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Deep Learning · Representation Learning

Ameya Prabhu, Shiven Sinha, Ponnurangam Kumaraguru, Philip Torr, Ozan Sener, Puneet Dokania

Continual learning has primarily focused on the issue of catastrophic forgetting and the associated stability-plasticity tradeoffs. However, little attention has been paid to the efficacy of continually learned representations, as representations are learned alongside classifiers throughout the learning process. Our primary contribution is empirically demonstrating that existing online continually trained deep networks produce inferior representations compared to a simple pre-defined random transforms. Our approach embeds raw pixels using a fixed random transform, approximating an RBF-Kernel initialized before any data is seen. We then train a simple linear classifier on top without storing any exemplars, processing one sample at a time in an online continual learning setting. This method, called RanDumb, significantly outperforms state-of-the-art continually learned representations across all standard online continual learning benchmarks. Our study reveals the significant limitations of representation learning, particularly in low-exemplar and online continual learning scenarios. Extending our investigation to popular exemplar-free scenarios with pretrained models, we find that training only a linear classifier on top of pretrained representations surpasses most continual fine-tuning and prompt-tuning strategies. Overall, our investigation challenges the prevailing assumptions about effective representation learning in the online continual learning.

Applications

Wenbing Li, Hang Zhou, Junqing Yu, Zikai Song, Wei Yang

The essence of multi-modal fusion lies in exploiting the complementary information inherent in diverse modalities.However, most prevalent fusion methods rely on traditional neural architectures and are inadequately equipped to capture the dynamics of interactions across modalities, particularly in presence of complex intra- and inter-modality correlations.Recent advancements in State Space Models (SSMs), notably exemplified by the Mamba model, have emerged as promising contenders. Particularly, its state evolving process implies stronger modality fusion paradigm, making multi-modal fusion on SSMs an appealing direction. However, fusing multiple modalities is challenging for SSMs due to its hardware-aware parallelism designs. To this end, this paper proposes the Coupled SSM model, for coupling state chains of multiple modalities while maintaining independence of intra-modality state processes. Specifically, in our coupled scheme, we devise an inter-modal hidden states transition scheme, in which the current state is dependent on the states of its own chain and that of the neighbouring chains at the previous time-step. To fully comply with the hardware-aware parallelism, we obtain the global convolution kernel by deriving the state equation while introducing the historical state.Extensive experiments on CMU-MOSEI, CH-SIMS, CH-SIMSV2 through multi-domain input verify the effectiveness of our model compared to current state-of-the-art methods, improved F1-Score by 0.4%, 0.9%, and 2.3% on the three datasets respectively, 49% faster inference and 83.7% GPU memory save. The results demonstrate that Coupled Mamba model is capable of enhanced multi-modal fusion.

Miscellaneous Aspects of Machine Learning · Supervised Learning

Jin-Hong Du, Pratik Patil

We study the implicit regularization effects induced by (observation) weighting of pretrained features.For weight and feature matrices of bounded operator norms that are infinitesimally free with respect to (normalized) trace functionals, we derive equivalence paths connecting different weighting matrices and ridge regularization levels.Specifically, we show that ridge estimators trained on weighted features along the same path are asymptotically equivalent when evaluated against test vectors of bounded norms.These paths can be interpreted as matching the effective degrees of freedom of ridge estimators fitted with weighted features.For the special case of subsampling without replacement, our results apply to independently sampled random features and kernel features and confirm recent conjectures (Conjectures 7 and 8) of the authors on the existence of such paths in Patil and Du (2023).We also present an additive risk decomposition for ensembles of weighted estimators and show that the risks are equivalent along the paths when the ensemble size goes to infinity.As a practical consequence of the path equivalences, we develop an efficient cross-validation method for tuning and apply it to subsampled pretrained representations across several models (e.g., ResNet-50) and datasets (e.g., CIFAR-100).

