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Optimization · Convex

Aaron Mishkin, Ahmed Khaled, Yuanhao Wang, Aaron Defazio, Robert Gower

We develop new sub-optimality bounds for gradient descent (GD) that depend on the conditioning of the objective along the path of optimization, rather than on global, worst-case constants. Key to our proofs is directional smoothness, a measure of gradient variation that we use to develop upper-bounds on the objective. Minimizing these upper-bounds requires solving implicit equations to obtain a sequence of strongly adapted step-sizes; we show that these equations are straightforward to solve for convex quadratics and lead to new guarantees for two classical step-sizes. For general functions, we prove that the Polyak step-size and normalized GD obtain fast, path-dependent rates despite using no knowledge of the directional smoothness. Experiments on logistic regression show our convergence guarantees are tighter than the classical theory based on $L$-smoothness.

Optimization

Aaron Defazio, Xingyu Yang, Ahmed Khaled, Konstantin Mishchenko, Harsh Mehta, Ashok Cutkosky

Existing learning rate schedules that do not require specification of the optimization stopping step $T$ are greatly out-performed by learning rate schedules that depend on $T$. We propose an approach that avoids the need for this stopping time by eschewing the use of schedules entirely, while exhibiting state-of-the-art performance compared to schedules across a wide family of problems ranging from convex problems to large-scale deep learning problems. Our Schedule-Free approach introduces no additional hyper-parameters over standard optimizers with momentum. Our method is a direct consequence of a new theory we develop that unifies scheduling and iterate averaging. An open source implementation of our method is available at https://github.com/facebookresearch/schedule_free. Schedule-Free AdamW is the core algorithm behind our winning entry to the MLCommons 2024 AlgoPerf Algorithmic Efficiency Challenge Self-Tuning track.

Probabilistic Methods · Monte Carlo and Sampling Methods

Yuzhou Gu, Nikki Lijing Kuang, Yian Ma, Zhao Song, Lichen Zhang

We consider the problem of sampling from a $d$-dimensional log-concave distribution $\pi(\theta) \propto \exp(-f(\theta))$ for $L$-Lipschitz $f$, constrained to a convex body (described by $n$ hyperplanes) equipped with a barrier function, contained in a ball of radius $R$ with a $w$-warm start. We propose a \emph{robust} sampling framework that computes spectral approximations to the Hessian of the barrier functions in each iteration. We prove that for the polytope constraints, sampling with the Lee-Sidford barrier function mixes within $\widetilde O((d^2+dL^2R^2)\log(w/\delta))$ steps with a per step cost of $\widetilde O(nd^{\omega-1})$, where $\omega\approx 2.37$ is the fast matrix multiplication exponent. Compared to the prior work of Mangoubi and Vishnoi, our approach gives faster mixing time as we are able to design a generalized soft-threshold Dikin walk beyond log-barrier.We further extend our result to show how to sample from a $d$-dimensional spectrahedron, the constrained set of a semidefinite program, specified by the set $\{x\in \mathbb{R}^d: \sum_{i=1}^d x_i A_i \succeq C \}$ where $A_1,\ldots,A_d, C$ are $n\times n$ real symmetric matrices. We design a walk that mixes in $\widetilde O((nd+dL^2R^2)\log(w/\delta))$ steps with a per iteration cost of $\widetilde O(n^\omega+n^2d^{3\omega-5})$. We improve the mixing time bound of prior best Dikin walk due to Narayanan and Rakhlin that mixes in $\widetilde O((n^2d^3+n^2dL^2R^2)\log(w/\delta))$ steps.

Computer Vision

Sanghyun Son, Matheus Gadelha, Yang Zhou, Zexiang Xu, Ming Lin, Yi Zhou

We present a differentiable representation, DMesh, for general 3D triangular meshes. DMesh considers both the geometry and connectivity information of a mesh. In our design, we first get a set of convex tetrahedra that compactly tessellates the domain based on Weighted Delaunay Triangulation (WDT), and select triangular faces on the tetrahedra to define the final mesh. We formulate probability of faces to exist on the actual surface in a differentiable manner based on the WDT. This enables DMesh to represent meshes of various topology in a differentiable way, and allows us to reconstruct the mesh under various observations, such as point clouds and multi-view images using gradient-based optimization. We publicize the source code and supplementary material at our project page (https://sonsang.github.io/dmesh-project).

