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Deep Learning · Self-Supervised Learning

Panagiotis Koromilas, Giorgos Bouritsas, Theodoros Giannakopoulos, Mihalis Nicolaou, Yannis Panagakis

What do different contrastive learning (CL) losses actually optimize for? Although multiple CL methods have demonstrated remarkable representation learning capabilities, the differences in their inner workings remain largely opaque. In this work, we analyse several CL families and prove that, under certain conditions, they admit the same minimisers when optimizing either their batch-level objectives or their expectations asymptotically. In both cases, an intimate connection with the hyperspherical energy minimisation (HEM) problem resurfaces. Drawing inspiration from this, we introduce a novel CL objective, coined Decoupled Hyperspherical Energy Loss (DHEL). DHEL simplifies the problem by decoupling the target hyperspherical energy from the alignment of positive examples while preserving the same theoretical guarantees. Going one step further, we show the same results hold for another relevant CL family, namely kernel contrastive learning (KCL), with the additional advantage of the expected loss being independent of batch size, thus identifying the minimisers in the non-asymptotic regime. Empirical results demonstrate improved downstream performance and robustness across combinations of different batch sizes and hyperparameters and reduced dimensionality collapse, on several computer vision datasets.

Miscellaneous Aspects of Machine Learning · Kernel methods

George Chen

Kernel survival analysis models estimate individual survival distributions with the help of a kernel function, which measures the similarity between any two data points. Such a kernel function can be learned using deep kernel survival models. In this paper, we present a new deep kernel survival model called a survival kernet, which scales to large datasets in a manner that is amenable to model interpretation and also theoretical analysis. Specifically, the training data are partitioned into clusters based on a recently developed training set compression scheme for classification and regression called kernel netting that we extend to the survival analysis setting. At test time, each data point is represented as a weighted combination of these clusters, and each such cluster can be visualized. For a special case of survival kernets, we establish a finite-sample error bound on predicted survival distributions that is, up to a log factor, optimal. Whereas scalability at test time is achieved using the aforementioned kernel netting compression strategy, scalability during training is achieved by a warm-start procedure based on tree ensembles such as XGBoost and a heuristic approach to accelerating neural architecture search. On four standard survival analysis datasets of varying sizes (up to roughly 3 million data points), we show that survival kernets are highly competitive compared to various baselines tested in terms of time-dependent concordance index. Our code is available at: https://github.com/georgehc/survival-kernets

Hanting Chen, Liuzhicheng Liuzhicheng, Xutao Wang, Yuchuan Tian, Yunhe Wang

In an effort to reduce the computational load of Transformers, research on linear attention has gained significant momentum. However, the improvement strategies for attention mechanisms typically necessitate extensive retraining, which is impractical for large language models with a vast array of parameters. In this paper, we present DiJiang, a novel Frequency Domain Kernelization approach that enables the transformation of a pre-trained vanilla Transformer into a linear complexity model with little training costs. By employing a weighted Quasi-Monte Carlo method for sampling, the proposed approach theoretically offers superior approximation efficiency. To further reduce the training computational complexity, our kernelization is based on Discrete Cosine Transform (DCT) operations. Extensive experiments demonstrate that the proposed method achieves comparable performance to the original Transformer, but with significantly reduced training costs and much faster inference speeds. Our DiJiang-7B achieves comparable performance with LLaMA2-7B on various benchmark while requires only about 1/50 training cost. Code is available at https://github.com/YuchuanTian/DiJiang.

Miscellaneous Aspects of Machine Learning · Kernel methods

Pierre Glaser, Steffan Paul, Alissa M. Hummer, Charlotte Deane, Debora Marks, Alan Amin

We propose a set of kernel-based tools to evaluate the designs and tune the hyperparameters of conditional sequence models, with a focus on problems in computational biology. The backbone of our tools is a new measure of discrepancy between the true conditional distribution and the model's estimate, called the Augmented Conditional Maximum Mean Discrepancy (ACMMD). Provided that the model can be sampled from, the ACMMD can be estimated unbiasedly from data to quantify absolute model fit, integrated within hypothesis tests, and used to evaluate model reliability. We demonstrate the utility of our approach by analyzing a popular protein design model, ProteinMPNN. We are able to reject the hypothesis that ProteinMPNN fits its data for various protein families, and tune the model's temperature hyperparameter to achieve a better fit.