Probabilistic Methods · Monte Carlo and Sampling Methods

Alberto Cabezas, Louis Sharrock, Christopher Nemeth

Continuous normalizing flows (CNFs) learn the probability path between a reference distribution and a target distribution by modeling the vector field generating said path using neural networks. Recently, Lipman et al. (2022) introduced a simple and inexpensive method for training CNFs in generative modeling, termed flow matching (FM). In this paper, we repurpose this method for probabilistic inference by incorporating Markovian sampling methods in evaluating the FM objective, and using the learned CNF to improve Monte Carlo sampling. Specifically, we propose an adaptive Markov chain Monte Carlo (MCMC) algorithm, which combines a local Markov transition kernel with a non-local, flow-informed transition kernel, defined using a CNF. This CNF is adapted on-the-fly using samples from the Markov chain, which are used to specify the probability path for the FM objective. Our method also includes an adaptive tempering mechanism that allows the discovery of multiple modes in the target distribution. Under mild assumptions, we establish convergence of our method to a local optimum of the FM objective. We then benchmark our approach on several synthetic and real-world examples, achieving similar performance to other state-of-the-art methods but often at a significantly lower computational cost.

Chao Wang, Xin HE, Yuwen Wang, Junhui Wang

This paper investigates the impact of alignment between the target function of interest and the kernel matrix on a variety of kernel-based methods based on a general loss belonging to a rich loss function family, which covers many commonly used methods in regression and classification problems. We consider the truncated kernel-based method (TKM) which is estimated within a reduced function space constructed by using the spectral truncation of the kernel matrix and compare its theoretical behavior to that of the standard kernel-based method (KM) under various settings. By using the kernel complexity function that quantifies the complexity of the induced function space, we derive the upper bounds for both TKM and KM, and further reveal their dependencies on the degree of target-kernel alignment. Specifically, for the alignment with polynomial decay, the established results indicate that under the just-aligned and weakly-aligned regimes, TKM and KM share the same learning rate. Yet, under the strongly-aligned regime, KM suffers the saturation effect, while TKM can be continuously improved as the alignment becomes stronger. This further implies that TKM has a strong ability to capture the strong alignment and provide a theoretically guaranteed solution to eliminate the phenomena of saturation effect. The minimax lower bound is also established for the squared loss to confirm the optimality of TKM. Extensive numerical experiments further support our theoretical findings. The Python code for reproducing the numerical experiments is available at https://github.com/wywangen.

Miscellaneous Aspects of Machine Learning · Kernel methods

Alexander Modell

In this paper, we derive *entrywise* error bounds for low-rank approximations of kernel matrices obtained using the truncated eigen-decomposition (or singular value decomposition). While this approximation is well-known to be optimal with respect to the spectral and Frobenius norm error, little is known about the statistical behaviour of individual entries. Our error bounds fill this gap. A key technical innovation is a delocalisation result for the eigenvectors of the kernel matrix corresponding to small eigenvalues, which takes inspiration from the field of Random Matrix Theory. Finally, we validate our theory with an empirical study of a collection of synthetic and real-world datasets.

Generative Models · In Context Learning

Ruifeng Ren, Yong Liu

Pre-trained large language models based on Transformers have demonstrated remarkable in-context learning (ICL) abilities. With just a few demonstration examples, the models can implement new tasks without any parameter updates. However, it is still an open question to understand the mechanism of ICL. In this paper, we attempt to explore the ICL process in Transformers through a lens of representation learning. Initially, leveraging kernel methods, we figure out a dual model for one softmax attention layer. The ICL inference process of the attention layer aligns with the training procedure of its dual model, generating token representation predictions that are equivalent to the dual model's test outputs. We delve into the training process of this dual model from a representation learning standpoint and further derive a generalization error bound related to the quantity of demonstration tokens. Subsequently, we extend our theoretical conclusions to more complicated scenarios, including one Transformer layer and multiple attention layers. Furthermore, drawing inspiration from existing representation learning methods especially contrastive learning, we propose potential modifications for the attention layer. Finally, experiments are designed to support our findings.

Generative Models · New Approaches

Richard Nock, Mathieu Guillame-Bert

We focus on generative AI for a type of data that still represent one of the most prevalent form of data: tabular data. We introduce a new powerful class of forest-based models fit for such tasks and a simple training algorithm with strong convergence guarantees in a boosting model that parallels that of the original weak / strong supervised learning setting. This algorithm can be implemented by a few tweaks to the most popular induction scheme for decision tree induction (*i.e. supervised learning*) with two classes. Experiments on the quality of generated data display substantial improvements compared to the state of the art. The losses our algorithm minimize and the structure of our models make them practical for related tasks that require fast estimation of a density given a generative model and an observation (even partially specified): such tasks include missing data imputation and density estimation. Additional experiments on these tasks reveal that our models can be notably good contenders to diverse state of the art methods, relying on models as diverse as (or mixing elements of) trees, neural nets, kernels or graphical models.