Deep Learning · Theory

Hugo Cui, Freya Behrens, Florent Krzakala, Lenka Zdeborová

Many empirical studies have provided evidence for the emergence of algorithmic mechanisms (abilities) in the learning of language models, that lead to qualitative improvements of the model capabilities. Yet, a theoretical characterization of how such mechanisms emerge remains elusive. In this paper, we take a step in this direction by providing a tight theoretical analysis of the emergence of semantic attention in a solvable model of dot-product attention. More precisely, we consider a non-linear self-attention layer with trainable tied and low-rank query and key matrices. In the asymptotic limit of high-dimensional data and a comparably large number of training samples we provide a tight closed-form characterization of the global minimum of the non-convex empirical loss landscape. We show that this minimum corresponds to either a positional attention mechanism (with tokens attending to each other based on their respective positions) or a semantic attention mechanism (with tokens attending to each other based on their meaning), and evidence an emergent phase transition from the former to the latter with increasing sample complexity. Finally, we compare the dot-product attention layer to a linear positional baseline, and show that it outperforms the latter using the semantic mechanism provided it has access to sufficient data.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Yunjuan Wang, Raman Arora

We focus on developing a theoretical understanding of meta-learning. Given multiple tasks drawn i.i.d. from some (unknown) task distribution, the goal is to find a good pre-trained model that can be adapted to a new, previously unseen, task with little computational and statistical overhead. We introduce a novel notion of stability for meta-learning algorithms, namely *uniform meta-stability*. We instantiate two uniformly meta-stable learning algorithms based on regularized empirical risk minimization and gradient descent and give explicit generalization bounds for convex learning problems with smooth losses and for weakly convex learning problems with non-smooth losses. Finally, we extend our results to stochastic and adversarially robust variants of our meta-learning algorithm.

Online Learning

Yan-Feng Xie, Peng Zhao, Zhi-Hua Zhou

Gradient-variation online learning aims to achieve regret guarantees that scale with variations in the gradients of online functions, which is crucial for attaining fast convergence in games and robustness in stochastic optimization, hence receiving increased attention. Existing results often require the smoothness condition by imposing a fixed bound on gradient Lipschitzness, which may be unrealistic in practice. Recent efforts in neural network optimization suggest a generalized smoothness condition, allowing smoothness to correlate with gradient norms. In this paper, we systematically study gradient-variation online learning under generalized smoothness. We extend the classic optimistic mirror descent algorithm to derive gradient-variation regret by analyzing stability over the optimization trajectory and exploiting smoothness locally. Then, we explore universal online learning, designing a single algorithm with the optimal gradient-variation regrets for convex and strongly convex functions simultaneously, without requiring prior knowledge of curvature. This algorithm adopts a two-layer structure with a meta-algorithm running over a group of base-learners. To ensure favorable guarantees, we design a new Lipschitz-adaptive meta-algorithm, capable of handling potentially unbounded gradients while ensuring a second-order bound to effectively ensemble the base-learners. Finally, we provide the applications for fast-rate convergence in games and stochastic extended adversarial optimization.

Optimization · Non-Convex

Nikos Tsikouras, Constantine Caramanis, Christos Tzamos

Embeddings play a pivotal role across various disciplines, offering compact representations of complex data structures. Randomized methods like Johnson-Lindenstrauss (JL) provide state-of-the-art and essentially unimprovable theoretical guarantees for achieving such representations. These guarantees are worst-case and in particular, neither the analysis, ${\textit{nor the algorithm}}$, takes into account any potential structural information of the data. The natural question is: must we randomize? Could we instead use an optimization-based approach, working directly with the data? A first answer is no: as we show, the distance-preserving objective of JL has a non-convex landscape over the space of projection matrices, with many bad stationary points. But this is not the final answer. We present a novel method motivated by diffusion models, that circumvents this fundamental challenge: rather than performing optimization directly over the space of projection matrices, we use optimization over the larger space of $\textit{random solution samplers}$, gradually reducing the variance of the sampler. We show that by moving through this larger space, our objective converges to a deterministic (zero variance) solution, avoiding bad stationary points. This method can also be seen as an optimization-based derandomization approach, and is an idea and method that we believe can be applied to many other problems.