Optimization · Optimization and Learning under Uncertainty

Yucen Li, Tim G. J. Rudner, Andrew Gordon Wilson

Bayesian optimization is a highly efficient approach to optimizing objective functions which are expensive to query. These objectives are typically represented by Gaussian process (GP) surrogate models which are easy to optimize and support exact inference. While standard GP surrogates have been well-established in Bayesian optimization, Bayesian neural networks (BNNs) have recently become practical function approximators, with many benefits over standard GPs such as the ability to naturally handle non-stationarity and learn representations for high-dimensional data. In this paper, we study BNNs as alternatives to standard GP surrogates for optimization. We consider a variety of approximate inference procedures for finite-width BNNs, including high-quality Hamiltonian Monte Carlo, low-cost stochastic MCMC, and heuristics such as deep ensembles. We also consider infinite-width BNNs, linearized Laplace approximations, and partially stochastic models such as deep kernel learning. We evaluate this collection of surrogate models on diverse problems with varying dimensionality, number of objectives, non-stationarity, and discrete and continuous inputs. We find: (i) the ranking of methods is highly problem dependent, suggesting the need for tailored inductive biases; (ii) HMC is the most successful approximate inference procedure for fully stochastic BNNs; (iii) full stochasticity may be unnecessary as deep kernel learning is relatively competitive; (iv) deep ensembles perform relatively poorly; (v) infinite-width BNNs are particularly promising, especially in high dimensions.

Applications · Neuroscience, Cognitive Science

Ilyass Hammouamri, Ismail Khalfaoui Hassani, Timothée Masquelier

Spiking Neural Networks (SNNs) are a promising research direction for building power-efficient information processing systems, especially for temporal tasks such as speech recognition. In SNNs, delays refer to the time needed for one spike to travel from one neuron to another. These delays matter because they influence the spike arrival times, and it is well-known that spiking neurons respond more strongly to coincident input spikes. More formally, it has been shown theoretically that plastic delays greatly increase the expressivity in SNNs. Yet, efficient algorithms to learn these delays have been lacking. Here, we propose a new discrete-time algorithm that addresses this issue in deep feedforward SNNs using backpropagation, in an offline manner. To simulate delays between consecutive layers, we use 1D convolutions across time. The kernels contain only a few non-zero weights – one per synapse – whose positions correspond to the delays. These positions are learned together with the weights using the recently proposed Dilated Convolution with Learnable Spacings (DCLS). We evaluated our method on three datasets: the Spiking Heidelberg Dataset (SHD), the Spiking Speech Commands (SSC) and its non spiking version Google Speech Commands v0.02 (GSC) benchmarks, which require detecting temporal patterns. We used feedforward SNNs with two or three hidden fully connected layers, and vanilla leaky integrate-and-fire neurons. We showed that fixed random delays help and that learning them helps even more. Furthermore, our method outperformed the state-of-the-art in the three datasets without using recurrent connections and with substantially fewer parameters. Our work demonstrates the potential of delay learning in developing accurate and precise models for temporal data processing. Our code is based on PyTorch / SpikingJelly and available at: https://github.com/Thvnvtos/SNN-delays

Suyu Ge, Yunan Zhang, Liyuan Liu, Minjia Zhang, Jiawei Han, Jianfeng Gao

In this study, we introduce adaptive KV cache compression, a plug-and-play method that reduces the memory footprint of generative inference for Large Language Models (LLMs). Different from the conventional KV cache that retains key and value vectors for all context tokens, we conduct targeted profiling to discern the intrinsic structure of attention modules. Based on the recognized structure, we then construct the KV cache in an adaptive manner: evicting long-range contexts on attention heads emphasizing local contexts, discarding non-special tokens on attention heads centered on special tokens, and only employing the standard KV cache for attention heads that broadly attend to all tokens. Moreover, with the lightweight attention profiling used to guide the construction of the adaptive KV cache, FastGen can be deployed without resource-intensive fine-tuning or re-training. In our experiments across various asks, FastGen demonstrates substantial reduction on GPU memory consumption with negligible generation quality loss. We will release our code and the compatible CUDA kernel for reproducibility.