Graph Neural Networks

Aref Einizade, Fragkiskos Malliaros, Jhony H. Giraldo

Processing multidomain data defined on multiple graphs holds significant potential in various practical applications in computer science. However, current methods are mostly limited to discrete graph filtering operations. Tensorial partial differential equations on graphs (TPDEGs) provide a principled framework for modeling structured data across multiple interacting graphs, addressing the limitations of the existing discrete methodologies. In this paper, we introduce Continuous Product Graph Neural Networks (CITRUS) that emerge as a natural solution to the TPDEG. CITRUS leverages the separability of continuous heat kernels from Cartesian graph products to efficiently implement graph spectral decomposition. We conduct thorough theoretical analyses of the stability and over-smoothing properties of CITRUS in response to domain-specific graph perturbations and graph spectra effects on the performance. We evaluate CITRUS on well-known traffic and weather spatiotemporal forecasting datasets, demonstrating superior performance over existing approaches. The implementation codes are available at https://github.com/ArefEinizade2/CITRUS.

Reinforcement Learning · Planning

Dillon Chen, Sylvie Thiebaux

Graph learning is naturally well suited for use in symbolic, object-centric planning due to its ability to exploit relational structures exhibited in planning domains and to take as input planning instances with arbitrary number of objects. Numeric planning is an extension of symbolic planning in which states may now also exhibit numeric variables. In this work, we propose data-efficient and interpretable machine learning models for learning to solve numeric planning tasks. This involves constructing a new graph kernel for graphs with both continuous and categorical attributes, as well as new optimisation methods for learning heuristic functions for numeric planning. Experiments show that our graph kernels are vastly more efficient and generalise better than graph neural networks for numeric planning, and also yield competitive coverage performance over domain-independent numeric planners.

Theory · Reinforcement Learning and Planning

Sattar Vakili, Julia Olkhovskaya

Reinforcement Learning (RL) utilizing kernel ridge regression to predict the expected value function represents a powerful method with great representational capacity. This setting is a highly versatile framework amenable to analytical results. We consider kernel-based function approximation for RL in the infinite horizon average reward setting, also referred to as the undiscounted setting. We propose an *optimistic* algorithm, similar to acquisition function based algorithms in the special case of bandits. We establish novel *no-regret* performance guarantees for our algorithm, under kernel-based modelling assumptions. Additionally, we derive a novel confidence interval for the kernel-based prediction of the expected value function, applicable across various RL problems.

Deep Learning · Algorithms

Gavia Gray, aman tiwari, Shane Bergsma, Joel Hestness

Per-example gradient norms are a vital ingredient for estimating gradient noise scale (GNS) with minimal variance. Observing the tensor contractions required to compute them, we propose a method with minimal FLOPs in 3D or greater tensor regimes by simultaneously computing the norms while computing the parameter gradients. Using this method we are able to observe the GNS of different layers at higher accuracy than previously possible. We find that the total GNS of contemporary transformer models is predicted well by the GNS of only the normalization layers. As a result, focusing only on the normalization layer, we develop a custom kernel to compute the per-example gradient norms while performing the LayerNorm backward pass with zero throughput overhead. Tracking GNS on only those layers, we are able to guide a practical batch size schedule that reduces training time by 18% on a Chinchilla-optimal language model.

