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We prove that finding all globally optimal two-layer ReLU neural networks can be performed by solving a convex optimization program with cone constraints. Our analysis is novel, characterizes all optimal solutions, and does not leverage duality-based analysis which was recently used to lift neural network training into convex spaces. Given the set of solutions of our convex optimization program, we show how to construct exactly the entire set of optimal neural networks. We provide a detailed characterization of this optimal set and its invariant transformations. As additional consequences of our convex perspective, (i) we establish that Clarke stationary points found by stochastic gradient descent correspond to the global optimum of a subsampled convex problem (ii) we provide a polynomial-time algorithm for checking if a neural network is a global minimum of the training loss (iii) we provide an explicit construction of a continuous path between any neural network and the global minimum of its sublevel set and (iv) characterize the minimal size of the hidden layer so that the neural network optimization landscape has no spurious valleys.Overall, we provide a rich framework for studying the landscape of neural network training loss through convexity.
When transferring a pretrained model to a downstream task, two popular methods are full fine-tuning (updating all the model parameters) and linear probing (updating only the last linear layer---the "head"). It is well known that fine-tuning leads to better accuracy in-distribution (ID). However, in this paper, we find that fine-tuning can achieve worse accuracy than linear probing out-of-distribution (OOD) when the pretrained features are good and the distribution shift is large. On 10 distribution shift datasets (BREEDS-Living17, BREEDS-Entity30, DomainNet, CIFAR $\to$ STL, CIFAR-10.1, FMoW, ImageNetV2, ImageNet-R, ImageNet-A, ImageNet-Sketch), fine-tuning obtains on average 2% higher accuracy ID but 7% lower accuracy OOD than linear probing. We show theoretically that this tradeoff between ID and OOD accuracy arises even in a simple setting: fine-tuning overparameterized two-layer linear networks. We prove that the OOD error of fine-tuning is high when we initialize with a fixed or random head---this is because while fine-tuning learns the head, the lower layers of the neural network change simultaneously and distort the pretrained features. Our analysis suggests that the easy two-step strategy of linear probing then full fine-tuning (LP-FT), sometimes used as a fine-tuning heuristic, combines the benefits of both fine-tuning and linear probing. Empirically, LP-FT outperforms both fine-tuning and linear probing on the above datasets (1% better ID, 10% better OOD than full fine-tuning).
This paper studies node classification in the inductive setting, i.e., aiming to learn a model on labeled training graphs and generalize it to infer node labels on unlabeled test graphs. This problem has been extensively studied with graph neural networks (GNNs) by learning effective node representations, as well as traditional structured prediction methods for modeling the structured output of node labels, e.g., conditional random fields (CRFs). In this paper, we present a new approach called the Structured Proxy Network (SPN), which combines the advantages of both worlds. SPN defines flexible potential functions of CRFs with GNNs. However, learning such a model is nontrivial as it involves optimizing a maximin game with high-cost inference. Inspired by the underlying connection between joint and marginal distributions defined by Markov networks, we propose to solve an approximate version of the optimization problem as a proxy, which yields a near-optimal solution, making learning more efficient. Extensive experiments on two settings show that our approach outperforms many competitive baselines.
A central goal of sequence modeling is designing a single principled model that can address sequence data across a range of modalities and tasks, particularly on long-range dependencies. Although conventional models including RNNs, CNNs, and Transformers have specialized variants for capturing long dependencies, they still struggle to scale to very long sequences of $10000$ or more steps. A promising recent approach proposed modeling sequences by simulating the fundamental state space model (SSM) \( x'(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t) \), and showed that for appropriate choices of the state matrix \( A \), this system could handle long-range dependencies mathematically and empirically. However, this method has prohibitive computation and memory requirements, rendering it infeasible as a general sequence modeling solution. We propose the Structured State Space sequence model (S4) based on a new parameterization for the SSM, and show that it can be computed much more efficiently than prior approaches while preserving their theoretical strengths. Our technique involves conditioning \( A \) with a low-rank correction, allowing it to be diagonalized stably and reducing the SSM to the well-studied computation of a Cauchy kernel. S4 achieves strong empirical results across a diverse range of established benchmarks, including (i) 91\% accuracy on sequential CIFAR-10 with no data augmentation or auxiliary losses, on par with a larger 2-D ResNet, (ii) substantially closing the gap to Transformers on image and language modeling tasks, while performing generation $60\times$ faster (iii) SoTA on every task from the Long Range Arena benchmark, including solving the challenging Path-X task of length 16k that all prior work fails on, while being as efficient as all competitors.
