NeurIPS 2024 Papers

The Gromov-Wasserstein (GW) distance is an extension of the optimal transport problem that allows one to match objects between incomparable spaces. At its core, the GW distance is specified as the solution of a non-convex quadratic program and is not known to be tractable to solve. In particular, existing solvers for the GW distance are only able to find locally optimal solutions. In this work, we propose a semi-definite programming (SDP) relaxation of the GW distance. The relaxation can be viewed as the Lagrangian dual of the GW distance augmented with constraints that relate to the linear and quadratic terms of transportation plans. In particular, our relaxation provides a tractable (polynomial-time) algorithm to compute globally optimal transportation plans (in some instances) together with an accompanying proof of global optimality. Our numerical experiments suggest that the proposed relaxation is strong in that it frequently computes the globally optimal solution. Our Python implementation is available at https://github.com/tbng/gwsdp.

In this paper, we explore two fundamental first-order algorithms in convex optimization, namely, gradient descent (GD) and proximal gradient method (ProxGD). Our focus is on making these algorithms entirely adaptive by leveraging local curvature information of smooth functions. We propose adaptive versions of GD and ProxGD that are based on observed gradient differences and, thus, have no added computational costs. Moreover, we prove convergence of our methods assuming only local Lipschitzness of the gradient. In addition, the proposed versions allow for even larger stepsizes than those initially suggested in [MM20].

Stochastic gradient descent (SGD) is perhaps the most prevalent optimization method in modern machine learning. Contrary to the empirical practice of sampling from the datasets \emph{without replacement} and with (possible) reshuffling at each epoch, the theoretical counterpart of SGD usually relies on the assumption of \emph{sampling with replacement}. It is only very recently that SGD using sampling without replacement -- shuffled SGD -- has been analyzed with matching upper and lower bounds. However, we observe that those bounds are too pessimistic to explain often superior empirical performance of data permutations (sampling without replacement) over vanilla counterparts (sampling with replacement) on machine learning problems. Through fine-grained analysis in the lens of primal-dual cyclic coordinate methods and the introduction of novel smoothness parameters, we present several results for shuffled SGD on smooth and non-smooth convex losses, where our novel analysis framework provides tighter convergence bounds over all popular shuffling schemes (IG, SO, and RR). Notably, our new bounds predict faster convergence than existing bounds in the literature -- by up to a factor of $O(\sqrt{n})$, mirroring benefits from tighter convergence bounds using component smoothness parameters in randomized coordinate methods. Lastly, we numerically demonstrate on common machine learning datasets that our bounds are indeed much tighter, thus offering a bridge between theory and practice.

Large language models have the ability to generate text that mimics patterns in their inputs. We introduce a simple Markov Chain sequence modeling task in order to study how this in-context learning capability emerges. In our setting, each example is sampled from a Markov chain drawn from a prior distribution over Markov chains. Transformers trained on this task form \emph{statistical induction heads} which compute accurate next-token probabilities given the bigram statistics of the context. During the course of training, models pass through multiple phases: after an initial stage in which predictions are uniform, they learn to sub-optimally predict using in-context single-token statistics (unigrams); then, there is a rapid phase transition to the correct in-context bigram solution. We conduct an empirical and theoretical investigation of this multi-phase process, showing how successful learning results from the interaction between the transformer's layers, and uncovering evidence that the presence of the simpler unigram solution may delay formation of the final bigram solution. We examine how learning is affected by varying the prior distribution over Markov chains, and consider the generalization of our in-context learning of Markov chains (ICL-MC) task to $n$-grams for $n > 2$.

Combinatorial optimization problems are widespread but inherently challenging due to their discrete nature. The primary limitation of existing methods is that they can only access a small fraction of the solution space at each iteration, resulting in limited efficiency for searching the global optimal. To overcome this challenge, diverging from conventional efforts of expanding the solver's search scope, we focus on enabling information to actively propagate to the solver through heat diffusion. By transforming the target function while preserving its optima, heat diffusion facilitates information flow from distant regions to the solver, providing more efficient navigation. Utilizing heat diffusion, we propose a framework for solving general combinatorial optimization problems. The proposed methodology demonstrates superior performance across a range of the most challenging and widely encountered combinatorial optimizations. Echoing recent advancements in harnessing thermodynamics for generative artificial intelligence, our study further reveals its significant potential in advancing combinatorial optimization.