Junzhe Zhu, Peiye Zhuang, Sanmi Koyejo

The advancements in automatic text-to-3D generation have been remarkable. Most existing methods use pre-trained text-to-image diffusion models to optimize 3D representations like Neural Radiance Fields (NeRFs) via latent-space denoising score matching. Yet, these methods often result in artifacts and inconsistencies across different views due to their suboptimal optimization approaches and limited understanding of 3D geometry. Moreover, the inherent constraints of NeRFs in rendering crisp geometry and stable textures usually lead to a two-stage optimization to attain high-resolution details. This work proposes holistic sampling and smoothing approaches to achieve high-quality text-to-3D generation, all in a single-stage optimization. We compute denoising scores in the text-to-image diffusion model's latent and image spaces. Instead of randomly sampling timesteps (also referred to as noise levels in denoising score matching), we introduce a novel timestep annealing approach that progressively reduces the sampled timestep throughout optimization. To generate high-quality renderings in a single-stage optimization, we propose regularization for the variance of z-coordinates along NeRF rays. To address texture flickering issues in NeRFs, we introduce a kernel smoothing technique that refines importance sampling weights coarse-to-fine, ensuring accurate and thorough sampling in high-density regions. Extensive experiments demonstrate the superiority of our method over previous approaches, enabling the generation of highly detailed and view-consistent 3D assets through a single-stage training process.

Optimization · Global Optimization

Carl Hvarfner, Frank Hutter, Luigi Nardi

The optimization of expensive-to-evaluate black-box functions is prevalent in various scientific disciplines. Bayesian optimization is an automatic, general and sample-efficient method to solve these problems with minimal knowledge of the the underlying function dynamics. However, the ability of Bayesian optimization to incorporate prior knowledge or beliefs about the function at hand in order to accelerate the optimization is limited, which reduces its appeal for knowledgeable practitioners with tight budgets. To allow domain experts to customize the optimization routine, we propose ColaBO, the first Bayesian-principled framework for incorporating prior beliefs beyond the typical kernel structure, such as the likely location of the optimizer or the optimal value. The generality of ColaBO makes it applicable across different Monte Carlo acquisition functions and types of user beliefs. We empirically demonstrate ColaBO's ability to substantially accelerate optimization when the prior information is accurate, and to retain approximately default performance when it is misleading.

Probabilistic Methods · Monte Carlo and Sampling Methods

Isaac Reid, Krzysztof Choromanski, Eli Berger, Adrian Weller

We propose a novel random walk-based algorithm for unbiased estimation of arbitrary functions of a weighted adjacency matrix, coined general graph random features (g-GRFs). This includes many of the most popular examples of kernels defined on the nodes of a graph. Our algorithm enjoys subquadratic time complexity with respect to the number of nodes, overcoming the notoriously prohibitive cubic scaling of exact graph kernel evaluation. It can also be trivially distributed across machines, permitting learning on much larger networks. At the heart of the algorithm is a modulation function which upweights or downweights the contribution from different random walks depending on their lengths. We show that by parameterising it with a neural network we can obtain g-GRFs that give higher-quality kernel estimates or perform efficient, scalable kernel learning. We provide robust theoretical analysis and support our findings with experiments including pointwise estimation of fixed graph kernels, solving non-homogeneous graph ordinary differential equations, node clustering and kernel regression on triangular meshes.

