论文检索

Deep Learning · Large Language Models

Yunhao Tang, Zhaohan Guo, Zeyu Zheng, Daniele Calandriello, REMI MUNOS, Mark Rowland, Pierre Richemond, Michal Valko, Bernardo Avila Pires, Bilal Piot

Offline preference optimization allows fine-tuning large models directly from offline data, and has proved effective in recent alignment practices. We propose generalized preference optimization (GPO), a family of offline losses parameterized by a general class of convex functions. GPO enables a unified view over preference optimization, encompassing existing algorithms such as DPO, IPO and SLiC as special cases, while naturally introducing new variants. The GPO framework also sheds light on how offline algorithms enforce regularization, through the design of the convex function that defines the loss. Our analysis and experiments reveal the connections and subtle differences between the offline regularization and the KL divergence regularization intended by the canonical RLHF formulation. In a controlled setting akin to Gao et al 2023, we also show that different GPO variants achieve similar trade-offs between regularization and performance, though the optimal values of hyper-parameter might differ as predicted by theory. In all, our results present new algorithmic toolkits and empirical insights to alignment practitioners.

Theory · Learning Theory

Batiste Le Bars, Aurélien Bellet, Marc Tommasi, Kevin Scaman, Giovanni Neglia

This paper presents a new generalization error analysis for Decentralized Stochastic Gradient Descent (D-SGD) based on algorithmic stability. The obtained results overhaul a series of recent works that suggested an increased instability due to decentralization and a detrimental impact of poorly-connected communication graphs on generalization. On the contrary, we show, for convex, strongly convex and non-convex functions, that D-SGD can always recover generalization bounds analogous to those of classical SGD, suggesting that the choice of graph does not matter. We then argue that this result is coming from a worst-case analysis, and we provide a refined optimization-dependent generalization bound for general convex functions. This new bound reveals that the choice of graph can in fact improve the worst-case bound in certain regimes, and that surprisingly, a poorly-connected graph can even be beneficial for generalization.

Deep Learning

Nina Vesseron, Marco Cuturi

In 1991, Brenier proved a theorem that generalizes the polar decomposition for square matrices -- factored as PSD $\times$ unitary -- to any vector field $F:\mathbb{R}^d\rightarrow \mathbb{R}^d$. The theorem, known as the polar factorization theorem, states that any field $F$ can be recovered as the composition of the gradient of a convex function $u$ with a measure-preserving map $M$, namely $F=\nabla u \circ M$. We propose a practical implementation of this far-reaching theoretical result, and explore possible uses within machine learning. The theorem is closely related to optimal transport (OT) theory, and we borrow from recent advances in the field of neural optimal transport to parameterize the potential $u$ as an input convex neural network. The map $M$ can be either evaluated pointwise using $u^*$, the convex conjugate of $u$, through the identity $M=\nabla u^* \circ F$, or learned as an auxiliary network. Because $M$ is, in general, not injective, we consider the additional task of estimating the ill-posed inverse map that can approximate the pre-image measure $M^{-1}$ using a stochastic generator. We illustrate possible applications of Brenier's polar factorization to non-convex optimization problems, as well as sampling of densities that are not log-concave.

Optimization

William Powell, Hanbaek Lyu

For obtaining optimal first-order convergence guarantees for stochastic optimization, it is necessary to use a recurrent data sampling algorithm that samples every data point with sufficient frequency. Most commonly used data sampling algorithms (e.g., i.i.d., MCMC, random reshuffling) are indeed recurrent under mild assumptions. In this work, we show that for a particular class of stochastic optimization algorithms, we do not need any further property (e.g., independence, exponential mixing, and reshuffling) beyond recurrence in data sampling to guarantee optimal rate of first-order convergence. Namely, using regularized versions of Minimization by Incremental Surrogate Optimization (MISO), we show that for non-convex and possibly non-smooth objective functions with constraints, the expected optimality gap converges at an optimal rate $O(n^{-1/2})$ under general recurrent sampling schemes. Furthermore, the implied constant depends explicitly on the 'speed of recurrence', measured by the expected amount of time to visit a data point, either averaged ('target time') or supremized ('hitting time') over the starting locations. We discuss applications of our general framework to decentralized optimization and distributed non-negative matrix factorization.

