One crucial factor behind the success of deep learning lies in the implicit bias induced by noise inherent in gradient-based training algorithms. Motivated by empirical observations that training with noisy labels improves model generalization, we delve into the underlying mechanisms behind stochastic gradient descent (SGD) with label noise. Focusing on a two-layer over-parameterized linear network, we analyze the learning dynamics of label noise SGD, unveiling a two-phase learning behavior. In Phase I, the magnitudes of model weights progressively diminish, and the model escapes the lazy regime; enters the rich regime. In Phase II, the alignment between model weights and the ground-truth interpolator increases, and the model eventually converges. Our analysis highlights the critical role of label noise in driving the transition from the lazy to the rich regime and minimally explains its empirical success. Furthermore, we extend these insights to Sharpness-Aware Minimization (SAM), showing that the principles governing label noise SGD also apply to broader optimization algorithms. Extensive experiments, conducted under both synthetic and real-world setups, strongly support our theory.

Graph neural networks (GNNs) have demonstrated remarkable capabilities in learning from graph-structured data, often outperforming traditional Multilayer Perceptrons (MLPs) in numerous graph-based tasks. Although existing works have demonstrated the benefits of graph convolution through Laplacian smoothing, expressivity or separability, there remains a lack of quantitative analysis comparing GNNs and MLPs from an optimization and generalization perspective. This study aims to address this gap by examining the role of graph convolution through feature learning theory. Using a signal-noise data model, we conduct a comparative analysis of the optimization and generalization between two-layer graph convolutional networks (GCNs) and their MLP counterparts. Our approach tracks the trajectory of signal learning and noise memorization in GNNs, characterizing their post-training generalization. We reveal that GNNs significantly prioritize signal learning, thus enhancing the regime of {low test error} over MLPs by $D^{q-2}$ times, where $D$ denotes a node’s expected degree and $q$ is the power of ReLU activation function with $q>2$. This finding highlights a substantial and quantitative discrepancy between GNNs and MLPs in terms of optimization and generalization, a conclusion further supported by our empirical simulations on both synthetic and real-world datasets.

This study extends the problem settings of Invariant Risk Minimization(IRM) for Out-of-Distribution generalization problems to unknown clustered environments settings. In this scenario, where a given set of environments exhibits an unknown clustered structure, our objective is to identify a single invariant feature extractor and per-cluster regressors (or classifiers) built on top of the feature extractor. To achieve this, we propose a new framework called Clustered IRM for simultaneously identifying the cluster structure and the invariant features. Our theoretical analysis demonstrates that the required number of training environments for such identification is only $O(d_\mathrm{sp} + K^2)$, where $d_\mathrm{sp}$ represents the dimensionality of the spurious features, and $K$ is the number of clusters. Numerical experiments validate the effectiveness of our proposed framework.

Flow matching (FM) has gained significant attention as a simulation-free generative model. Unlike diffusion models, which are based on stochastic differential equations, FM employs a simpler approach by solving an ordinary differential equation with an initial condition from a normal distribution, thus streamlining the sample generation process. This paper discusses the convergence properties of FM in terms of the $p$-Wasserstein distance, a measure of distributional discrepancy. We establish that FM can achieve an almost minimax optimal convergence rate for $1 \leq p \leq 2$, presenting the first theoretical evidence that FM can reach convergence rates comparable to those of diffusion models. Our analysis extends existing frameworks by examining a broader class of mean and variance functions for the vector fields and identifies specific conditions necessary to attain these optimal rates.