Josh Alman, Zhao Song

In the classical transformer attention scheme, we are given three $n \times d$ size matrices $Q, K, V$ (the query, key, and value tokens), and the goal is to compute a new $n \times d$ size matrix $D^{-1} \exp(QK^\top) V$ where $D = \mathrm{diag}( \exp(QK^\top) {\bf 1}_n )$. Here, $\exp()$ is applied entry-wise and ${\bf 1}_n$ denotes a length-$n$ vector whose entries are all ones.Intuitively, attention computation captures pairwise information between words in a sentence, but not higher-order information. Indeed, recent work \cite{sht23} has shown that attention units cannot solve simple problems about detecting triples of connected words.In this work, we study a generalization of attention which captures triple-wise correlations. The generalization is based on computations involving tensors defined by tuples of words. More formally, given five $n \times d$ size matrices $Q, K_1, K_2, V_1$ and $V_2$ (generalized query, key, and value tokens), our new goal is to compute an $n \times d$ size matrix $D^{-1} \exp( Q ( K_1 \oslash K_2)^\top ) (V_1 \oslash V_2) $ where $D = \mathrm{diag}( \exp( Q ( K_1 \oslash K_2)^\top ) {\bf 1}_{n^2} )$ and $K_1 \oslash K_2 \in \mathbb{R}^{n^2 \times d}$ denotes the column-wise Kronecker product of $K_1$ and $K_2$. This generalization is indeed able to solve problems about detecting triple-wise connections that were shown to be impossible for transformers.The potential downside of this generalization is that it appears as though computations are even more difficult, since the straightforward algorithm requires cubic time in $n$. However, we show that in the bounded-entry setting (which arises in practice, and which is well-studied in both theory and practice), there is actually a near-linear time algorithm. More precisely, we show that bounded entries are both necessary and sufficient for quickly performing generalized computations:$\bullet$ On the positive side, if all entries of the input matrices are bounded above by $o(\sqrt[3]{\log n})$ then we show how to approximate the ``tensor-type'' attention matrix in $n^{1+o(1)}$ time.$\bullet$ On the negative side, we show that if the entries of the input matrices may be as large as $\Omega(\sqrt[3]{\log n})$, then there is no algorithm that runs faster than $n^{3-o(1)}$ (assuming the Strong Exponential Time Hypothesis from fine-grained complexity theory).We also show that our construction, algorithms, and lower bounds naturally generalize to higher-order tensors and correlations. Interestingly, the higher the order of the tensors, the lower the bound on the entries needs to be for an efficient algorithm. Our results thus yield a natural tradeoff between the boundedness of the entries, and order of the tensor one may use for more expressive, efficient attention computation.Our constructions make use of a novel connection with a higher-order variant on the kernel density estimation problem. They combine a number of technical tools, including the polynomial method, algebraic geometry codes, and multiparty Merlin-Arthur communication protocols.

Deep Learning

Mircea Mironenco, Patrick Forré

Invariance and equivariance to geometrical transformations have proven to be very useful inductive biases when training (convolutional) neural network models, especially in the low-data regime.Much work has focused on the case where the symmetry group employed is compact or abelian, or both.Recent work has explored enlarging the class of transformations used to the case of Lie groups, principally through the use of their Lie algebra, as well as the group exponential and logarithm maps.The applicability of such methods to larger transformation groups is limited by the fact that depending on the group of interest $G$, the exponential map may not be surjective.Further limitations are encountered when $G$ is neither compact nor abelian.Using the structure and geometry of Lie groups and their homogeneous spaces, we present a framework by which it is possible to work with such groups primarily focusing on the Lie groups $G = \textnormal{GL}^{+}(n, \mathbb{R})$ and $G = \textnormal{SL}(n, \mathbb{R})$, as well as their representation as affine transformations $\mathbb{R}^{n} \rtimes G$.Invariant integration as well as a global parametrization is realized by decomposing the "larger" groups into subgroups and submanifolds which can be handled individually.Under this framework, we show how convolution kernels can be parametrized to build models equivariant with respect to affine transformations.We evaluate the robustness and out-of-distribution generalisation capability of our model on the standard affine-invariant benchmark classification task, where we outperform all previous equivariant models as well as all Capsule Network proposals.