Deep Learning · Attention Mechanisms

Hongkang Li, Meng Wang, Songtao Lu, Xiaodong Cui, Pin-Yu Chen

Transformer-based large language models have displayed impressive in-context learning capabilities, where a pre-trained model can handle new tasks without fine-tuning by simply augmenting the query with some input-output examples from that task. Despite the empirical success, the mechanics of how to train a Transformer to achieve ICL and the corresponding ICL capacity is mostly elusive due to the technical challenges of analyzing the nonconvex training problems resulting from the nonlinear self-attention and nonlinear activation in Transformers. To the best of our knowledge, this paper provides the first theoretical analysis of the training dynamics of Transformers with nonlinear self-attention and nonlinear MLP, together with the ICL generalization capability of the resulting model. Focusing on a group of binary classification tasks, we train Transformers using data from a subset of these tasks and quantify the impact of various factors on the ICL generalization performance on the remaining unseen tasks with and without data distribution shifts. We also analyze how different components in the learned Transformers contribute to the ICL performance. Furthermore, we provide the first theoretical analysis of how model pruning affects ICL performance and prove that proper magnitude-based pruning can have a minimal impact on ICL while reducing inference costs. These theoretical findings are justified through numerical experiments.

Deep Learning · Graph Neural Networks

Hongkang Li, Meng Wang, Tengfei Ma, Sijia Liu, Zaixi Zhang, Pin-Yu Chen

Graph Transformers, which incorporate self-attention and positional encoding, have recently emerged as a powerful architecture for various graph learning tasks. Despite their impressive performance, the complex non-convex interactions across layers and the recursive graph structure have made it challenging to establish a theoretical foundation for learning and generalization. This study introduces the first theoretical investigation of a shallow Graph Transformer for semi-supervised node classification, comprising a self-attention layer with relative positional encoding and a two-layer perception. Focusing on a graph data model with discriminative nodes that determine node labels and non-discriminative nodes that are class-irrelevant, we characterize the sample complexity required to achieve a desirable generalization error by training with stochastic gradient descent (SGD). This paper provides the quantitative characterization of the sample complexity and number of iterations for convergence dependent on the fraction of discriminative nodes, the dominant patterns, and the initial model errors. Furthermore, we demonstrate that self-attention and positional encoding enhance generalization by making the attention map sparse and promoting the core neighborhood during training, which explains the superior feature representation of Graph Transformers. Our theoretical results are supported by empirical experiments on synthetic and real-world benchmarks.

Optimization

Rudrajit Das, Xi Chen, Bertram Ieong, Parikshit Bansal, Sujay Sanghavi

It is well known that selecting samples with large losses/gradients can significantly reduce the number of training steps. However, the selection overhead is often too high to yield any meaningful gains in terms of overall training time. In this work, we focus on the greedy approach of selecting samples with large *approximate losses* instead of exact losses in order to reduce the selection overhead. For smooth convex losses, we show that such a greedy strategy can converge to a constant factor of the minimum value of the average loss in fewer iterations than the standard approach of random selection. We also theoretically quantify the effect of the approximation level. We then develop SIFT which uses early exiting to obtain approximate losses with an intermediate layer's representations for sample selection. We evaluate SIFT on the task of training a 110M parameter 12 layer BERT base model, and show significant gains (in terms of training hours and number of backpropagation steps) without any optimized implementation over vanilla training. For e.g., to reach 64% validation accuracy, SIFT with exit at the first layer takes $\sim$ 43 hours compared to $\sim$ 57 hours of vanilla training.

Deep Learning · Sequential Models, Time series

Yara Shamshoum, Nitzan Hodos, Yuval Sieradzki, Assaf Schuster

Many recent works use machine learning models to solve various complex algorithmic problems. However, these models attempt to reach a solution without considering the problem's required computational complexity, which can be detrimental to their ability to solve it correctly. In this work we investigate the effect of computational time and memory on generalization of implicit algorithmic solvers. To do so, we focus on the Differentiable Neural Computer (DNC), a general problem solver that also lets us reason directly about its usage of time and memory. In this work, we argue that the number of planning steps the model is allowed to take, which we call ”planning budget”, is a constraint that can cause the model to generalize poorly and hurt its ability to fully utilize its external memory. We evaluate our method on Graph Shortest Path, Convex Hull, Graph MinCut and Associative Recall, and show how the planning budget can drastically change the behavior of the learned algorithm, in terms of learned time complexity, training time, stability and generalization to inputs larger than those seen during training.

