论文检索

Deep Learning/Attention Mechanisms

Davoud Ataee Tarzanagh, Yingcong Li, Xuechen Zhang, Samet Oymak

Attention mechanism is a central component of the transformer architecture which led to the phenomenal success of large language models. However, the theoretical principles underlying the attention mechanism are poorly understood, especially its nonconvex optimization dynamics. In this work, we explore the seminal softmax-attention model $f(X)=\langle Xv, \texttt{softmax}(XWp)\rangle$, where $X$ is the token sequence and $(v,W,p)$ are trainable parameters. We prove that running gradient descent on $p$, or equivalently $W$, converges in direction to a max-margin solution that separates *locally-optimal* tokens from non-optimal ones. This clearly formalizes attention as an optimal token selection mechanism. Remarkably, our results are applicable to general data and precisely characterize *optimality* of tokens in terms of the value embeddings $Xv$ and problem geometry. We also provide a broader regularization path analysis that establishes the margin maximizing nature of attention even for nonlinear prediction heads. When optimizing $v$ and $p$ simultaneously with logistic loss, we identify conditions under which the regularization paths directionally converge to their respective hard-margin SVM solutions where $v$ separates the input features based on their labels. Interestingly, the SVM formulation of $p$ is influenced by the support vector geometry of $v$. Finally, we verify our theoretical findings via numerical experiments and provide insights.

Deep Learning/Theory

Tolga Ergen, Mert Pilanci

Understanding the fundamental principles behind the success of deep neural networks is one of the most important open questions in the current literature. To this end, we study the training problem of deep neural networks and introduce an analytic approach to unveil hidden convexity in the optimization landscape. We consider a deep parallel ReLU network architecture, which also includes standard deep networks and ResNets as its special cases. We then show that pathwise regularized training problems can be represented as an exact convex optimization problem. We further prove that the equivalent convex problem is regularized via a group sparsity inducing norm. Thus, a path regularized parallel ReLU network can be viewed as a parsimonious convex model in high dimensions. More importantly, since the original training problem may not be trainable in polynomial-time, we propose an approximate algorithm with a fully polynomial-time complexity in all data dimensions. Then, we prove strong global optimality guarantees for this algorithm. We also provide experiments corroborating our theory.

Applications/Neuroscience, Cognitive Science

Fatih Dinc, Adam Shai, Mark Schnitzer, Hidenori Tanaka

Advances in optical and electrophysiological recording technologies have made it possible to record the dynamics of thousands of neurons, opening up new possibilities for interpreting and controlling large neural populations in behaving animals. A promising way to extract computational principles from these large datasets is to train data-constrained recurrent neural networks (dRNNs). Performing this training in real-time could open doors for research techniques and medical applications to model and control interventions at single-cell resolution and drive desired forms of animal behavior. However, existing training algorithms for dRNNs are inefficient and have limited scalability, making it a challenge to analyze large neural recordings even in offline scenarios. To address these issues, we introduce a training method termed Convex Optimization of Recurrent Neural Networks (CORNN). In studies of simulated recordings, CORNN attained training speeds $\sim$100-fold faster than traditional optimization approaches while maintaining or enhancing modeling accuracy. We further validated CORNN on simulations with thousands of cells that performed simple computations such as those of a 3-bit flip-flop or the execution of a timed response. Finally, we showed that CORNN can robustly reproduce network dynamics and underlying attractor structures despite mismatches between generator and inference models, severe subsampling of observed neurons, or mismatches in neural time-scales. Overall, by training dRNNs with millions of parameters in subminute processing times on a standard computer, CORNN constitutes a first step towards real-time network reproduction constrained on large-scale neural recordings and a powerful computational tool for advancing the understanding of neural computation.