This paper considers the distributed convex-concave minimax optimization under the second-order similarity. We propose stochastic variance-reduced optimistic gradient sliding (SVOGS) method, which takes the advantage of the finite-sum structure in the objective by involving the mini-batch client sampling and variance reduction. We prove SVOGS can achieve the $\varepsilon$-duality gap within communication rounds of ${\mathcal O}(\delta D^2/\varepsilon)$, communication complexity of ${\mathcal O}(n+\sqrt{n}\delta D^2/\varepsilon)$, and local gradient calls of $\tilde{\mathcal O}(n+(\sqrt{n}\delta+L)D^2/\varepsilon\log(1/\varepsilon))$, where $n$ is the number of nodes, $\delta$ is the degree of the second-order similarity, $L$ is the smoothness parameter and $D$ is the diameter of the constraint set. We can verify that all of above complexity (nearly) matches the corresponding lower bounds. For the specific $\mu$-strongly-convex-$\mu$-strongly-convex case, our algorithm has the upper bounds on communication rounds, communication complexity, and local gradient calls of $\mathcal O(\delta/\mu\log(1/\varepsilon))$, ${\mathcal O}((n+\sqrt{n}\delta/\mu)\log(1/\varepsilon))$, and $\tilde{\mathcal O}(n+(\sqrt{n}\delta+L)/\mu)\log(1/\varepsilon))$ respectively, which are also nearly tight. Furthermore, we conduct the numerical experiments to show the empirical advantages of proposed method.

The recently-proposed framework of Predict+Optimize tackles optimization problems with parameters that are unknown at solving time, in a supervised learning setting. Prior frameworks consider only the scenario where all unknown parameters are (eventually) revealed simultaneously. In this work, we propose Multi-Stage Predict+Optimize, a novel extension catering to applications where unknown parameters are revealed in sequential stages, with optimization decisions made in between. We further develop three training algorithms for neural networks (NNs) for our framework as proof of concept, both of which handle all mixed integer linear programs. The first baseline algorithm is a natural extension of prior work, training a single NN which makes a single prediction of unknown parameters. The second and third algorithms instead leverage the possibility of updating parameter predictions between stages, and trains one NN per stage. To handle the interdependency between the neural networks, we adopt sequential and parallelized versions of coordinate descent for training. Experimentation on three benchmarks demonstrates the superior learning performance of our methods over classical approaches.

Sharpness Aware Minimization (SAM) enhances performance across various neural architectures and datasets. As models are continually scaled up to improve performance, a rigorous understanding of SAM’s scaling behaviour is paramount. To this end, we study the infinite-width limit of neural networks trained with SAM, using the Tensor Programs framework. Our findings reveal that the dynamics of standard SAM effectively reduce to applying SAM solely in the last layer in wide neural networks, even with optimal hyperparameters. In contrast, we identify a stable parameterization with layerwise perturbation scaling, which we call *Maximal Update and Perturbation Parameterization* ($\mu$P$^2$), that ensures all layers are both feature learning and effectively perturbed in the limit. Through experiments with MLPs, ResNets and Vision Transformers, we empirically demonstrate that $\mu$P$^2$ is the first parameterization to achieve hyperparameter transfer of the joint optimum of learning rate and perturbation radius across model scales. Moreover, we provide an intuitive condition to derive $\mu$P$^2$ for other perturbation rules like Adaptive SAM and SAM-ON, also ensuring balanced perturbation effects across all layers.

Models are expected to engage in invariance learning, which involves distinguishing the core relations that remain consistent across varying environments to ensure the predictions are safe, robust and fair. While existing works consider specific algorithms to realize invariance learning, we show that model has the potential to learn invariance through standard training procedures. In other words, this paper studies the implicit bias of Stochastic Gradient Descent (SGD) over heterogeneous data and shows that the implicit bias drives the model learning towards an invariant solution. We call the phenomenon the implicit invariance learning. Specifically, we theoretically investigate the multi-environment low-rank matrix sensing problem where in each environment, the signal comprises (i) a lower-rank invariant part shared across all environments; and (ii) a significantly varying environment-dependent spurious component. The key insight is, through simply employing the large step size large-batch SGD sequentially in each environment without any explicit regularization, the oscillation caused by heterogeneity can provably prevent model learning spurious signals. The model reaches the invariant solution after certain iterations. In contrast, model learned using pooled SGD over all data would simultaneously learn both the invariant and spurious signals. Overall, we unveil another implicit bias that is a result of the symbiosis between the heterogeneity of data and modern algorithms, which is, to the best of our knowledge, first in the literature.