Social Aspects · Trustworthy Machine Learning

Shaopeng Fu, Di Wang

Adversarial training (AT) is a canonical method for enhancing the robustness of deep neural networks (DNNs). However, recent studies empirically demonstrated that it suffers from robust overfitting, i.e., a long time AT can be detrimental to the robustness of DNNs. This paper presents a theoretical explanation of robust overfitting for DNNs. Specifically, we non-trivially extend the neural tangent kernel (NTK) theory to AT and prove that an adversarially trained wide DNN can be well approximated by a linearized DNN. Moreover, for squared loss, closed-form AT dynamics for the linearized DNN can be derived, which reveals a new AT degeneration phenomenon: a long-term AT will result in a wide DNN degenerates to that obtained without AT and thus cause robust overfitting. Based on our theoretical results, we further design a method namely Adv-NTK, the first AT algorithm for infinite-width DNNs. Experiments on real-world datasets show that Adv-NTK can help infinite-width DNNs enhance comparable robustness to that of their finite-width counterparts, which in turn justifies our theoretical findings. The code is available at https://github.com/fshp971/adv-ntk.

Probabilistic Methods · Bayesian Models and Methods

Edward Milsom, Ben Anson, Laurence Aitchison

Standard infinite-width limits of neural networks sacrifice the ability for intermediate layers to learn representations from data. Recent work (“A theory of representation learning gives a deep generalisation of kernel methods”, Yang et al. 2023) modified the Neural Network Gaussian Process (NNGP) limit of Bayesian neural networks so that representation learning is retained. Furthermore, they found that applying this modified limit to a deep Gaussian process gives a practical learning algorithm which they dubbed the “deep kernel machine” (DKM). However, they only considered the simplest possible setting: regression in small, fully connected networks with e.g. 10 input features. Here, we introduce convolutional deep kernel machines. This required us to develop a novel inter-domain inducing point approximation, as well as introducing and experimentally assessing a number of techniques not previously seen in DKMs, including analogues to batch normalisation, different likelihoods, and different types of top-layer. The resulting model trains in roughly 77 GPU hours, achieving around 99\% test accuracy on MNIST, 72\% on CIFAR-100, and 92.7\% on CIFAR-10, which is SOTA for kernel methods.

Probabilistic Methods · Monte Carlo and Sampling Methods

Isaac Reid, Eli Berger, Krzysztof Choromanski, Adrian Weller

We present a novel quasi-Monte Carlo mechanism to improve graph-based sampling, coined repelling random walks. By inducing correlations between the trajectories of an interacting ensemble such that their marginal transition probabilities are unmodified, we are able to explore the graph more efficiently, improving the concentration of statistical estimators whilst leaving them unbiased. The mechanism has a trivial drop-in implementation. We showcase the effectiveness of repelling random walks in a range of settings including estimation of graph kernels, the PageRank vector and graphlet concentrations. We provide detailed experimental evaluation and robust theoretical guarantees. To our knowledge, repelling random walks constitute the first rigorously studied quasi-Monte Carlo scheme correlating the directions of walkers on a graph, inviting new research in this exciting nascent domain.

Theory · Learning Theory

Maksim Velikanov, Maxim Panov, Dmitry Yarotsky

The asymptotically precise estimation of the generalization of kernel methods has recently received attention due to the parallels between neural networks and their associated kernels. However, prior works derive such estimates for training by kernel ridge regression (KRR), whereas neural networks are typically trained with gradient descent (GD). In the present work, we consider the training of kernels with a family of \emph{spectral algorithms} specified by profile $h(\lambda)$, and including KRR and GD as special cases. Then, we derive the generalization error as a functional of learning profile $h(\lambda)$ for two data models: high-dimensional Gaussian and low-dimensional translation-invariant model. Under power-law assumptions on the spectrum of the kernel and target, we use our framework to (i) give full loss asymptotics for both noisy and noiseless observations (ii) show that the loss localizes on certain spectral scales, giving a new perspective on the KRR saturation phenomenon (iii) conjecture, and demonstrate for the considered data models, the universality of the loss w.r.t. non-spectral details of the problem, but only in case of noisy observation.