We introduce a novel alignment method for diffusion models from distribution optimization perspectives while providing rigorous convergence guarantees. We first formulate the problem as a generic regularized loss minimization over probability distributions and directly optimize the distribution using the Dual Averaging method. Next, we enable sampling from the learned distribution by approximating its score function via Doob's $h$-transform technique. The proposed framework is supported by rigorous convergence guarantees and an end-to-end bound on the sampling error, which imply that when the original distribution's score is known accurately, the complexity of sampling from shifted distributions is independent of isoperimetric conditions. This framework is broadly applicable to general distribution optimization problems, including alignment tasks in Reinforcement Learning with Human Feedback (RLHF), Direct Preference Optimization (DPO), and Kahneman-Tversky Optimization (KTO). We empirically validate its performance on synthetic and image datasets using the DPO objective.

We provide a convergence analysis of \emph{deep feature instrumental variable} (DFIV) regression (Xu et al., 2021), a nonparametric approach to IV regression using data-adaptive features learned by deep neural networks in two stages. We prove that the DFIV algorithm achieves the minimax optimal learning rate when the target structural function lies in a Besov space. This is shown under standard nonparametric IV assumptions, and an additional smoothness assumption on the regularity of the conditional distribution of the covariate given the instrument, which controls the difficulty of Stage 1. We further demonstrate that DFIV, as a data-adaptive algorithm, is superior to fixed-feature (kernel or sieve) IV methods in two ways. First, when the target function possesses low spatial homogeneity (i.e., it has both smooth and spiky/discontinuous regions), DFIV still achieves the optimal rate, while fixed-feature methods are shown to be strictly suboptimal. Second, comparing with kernel-based two-stage regression estimators, DFIV is provably more data efficient in the Stage 1 samples.

This work provides the first theoretical analysis of training transformers to solve complex problems by recursively generating intermediate states, analogous to fine-tuning for chain-of-thought (CoT) reasoning. We consider training a one-layer transformer to solve the fundamental $k$-parity problem, extending the work on RNNs by \citet{Wies23}. We establish three key results: (1) any finite-precision gradient-based algorithm, without intermediate supervision, requires substantial iterations to solve parity with finite samples. (2) In contrast, when intermediate parities are incorporated into the loss function, our model can learn parity in one gradient update when aided by \emph{teacher forcing}, where ground-truth labels of the reasoning chain are provided at each generation step. (3) Even without teacher forcing, where the model must generate CoT chains end-to-end, parity can be learned efficiently if augmented data is employed to internally verify the soundness of intermediate steps. Our findings, supported by numerical experiments, show that task decomposition and stepwise reasoning naturally arise from optimizing transformers with CoT; moreover, self-consistency checking can improve multi-step reasoning ability, aligning with empirical studies of CoT.

The Adam optimizer is widely used for transformer optimization in practice, which makes understanding the underlying optimization mechanisms an important problem. However, due to the Adam's complexity, theoretical analysis of how it optimizes transformers remains a challenging task. Fortunately, Sign Gradient Descent (SignGD) serves as an effective surrogate for Adam. Despite its simplicity, theoretical understanding of how SignGD optimizes transformers still lags behind. In this work, we study how SignGD optimizes a two-layer transformer -- consisting of a softmax attention layer with trainable query-key parameterization followed by a linear layer -- on a linearly separable noisy dataset. We identify four stages in the training dynamics, each exhibiting intriguing behaviors. Based on the training dynamics, we prove the fast convergence but poor generalization of the learned transformer on the noisy dataset. We also show that Adam behaves similarly to SignGD in terms of both optimization and generalization in this setting. Additionally, we find that the poor generalization of SignGD is not solely due to data noise, suggesting that both SignGD and Adam requires high-quality data for real-world tasks. Finally, experiments on synthetic and real-world datasets empirically support our theoretical results.

Deep neural networks based on state space models (SSMs) are attracting significant attention in sequence modeling since their computational cost is much smaller than that of Transformers. While the capabilities of SSMs have been demonstrated through experiments in various tasks, theoretical understanding of SSMs is still limited. In particular, most theoretical studies discuss the capabilities of SSM layers without nonlinear layers, and there is a lack of discussion on their combination with nonlinear layers. In this paper, we explore the capabilities of SSMs combined with fully connected neural networks, and show that they are comparable to Transformers in extracting the essential tokens depending on the input. As concrete examples, we consider two synthetic tasks, which are challenging for a single SSM layer, and demonstrate that SSMs combined with nonlinear layers can efficiently solve these tasks. Furthermore, we study the nonparametric regression task, and prove that the ability of SSMs is equivalent to that of Transformers in estimating functions belonging to a certain class.