Reinforcement Learning · Batch Offline

Haobo Zhang, Xiyue Peng, Honghao Wei, Xin Liu

Offline safe reinforcement learning (RL) aims to find an optimal policy using a pre-collected dataset when data collection is impractical or risky. We propose a novel linear programming (LP) based primal-dual algorithm for convex MDPs that incorporates ``uncertainty'' parameters to improve data efficiency while requiring only partial data coverage assumption. Our theoretical results achieve a sample complexity of $\mathcal{O}(1/(1-\gamma)\sqrt{n})$ under general function approximation, improving the current state-of-the-art by a factor of $1/(1-\gamma)$, where $n$ is the number of data samples in an offline dataset, and $\gamma$ is the discount factor. The numerical experiments validate our theoretical findings, demonstrating the practical efficacy of our approach in achieving improved safety and learning efficiency in safe offline settings.

Generative Models · Diffusion models

Jacob K Christopher, Stephen Baek, Nando Fioretto

This paper introduces an approach to endow generative diffusion processes the ability to satisfy and certify compliance with constraints and physical principles. The proposed method recast the traditional sampling process of generative diffusion models as a constrained optimization problem, steering the generated data distribution to remain within a specified region to ensure adherence to the given constraints.These capabilities are validated on applications featuring both convex and challenging, non-convex, constraints as well as ordinary differential equations, in domains spanning from synthesizing new materials with precise morphometric properties, generating physics-informed motion, optimizing paths in planning scenarios, and human motion synthesis.

Optimization · Zero-order and Black-box Optimization

Felix Petersen, Christian Borgelt, Tobias Sutter, Hilde Kuehne, Oliver Deussen, Stefano Ermon

When training neural networks with custom objectives, such as ranking losses and shortest-path losses, a common problem is that they are, per se, non-differentiable. A popular approach is to continuously relax the objectives to provide gradients, enabling learning. However, such differentiable relaxations are often non-convex and can exhibit vanishing and exploding gradients, making them (already in isolation) hard to optimize. Here, the loss function poses the bottleneck when training a deep neural network. We present Newton Losses, a method for improving the performance of existing hard to optimize losses by exploiting their second-order information via their empirical Fisher and Hessian matrices. Instead of training the neural network with second-order techniques, we only utilize the loss function's second-order information to replace it by a Newton Loss, while training the network with gradient descent. This makes our method computationally efficient. We apply Newton Losses to eight differentiable algorithms for sorting and shortest-paths, achieving significant improvements for less-optimized differentiable algorithms, and consistent improvements, even for well-optimized differentiable algorithms.

Generative Models · In Context Learning

Dake Bu, Wei Huang, Andi Han, Atsushi Nitanda, Taiji Suzuki, Qingfu Zhang, Hau-San Wong

Transformer-based large language models (LLMs) have displayed remarkable creative prowess and emergence capabilities. Existing empirical studies have revealed a strong connection between these LLMs' impressive emergence abilities and their in-context learning (ICL) capacity, allowing them to solve new tasks using only task-specific prompts without further fine-tuning. On the other hand, existing empirical and theoretical studies also show that there is a linear regularity of the multi-concept encoded semantic representation behind transformer-based LLMs. However, existing theoretical work fail to build up an understanding of the connection between this regularity and the innovative power of ICL. Additionally, prior work often focuses on simplified, unrealistic scenarios involving linear transformers or unrealistic loss functions, and they achieve only linear or sub-linear convergence rates. In contrast, this work provides a fine-grained mathematical analysis to show how transformers leverage the multi-concept semantics of words to enable powerful ICL and excellent out-of-distribution ICL abilities, offering insights into how transformers innovate solutions for certain unseen tasks encoded with multiple cross-concept semantics. Inspired by empirical studies on the linear latent geometry of LLMs, the analysis is based on a concept-based low-noise sparse coding prompt model. Leveraging advanced techniques, this work showcases the exponential 0-1 loss convergence over the highly non-convex training dynamics, which pioneeringly incorporates the challenges of softmax self-attention, ReLU-activated MLPs, and cross-entropy loss. Empirical simulations corroborate the theoretical findings.