The recently discovered Neural Collapse (NC) phenomenon occurs pervasively in today's deep net training paradigm of driving cross-entropy (CE) loss towards zero. During NC, last-layer features collapse to their class-means, both classifiers and class-means collapse to the same Simplex Equiangular Tight Frame, and classifier behavior collapses to the nearest-class-mean decision rule. Recent works demonstrated that deep nets trained with mean squared error (MSE) loss perform comparably to those trained with CE. As a preliminary, we empirically establish that NC emerges in such MSE-trained deep nets as well through experiments on three canonical networks and five benchmark datasets. We provide, in a Google Colab notebook, PyTorch code for reproducing MSE-NC and CE-NC: https://colab.research.google.com/github/neuralcollapse/neuralcollapse/blob/main/neuralcollapse.ipynb. The analytically-tractable MSE loss offers more mathematical opportunities than the hard-to-analyze CE loss, inspiring us to leverage MSE loss towards the theoretical investigation of NC. We develop three main contributions: (I) We show a new decomposition of the MSE loss into (A) terms directly interpretable through the lens of NC and which assume the last-layer classifier is exactly the least-squares classifier; and (B) a term capturing the deviation from this least-squares classifier. (II) We exhibit experiments on canonical datasets and networks demonstrating that term-(B) is negligible during training. This motivates us to introduce a new theoretical construct: the central path, where the linear classifier stays MSE-optimal for feature activations throughout the dynamics. (III) By studying renormalized gradient flow along the central path, we derive exact dynamics that predict NC.
In this paper, we introduce Target-Aware Weighted Training (TAWT), a weighted training algorithm for cross-task learning based on minimizing a representation-based task distance between the source and target tasks. We show that TAWT is easy to implement, is computationally efficient, requires little hyperparameter tuning, and enjoys non-asymptotic learning-theoretic guarantees. The effectiveness of TAWT is corroborated through extensive experiments with BERT on four sequence tagging tasks in natural language processing (NLP), including part-of-speech (PoS) tagging, chunking, predicate detection, and named entity recognition (NER). As a byproduct, the proposed representation-based task distance allows one to reason in a theoretically principled way about several critical aspects of cross-task learning, such as the choice of the source data and the impact of fine-tuning.
Neural networks and other machine learning models compute continuous representations, while humans communicate mostly through discrete symbols. Reconciling these two forms of communication is desirable for generating human-readable interpretations or learning discrete latent variable models, while maintaining end-to-end differentiability. Some existing approaches (such as the Gumbel-Softmax transformation) build continuous relaxations that are discrete approximations in the zero-temperature limit, while others (such as sparsemax transformations and the Hard Concrete distribution) produce discrete/continuous hybrids. In this paper, we build rigorous theoretical foundations for these hybrids, which we call "mixed random variables.'' Our starting point is a new "direct sum'' base measure defined on the face lattice of the probability simplex. From this measure, we introduce new entropy and Kullback-Leibler divergence functions that subsume the discrete and differential cases and have interpretations in terms of code optimality. Our framework suggests two strategies for representing and sampling mixed random variables, an extrinsic ("sample-and-project'’) and an intrinsic one (based on face stratification). We experiment with both approaches on an emergent communication benchmark and on modeling MNIST and Fashion-MNIST data with variational auto-encoders with mixed latent variables.
Learning causal relationships in high-dimensional data (images, videos) is a hard task, as they are often defined on low dimensional manifolds and must be extracted from complex signals dominated by appearance, lighting, textures and also spurious correlations in the data. We present a method for learning counterfactual reasoning of physical processes in pixel space, which requires the prediction of the impact of interventions on initial conditions. Going beyond the identification of structural relationships, we deal with the challenging problem of forecasting raw video over long horizons. Our method does not require the knowledge or supervision of any ground truth positions or other object or scene properties. Our model learns and acts on a suitable hybrid latent representation based on a combination of dense features, sets of 2D keypoints and an additional latent vector per keypoint. We show that this better captures the dynamics of physical processes than purely dense or sparse representations. We introduce a new challenging and carefully designed counterfactual benchmark for predictions in pixel space and outperform strong baselines in physics-inspired ML and video prediction.