Applications/Language, Speech and Dialog

David Skrill, Samuel Norman-Haignere

Modern language models excel at integrating across long temporal scales needed to encode linguistic meaning and show non-trivial similarities to biological neural systems. Prior work suggests that human brain responses to language exhibit hierarchically organized "integration windows" that substantially constrain the overall influence of an input token (e.g., a word) on the neural response. However, little prior work has attempted to use integration windows to characterize computations in large language models (LLMs). We developed a simple word-swap procedure for estimating integration windows from black-box language models that does not depend on access to gradients or knowledge of the model architecture (e.g., attention weights). Using this method, we show that trained LLMs exhibit stereotyped integration windows that are well-fit by a convex combination of an exponential and a power-law function, with a partial transition from exponential to power-law dynamics across network layers. We then introduce a metric for quantifying the extent to which these integration windows vary with structural boundaries (e.g., sentence boundaries), and using this metric, we show that integration windows become increasingly yoked to structure at later network layers. None of these findings were observed in an untrained model, which as expected integrated uniformly across its input. These results suggest that LLMs learn to integrate information in natural language using a stereotyped pattern: integrating across position-yoked, exponential windows at early layers, followed by structure-yoked, power-law windows at later layers. The methods we describe in this paper provide a general-purpose toolkit for understanding temporal integration in language models, facilitating cross-disciplinary research at the intersection of biological and artificial intelligence.

Optimization

Lam Nguyen, Trang H. Tran

Stochastic gradient descent (SGD) algorithm is the method of choice in many machine learning tasks thanks to its scalability and efficiency in dealing with large-scale problems. In this paper, we focus on the shuffling version of SGD which matches the mainstream practical heuristics. We show the convergence to a global solution of shuffling SGD for a class of non-convex functions under over-parameterized settings. Our analysis employs more relaxed non-convex assumptions than previous literature. Nevertheless, we maintain the desired computational complexity as shuffling SGD has achieved in the general convex setting.

Social Aspects/Privacy-preserving Statistics and Machine Learning

Rachel Redberg, Antti Koskela, Yu-Xiang Wang

In the arena of privacy-preserving machine learning, differentially private stochastic gradient descent (DP-SGD) has outstripped the objective perturbation mechanism in popularity and interest. Though unrivaled in versatility, DP-SGD requires a non-trivial privacy overhead (for privately tuning the model’s hyperparameters) and a computational complexity which might be extravagant for simple models such as linear and logistic regression. This paper revamps the objective perturbation mechanism with tighter privacy analyses and new computational tools that boost it to perform competitively with DP-SGD on unconstrained convex generalized linear problems.

Optimization/Learning for Optimization

Ashok Cutkosky, Aaron Defazio, Harsh Mehta

We introduce a technique for tuning the learning rate scale factor of any base optimization algorithm and schedule automatically, which we call Mechanic. Our method provides a practical realization of recent theoretical reductions for accomplishing a similar goal in online convex optimization. We rigorously evaluate Mechanic on a range of large scale deep learning tasks with varying batch sizes, schedules, and base optimization algorithms. These experiments demonstrate that depending on the problem, Mechanic either comes very close to, matches or even improves upon manual tuning of learning rates.

Optimization/Bilevel optimization

Quan Xiao, Songtao Lu, Tianyi Chen

Bilevel optimization has recently regained interest owing to its applications in emerging machine learning fields such as hyperparameter optimization, meta-learning, and reinforcement learning. Recent results have shown that simple alternating (implicit) gradient-based algorithms can match the convergence rate of single-level gradient descent (GD) when addressing bilevel problems with a strongly convex lower-level objective. However, it remains unclear whether this result can be generalized to bilevel problems beyond this basic setting. In this paper, we first introduce a stationary metric for the considered bilevel problems, which generalizes the existing metric, for a nonconvex lower-level objective that satisfies the Polyak-Łojasiewicz (PL) condition. We then propose a Generalized ALternating mEthod for bilevel opTimization (GALET) tailored to BLO with convex PL LL problem and establish that GALET achieves an $\epsilon$-stationary point for the considered problem within $\tilde{\cal O}(\epsilon^{-1})$ iterations, which matches the iteration complexity of GD for single-level smooth nonconvex problems.