In typical multimodal contrastive learning, such as CLIP, encoders produce one point in the latent representation space for each input. However, one-point representation has difficulty in capturing the relationship and the similarity structure of a huge amount of instances in the real world. For richer classes of the similarity, we propose the use of weighted point sets, namely, sets of pairs of weight and vector, as representations of instances. In this work, we theoretically show the benefit of our proposed method through a new understanding of the contrastive loss of CLIP, which we call symmetric InfoNCE. We clarify that the optimal similarity that minimizes symmetric InfoNCE is the pointwise mutual information, and show an upper bound of excess risk on downstream classification tasks of representations that achieve the optimal similarity. In addition, we show that our proposed similarity based on weighted point sets consistently achieves the optimal similarity. To verify the effectiveness of our proposed method, we demonstrate pretraining of text-image representation models and classification tasks on common benchmarks.

In-context learning (ICL) has garnered significant attention for its ability to grasp functions/tasks from demonstrations. Recent studies suggest the presence of a latent task/function vector in LLMs during ICL. Merullo et al. (2024) showed that LLMs leverage this vector alongside the residual stream for Word2Vec-like vector arithmetic, solving factual-recall ICL tasks. Additionally, recent work empirically highlighted the key role of Question-Answer data in enhancing factual-recall capabilities. Despite these insights, a theoretical explanation remains elusive. To move one step forward, we propose a theoretical framework building on empirically grounded hierarchical concept modeling. We develop an optimization theory, showing how nonlinear residual transformers trained via gradient descent on cross-entropy loss perform factual-recall ICL tasks via vector arithmetic. We prove 0-1 loss convergence and show the strong generalization, including robustness to concept recombination and distribution shifts. These results elucidate the advantages of transformers over static embedding predecessors. Empirical simulations corroborate our theoretical insights.

Deep learning with noisy labels presents significant challenges. In this work, we theoretically characterize the role of label noise from a feature learning perspective. Specifically, we consider a signal-noise data distribution, where each sample comprises a label-dependent signal and label-independent noise, and rigorously analyze the training dynamics of a two-layer convolutional neural network under this data setup, along with the presence of label noise. Our analysis identifies two key stages. In Stage I, the model perfectly fits all the clean samples (i.e., samples without label noise) while ignoring the noisy ones (i.e., samples with noisy labels). During this stage, the model learns the signal from the clean samples, which generalizes well on unseen data. In Stage II, as the training loss converges, the gradient in the direction of noise surpasses that of the signal, leading to overfitting on noisy samples. Eventually, the model memorizes the noise present in the noisy samples and degrades its generalization ability. Furthermore, our analysis provides a theoretical basis for two widely used techniques for tackling label noise: early stopping and sample selection. Experiments on both synthetic and real-world setups validate our theory.

Aligning large language models (LLMs) with human preferences is crucial for safe deployment, yet existing methods assume specific preference models like Bradley-Terry model. This assumption leads to statistical inconsistency, where more data doesn’t guarantee convergence to true human preferences. To address this critical gap, we introduce a novel alignment method Direct Density Ratio Optimization (DDRO). DDRO directly estimates the density ratio between preferred and unpreferred output distributions, circumventing the need for explicit human preference modeling. We theoretically prove that DDRO is statistically consistent, ensuring convergence to the true preferred distribution as the data size grows, regardless of the underlying preference structure. Experiments demonstrate that DDRO achieves superior performance compared to existing methods, showcasing its effectiveness and potential for significant improvement. DDRO unlocks the potential for truly data-driven alignment, paving the way for more reliable and human-aligned LLMs.