Deep Learning · Theory

Flavio Martinelli, Berfin Simsek, Wulfram Gerstner, Johanni Brea

Can we identify the weights of a neural network by probing its input-output mapping? At first glance, this problem seems to have many solutions because of permutation, overparameterisation and activation function symmetries. Yet, we show that the incoming weight vector of each neuron is identifiable up to sign or scaling, depending on the activation function. Our novel method 'Expand-and-Cluster’ can identify layer sizes and weights of a target network for all commonly used activation functions. Expand-and-Cluster consists of two phases: (i) to relax the non-convex optimisation problem, we train multiple overparameterised student networks to best imitate the target function; (ii) to reverse engineer the target network's weights, we employ an ad-hoc clustering procedure that reveals the learnt weight vectors shared between students -- these correspond to the target weight vectors. We demonstrate successful weights and size recovery of trained shallow and deep networks with less than 10% overhead in the layer size and describe an 'ease-of-identifiability' axis by analysing 150 synthetic problems of variable difficulty.

Social Aspects · Privacy-preserving Statistics and Machine Learning

Naman Agarwal, Satyen Kale, Karan Singh, Abhradeep Guha Thakurta

We design differentially private regret-minimizing algorithms in the online convex optimization (OCO) framework. Unlike recent results, our algorithms and analyses do not require smoothness, thus yielding the first private regret bounds with an optimal leading-order term for non-smooth loss functions. Additionally, even for smooth losses, the resulting regret guarantees improve upon previous results in terms their dependence of dimension. Our results provide the best known rates for DP-OCO in all practical regimes of the privacy parameter, barring when it is exceptionally small. The principal innovation in our algorithm design is the use of sampling from strongly log-concave densities which satisfy the Log-Sobolev Inequality. The resulting concentration of measure allows us to obtain a better trade-off for the dimension factors than prior work, leading to improved results. Following previous works on DP-OCO, the proposed algorithm explicitly limits the number of switches via rejection sampling. Thus, independently of privacy constraints, the algorithm also provides improved results for online convex optimization with a switching budget.

Optimization · First-order

Kyuwon Kim, Donghwan Kim

In nonconvex-nonconcave minimax optimization, two-timescale gradient methods have shown their potential to find local minimax (optimal) points, provided that the timescale separation between the min and the max player is sufficiently large. However, existing two-timescale variants of gradient descent ascent and extragradient methods face two shortcomings, especially when we search for non-strict local minimax points that are prevalent in modern overparameterized setting. In specific, (1) these methods can be unstable at some non-strict local minimax points even with sufficiently large timescale separation, and even (2) computing a proper amount of timescale separation is infeasible in practice. To remedy these two issues, we propose to incorporate two simple but provably effective schemes, double-step alternating update and increasing timescale separation, into the two-timescale extragradient method, respectively. Under mild conditions, we show that the proposed methods converge to non-strict local minimax points that all existing two-timescale methods fail to converge.

Ezgi Korkmaz

Deep neural policies have recently been installed in a diverse range of settings, from biotechnology to automated financial systems. However, the utilization of deep neural networks to approximate the value function leads to concerns on the decision boundary stability, in particular, with regard to the sensitivity of policy decision making to indiscernible, non-robust features due to highly non-convex and complex deep neural manifolds. These concerns constitute an obstruction to understanding the reasoning made by deep neural policies, and their foundational limitations. Hence, it is crucial to develop techniques that aim to understand the sensitivities in the learnt representations of neural network policies. To achieve this we introduce a theoretically founded method that provides a systematic analysis of the unstable directions in the deep neural policy decision boundary across both time and space. Through experiments in the Arcade Learning Environment (ALE), we demonstrate the effectiveness of our technique for identifying correlated directions of instability, and for measuring how sample shifts remold the set of sensitive directions in the neural policy landscape. Most importantly, we demonstrate that state-of-the-art robust training techniques yield learning of disjoint unstable directions, with dramatically larger oscillations over time, when compared to standard training. We believe our results reveal the fundamental properties of the decision process made by reinforcement learning policies, and can help in constructing reliable and robust deep neural policies.