Modern language models can generate high-quality short texts. However, they often meander or are incoherent when generating longer texts. These issues arise from the next-token-only language modeling objective. To address these issues, we introduce Time Control (TC), a language model that implicitly plans via a latent stochastic process. TC does this by learning a representation which maps the dynamics of how text changes in a document to the dynamics of a stochastic process of interest. Using this representation, the language model can generate text by first implicitly generating a document plan via a stochastic process, and then generating text that is consistent with this latent plan. Compared to domain-specific methods and fine-tuning GPT2 across a variety of text domains, TC improves performance on text infilling and discourse coherence. On long text generation settings, TC preserves the text structure both in terms of ordering (up to +40% better) and text length consistency (up to +17% better). Human evaluators also prefer TC's output 28.6% more than the baselines.
Characterizing the separation power of graph neural networks (GNNs) provides an understanding of their limitations for graph learning tasks. Results regarding separation power are, however, usually geared at specific GNNs architectures, and tools for understanding arbitrary GNN architectures are generally lacking. We provide an elegant way to easily obtain bounds on the separation power of GNNs in terms of the Weisfeiler-Leman (WL) tests, which have become the yardstick to measure the separation power of GNNs. The crux is to view GNNs as expressions in a procedural tensor language describing the computations in the layers of the GNNs. Then, by a simple analysis of the obtained expressions, in terms of the number of indexes used and the nesting depth of summations, bounds on the separation power in terms of the WL-tests readily follow. We use tensor language to define Higher-Order Message-Passing Neural Networks (or k-MPNNs), a natural extension of MPNNs. Furthermore, the tensor language point of view allows for the derivation of universality results for classes of GNNs in a natural way. Our approach provides a toolbox with which GNN architecture designers can analyze the separation power of their GNNs, without needing to know the intricacies of the WL-tests. We also provide insights in what is needed to boost the separation power of GNNs.
As Artificial Intelligence as a Service gains popularity, protecting well-trained models as intellectual property is becoming increasingly important. There are two common types of protection methods: ownership verification and usage authorization. In this paper, we propose Non-Transferable Learning (NTL), a novel approach that captures the exclusive data representation in the learned model and restricts the model generalization ability to certain domains. This approach provides effective solutions to both model verification and authorization. Specifically: 1) For ownership verification, watermarking techniques are commonly used but are often vulnerable to sophisticated watermark removal methods. By comparison, our NTL-based ownership verification provides robust resistance to state-of-the-art watermark removal methods, as shown in extensive experiments with 6 removal approaches over the digits, CIFAR10 & STL10, and VisDA datasets. 2) For usage authorization, prior solutions focus on authorizing specific users to access the model, but authorized users can still apply the model to any data without restriction. Our NTL-based authorization approach instead provides data-centric protection, which we call applicability authorization, by significantly degrading the performance of the model on unauthorized data. Its effectiveness is also shown through experiments on aforementioned datasets.
Continual learning (CL) aims to learn a sequence of tasks without forgetting the previously acquired knowledge. However, recent CL advances are restricted to supervised continual learning (SCL) scenarios. Consequently, they are not scalable to real-world applications where the data distribution is often biased and unannotated. In this work, we focus on unsupervised continual learning (UCL), where we learn the feature representations on an unlabelled sequence of tasks and show that reliance on annotated data is not necessary for continual learning. We conduct a systematic study analyzing the learned feature representations and show that unsupervised visual representations are surprisingly more robust to catastrophic forgetting, consistently achieve better performance, and generalize better to out-of-distribution tasks than SCL. Furthermore, we find that UCL achieves a smoother loss landscape through qualitative analysis of the learned representations and learns meaningful feature representations. Additionally, we propose Lifelong Unsupervised Mixup (Lump), a simple yet effective technique that interpolates between the current task and previous tasks' instances to alleviate catastrophic forgetting for unsupervised representations.