Optimization

Mohammad Pedramfar, Vaneet Aggarwal

This paper introduces the notion of upper-linearizable/quadratizable functions, a class that extends concavity and DR-submodularity in various settings, including monotone and non-monotone cases over different types of convex sets. A general meta-algorithm is devised to convert algorithms for linear/quadratic maximization into ones that optimize upper-linearizable/quadratizable functions, offering a unified approach to tackling concave and DR-submodular optimization problems. The paper extends these results to multiple feedback settings, facilitating conversions between semi-bandit/first-order feedback and bandit/zeroth-order feedback, as well as between first/zeroth-order feedback and semi-bandit/bandit feedback. Leveraging this framework, new algorithms are derived using existing results as base algorithms for convex optimization, improving upon state-of-the-art results in various cases. Dynamic and adaptive regret guarantees are obtained for DR-submodular maximization, marking the first algorithms to achieve such guarantees in these settings. Notably, the paper achieves these advancements with fewer assumptions compared to existing state-of-the-art results, underscoring its broad applicability and theoretical contributions to non-convex optimization.

Optimization · Convex

Tianbo Li, Zekun Shi, Jiaxi Zhao, Min Lin

Performing eigendecomposition during neural network training is essential for tasks such as dimensionality reduction, network compression, image denoising, and graph learning. However, eigendecomposition is computationally expensive as it is orders of magnitude slower than other neural network operations. To address this challenge, we propose a novel approach called "amortized eigendecomposition" that relaxes the exact eigendecomposition by introducing an additional loss term called eigen loss. Our approach offers significant speed improvements by replacing the computationally expensive eigendecomposition with a more affordable QR decomposition at each iteration. Theoretical analysis guarantees that the desired eigenpair is attained as optima of the eigen loss. Empirical studies on nuclear norm regularization, latent-space principal component analysis, and graphs adversarial learning demonstrate significant improvements in training efficiency while producing nearly identical outcomes to conventional approaches. This novel methodology promises to integrate eigendecomposition efficiently into neural network training, overcoming existing computational challenges and unlocking new potential for advanced deep learning applications.

Theory · Reinforcement Learning and Planning

Bianca Marin Moreno, Margaux Brégère, Pierre Gaillard, Nadia Oudjane

We explore online learning in episodic loop-free Markov decision processes on non-stationary environments (changing losses and probability transitions). Our focus is on the Concave Utility Reinforcement Learning problem (CURL), an extension of classical RL for handling convex performance criteria in state-action distributions induced by agent policies. While various machine learning problems can be written as CURL, its non-linearity invalidates traditional Bellman equations. Despite recent solutions to classical CURL, none address non-stationary MDPs. This paper introduces MetaCURL, the first CURL algorithm for non-stationary MDPs. It employs a meta-algorithm running multiple black-box algorithms instances over different intervals, aggregating outputs via a sleeping expert framework. The key hurdle is partial information due to MDP uncertainty. Under partial information on the probability transitions (uncertainty and non-stationarity coming only from external noise, independent of agent state-action pairs), we achieve optimal dynamic regret without prior knowledge of MDP changes. Unlike approaches for RL, MetaCURL handles full adversarial losses, not just stochastic ones. We believe our approach for managing non-stationarity with experts can be of interest to the RL community.

Graph Neural Networks

Morris Yau, Nikolaos Karalias, Eric Lu, Jessica Xu, Stefanie Jegelka

In this work we design graph neural network architectures that capture optimalapproximation algorithms for a large class of combinatorial optimization problems,using powerful algorithmic tools from semidefinite programming (SDP). Concretely, we prove that polynomial-sized message-passing algorithms can representthe most powerful polynomial time algorithms for Max Constraint SatisfactionProblems assuming the Unique Games Conjecture. We leverage this result toconstruct efficient graph neural network architectures, OptGNN, that obtain high quality approximate solutions on landmark combinatorial optimization problemssuch as Max-Cut, Min-Vertex-Cover, and Max-3-SAT. Our approach achievesstrong empirical results across a wide range of real-world and synthetic datasetsagainst solvers and neural baselines. Finally, we take advantage of OptGNN’sability to capture convex relaxations to design an algorithm for producing boundson the optimal solution from the learned embeddings of OptGNN.