Reinforcement Learning

Firas Al-Hafez, Guoping Zhao, Jan Peters, Davide Tateo

Stateful policies play an important role in reinforcement learning, such as handling partially observable environments, enhancing robustness, or imposing an inductive bias directly into the policy structure. The conventional method for training stateful policies is Backpropagation Through Time (BPTT), which comes with significant drawbacks, such as slow training due to sequential gradient propagation and the occurrence of vanishing or exploding gradients. The gradient is often truncated to address these issues, resulting in a biased policy update. We present a novel approach for training stateful policies by decomposing the latter into a stochastic internal state kernel and a stateless policy, jointly optimized by following the stateful policy gradient. We introduce different versions of the stateful policy gradient theorem, enabling us to easily instantiate stateful variants of popular reinforcement learning and imitation learning algorithms. Furthermore, we provide a theoretical analysis of our new gradient estimator and compare it with BPTT. We evaluate our approach on complex continuous control tasks, e.g. humanoid locomotion, and demonstrate that our gradient estimator scales effectively with task complexity while offering a faster and simpler alternative to BPTT.

Theory · Learning Theory

Runtian Zhai, Rattana Pukdee, Roger Jin, Nina Balcan, Pradeep K Ravikumar

Unlabeled data is a key component of modern machine learning. In general, the roleof unlabeled data is to impose a form of smoothness, usually from the similarityinformation encoded in a base kernel, such as the ϵ-neighbor kernel or the adjacencymatrix of a graph. This work revisits the classical idea of spectrally transformedkernel regression (STKR), and provides a new class of general and scalable STKRestimators able to leverage unlabeled data. Intuitively, via spectral transformation,STKR exploits the data distribution for which unlabeled data can provide additionalinformation. First, we show that STKR is a principled and general approach,by characterizing a universal type of “target smoothness”, and proving that anysufficiently smooth function can be learned by STKR. Second, we provide scalableSTKR implementations for the inductive setting and a general transformationfunction, while prior work is mostly limited to the transductive setting. Third, wederive statistical guarantees for two scenarios: STKR with a known polynomialtransformation, and STKR with kernel PCA when the transformation is unknown.Overall, we believe that this work helps deepen our understanding of how to workwith unlabeled data, and its generality makes it easier to inspire new methods.

Reinforcement Learning

Christian Gumbsch, Noor Sajid, Georg Martius, Martin V. Butz

Hierarchical world models can significantly improve model-based reinforcement learning (MBRL) and planning by enabling reasoning across multiple time scales. Nonetheless, the majority of state-of-the-art MBRL methods employ flat, non-hierarchical models. We propose Temporal Hierarchies from Invariant Context Kernels (THICK), an algorithm that learns a world model hierarchy via discrete latent dynamics. The lower level of THICK updates parts of its latent state sparsely in time, forming invariant contexts. The higher level exclusively predicts situations involving context changes. Our experiments demonstrate that THICK learns categorical, interpretable, temporal abstractions on the high level, while maintaining precise low-level predictions. Furthermore, we show that the emergent hierarchical predictive model seamlessly enhances the abilities of MBRL or planning methods. We believe that THICK contributes to the further development of hierarchical agents capable of more sophisticated planning and reasoning abilities.

Deep Learning · Theory

Yuto Nishimura, Taiji Suzuki

Recent deep learning applications, exemplified by text-to-image tasks, often involve high-dimensional inputs and outputs. While several studies have investigated the function estimation capabilities of deep learning, research on dilated convolutional neural networks (CNNs) has mainly focused on cases where input dimensions are infinite but output dimensions are one-dimensional, similar to many other studies. However, many practical deep learning tasks involve high-dimensional (or even infinite dimensional) inputs and outputs.In this paper, we investigate the optimality of dilated CNNs for estimating a map between infinite-dimensional input and output spaces by analyzing their approximation and estimation abilities. For that purpose, we first show that approximation and estimation errors depend only on the smoothness and decay rate with respect to the infinity norm of the output, and their estimation accuracy actually achieve the {\it minimax optimal} rate of convergence.Second, we demonstrate that the dilated CNNs outperform {\it any} linear estimators including kernel ridge regression and $k$-NN estimators in a minimax error sense, highlighting the usefulness of feature learning realized by deep neural networks.Our theoretical analysis particularly explains the success of deep learning in recent high-dimensional input-output tasks.