Mixture of Experts (MoE), an ensemble of specialized models equipped with a router that dynamically distributes each input to appropriate experts, has achieved successful results in the field of machine learning. However, theoretical understanding of this architecture is falling behind due to its inherent complexity. In this paper, we theoretically study the sample and runtime complexity of MoE following the stochastic gradient descent when learning a regression task with an underlying cluster structure of single index models. On the one hand, we show that a vanilla neural network fails in detecting such a latent organization as it can only process the problem as a whole. This is intrinsically related to the concept of *information exponent* which is low for each cluster, but increases when we consider the entire task. On the other hand, with a MoE, we show that it succeeds in dividing the problem into easier subproblems by leveraging the ability of each expert to weakly recover the simpler function corresponding to an individual cluster. To the best of our knowledge, this work is among the first to explore the benefits of the MoE framework by examining its SGD dynamics in the context of nonlinear regression.

A key paradigm to improve the reasoning capabilities of large language models (LLMs) is to allocate more inference-time compute to search against a verifier or reward model. This process can then be utilized to refine the pretrained model or distill its reasoning patterns into more efficient models. In this paper, we study inference-time computation by viewing chain-of-thought (CoT) generation as a metastable Markov process: easy reasoning steps (e.g., algebraic manipulations) form densely connected clusters, while hard reasoning steps (e.g., applying a relevant theorem) create sparse, low-probability edges between clusters, leading to phase transitions at longer timescales. Under this framework, we prove that implementing a search protocol that rewards sparse edges improves CoT by decreasing the expected number of steps to reach different clusters. In contrast, we establish a limit on reasoning capability when the model is restricted to local information of the pretrained graph. We also show that the information gained by search can be utilized to obtain a better reasoning model: (1) the pretrained model can be directly finetuned to favor sparse edges via policy gradient methods, and moreover (2) a compressed metastable representation of the reasoning dynamics can be distilled into a smaller, more efficient model.

Pretrained transformers have demonstrated the ability to implement various algorithms at inference time without parameter updates. While theoretical works have established this capability through constructions and approximation guarantees, the optimization and statistical efficiency aspects remain understudied. In this work, we investigate how transformers learn features in-context – a key mechanism underlying their inference-time adaptivity. We focus on the in-context learning of single-index models $y=\sigma_*(⟨\\boldsymbol{x},\\boldsymbol{\beta}⟩)$, which are low-dimensional nonlinear functions parameterized by feature vector $\\boldsymbol\beta$. We prove that transformers pretrained by gradient-based optimization can perform inference-time feature learning, i.e., extract information of the target features $\\boldsymbol{\beta}$ solely from test prompts (despite $\\boldsymbol {\beta}$ varying across different prompts), hence achieving an in-context statistical efficiency that surpasses any non-adaptive (fixed-basis) algorithms such as kernel methods. Moreover, we show that the inference-time sample complexity surpasses the Correlational Statistical Query (CSQ) lower bound, owing to nonlinear label transformations naturally induced by the Softmax self-attention mechanism.

Mean-field Langevin dynamics (MFLD) is an optimization method derived by taking the mean-field limit of noisy gradient descent for two-layer neural networks in the mean-field regime. Recently, the propagation of chaos (PoC) for MFLD has gained attention as it provides a quantitative characterization of the optimization complexity in terms of the number of particles and iterations. A remarkable progress by Chen et al. (2022) showed that the approximation error due to finite particles remains uniform in time and diminishes as the number of particles increases. In this paper, by refining the defective log-Sobolev inequality---a key result from that earlier work---under the neural network training setting, we establish an improved PoC result for MFLD, which removes the exponential dependence on the regularization coefficient from the particle approximation term of the optimization complexity. As an application, we propose a PoC-based model ensemble strategy with theoretical guarantees.