Applications · Physics

Haoyu Yang, Mark Ren

Lithography, transferring chip design masks to the silicon wafer, is the most important phase in modern semiconductor manufacturing flow. Due to the limitations of lithography systems, Extensive design optimizations are required to tackle the design and silicon mismatch. Inverse lithography technology (ILT) is one of the promising solutions to perform pre-fabrication optimization, termed mask optimization. Because of mask optimization problems’ constrained non-convexity, numerical ILT solvers rely heavily on good initialization to avoid getting stuck on sub-optimal solutions. Machine learning (ML) techniques are hence proposed to generate mask initialization for ILT solvers with one-shot inference, targeting faster and better convergence during ILT. This paper addresses the question of whether ML models can directly generate high-quality optimized masks without engaging ILT solvers in the loop. We propose an implicit learning ILT framework: ILILT, which leverages the implicit layer learning method and lithography-conditioned inputs to ground the model. Trained to understand the ILT optimization procedure, ILILT can outperform the state-of-the-art machine learning solutions, significantly improving efficiency and quality.

Optimization · Non-Convex

Qinzi Zhang, Ashok Cutkosky

Training neural networks requires optimizing a loss function that may be highly irregular, and in particular neither convex nor smooth. Popular training algorithms are based on stochastic gradient descent with momentum (SGDM), for which classical analysis applies only if the loss is either convex or smooth. We show that a very small modification to SGDM closes this gap: simply scale the update at each time point by an exponentially distributed random scalar. The resulting algorithm achieves optimal convergence guarantees. Intriguingly, this result is not derived by a specific analysis of SGDM: instead, it falls naturally out of a more general framework for converting online convex optimization algorithms to non-convex optimization algorithms.

Miscellaneous Aspects of Machine Learning · Online Learning, Active Learning and Bandits

Zhiyu Zhang, David Bombara, Heng Yang

We study online learning in adversarial nonstationary environments. Since the future can be very different from the past, a critical challenge is to gracefully forget the history while new data comes in. To formalize this intuition, we revisit the discounted regret in online convex optimization, and propose an adaptive (i.e., instance optimal), FTRL-based algorithm that improves the widespread non-adaptive baseline -- gradient descent with a constant learning rate. From a practical perspective, this refines the classical idea of regularization in lifelong learning: we show that designing better regularizers can be guided by the principled theory of adaptive online optimization. Complementing this result, we also consider the (Gibbs & Candes, 2021)-style online conformal prediction problem, where the goal is to sequentially predict the uncertainty sets of a black-box machine learning model. We show that the FTRL nature of our algorithm can simplify the conventional gradient-descent-based analysis, leading to instance-dependent performance guarantees.

Optimization · Stochastic

Lijun Zhang, Haomin Bai, Wei-Wei Tu, Ping Yang, Yao Hu

While traditional distributionally robust optimization (DRO) aims to minimize the maximal risk over a set of distributions, Agarwal & Zhang (2022) recently proposed a variant that replaces risk with *excess risk*. Compared to DRO, the new formulation—minimax excess risk optimization (MERO) has the advantage of suppressing the effect of heterogeneous noise in different distributions. However, the choice of excess risk leads to a very challenging minimax optimization problem, and currently there exists only an inefficient algorithm for empirical MERO. In this paper, we develop efficient stochastic approximation approaches which directly target MERO. Specifically, we leverage techniques from stochastic convex optimization to estimate the minimal risk of every distribution, and solve MERO as a stochastic convex-concave optimization (SCCO) problem with biased gradients. The presence of bias makes existing theoretical guarantees of SCCO inapplicable, and fortunately, we demonstrate that the bias, caused by the estimation error of the minimal risk, is under-control. Thus, MERO can still be optimized with a nearly optimal convergence rate. Moreover, we investigate a practical scenario where the quantity of samples drawn from each distribution may differ, and propose a stochastic approach that delivers *distribution-dependent* convergence rates.

