Data valuation has found various applications in machine learning, such as data filtering, efficient learning and incentives for data sharing. The most popular current approach to data valuation is the Shapley value. While popular for its various applications, Shapley value is computationally expensive even to approximate, as it requires repeated iterations of training models on different subsets of data. In this paper we show that the Shapley value of data points can be approximated more efficiently by leveraging the structural properties of machine learning problems. We derive convergence guarantees on the accuracy of the approximate Shapley value for different learning settings including Stochastic Gradient Descent with convex and non-convex loss functions. Our analysis suggests that in fact models trained on small subsets are more important in the context of data valuation. Based on this idea, we describe $\delta$-Shapley -- a strategy of only using small subsets for the approximation. Experiments show that this approach preserves approximate value and rank of data, while achieving speedup of up to 9.9x. In pre-trained networks the approach is found to bring more efficiency in terms of accurate evaluation using small subsets.

The connections between (convex) optimization and (logconcave) sampling have been considerably enriched in the past decade with many conceptual and mathematical analogies. For instance, the Langevin algorithm can be viewed as a sampling analogue of gradient descent and has condition-number-dependent guarantees on its performance. In the early 1990s, Nesterov and Nemirovski developed the Interior-Point Method (IPM) for convex optimization based on self-concordant barriers, providing efficient algorithms for structured convex optimization, often faster than the general method. This raises the following question: can we develop an analogous IPM for structured sampling problems? In 2012, Kannan and Narayanan proposed the Dikin walk for uniformly sampling polytopes, and an improved analysis was given in 2020 by Laddha-Lee-Vempala. The Dikin walk uses a local metric defined by a self-concordant barrier for linear constraints. Here we generalize this approach by developing and adapting IPM machinery together with the Dikin walk for poly-time sampling algorithms. Our IPM-based sampling framework provides an efficient warm start and goes beyond uniform distributions and linear constraints. We illustrate the approach on important special cases, in particular giving the fastest algorithms to sample uniform, exponential, or Gaussian distributions on a truncated PSD cone. The framework is general and can be applied to other sampling algorithms.

Several recent works demonstrate that transformers can implement algorithms like gradient descent. By a careful construction of weights, these works show that multiple layers of transformers are expressive enough to simulate iterations of gradient descent. Going beyond the question of expressivity, we ask: Can transformers learn to implement such algorithms by training over random problem instances? To our knowledge, we make the first theoretical progress on this question via an analysis of the loss landscape for linear transformers trained over random instances of linear regression. For a single attention layer, we prove the global minimum of the training objective implements a single iteration of preconditioned gradient descent. Notably, the preconditioning matrix not only adapts to the input distribution but also to the variance induced by data inadequacy. For a transformer with $L$ attention layers, we prove certain critical points of the training objective implement $L$ iterations of preconditioned gradient descent. Our results call for future theoretical studies on learning algorithms by training transformers.

Over the past decade, deep neural networks have demonstrated significant success using the training scheme that involves mini-batch stochastic gradient descent on extensive datasets. Expanding upon this accomplishment, there has been a surge in research exploring the application of neural networks in other learning scenarios. One notable framework that has garnered significant attention is meta-learning. Often described as "learning to learn," meta-learning is a data-driven approach to optimize the learning algorithm. Other branches of interest are continual learning and online learning, both of which involve incrementally updating a model with streaming data. While these frameworks were initially developed independently, recent works have started investigating their combinations, proposing novel problem settings and learning algorithms. However, due to the elevated complexity and lack of unified terminology, discerning differences between the learning frameworks can be challenging even for experienced researchers. To facilitate a clear understanding, this paper provides a comprehensive survey that organizes various problem settings using consistent terminology and formal descriptions. By offering an overview of these learning paradigms, our work aims to foster further advancements in this promising area of research.

Decentralized stochastic gradient descent (D-SGD) allows collaborative learning on massive devices simultaneously without the control of a central server. However, existing theories claim that decentralization invariably undermines generalization. In this paper, we challenge the conventional belief and present a completely new perspective for understanding decentralized learning. We prove that D-SGD implicitly minimizes the loss function of an average-direction Sharpness-aware minimization (SAM) algorithm under general non-convex non-$\beta$-smooth settings. This surprising asymptotic equivalence reveals an intrinsic regularization-optimization trade-off and three advantages of decentralization: (1) there exists a free uncertainty evaluation mechanism in D-SGD to improve posterior estimation; (2) D-SGD exhibits a gradient smoothing effect; and (3) the sharpness regularization effect of D-SGD does not decrease as total batch size increases, which justifies the potential generalization benefit of D-SGD over centralized SGD (C-SGD) in large-batch scenarios. The code is available at https://github.com/Raiden-Zhu/ICML-2023-DSGD-and-SAM.