As the space of causal sequence modeling architectures continues to grow, the need to develop a general framework for their analysis becomes increasingly important. With this aim, we draw insights from classical signal processing and control theory, to develop a quantitative measure of memory utilization: the internal mechanisms through which a model stores past information to produce future outputs. This metric, which we call effective state-size (ESS), is tailored to the fundamental class of systems with input-invariant and input-varying linear operators, encompassing a variety of computational units such as variants of attention, convolutions, and recurrences. Unlike prior work on memory utilization, which either relies on raw operator visualizations (e.g. attention maps), or simply the total memory capacity (i.e. cache size) of a model, our metrics provide highly interpretable and actionable measurements. In particular, we show how ESS can be leveraged to improve initialization strategies, inform novel regularizers and advance the performance-efficiency frontier through model distillation. Furthermore, we demonstrate that the effect of context delimiters (such as end-of-speech tokens) on ESS highlights cross-architectural differences in how large language models utilize their available memory to recall information. Overall, we find that ESS provides valuable insights into the dynamics that dictate memory utilization, enabling the design of more efficient and effective sequence models.

Gradient-based optimization methods have shown remarkable empirical success, yet their theoretical generalization properties remain only partially understood. In this paper, we establish a generalization bound for gradient flow that aligns with the classical Rademacher complexity bounds for kernel methods—specifically those based on the RKHS norm and kernel trace—through a data-dependent kernel called the loss path kernel (LPK). Unlike static kernels such as NTK, the LPK captures the entire training trajectory, adapting to both data and optimization dynamics, leading to tighter and more informative generalization guarantees. Moreover, the bound highlights how the norm of the training loss gradients along the optimization trajectory influences the final generalization performance. The key technical ingredients in our proof combine stability analysis of gradient flow with uniform convergence via Rademacher complexity. Our bound recovers existing kernel regression bounds for overparameterized neural networks and shows the feature learning capability of neural networks compared to kernel methods. Numerical experiments on real-world datasets validate that our bounds correlate well with the true generalization gap.

The capacity of deep learning models is often large enough to both learn the underlying statistical signal and overfit to noise in the training set. This noise memorization can be harmful especially for data with a low signal-to-noise ratio (SNR), leading to poor generalization. Inspired by prior observations that label noise provides implicit regularization that improves generalization, in this work, we investigate whether introducing label noise to the gradient updates can enhance the test performance of neural network (NN) in the low SNR regime. Specifically, we consider training a two-layer NN with a simple label noise gradient descent (GD) algorithm, in an idealized signal-noise data setting. We prove that adding label noise during training suppresses noise memorization, preventing it from dominating the learning process; consequently, label noise GD enjoys rapid signal growth while the overfitting remains controlled, thereby achieving good generalization despite the low SNR. In contrast, we also show that NN trained with standard GD tends to overfit to noise in the same low SNR setting and establish a non-vanishing lower bound on its test error, thus demonstrating the benefit of introducing label noise in gradient-based training.

State-space models (SSMs), particularly Mamba, emerge as an efficient Transformer alternative with linear complexity for long-sequence modeling. Recent empirical works demonstrate Mamba's in-context learning (ICL) capabilities competitive with Transformers, a critical capacity for large foundation models. However, theoretical understanding of Mamba’s ICL remains limited, restricting deeper insights into its underlying mechanisms. Even fundamental tasks such as linear regression ICL, widely studied as a standard theoretical benchmark for Transformers, have not been thoroughly analyzed in the context of Mamba. To address this gap, we study the training dynamics of Mamba on the linear regression ICL task. By developing novel techniques tackling non-convex optimization with gradient descent related to Mamba's structure, we establish an exponential convergence rate to ICL solution, and derive a loss bound that is comparable to Transformer's. Importantly, our results reveal that Mamba can perform a variant of \textit{online gradient descent} to learn the latent function in context. This mechanism is different from that of Transformer, which is typically understood to achieve ICL through gradient descent emulation. The theoretical results are verified by experimental simulation.