Theory · Game Theory

Phuong Nam Tran, The Anh Ta, shuqing shi, Debmalya Mandal, Yali Du, Long Tran-Thanh

Reward allocation, also known as the credit assignment problem, has been an important topic in economics, engineering, and machine learning. An important concept in reward allocation is the core, which is the set of stable allocations where no agent has the motivation to deviate from the grand coalition. In previous works, computing the core requires either knowledge of the reward function in deterministic games or the reward distribution in stochastic games. However, this is unrealistic, as the reward function or distribution is often only partially known and may be subject to uncertainty. In this paper, we consider the core learning problem in stochastic cooperative games, where the reward distribution is unknown. Our goal is to learn the expected core, that is, the set of allocations that are stable in expectation, given an oracle that returns a stochastic reward for an enquired coalition each round. Within the class of strictly convex games, we present an algorithm named \texttt{Common-Points-Picking} that returns a point in the expected core given a polynomial number of samples, with high probability. To analyse the algorithm, we develop a new extension of the separation hyperplane theorem for multiple convex sets.t.

Ruiqi Zhang, Jingfeng Wu, Peter Bartlett

We study the \emph{in-context learning} (ICL) ability of a \emph{Linear Transformer Block} (LTB) that combines a linear attention component and a linear multi-layer perceptron (MLP) component. For ICL of linear regression with a Gaussian prior and a \emph{non-zero mean}, we show that LTB can achieve nearly Bayes optimal ICL risk. In contrast, using only linear attention must incur an irreducible additive approximation error. Furthermore, we establish a correspondence between LTB and one-step gradient descent estimators with learnable initialization ($\mathsf{GD}-\beta$), in the sense that every $\mathsf{GD}-\beta$ estimator can be implemented by an LTB estimator and every optimal LTB estimator that minimizes the in-class ICL risk is effectively a $\mathsf{GD}-\beta$ estimator.Finally, we show that $\mathsf{GD}-\beta$ estimators can be efficiently optimized with gradient flow, despite a non-convex training objective.Our results reveal that LTB achieves ICL by implementing $\mathsf{GD}-\beta$, and they highlight the role of MLP layers in reducing approximation error.

Online Learning

Taira Tsuchiya, Shinji Ito

In this work, we explore online convex optimization (OCO) and introduce a new condition and analysis that provides fast rates by exploiting the curvature of feasible sets. In online linear optimization, it is known that if the average gradient of loss functions exceeds a certain threshold, the curvature of feasible sets can be exploited by the follow-the-leader (FTL) algorithm to achieve a logarithmic regret. This study reveals that algorithms adaptive to the curvature of loss functions can also leverage the curvature of feasible sets. In particular, we first prove that if an optimal decision is on the boundary of a feasible set and the gradient of an underlying loss function is non-zero, then the algorithm achieves a regret bound of $O(\rho \log T)$ in stochastic environments. Here, $\rho > 0$ is the radius of the smallest sphere that includes the optimal decision and encloses the feasible set. Our approach, unlike existing ones, can work directly with convex loss functions, exploiting the curvature of loss functions simultaneously, and can achieve the logarithmic regret only with a local property of feasible sets. Additionally, the algorithm achieves an $O(\sqrt{T})$ regret even in adversarial environments, in which FTL suffers an $\Omega(T)$ regret, and achieves an $O(\rho \log T + \sqrt{C \rho \log T})$ regret in corrupted stochastic environments with corruption level $C$. Furthermore, by extending our analysis, we establish a matching regret upper bound of $O\Big(T^{\frac{q-2}{2(q-1)}} (\log T)^{\frac{q}{2(q-1)}}\Big)$ for $q$-uniformly convex feasible sets, where uniformly convex sets include strongly convex sets and $\ell_p$-balls for $p \in [2,\infty)$. This bound bridges the gap between the $O(\log T)$ bound for strongly convex sets~($q=2$) and the $O(\sqrt{T})$ bound for non-curved sets~($q\to\infty$).

Optimization · Convex

Junyu Chen, Binh T. Nguyen, Shang Koh, Yong Sheng Soh

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.