The ability to identify whether or not a test sample belongs to one of the semantic classes in a classifier's training set is critical to practical deployment of the model. This task is termed open-set recognition (OSR) and has received significant attention in recent years. In this paper, we first demonstrate that the ability of a classifier to make the 'none-of-above' decision is highly correlated with its accuracy on the closed-set classes. We find that this relationship holds across loss objectives and architectures, and further demonstrate the trend both on the standard OSR benchmarks as well as on a large-scale ImageNet evaluation. Second, we use this correlation to boost the performance of the maximum softmax probability OSR 'baseline' by improving its closed-set accuracy, and with this strong baseline achieve state-of-the-art on a number of OSR benchmarks. Similarly, we boost the performance of the existing state-of-the-art method by improving its closed-set accuracy, but the resulting discrepancy with the strong baseline is marginal. Our third contribution is to present the 'Semantic Shift Benchmark' (SSB), which better respects the task of detecting semantic novelty, as opposed to low-level distributional shifts as tackled by neighbouring machine learning fields. On this new evaluation, we again demonstrate that there is negligible difference between the strong baseline and the existing state-of-the-art. Code available at: https://github.com/sgvaze/osr_closed_set_all_you_need.
In this paper, we perform an in-depth study of the properties and applications of aligned generative models.We refer to two models as aligned if they share the same architecture, and one of them (the child) is obtained from the other (the parent) via fine-tuning to another domain, a common practice in transfer learning. Several works already utilize some basic properties of aligned StyleGAN models to perform image-to-image translation. Here, we perform the first detailed exploration of model alignment, also focusing on StyleGAN. First, we empirically analyze aligned models and provide answers to important questions regarding their nature. In particular, we find that the child model's latent spaces are semantically aligned with those of the parent, inheriting incredibly rich semantics, even for distant data domains such as human faces and churches. Second, equipped with this better understanding, we leverage aligned models to solve a diverse set of tasks. In addition to image translation, we demonstrate fully automatic cross-domain image morphing. We further show that zero-shot vision tasks may be performed in the child domain, while relying exclusively on supervision in the parent domain. We demonstrate qualitatively and quantitatively that our approach yields state-of-the-art results, while requiring only simple fine-tuning and inversion.
Deep learning has seen a movement away from representing examples with a monolithic hidden state towards a richly structured state. For example, Transformers segment by position, and object-centric architectures decompose images into entities. In all these architectures, interactions between different elements are modeled via pairwise interactions: Transformers make use of self-attention to incorporate information from other positions and object-centric architectures make use of graph neural networks to model interactions among entities. We consider how to improve on pairwise interactions in terms of global coordination and a coherent, integrated representation that can be used for downstream tasks. In cognitive science, a global workspace architecture has been proposed in which functionally specialized components share information through a common, bandwidth-limited communication channel. We explore the use of such a communication channel in the context of deep learning for modeling the structure of complex environments. The proposed method includes a shared workspace through which communication among different specialist modules takes place but due to limits on the communication bandwidth, specialist modules must compete for access. We show that capacity limitations have a rational basis in that (1) they encourage specialization and compositionality and (2) they facilitate the synchronization of otherwise independent specialists.
Tensor computations underlie modern scientific computing and deep learning.A number of tensor frameworks emerged varying in execution model, hardware support, memory management, model definition, etc.However, tensor operations in all frameworks follow the same paradigm.Recent neural network architectures demonstrate demand for higher expressiveness of tensor operations.The current paradigm is not suited to write readable, reliable, or easy-to-modify code for multidimensional tensor manipulations. Moreover, some commonly used operations do not provide sufficient checks and can break a tensor structure.These mistakes are elusive as no tools or tests can detect them.Independently, API discrepancies complicate code transfer between frameworks.We propose einops notation: a uniform and generic way to manipulate tensor structure, that significantly improves code readability and flexibility by focusing on the structure of input and output tensors.We implement einops notation in a Python package that efficiently supports multiple widely used frameworks and provides framework-independent minimalist API for tensor manipulations.