Gradient clipping is a popular modification to standard (stochastic) gradient descent, at every iteration limiting the gradient norm to a certain value $c >0$. It is widely used for example for stabilizing the training of deep learning models (Goodfellow et al., 2016), or for enforcing differential privacy (Abadi et al., 2016). Despite popularity and simplicity of the clipping mechanism, its convergence guarantees often require specific values of $c$ and strong noise assumptions. In this paper, we give convergence guarantees that show precise dependence on arbitrary clipping thresholds $c$ and show that our guarantees are tight with both deterministic and stochastic gradients. In particular, we show that (i) for deterministic gradient descent, the clipping threshold only affects the higher-order terms of convergence, (ii) in the stochastic setting convergence to the true optimum cannot be guaranteed under the standard noise assumption, even under arbitrary small step-sizes. We give matching upper and lower bounds for convergence of the gradient norm when running clipped SGD, and illustrate these results with experiments.

We propose a new dataset distillation algorithm using reparameterization and convexification of implicit gradients (RCIG), that substantially improves the state-of-the-art. To this end, we first formulate dataset distillation as a bi-level optimization problem. Then, we show how implicit gradients can be effectively used to compute meta-gradient updates. We further equip the algorithm with a convexified approximation that corresponds to learning on top of a frozen finite-width neural tangent kernel. Finally, we improve bias in implicit gradients by parameterizing the neural network to enable analytical computation of final-layer parameters given the body parameters. RCIG establishes the new state-of-the-art on a diverse series of dataset distillation tasks. Notably, with one image per class, on resized ImageNet, RCIG sees on average a 108\% improvement over the previous state-of-the-art distillation algorithm. Similarly, we observed a 66\% gain over SOTA on Tiny-ImageNet and 37\% on CIFAR-100.

We study benign overfitting in two-layer ReLU networks trained using gradient descent and hinge loss on noisy data for binary classification. In particular, we consider linearly separable data for which a relatively small proportion of labels are corrupted or flipped. We identify conditions on the margin of the clean data that give rise to three distinct training outcomes: benign overfitting, in which zero loss is achieved and with high probability test data is classified correctly; non-benign overfitting, in which zero loss is achieved but test data is misclassified with probability lower bounded by a constant; and non-overfitting, in which clean points, but not corrupt points, achieve zero loss and again with high probability test data is classified correctly. Our analysis provides a fine-grained description of the dynamics of neurons throughout training and reveals two distinct phases: in the first phase clean points achieve close to zero loss, in the second phase clean points oscillate on the boundary of zero loss while corrupt points either converge towards zero loss or are eventually zeroed by the network. We prove these results using a combinatorial approach that involves bounding the number of clean versus corrupt updates across these phases of training.

We consider a generic decentralized constrained optimization problem over static, directed communication networks, where each agent has exclusive access to only one convex, differentiable, local objective term and one convex constraint set. For this setup, we propose a novel decentralized algorithm, called DAGP (Double Averaging and Gradient Projection), based on local gradients, projection onto local constraints, and local averaging. We achieve global optimality through a novel distributed tracking technique we call distributed null projection. Further, we show that DAGP can be used to solve unconstrained problems with non-differentiable objective terms with a problem reduction scheme. Assuming only smoothness of the objective terms, we study the convergence of DAGP and establish sub-linear rates of convergence in terms of feasibility, consensus, and optimality, with no extra assumption (e.g. strong convexity). For the analysis, we forego the difficulties of selecting Lyapunov functions by proposing a new methodology of convergence analysis in optimization problems, which we refer to as aggregate lower-bounding. To demonstrate the generality of this method, we also provide an alternative convergence proof for the standard gradient descent algorithm with smooth functions. Finally, we present numerical results demonstrating the effectiveness of our proposed method in both constrained and unconstrained problems. In particular, we propose a distributed scheme by DAGP for the optimal transport problem with superior performance and speed.

Preserving training dynamics across batch sizes is an important tool for practical machine learning as it enables the trade-off between batch size and wall-clock time. This trade-off is typically enabled by a scaling rule, for example, in stochastic gradient descent, one should scale the learning rate linearly with the batch size. Another important machine learning tool is the model EMA, a functional copy of a target model, whose parameters move towards those of its target model according to an Exponential Moving Average (EMA) at a rate parameterized by a momentum hyperparameter. This model EMA can improve the robustness and generalization of supervised learning, stabilize pseudo-labeling, and provide a learning signal for Self-Supervised Learning (SSL). Prior works have not considered the optimization of the model EMA when performing scaling, leading to different training dynamics across batch sizes and lower model performance. In this work, we provide a scaling rule for optimization in the presence of a model EMA and demonstrate the rule's validity across a range of architectures, optimizers, and data modalities. We also show the rule's validity where the model EMA contributes to the optimization of the target model, enabling us to train EMA-based pseudo-labeling and SSL methods at small and large batch sizes. For SSL, we enable training of BYOL up to batch size 24,576 without sacrificing performance, a 6$\times$ wall-clock time reduction under idealized hardware settings.