Optimization over permutations is typically an NP-hard problem that arises extensively in ranking, matching, tracking, etc. Birkhoff polytope-based relaxation methods have made significant advancements, particularly in penalty-free optimization and probabilistic inference. Relaxation onto the orthogonal group offers unique potential advantages such as a lower representation dimension and preservation of inner products; however, equally effective approaches remain unexplored. To bridge the gap, we present a temperature-controlled differentiable transformation that maps unconstrained vector space to the orthogonal group, where the temperature, in the limit, concentrates orthogonal matrices near permutation matrices. This transformation naturally implements a parameterization for the relaxation of permutation matrices, allowing for gradient-based optimization of problems involving permutations. Additionally, by deriving a re-parameterized gradient estimator, this transformation also provides efficient stochastic optimization over the latent permutations. Extensive experiments involving the optimization over permutation matrices validate the effectiveness of the proposed method.

In this paper, we introduce faster accelerated primal-dual algorithms for minimizing a convex function subject to strongly convex function constraints. Prior to our work, the best complexity bound was $\mathcal{O}(1/{\varepsilon})$, regardless of the strong convexity of the constraint function. It is unclear whether the strong convexity assumption can enable even better convergence results. To address this issue, we have developed novel techniques to progressively estimate the strong convexity of the Lagrangian function. Our approach, for the first time, effectively leverages the constraint strong convexity, obtaining an improved complexity of $\mathcal{O}(1/\sqrt{\varepsilon})$. This rate matches the complexity lower bound for strongly-convex-concave saddle point optimization and is therefore order-optimal. We show the superior performance of our methods in sparsity-inducing constrained optimization, notably Google's personalized PageRank problem. Furthermore, we show that a restarted version of the proposed methods can effectively identify the optimal solution's sparsity pattern within a finite number of steps, a result that appears to have independent significance.

The Sharpe ratio is an important and widely-used risk-adjusted return in financial engineering. In modern portfolio management, one may require an m-sparse (no more than m active assets) portfolio to save managerial and financial costs. However, few existing methods can optimize the Sharpe ratio with the m-sparse constraint, due to the nonconvexity and the complexity of this constraint. We propose to convert the m-sparse fractional optimization problem into an equivalent m-sparse quadratic programming problem. The semi-algebraic property of the resulting objective function allows us to exploit the Kurdyka-Lojasiewicz property to develop an efficient Proximal Gradient Algorithm (PGA) that leads to a portfolio which achieves the globally optimal m-sparse Sharpe ratio under certain conditions. The convergence rates of PGA are also provided. To the best of our knowledge, this is the first proposal that achieves a globally optimal m-sparse Sharpe ratio with a theoretically-sound guarantee.

The quadratic unconstrained binary optimization (QUBO) is a well-known NP-hard problem that takes an $n\times n$ matrix $Q$ as input and decides an $n$-dimensional 0-1 vector $x$, to optimize a quadratic function. Existing learning-based models that always formulate the solution process as sequential decisions suffer from high computational overload. To overcome this issue, we propose a neural solver called the Value Classification Model (VCM) that formulates the solution process from a classification perspective. It applies a Depth Value Network (DVN) based on graph convolution that exploits the symmetry property in $Q$ to auto-grasp value features. These features are then fed into a Value Classification Network (VCN) which directly generates classification solutions. Trained by a highly efficient model-tailored Greedy-guided Self Trainer (GST) which does not require any priori optimal labels, VCM significantly outperforms competitors in both computational efficiency and solution quality with a remarkable generalization ability. It can achieve near-optimal solutions in milliseconds with an average optimality gap of just 0.362\% on benchmarks with up to 2500 variables. Notably, a VCM trained at a specific DVN depth can steadily find better solutions by simply extending the testing depth, which narrows the gap to 0.034\% on benchmarks. To our knowledge, this is the first learning-based model to reach such a performance.

Second-order optimization has been shown to accelerate the training of deep neural networks in many applications, often yielding faster progress per iteration on the training loss compared to first-order optimizers. However, the generalization properties of second-order methods are still being debated. Theoretical investigations have proved difficult to carry out outside the tractable settings of heavily simplified model classes - thus, the relevance of existing theories to practical deep learning applications remains unclear. Similarly, empirical studies in large-scale models and real datasets are significantly confounded by the necessity to approximate second-order updates in practice. It is often unclear whether the observed generalization behaviour arises specifically from the second-order nature of the parameter updates, or instead reflects the specific structured (e.g. Kronecker) approximations used or any damping-based interpolation towards first-order updates. Here, we show for the first time that exact Gauss-Newton (GN) updates take on a tractable form in a class of deep reversible architectures that are sufficiently expressive to be meaningfully applied to common benchmark datasets. We exploit this novel setting to study the training and generalization properties of the GN optimizer. We find that exact GN generalizes poorly. In the mini-batch training setting, this manifests as rapidly saturating progress even on the training loss, with parameter updates found to overfit each mini-batch without producing the features that would support generalization to other mini-batches. In contrast to previous work, we show that our experiments run in the feature learning regime, in which the neural tangent kernel (NTK) changes during the course of training. However, changes in the NTK are not associated with any significant change in neural representations, explaining the lack of generalization.

