of non-negative functions. We call a kernel function that offers these attributes an *inverse M-kernel*; it is reminiscent of the inverse M-matrix. Furthermore, we show that for a one-dimensional input space, universal exponential/Abel kernels are inverse M-kernels and construct linear universal approximators of non-negative functions. To the best of our knowledge, it is the first time that the existence of linear universal approximators of non-negative functions has been elucidated. We confirm the effectiveness of our results by experiments on the problems of non-negativity-constrained regression, density estimation, and intensity estimation. Finally, we discuss issues and perspectives on multi-dimensional input settings.

Deep Learning · Algorithms

Jianwei Zheng, Wei Li, Ni Xu, Junwei Zhu, XiaoxuLin, Xiaoqin Zhang

Benefiting from the booming deep learning techniques, neural operators (NO) are considered as an ideal alternative to break the traditions of solving Partial Differential Equations (PDE) with expensive cost.Yet with the remarkable progress, current solutions concern little on the holistic function features--both global and local information-- during the process of solving PDEs.Besides, a meticulously designed kernel integration to meet desirable performance often suffers from a severe computational burden, such as GNO with $O(N(N-1))$, FNO with $O(NlogN)$, and Transformer-based NO with $O(N^2)$.To counteract the dilemma, we propose a mamba neural operator with $O(N)$ computational complexity, namely MambaNO.Functionally, MambaNO achieves a clever balance between global integration, facilitated by state space model of Mamba that scans the entire function, and local integration, engaged with an alias-free architecture. We prove a property of continuous-discrete equivalence to show the capability ofMambaNO in approximating operators arising from universal PDEs to desired accuracy. MambaNOs are evaluated on a diverse set of benchmarks with possibly multi-scale solutions and set new state-of-the-art scores, yet with fewer parameters and better efficiency.

Optimization · Discrete and Combinatorial Optimization

Justin Dumouchelle, Esther Julien, Jannis Kurtz, Elias Khalil

Bilevel optimization deals with nested problems in which *leader* takes the first decision to minimize their objective function while accounting for a *follower*'s best-response reaction. Constrained bilevel problems with integer variables are particularly notorious for their hardness. While exact solvers have been proposed for mixed-integer *linear* bilevel optimization, they tend to scale poorly with problem size and are hard to generalize to the non-linear case. On the other hand, problem-specific algorithms (exact and heuristic) are limited in scope. Under a data-driven setting in which similar instances of a bilevel problem are solved routinely, our proposed framework, Neur2BiLO, embeds a neural network approximation of the leader's or follower's value function, trained via supervised regression, into an easy-to-solve mixed-integer program. Neur2BiLO serves as a heuristic that produces high-quality solutions extremely fast for four applications with linear and non-linear objectives and pure and mixed-integer variables.

Deep Learning · Representation Learning

Christopher Scarvelis, Justin Solomon

Penalizing the nuclear norm of a function's Jacobian encourages it to locally behave like a low-rank linear map. Such functions vary locally along only a handful of directions, making the Jacobian nuclear norm a natural regularizer for machine learning problems. However, this regularizer is intractable for high-dimensional problems, as it requires computing a large Jacobian matrix and taking its SVD. We show how to efficiently penalize the Jacobian nuclear norm using techniques tailor-made for deep learning. We prove that for functions parametrized as compositions $f = g \circ h$, one may equivalently penalize the average squared Frobenius norm of $Jg$ and $Jh$. We then propose a denoising-style approximation that avoids the Jacobian computations altogether. Our method is simple, efficient, and accurate, enabling Jacobian nuclear norm regularization to scale to high-dimensional deep learning problems. We complement our theory with an empirical study of our regularizer's performance and investigate applications to denoising and representation learning.

Optimization

Kexin Jin, Jonas Latz, Chenguang Liu, Carola-Bibiane Schönlieb

Optimization problems with continuous data appear in, e.g., robust machine learning, functional data analysis, and variational inference. Here, the target function is given as an integral over a family of (continuously) indexed target functions---integrated with respect to a probability measure. Such problems can often be solved by stochastic optimization methods: performing optimization steps with respect to the indexed target function with randomly switched indices. In this work, we study a continuous-time variant of the stochastic gradient descent algorithm for optimization problems with continuous data. This so-called stochastic gradient process consists in a gradient flow minimizing an indexed target function that is coupled with a continuous-time index process determining the index. Index processes are, e.g., reflected diffusions, pure jump processes, or other Lévy processes on compact spaces. Thus, we study multiple sampling patterns for the continuous data space and allow for data simulated or streamed at runtime of the algorithm. We analyze the approximation properties of the stochastic gradient process and study its longtime behavior and ergodicity under constant and decreasing learning rates. We end with illustrating the applicability of the stochastic gradient process in a polynomial regression problem with noisy functional data, as well as in a physics-informed neural network.

Theory · Active Learning and Interactive Learning

Tavor Baharav, Ryan Kang, Colin Sullivan, Mo Tiwari, Eric Luxenberg, David Tse, Mert Pilanci

The softmax function is ubiquitous in machine learning and optimization applications. Computing the full softmax evaluation of a matrix-vector product can be computationally expensive in high-dimensional settings. In many applications, however, it is sufficient to calculate only the top few outputs of the softmax function. In this work, we present an algorithm, dubbed AdaptiveSoftmax, that adaptively computes the top k softmax values more efficiently than the full softmax computation, with probabilistic guarantees. We demonstrate the sample efficiency improvements afforded by AdaptiveSoftmax on real and synthetic data to corroborate our theoretical results. AdaptiveSoftmax yields >10x gain over full softmax computation on most datasets, yielding up to 30x improvement for Mistral7B evaluated on the Wikitext dataset. The adaptive method we propose for estimating the partition function (the softmax denominator) is of independent interest and can be used in other applications such as kernel density estimation.

Theory · Game Theory

George Christodoulou, Alkmini Sgouritsa, Ioannis Vlachos

Our work revisits the design of mechanisms via the learning-augmented framework. In this model, the algorithm is enhanced with imperfect (machine-learned) information concerning the input, usually referred to as prediction. The goal is to design algorithms whose performance degrades gently as a function of the prediction error and, in particular, perform well if the prediction is accurate, but also provide a worst-case guarantee under any possible error. This framework has been successfully applied recently to various mechanism design settings, where in most cases the mechanism is provided with a prediction about the types of the players.We adopt a perspective in which the mechanism is provided with an output recommendation. We make no assumptions about the quality of the suggested outcome, and the goal is to use the recommendation to design mechanisms with low approximation guarantees whenever the recommended outcome is reasonable, but at the same time to provide worst-case guarantees whenever the recommendation significantly deviates from the optimal one. We propose a generic, universal measure, which we call quality of recommendation, to evaluate mechanisms across various information settings. We demonstrate how this new metric can provide refined analysis in existing results.This model introduces new challenges, as the mechanism receives limited information comparing to settings that use predictions about the types of the agents. We study, through this lens, several well-studied mechanism design paradigms, devising new mechanisms, but also providing refined analysis for existing ones, using as a metric the quality of recommendation. We complement our positive results, by exploring the limitations of known classes of strategyproof mechanisms that can be devised using output recommendation.