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

Abishek Sankararaman, Balakrishnan Narayanaswamy

The real-time estimation of time-varying parameters from high-dimensional, heavy-tailed and corrupted data-streams is a common sub-routine in systems ranging from those for network monitoring and anomaly detection to those for traffic scheduling in data-centers. For estimation tasks that can be cast as minimizing a strongly convex loss function, we prove that an appropriately tuned version of the {\ttfamily clipped Stochastic Gradient Descent} (SGD) is simultaneously {\em(i)} adaptive to drift, {\em (ii)} robust to heavy-tailed inliers and arbitrary corruptions, {\em(iii)} requires no distributional knowledge and {\em (iv)} can be implemented in an online streaming fashion. All prior estimation algorithms have only been proven to posses a subset of these practical desiderata. A observation we make is that, neither the $\mathcal{O}\left(\frac{1}{t}\right)$ learning rate for {\ttfamily clipped SGD} known to be optimal for strongly convex loss functions of a \emph{stationary} data-stream, nor the $\mathcal{O}(1)$ learning rate known to be optimal for being adaptive to drift in a \emph{noiseless} environment can be used. Instead, a learning rate of $T^{-\alpha}$ for $ \alpha < 1$ where $T$ is the stream-length is needed to balance adaptivity to potential drift and to combat noise. We develop a new inductive argument and combine it with a martingale concentration result to derive high-probability under \emph{any learning rate} on data-streams exhibiting \emph{arbitrary distribution shift} - a proof strategy that may be of independent interest. Further, using the classical doubling-trick, we relax the knowledge of the stream length $T$. Ours is the first online estimation algorithm that is provably robust to heavy-tails, corruptions and distribution shift simultaneously. We complement our theoretical results empirically on synthetic and real data.

Theory/Online Learning and Bandits

Zihao Hu, Guanghui Wang, Jacob Abernethy

The projection operation is a critical component in a wide range of optimization algorithms, such as online gradient descent (OGD), for enforcing constraints and achieving optimal regret bounds. However, it suffers from computational complexity limitations in high-dimensional settings or when dealing with ill-conditioned constraint sets. Projection-free algorithms address this issue by replacing the projection oracle with more efficient optimization subroutines. But to date, these methods have been developed primarily in the Euclidean setting, and while there has been growing interest in optimization on Riemannian manifolds, there has been essentially no work in trying to utilize projection-free tools here. An apparent issue is that non-trivial affine functions are generally non-convex in such domains. In this paper, we present methods for obtaining sub-linear regret guarantees in online geodesically convex optimization on curved spaces for two scenarios: when we have access to (a) a separation oracle or (b) a linear optimization oracle. For geodesically convex losses, and when a separation oracle is available, our algorithms achieve $O(T^{\frac{1}{2}})$, $O(T^{\frac{3}{4}})$ and $O(T^{\frac{1}{2}})$ adaptive regret guarantees in the full information setting, the bandit setting with one-point feedback and the bandit setting with two-point feedback, respectively. When a linear optimization oracle is available, we obtain regret rates of $O(T^{\frac{3}{4}})$ for geodesically convex losses and $O(T^{\frac{2}{3}}\log T)$ for strongly geodesically convex losses.

Applications/Time Series

yifan zhang, Qingsong Wen, xue wang, Weiqi Chen, Liang Sun, Zhang Zhang, Liang Wang, Rong Jin, Tieniu Tan

Online updating of time series forecasting models aims to address the concept drifting problem by efficiently updating forecasting models based on streaming data. Many algorithms are designed for online time series forecasting, with some exploiting cross-variable dependency while others assume independence among variables. Given every data assumption has its own pros and cons in online time series modeling, we propose **On**line **e**nsembling **Net**work (**OneNet**). It dynamically updates and combines two models, with one focusing on modeling the dependency across the time dimension and the other on cross-variate dependency. Our method incorporates a reinforcement learning-based approach into the traditional online convex programming framework, allowing for the linear combination of the two models with dynamically adjusted weights. OneNet addresses the main shortcoming of classical online learning methods that tend to be slow in adapting to the concept drift. Empirical results show that OneNet reduces online forecasting error by more than $\mathbf{50}\\%$ compared to the State-Of-The-Art (SOTA) method.

Theory/Reinforcement Learning and Planning

Udaya Ghai, Arushi Gupta, Wenhan Xia, Karan Singh, Elad Hazan

We investigate robust model-free reinforcement learning algorithms designed for environments that may be dynamic or even adversarial. Traditional state-based policies often struggle to accommodate the challenges imposed by the presence of unmodeled disturbances in such settings. Moreover, optimizing linear state-based policies pose an obstacle for efficient optimization, leading to nonconvex objectives, even in benign environments like linear dynamical systems.Drawing inspiration from recent advancements in model-based control, we intro- duce a novel class of policies centered on disturbance signals. We define several categories of these signals, which we term pseudo-disturbances, and develop corresponding policy classes based on them. We provide efficient and practical algorithms for optimizing these policies.Next, we examine the task of online adaptation of reinforcement learning agents in the face of adversarial disturbances. Our methods seamlessly integrate with any black-box model-free approach, yielding provable regret guarantees when dealing with linear dynamics. These regret guarantees unconditionally improve the best-known results for bandit linear control in having no dependence on the state-space dimension. We evaluate our method over various standard RL benchmarks and demonstrate improved robustness.