In this work we analyze various scaling limits of the training dynamics of transformer models in the feature learning regime. We identify the set of parameterizations which admit well defined infinite width and depth limits that allow the attention layers to update throughout training, a relevant notion of feature learning in these models. We then use tools from dynamical mean field theory (DMFT) to analyze various infinite limits (infinite heads, infinite key/query dimension, and infinite depth) which have different statistical descriptions depending on which infinite limit is taken and how attention layers are scaled. We provide numerical evidence of convergence to the limits and show they maintain the correct scale of updates for both SGD and Adam.

The ultimate goal of multi-objective optimization (MO) is to assist human decision-makers (DMs) in identifying solutions of interest (SOI) that optimally reconcile multiple objectives according to their preferences. Preference-based evolutionary MO (PBEMO) has emerged as a promising framework that progressively approximates SOI by involving human in the optimization-cum-decision-making process. Yet, current PBEMO approaches are prone to be inefficient and misaligned with the DM’s true aspirations, especially when inadvertently exploiting mis-calibrated reward models. This is further exacerbated when considering the stochastic nature of human feedback. This paper proposes a novel framework that navigates MO to SOI by directly leveraging human feedback without being restricted by a predefined reward model nor cumbersome model selection. Specifically, we developed a clustering-based stochastic dueling bandits algorithm that strategically scales well to high-dimensional dueling bandits, and achieves a regret of $\mathcal{O}(K^2\log T)$, where $K$ is the number of clusters and $T$ is the number of rounds. The learned preferences are then transformed into a unified probabilistic format that can be readily adapted to prevalent EMO algorithms. This also leads to a principled termination criterion that strategically manages human cognitive loads and computational budget. Experiments on $48$ benchmark test problems, including synthetic problems, RNA inverse design and protein structure prediction, fully demonstrate the effectiveness of our proposed approach.

This paper investigates simple bilevel optimization problems where we minimize a convex upper-level objective over the optimal solution set of a convex lower-level objective. Existing methods for such problems either only guarantee asymptotic convergence, have slow sublinear rates, or require strong assumptions. To address these challenges, we propose a penalization framework that delineates the relationship between approximate solutions of the original problem and its reformulated counterparts. This framework accommodates varying assumptions regarding smoothness and convexity, enabling the application of specific methods with different complexity results. Specifically, when both upper- and lower-level objectives are composite convex functions, under an $\alpha$-Hölderian error bound condition and certain mild assumptions, our algorithm attains an $(\epsilon,\epsilon^{\beta})$-optimal solution of the original problem for any $\beta> 0$ within $\mathcal{O}\left(\sqrt{{1}/{\epsilon^{\max\\{\alpha,\beta\\}}}}\right)$ iterations. The result can be improved further if the smooth part of the upper-level objective is strongly convex. We also establish complexity results when the upper- and lower-level objectives are general nonsmooth functions. Numerical experiments demonstrate the effectiveness of our algorithms.

This work considers the problem of sampling from a probability distribution known up to a normalization constant while satisfying a set of statistical constraints specified by the expected values of general nonlinear functions. This problem finds applications in, e.g., Bayesian inference, where it can constrain moments to evaluate counterfactual scenarios or enforce desiderata such as prediction fairness. Methods developed to handle support constraints, such as those based on mirror maps, barriers, and penalties, are not suited for this task. This work therefore relies on gradient descent-ascent dynamics in Wasserstein space to put forward a discrete-time primal-dual Langevin Monte Carlo algorithm (PD-LMC) that simultaneously constrains the target distribution and samples from it. We analyze the convergence of PD-LMC under standard assumptions on the target distribution and constraints, namely (strong) convexity and log-Sobolev inequalities. To do so, we bring classical optimization arguments for saddle-point algorithms to the geometry of Wasserstein space. We illustrate the relevance and effectiveness of PD-LMC in several applications.

Districting is a complex combinatorial problem that consists in partitioning a geographical area into small districts. In logistics, it is a major strategic decision determining operating costs for several years. Solving districting problems using traditional methods is intractable even for small geographical areas and existing heuristics often provide sub-optimal results. We present a structured learning approach to find high-quality solutions to real-world districting problems in a few minutes. It is based on integrating a combinatorial optimization layer, the capacitated minimum spanning tree problem, into a graph neural network architecture. To train this pipeline in a decision-aware fashion, we show how to construct target solutions embedded in a suitable space and learn from target solutions. Experiments show that our approach outperforms existing methods as it can significantly reduce costs on real-world cities.