Transformers can implement both generalizable algorithms (e.g., induction heads) and simple positional shortcuts (e.g., memorizing fixed output positions). In this work, we study how the choice of pretraining data distribution steers a shallow transformer toward one behavior or the other. Focusing on a minimal trigger-output prediction task -- copying the token immediately following a special trigger upon its second occurrence -- we present a rigorous analysis of gradient-based training of a single-layer transformer. In both the infinite and finite sample regimes, we prove a transition in the learned mechanism: if input sequences exhibit sufficient diversity, measured by a low “max-sum” ratio of trigger-to-trigger distances, the trained model implements an induction head and generalizes to unseen contexts; by contrast, when this ratio is large, the model resorts to a positional shortcut and fails to generalize out-of-distribution (OOD). We also reveal a trade-off between the pretraining context length and OOD generalization, and derive the optimal pretraining distribution that minimizes computational cost per sample. Finally, we validate our theoretical predictions with controlled synthetic experiments, demonstrating that broadening context distributions robustly induces induction heads and enables OOD generalization. Our results shed light on the algorithmic biases of pretrained transformers and offer conceptual guidelines for data-driven control of their learned behaviors.

Linear attention has attracted interest as a computationally efficient approximation to softmax attention, especially for long sequences. Recent studies has explored distilling softmax attention in pre-trained Transformers into linear attention. However, a critical challenge remains: *how to choose the feature dimension that governs the approximation quality*. Existing methods fix this dimension uniformly across all attention layers, overlooking the diverse roles and complexities of them. In this paper, we propose a principled method to automatically determine the feature dimension in linear attention using the concept of statistical *degrees of freedom*, which represent the effective dimensionality of the inputs. We provide a theoretical bound on the approximation error and show that the dimension chosen by our method achieves smaller errors under a fixed computational budget. Furthermore, we introduce an efficient layerwise training strategy to learn nonlinear features tailored to each layer. Experiments on multiple pre-trained transformers demonstrate that our method improves the performance of distilled models compared to baselines without increasing the inference cost. Our findings also provide insight into how the complexity of the attention mechanism evolves across layers.

Diffusion models operating in discrete state spaces have emerged as powerful approaches, demonstrating remarkable efficacy across diverse domains, including reasoning tasks and molecular design. Despite their promising applications, the theoretical foundations of these models remain substantially underdeveloped, with the existing literature predominantly focusing on continuous-state diffusion models. A critical gap persists in the theoretical understanding of discrete diffusion modeling: the absence of a rigorous framework for quantifying estimation error with finite data. Consequently, the fundamental question of how precisely one can reconstruct the true underlying distribution from a limited training set remains unresolved. In this work, we analyze the estimation error induced by a score estimation of the discrete diffusion models. One of the main difficulties in the analysis stems from the fact that the cardinality of the state space can be exponentially large with respect to its dimension, which results in an intractable error bound by a naive approach. To overcome this difficulty, we make use of a property that the state space can be smoothly embedded in a continuous Euclidean space that enables us to derive a cardinality independent bound, which is more practical in real applications. In particular, we consider a setting where the state space is structured as a hypercube graph, and another where the induced graph Laplacian can be asymptotically well approximated by the ordinary Laplacian defined on the continuous space, and then derive state space size independent bounds.

Wasserstein gradient flow (WGF) is a common method to perform optimization over the space of probability measures. While WGF is guaranteed to converge to a first-order stationary point, for nonconvex functionals the converged solution does not necessarily satisfy the second-order optimality condition; i.e., it could converge to a saddle point. In this work, we propose a new algorithm for probability measure optimization, \emph{perturbed Wasserstein gradient flow} (PWGF), that achieves second-order optimality for general nonconvex objectives. PWGF enhances WGF by injecting noisy perturbations near saddle points via a Gaussian process-based scheme. By pushing the measure forward along a random vector field generated from a Gaussian process, PWGF helps the solution escape saddle points efficiently by perturbing the solution towards the smallest eigenvalue direction of the Wasserstein Hessian. We theoretically derive the computational complexity for PWGF to achieve a second-order stationary point. Furthermore, we prove that PWGF converges to a global optimum in polynomial time for strictly benign objectives.