Theory/Learning Theory

Futoshi Futami, Masahiro Fujisawa

We provide novel information-theoretic generalization bounds for stochastic gradient Langevin dynamics (SGLD) under the assumptions of smoothness and dissipativity, which are widely used in sampling and non-convex optimization studies.Our bounds are time-independent and decay to zero as the sample size increases, regardless of the number of iterations and whether the step size is fixed.Unlike previous studies, we derive the generalization error bounds by focusing on the time evolution of the Kullback--Leibler divergence, which is related to the stability of datasets and is the upper bound of the mutual information between output parameters and an input dataset.Additionally, we establish the first information-theoretic generalization bound when the training and test loss are the same by showing that a loss function of SGLD is sub-exponential.This bound is also time-independent and removes the problematic step size dependence in existing work, leading to an improved excess risk bound by combining our analysis with the existing non-convex optimization error bounds.

Optimization/Zero-order and Black-box Optimization

Shuang Li, Ke Li, Wei Li

Bayesian Optimization (BO) is a powerful method for tackling expensive black-box optimization problems. As a sequential model-based optimization strategy, BO iteratively explores promising solutions until a predetermined budget, either iterations or time, is exhausted. The decision on when to terminate BO significantly influences both the quality of solutions and its computational efficiency. In this paper, we propose a simple, yet theoretically grounded, two-step method for automatically terminating BO. Our core concept is to proactively identify if the search is within a convex region by examining previously observed samples. BO is halted once the local regret within this convex region falls below a predetermined threshold. To enhance numerical stability, we propose an approximation method for calculating the termination indicator by solving a bilevel optimization problem. We conduct extensive empirical studies on diverse benchmark problems, including synthetic functions, reinforcement learning, and hyperparameter optimization. Experimental results demonstrate that our proposed method saves up to $\approx 80\%$ computational budget yet is with an order of magnitude smaller performance degradation, comparing against the other peer methods. In addition, our proposed termination method is robust in terms of the setting of its termination criterion.

Optimization/Non-Convex

Scott Pesme, Nicolas Flammarion

In this paper we fully describe the trajectory of gradient flow over $2$-layer diagonal linear networks for the regression setting in the limit of vanishing initialisation. We show that the limiting flow successively jumps from a saddle of the training loss to another until reaching the minimum $\ell_1$-norm solution. We explicitly characterise the visited saddles as well as the jump times through a recursive algorithm reminiscent of the LARS algorithm used for computing the Lasso path. Starting from the zero vector, coordinates are successively activated until the minimum $\ell_1$-norm solution is recovered, revealing an incremental learning. Our proof leverages a convenient arc-length time-reparametrisation which enables to keep track of the transitions between the jumps. Our analysis requires negligible assumptions on the data, applies to both under and overparametrised settings and covers complex cases where there is no monotonicity of the number of active coordinates. We provide numerical experiments to support our findings.

Optimization/Control and Optimization

Ming Xu, Timothy L. Molloy, Stephen Gould

This paper proposes a new method for differentiating through optimal trajectories arising from non-convex, constrained discrete-time optimal control (COC) problems using the implicit function theorem (IFT). Previous works solve a differential Karush-Kuhn-Tucker (KKT) system for the trajectory derivative, and achieve this efficiently by solving an auxiliary Linear Quadratic Regulator (LQR) problem. In contrast, we directly evaluate the matrix equations which arise from applying variable elimination on the Lagrange multiplier terms in the (differential) KKT system. By appropriately accounting for the structure of the terms within the resulting equations, we show that the trajectory derivatives scale linearly with the number of timesteps. Furthermore, our approach allows for easy parallelization, significantly improved scalability with model size, direct computation of vector-Jacobian products and improved numerical stability compared to prior works. As an additional contribution, we unify prior works, addressing claims that computing trajectory derivatives using IFT scales quadratically with the number of timesteps. We evaluate our method on a both synthetic benchmark and four challenging, learning from demonstration benchmarks including a 6-DoF maneuvering quadrotor and 6-DoF rocket powered landing.