Miscellaneous Aspects of Machine Learning · Kernel methods

Anna Korba, Francis Bach, Clémentine CHAZAL

In this paper, we study the statistical and geometrical properties of the Kullback-Leibler divergence with kernel covariance operators (KKL) introduced by [Bach, 2022, Information Theory with Kernel Methods]. Unlike the classical Kullback-Leibler (KL) divergence that involves density ratios, the KKL compares probability distributions through covariance operators (embeddings) in a reproducible kernel Hilbert space (RKHS), and compute the Kullback-Leibler quantum divergence. This novel divergence hence shares parallel but different aspects with both the standard Kullback-Leibler between probability distributions and kernel embeddings metrics such as the maximum mean discrepancy. A limitation faced with the original KKL divergence is its inability to be defined for distributions with disjoint supports. To solve this problem, we propose in this paper a regularised variant that guarantees that divergence is well defined for all distributions. We derive bounds that quantify the deviation of the regularised KKL to the original one, as well as concentration bounds. In addition, we provide a closed-form expression for the regularised KKL, specifically applicable when the distributions consist of finite sets of points, which makes it implementable. Furthermore, we derive a Wasserstein gradient descent scheme of the KKL divergence in the case of discrete distributions, and study empirically its properties to transport a set of points to a target distribution.

Probabilistic Methods · Gaussian Processes

Mohit Yadav, Cameron Musco, Daniel Sheldon

Algorithms that utilize bandit feedback to optimize top-k recommendations are vital for online marketplaces, search engines, and content platforms. However, the combinatorial nature of this problem poses a significant challenge, as the possible number of ordered top-k recommendations from $n$ items grows exponentially with $k$. As a result, previous work often relies on restrictive assumptions about the reward or bandit feedback models, such as assuming that the feedback discloses rewards for each recommended item rather than a single scalar feedback for the entire set of top-k recommendations. We introduce a novel contextual bandit algorithm for top-k recommendations, leveraging a Gaussian process with a Kendall kernel to model the reward function.Our algorithm requires only scalar feedback from the top-k recommendations and does not impose restrictive assumptions on the reward structure. Theoretical analysis confirms that the proposed algorithm achieves sub-linear regret in relation to the number of rounds and arms. Additionally, empirical results using a bandit simulator demonstrate that the proposed algorithm outperforms other baselines across various scenarios.

Miscellaneous Aspects of Machine Learning · Unsupervised and Semi-supervised Learning

Zi-Hao Zhou, Siyuan Fang, Zi-Jing Zhou, Tong Wei, Yuanyu Wan, Min-Ling Zhang

Long-tailed semi-supervised learning poses a significant challenge in training models with limited labeled data exhibiting a long-tailed label distribution. Current state-of-the-art LTSSL approaches heavily rely on high-quality pseudo-labels for large-scale unlabeled data. However, these methods often neglect the impact of representations learned by the neural network and struggle with real-world unlabeled data, which typically follows a different distribution than labeled data. This paper introduces a novel probabilistic framework that unifies various recent proposals in long-tail learning. Our framework derives the class-balanced contrastive loss through Gaussian kernel density estimation. We introduce a continuous contrastive learning method, CCL, extending our framework to unlabeled data using *reliable* and *smoothed* pseudo-labels. By progressively estimating the underlying label distribution and optimizing its alignment with model predictions, we tackle the diverse distribution of unlabeled data in real-world scenarios. Extensive experiments across multiple datasets with varying unlabeled data distributions demonstrate that CCL consistently outperforms prior state-of-the-art methods, achieving over 4% improvement on the ImageNet-127 dataset. The supplementary material includes the source code for reproducibility.

Generative Models · Diffusion models

Siwei Tu, Weidong Yang, Ben Fei

Diffusion models have been widely utilized for image restoration. However, previous blind image restoration methods still need to assume the type of degradation model while leaving the parameters to be optimized, limiting their real-world applications. Therefore, we aim to tame generative diffusion prior for universal blind image restoration dubbed BIR-D, which utilizes an optimizable convolutional kernel to simulate the degradation model and dynamically update the parameters of the kernel in the diffusion steps, enabling it to achieve blind image restoration results even in various complex situations. Besides, based on mathematical reasoning, we have provided an empirical formula for the chosen of adaptive guidance scale, eliminating the need for a grid search for the optimal parameter. Experimentally, Our BIR-D has demonstrated superior practicality and versatility than off-the-shelf unsupervised methods across various tasks both on real-world and synthetic datasets, qualitatively and quantitatively. BIR-D is able to fulfill multi-guidance blind image restoration. Moreover, BIR-D can also restore images that undergo multiple and complicated degradations, demonstrating the practical applications. The code is available at https://github.com/Tusiwei/BIR-D.