We study the statistical and computational complexity of learning a target function $f_*:\R^d\to\R$ with \textit{additive structure}, that is, $f_*(x) = \frac{1}{\sqrt{M}}\sum_{m=1}^M f_m(⟨x, v_m⟩)$, where $f_1,f_2,...,f_M:\R\to\R$ are nonlinear link functions of single-index models (ridge functions) with diverse and near-orthogonal index features $\{v_m\}_{m=1}^M$, and the number of additive tasks $M$ grows with the dimensionality $M\asymp d^\gamma$ for $\gamma\ge 0$. This problem setting is motivated by the classical additive model literature, the recent representation learning theory of two-layer neural network, and large-scale pretraining where the model simultaneously acquires a large number of “skills” that are often \textit{localized} in distinct parts of the trained network. We prove that a large subset of polynomial $f_*$ can be efficiently learned by gradient descent training of a two-layer neural network, with a polynomial statistical and computational complexity that depends on the number of tasks $M$ and the \textit{information exponent} of $f_m$, despite the unknown link function and $M$ growing with the dimensionality. We complement this learnability guarantee with computational hardness result by establishing statistical query (SQ) lower bounds for both the correlational SQ and full SQ algorithms.

We propose a new bound for generalization of neural networks using Koopman operators. Whereas most of existing works focus on low-rank weight matrices, we focus on full-rank weight matrices. Our bound is tighter than existing norm-based bounds when the condition numbers of weight matrices are small. Especially, it is completely independent of the width of the network if the weight matrices are orthogonal. Our bound does not contradict to the existing bounds but is a complement to the existing bounds. As supported by several existing empirical results, low-rankness is not the only reason for generalization. Furthermore, our bound can be combined with the existing bounds to obtain a tighter bound. Our result sheds new light on understanding generalization of neural networks with full-rank weight matrices, and it provides a connection between operator-theoretic analysis and generalization of neural networks.

Federated Learning (FL) has attracted significant attention as an efficient privacy-preserving approach to distributed learning across multiple clients. Despite extensive empirical research and practical applications, a systematic way to theoretically understand the convergence and generalization properties in FL remains limited. This work aims to establish a unified theoretical foundation for understanding FL through feature learning theory. We focus on a scenario where each client employs a two-layer convolutional neural network (CNN) for local training on their own data. Many existing works analyze the convergence of Federated Averaging (FedAvg) under lazy training with linearizing assumptions in weight space. In contrast, our approach tracks the trajectory of signal learning and noise memorization in FL, eliminating the need for these assumptions. We further show that FedAvg can achieve near-zero test error by effectively increasing signal-to-noise ratio (SNR) in feature learning, while local training without communication achieves a large constant test error. This finding highlights the benefits of communication for generalization in FL. Moreover, our theoretical results suggest that a weighted FedAvg method, based on the similarity of input features across clients, can effectively tackle data heterogeneity issues in FL. Experimental results on both synthetic and real-world datasets verify our theoretical conclusions and emphasize the effectiveness of the weighted FedAvg approach.

In this paper, we extend mean-field Langevin dynamics to minimax optimization over probability distributions for the first time with symmetric and provably convergent updates. We propose \emph{mean-field Langevin averaged gradient} (MFL-AG), a single-loop algorithm that implements gradient descent ascent in the distribution spaces with a novel weighted averaging, and establish average-iterate convergence to the mixed Nash equilibrium. We also study both time and particle discretization regimes and prove a new uniform-in-time propagation of chaos result which accounts for the dependency of the particle interactions on all previous distributions. Furthermore, we propose \emph{mean-field Langevin anchored best response} (MFL-ABR), a symmetric double-loop algorithm based on best response dynamics with linear last-iterate convergence. Finally, we study applications to zero-sum Markov games and conduct simulations demonstrating long-term optimality.