Meta-learning enables algorithms to quickly learn a newly encountered task with just a few labeled examples by transferring previously learned knowledge. However, the bottleneck of current meta-learning algorithms is the requirement of a large number of meta-training tasks, which may not be accessible in real-world scenarios. To address the challenge that available tasks may not densely sample the space of tasks, we propose to augment the task set through interpolation. By meta-learning with task interpolation (MLTI), our approach effectively generates additional tasks by randomly sampling a pair of tasks and interpolating the corresponding features and labels. Under both gradient-based and metric-based meta-learning settings, our theoretical analysis shows MLTI corresponds to a data-adaptive meta-regularization and further improves the generalization. Empirically, in our experiments on eight datasets from diverse domains including image recognition, pose prediction, molecule property prediction, and medical image classification, we find that the proposed general MLTI framework is compatible with representative meta-learning algorithms and consistently outperforms other state-of-the-art strategies.
We introduce a self-supervised vision representation model BEiT, which stands for Bidirectional Encoder representation from Image Transformers. Following BERT developed in the natural language processing area, we propose a masked image modeling task to pretrain vision Transformers. Specifically, each image has two views in our pre-training, i.e., image patches (such as 16 x 16 pixels), and visual tokens (i.e., discrete tokens). We first ``tokenize'' the original image into visual tokens. Then we randomly mask some image patches and fed them into the backbone Transformer. The pre-training objective is to recover the original visual tokens based on the corrupted image patches. After pre-training BEiT, we directly fine-tune the model parameters on downstream tasks by appending task layers upon the pretrained encoder. Experimental results on image classification and semantic segmentation show that our model achieves competitive results with previous pre-training methods.
We are concerned with the problem of distributional prediction with incomplete features: The goal is to estimate the distribution of target variables given feature vectors with some of the elements missing. A typical approach to this problem is to perform missing-value imputation and regression, simultaneously or sequentially, which we call the generative approach. Another approach is to perform regression after appropriately encoding missing values into the feature, which we call the discriminative approach. In comparison, the generative approach is more robust to the feature corruption while the discriminative approach is more favorable to maximize the performance of prediction. In this study, we propose a hybrid method to take the best of both worlds. Our method utilizes the black-box variational inference framework so that it can be applied to a wide variety of modern machine learning models, including the variational autoencoders. We also confirmed the effectiveness of the proposed method empirically.
Most graph neural networks (GNNs) use the message passing paradigm, in which node features are propagated on the input graph. Recent works pointed to the distortion of information flowing from distant nodes as a factor limiting the efficiency of message passing for tasks relying on long-distance interactions. This phenomenon, referred to as 'over-squashing', has been heuristically attributed to graph bottlenecks where the number of $k$-hop neighbors grows rapidly with $k$. We provide a precise description of the over-squashing phenomenon in GNNs and analyze how it arises from bottlenecks in the graph. For this purpose, we introduce a new edge-based combinatorial curvature and prove that negatively curved edges are responsible for the over-squashing issue. We also propose and experimentally test a curvature-based graph rewiring method to alleviate the over-squashing.
Differentially Private (DP) learning has seen limited success for building large deep learning models of text, and straightforward attempts at applying Differentially Private Stochastic Gradient Descent (DP-SGD) to NLP tasks have resulted in large performance drops and high computational overhead.We show that this performance drop can be mitigated with (1) the use of large pretrained language models; (2) non-standard hyperparameters that suit DP optimization; and (3) fine-tuning objectives which are aligned with the pretraining procedure.With the above, we obtain NLP models that outperform state-of-the-art DP-trained models under the same privacy budget and strong non-private baselines---by directly fine-tuning pretrained models with DP optimization on moderately-sized corpora. To address the computational challenge of running DP-SGD with large Transformers, we propose a memory saving technique that allows clipping in DP-SGD to run without instantiating per-example gradients for any linear layer in the model. The technique enables privately training Transformers with almost the same memory cost as non-private training at a modest run-time overhead. Contrary to conventional wisdom that DP optimization fails at learning high-dimensional models (due to noise that scales with dimension) empirical results reveal that private learning with pretrained language models tends to not suffer from dimension-dependent performance degradation.Code to reproduce results can be found at https://github.com/lxuechen/private-transformers.