Deep Learning · Large Language Models

Rui Yang, Xiaoman Pan, Feng Luo, Shuang Qiu, Han Zhong, Dong Yu, Jianshu Chen

We consider the problem of multi-objective alignment of foundation models with human preferences, which is a critical step towards helpful and harmless AI systems. However, it is generally costly and unstable to fine-tune large foundation models using reinforcement learning (RL), and the multi-dimensionality, heterogeneity, and conflicting nature of human preferences further complicate the alignment process. In this paper, we introduce Rewards-in-Context (RiC), which conditions the response of a foundation model on multiple rewards in its prompt context and applies supervised fine-tuning for alignment. The salient features of RiC are simplicity and adaptivity, as it only requires supervised fine-tuning of a single foundation model and supports dynamic adjustment for user preferences during inference time. Inspired by the analytical solution of an abstracted convex optimization problem, our dynamic inference-time adjustment method approaches the Pareto-optimal solution for multiple objectives. Empirical evidence demonstrates the efficacy of our method in aligning both Large Language Models (LLMs) and diffusion models to accommodate diverse rewards with only around 10% GPU hours compared with multi-objective RL baseline.

Theory · Online Learning and Bandits

Wenhao Yang, Wei Jiang, Yibo Wang, Ping Yang, Yao Hu, Lijun Zhang

To deal with changing environments, adaptive regret has been proposed to minimize the regret over every interval. Previous studies have established a small-loss adaptive regret bound for general convex functions under the smoothness condition, offering the advantage of being much tighter than minimax rates for benign problems. However, it remains unclear whether similar bounds are attainable for other types of convex functions, such as exp-concave and strongly convex functions. In this paper, we first propose a novel algorithm that achieves a small-loss adaptive regret bound for exp-concave and smooth function. Subsequently, to address the limitation that existing algorithms can only handle one type of convex functions, we further design a universal algorithm capable of delivering small-loss adaptive regret bounds for general convex, exp-concave, and strongly convex functions simultaneously. That is challenging because the universal algorithm follows the meta-expert framework, and we need to ensure that upper bounds for both meta-regret and expert-regret are of small-loss types. Moreover, we provide a novel analysis demonstrating that our algorithms are also equipped with minimax adaptive regret bounds when functions are non-smooth.

Theory · Online Learning and Bandits

Yu-Hu Yan, Jing Wang, Peng Zhao

We investigate online Linear Quadratic Regulator (LQR) with bandit feedback and semi-adversarial disturbances. Previous works assume costs with *homogeneous* curvatures (i.e., with a uniform strong convexity lower bound), which can be hard to satisfy in many real scenarios and prohibits adapting to true curvatures for better performance. In this paper, we initiate the study of bandit LQR control with *heterogeneous* cost curvatures, aiming to strengthen the algorithm's adaptivity. To achieve this, we reduce the problem to bandit convex optimization with memory via a ``with-history'' reduction to avoid hard-to-control truncation errors. Then we provide a novel analysis for an important *stability* term that appeared in both regret and memory, using *Newton decrement* developed in interior-point methods. The analysis enables us to guarantee memory-related terms introduced in the reduction and also provide a simplified analysis for handling heterogeneous curvatures in bandit convex optimization. Finally, we achieve interpolated guarantees that can not only recover existing bounds for convex and quadratic costs but also attain new implications for cases of corrupted and decaying quadraticity.

Optimization · Non-Convex

Mengchu Xu, Zhang Yuxuan, Jian Wang

Sparse phase retrieval aims to reconstruct an $n$-dimensional $k$-sparse signal from its phaseless measurements. For most of the existing reconstruction algorithms, their sampling complexity is known to be dominated by the initialization stage. In this paper, in order to improve the sampling complexity for initialization, we propose a novel method termed exponential spectral pursuit (ESP). Theoretically, our method offers a tighter bound of sampling complexity compared to the state-of-the-art ones, such as the truncated power method. Moreover, it empirically outperforms the existing initialization methods for sparse phase retrieval.