Deep Learning/Theory

Jingfeng Wu, Vladimir Braverman, Jason Lee

Recent research has observed that in machine learning optimization, gradient descent (GD) often operates at the edge of stability (EoS) [Cohen et al., 2021], where the stepsizes are set to be large, resulting in non-monotonic losses induced by the GD iterates. This paper studies the convergence and implicit bias of constant-stepsize GD for logistic regression on linearly separable data in the EoS regime. Despite the presence of local oscillations, we prove that the logistic loss can be minimized by GD with any constant stepsize over a long time scale. Furthermore, we prove that with any constant stepsize, the GD iterates tend to infinity when projected to a max-margin direction (the hard-margin SVM direction) and converge to a fixed vector that minimizes a strongly convex potential when projected to the orthogonal complement of the max-margin direction. In contrast, we also show that in the EoS regime, GD iterates may diverge catastrophically under the exponential loss, highlighting the superiority of the logistic loss. These theoretical findings are in line with numerical simulations and complement existing theories on the convergence and implicit bias of GD for logistic regression, which are only applicable when the stepsizes are sufficiently small.

Probabilistic Methods/Variational Inference

Veit David Wild, Sahra Ghalebikesabi, Dino Sejdinovic, Jeremias Knoblauch

We establish the first mathematically rigorous link between Bayesian, variational Bayesian, and ensemble methods. A key step towards this it to reformulate the non-convex optimisation problem typically encountered in deep learning as a convex optimisation in the space of probability measures. On a technical level, our contribution amounts to studying generalised variational inference through the lense of Wasserstein gradient flows. The result is a unified theory of various seemingly disconnected approaches that are commonly used for uncertainty quantification in deep learning---including deep ensembles and (variational) Bayesian methods. This offers a fresh perspective on the reasons behind the success of deep ensembles over procedures based on parameterised variational inference, and allows the derivation of new ensembling schemes with convergence guarantees. We showcase this by proposing a family of interacting deep ensembles with direct parallels to the interactions of particle systems in thermodynamics, and use our theory to prove the convergence of these algorithms to a well-defined global minimiser on the space of probability measures.

Deep Learning/Robustness

Samuel Pfrommer, Brendon Anderson, Julien Piet, Somayeh Sojoudi

Real-world adversarial attacks on machine learning models often feature an asymmetric structure wherein adversaries only attempt to induce false negatives (e.g., classify a spam email as not spam). We formalize the asymmetric robustness certification problem and correspondingly present the feature-convex neural network architecture, which composes an input-convex neural network (ICNN) with a Lipschitz continuous feature map in order to achieve asymmetric adversarial robustness. We consider the aforementioned binary setting with one "sensitive" class, and for this class we prove deterministic, closed-form, and easily-computable certified robust radii for arbitrary $\ell_p$-norms. We theoretically justify the use of these models by characterizing their decision region geometry, extending the universal approximation theorem for ICNN regression to the classification setting, and proving a lower bound on the probability that such models perfectly fit even unstructured uniformly distributed data in sufficiently high dimensions. Experiments on Malimg malware classification and subsets of the MNIST, Fashion-MNIST, and CIFAR-10 datasets show that feature-convex classifiers attain substantial certified $\ell_1$, $\ell_2$, and $\ell_{\infty}$-radii while being far more computationally efficient than competitive baselines.

Miscellaneous Aspects of Machine Learning/Representation Learning

Shashanka Venkataramanan, Ewa Kijak, laurent amsaleg, Yannis Avrithis

Mixup refers to interpolation-based data augmentation, originally motivated as a way to go beyond empirical risk minimization (ERM). Its extensions mostly focus on the definition of interpolation and the space (input or feature) where it takes place, while the augmentation process itself is less studied. In most methods, the number of generated examples is limited to the mini-batch size and the number of examples being interpolated is limited to two (pairs), in the input space.We make progress in this direction by introducing MultiMix, which generates an arbitrarily large number of interpolated examples beyond the mini-batch size and interpolates the entire mini-batch in the embedding space. Effectively, we sample on the entire convex hull of the mini-batch rather than along linear segments between pairs of examples.On sequence data, we further extend to Dense MultiMix. We densely interpolate features and target labels at each spatial location and also apply the loss densely. To mitigate the lack of dense labels, we inherit labels from examples and weight interpolation factors by attention as a measure of confidence.Overall, we increase the number of loss terms per mini-batch by orders of magnitude at little additional cost. This is only possible because of interpolating in the embedding space. We empirically show that our solutions yield significant improvement over state-of-the-art mixup methods on four different benchmarks, despite interpolation being only linear. By analyzing the embedding space, we show that the classes are more tightly clustered and uniformly spread over the embedding space, thereby explaining the improved behavior.