In distributed learning, local SGD (also known as federated averaging) and its simple baseline minibatch SGD are widely studied optimization methods. Most existing analyses of these methods assume independent and unbiased gradient estimates obtained via with-replacement sampling. In contrast, we study shuffling-based variants: minibatch and local Random Reshuffling, which draw stochastic gradients without replacement and are thus closer to practice. For smooth functions satisfying the Polyak-Łojasiewicz condition, we obtain convergence bounds (in the large epoch regime) which show that these shuffling-based variants converge faster than their with-replacement counterparts. Moreover, we prove matching lower bounds showing that our convergence analysis is tight. Finally, we propose an algorithmic modification called synchronized shuffling that leads to convergence rates faster than our lower bounds in near-homogeneous settings.
Predicting molecular conformations from molecular graphs is a fundamental problem in cheminformatics and drug discovery. Recently, significant progress has been achieved with machine learning approaches, especially with deep generative models. Inspired by the diffusion process in classical non-equilibrium thermodynamics where heated particles will diffuse from original states to a noise distribution, in this paper, we propose a novel generative model named GeoDiff for molecular conformation prediction. GeoDiff treats each atom as a particle and learns to directly reverse the diffusion process (i.e., transforming from a noise distribution to stable conformations) as a Markov chain. Modeling such a generation process is however very challenging as the likelihood of conformations should be roto-translational invariant. We theoretically show that Markov chains evolving with equivariant Markov kernels can induce an invariant distribution by design, and further propose building blocks for the Markov kernels to preserve the desirable equivariance property. The whole framework can be efficiently trained in an end-to-end fashion by optimizing a weighted variational lower bound to the (conditional) likelihood. Experiments on multiple benchmarks show that GeoDiff is superior or comparable to existing state-of-the-art approaches, especially on large molecules.
Understanding how neural dynamics give rise to behaviour is one of the most fundamental questions in systems neuroscience. To achieve this, a common approach is to record neural populations in behaving animals, and model these data as emanating from a latent dynamical system whose state trajectories can then be related back to behavioural observations via some form of decoding. As recordings are typically performed in localized circuits that form only a part of the wider implicated network, it is important to simultaneously learn the local dynamics and infer any unobserved external input that might drive them. Here, we introduce iLQR-VAE, a novel control-based approach to variational inference in nonlinear dynamical systems, capable of learning both latent dynamics, initial conditions, and ongoing external inputs. As in recent deep learning approaches, our method is based on an input-driven sequential variational autoencoder (VAE). The main novelty lies in the use of the powerful iterative linear quadratic regulator algorithm (iLQR) in the recognition model. Optimization of the standard evidence lower-bound requires differentiating through iLQR solutions, which is made possible by recent advances in differentiable control. Importantly, having the recognition model be implicitly defined by the generative model greatly reduces the number of free parameters and allows for flexible, high-quality inference. This makes it possible for instance to evaluate the model on a single long trial after training on smaller chunks. We demonstrate the effectiveness of iLQR-VAE on a range of synthetic systems, with autonomous as well as input-driven dynamics. We further apply it to neural and behavioural recordings in non-human primates performing two different reaching tasks, and show that iLQR-VAE yields high-quality kinematic reconstructions from the neural data.
This work considers identifying parameters characterizing a physical system's dynamic motion directly from a video whose rendering configurations are inaccessible. Existing solutions require massive training data or lack generalizability to unknown rendering configurations. We propose a novel approach that marries domain randomization and differentiable rendering gradients to address this problem. Our core idea is to train a rendering-invariant state-prediction (RISP) network that transforms image differences into state differences independent of rendering configurations, e.g., lighting, shadows, or material reflectance. To train this predictor, we formulate a new loss on rendering variances using gradients from differentiable rendering. Moreover, we present an efficient, second-order method to compute the gradients of this loss, allowing it to be integrated seamlessly into modern deep learning frameworks. We evaluate our method in rigid-body and deformable-body simulation environments using four tasks: state estimation, system identification, imitation learning, and visuomotor control. We further demonstrate the efficacy of our approach on a real-world example: inferring the state and action sequences of a quadrotor from a video of its motion sequences. Compared with existing methods, our approach achieves significantly lower reconstruction errors and has better generalizability among unknown rendering configurations.