Optimization · Convex

Youyuan Long, Tolga Ok, Pedro Zattoni Scroccaro, Peyman Mohajerin Esfahani

Inverse Optimization (IO) is a framework for learning the unknown objective function of an expert decision-maker from a past dataset.In this paper, we extend the hypothesis class of IO objective functions to a reproducing kernel Hilbert space (RKHS), thereby enhancing feature representation to an infinite-dimensional space.We demonstrate that a variant of the representer theorem holds for a specific training loss, allowing the reformulation of the problem as a finite-dimensional convex optimization program.To address scalability issues commonly associated with kernel methods, we propose the Sequential Selection Optimization (SSO) algorithm to efficiently train the proposed Kernel Inverse Optimization (KIO) model.Finally, we validate the generalization capabilities of the proposed KIO model and the effectiveness of the SSO algorithm through learning-from-demonstration tasks on the MuJoCo benchmark.

Privacy

Bonwoo Lee, Jeongyoun Ahn, Cheolwoo Park

As the volume of data invested in statistical learning increases and concerns regarding privacy grow, the privacy leakage issue has drawn significant attention. Differential privacy has emerged as a widely accepted concept capable of mitigating privacy concerns, and numerous differentially private (DP) versions of machine learning algorithms have been developed. However, existing works on DP kernel learning algorithms have exhibited practical limitations, including scalability, restricted choice of kernels, or dependence on test data availability. We propose DP scalable kernel empirical risk minimization (ERM) algorithms and a DP kernel mean embedding (KME) release algorithm suitable for general kernels. Our approaches address the shortcomings of previous algorithms by employing Nyström methods, classical techniques in non-private scalable kernel learning. These methods provide data-dependent low-rank approximations of the kernel matrix for general kernels in a DP manner. We present excess empirical risk bounds and computational complexities for the scalable kernel DP ERM, KME algorithms, contrasting them with established methodologies. Furthermore, we develop a private data-generating algorithm capable of learning diverse kernel models. We conduct experiments to demonstrate the performance of our algorithms, comparing them with existing methods to highlight their superiority.

Miscellaneous Aspects of Machine Learning · Kernel methods

Sejun Park, Kihun Hong, Ganguk Hwang

Over the past decade, there is a growing interest in collaborative learning that can enhance AI models of multiple parties.However, it is still challenging to enhance performance them without sharing private data and models from individual parties.One recent promising approach is to develop distillation-based algorithms that exploit unlabeled public data but the results are still unsatisfactory in both theory and practice.To tackle this problem, we rigorously analyze a representative distillation-based algorithm in the view of kernel regression.This work provides the first theoretical results to prove the (nearly) minimax optimality of the nonparametric collaborative learning algorithm that does not directly share local data or models in massively distributed statistically heterogeneous environments.Inspired by our theoretical results, we also propose a practical distillation-based collaborative learning algorithm based on neural network architecture.Our algorithm successfully bridges the gap between our theoretical assumptions and practical settings with neural networks through feature kernel matching.We simulate various regression tasks to verify our theory and demonstrate the practical feasibility of our proposed algorithm.

Miscellaneous Aspects of Machine Learning · Kernel methods

Florian Kalinke, Zoltan Szabo

Kernel techniques are among the most influential approaches in data science and statistics. Under mild conditions, the reproducing kernel Hilbert space associated to a kernel is capable of encoding the independence of $M\ge2$ random variables. Probably the most widespread independence measure relying on kernels is the so-called Hilbert-Schmidt independence criterion (HSIC; also referred to as distance covariance in the statistics literature). Despite various existing HSIC estimators designed since its introduction close to two decades ago, the fundamental question of the rate at which HSIC can be estimated is still open. In this work, we prove that the minimax optimal rate of HSIC estimation on $\mathbb{R}^d$ for Borel measures containing the Gaussians with continuous bounded translation-invariant characteristic kernels is $\mathcal{O}\left(n^{-1/2}\right)$. Specifically, our result implies the optimality in the minimax sense of many of the most-frequently used estimators (including the U-statistic, the V-statistic, and the Nyström-based one) on $\mathbb{R}^d$.