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.

Recent works have shown that neural networks optimized by gradient-based methods can adapt to sparse or low-dimensional target functions through feature learning; an often studied target is the sparse parity function on the unit hypercube. However, such isotropic data setting does not capture the anisotropy and low intrinsic dimensionality exhibited in realistic datasets. In this work, we address this shortcoming by studying how gradient-based feature learning interacts with structured (anisotropic) input data: we consider the classification of $k$-sparse parity on high-dimensional orthotope where the feature coordinates have varying magnitudes, and analyze the learning complexity of the mean-field Langevin dynamics (MFLD), which describes the noisy gradient descent update on two-layer neural network. We show that the statistical complexity (i.e. sample size) and computational complexity (i.e. network width) of MFLD can both be improved when prominent directions of the anisotropic input data align with the support of the target function. Moreover, by employing a coordinate transform determined by the gradient covariance, the width can be made independent of the target degree $k$. Lastly, we demonstrate the benefit of feature learning by establishing a kernel lower bound on the classification error, which applies to neural networks in the lazy regime.

Deep neural networks with feature learning have shown surprising generalization performance in high dimensional settings, but it has not been fully understood how and when they enjoy the benefit of feature learning. In this paper, we theoretically analyze the statistical properties of the benefits from feature learning in a two-layer linear neural network with multiple outputs in a high-dimensional setting. For that purpose, we propose a new criterion that allows feature learning of a two-layer linear neural network in a high-dimensional setting. Interestingly, we can show that models with smaller values of the criterion generalize even in situations where normal ridge regression fails to generalize. This is because the proposed criterion contains a proper regularization for the feature mapping and acts as an upper bound on the predictive risk. As an important characterization of the criterion, the two-layer linear neural network that minimizes this criterion can achieve the optimal Bayes risk that is determined by the distribution of the true signals across the multiple outputs. To the best of our knowledge, this is the first study to specifically identify the conditions under which a model obtained by proper feature learning can outperform normal ridge regression in a high-dimensional multiple-output linear regression problem.

Neural Network-based active learning (NAL) is a cost-effective data selection technique that utilizes neural networks to select and train on a small subset of samples. While existing work successfully develops various effective or theory-justified NAL algorithms, the understanding of the two commonly used query criteria of NAL: uncertainty-based and diversity-based, remains in its infancy. In this work, we try to move one step forward by offering a unified explanation for the success of both query criteria-based NAL from a feature learning view. Specifically, we consider a feature-noise data model comprising easy-to-learn or hard-to-learn features disrupted by noise, and conduct analysis over 2-layer NN-based NALs in the pool-based scenario. We provably show that both uncertainty-based and diversity-based NAL are inherently amenable to one and the same principle, i.e., striving to prioritize samples that contain yet-to-be-learned features. We further prove that this shared principle is the key to their success-achieve small test error within a small labeled set. Contrastingly, the strategy-free passive learning exhibits a large test error due to the inadequate learning of yet-to-be-learned features, necessitating resort to a significantly larger label complexity for a sufficient test error reduction. Experimental results validate our findings.

This paper studies kernel ridge regression in high dimensions under covariate shifts and analyzes the role of importance re-weighting. We first derive the asymptotic expansion of high dimensional kernels under covariate shifts. By a bias-variance decomposition, we theoretically demonstrate that the re-weighting strategy allows for decreasing the variance. For bias, we analyze the regularization of the arbitrary or well-chosen scale, showing that the bias can behave very differently under different regularization scales. In our analysis, the bias and variance can be characterized by the spectral decay of a data-dependent regularized kernel: the original kernel matrix associated with an additional re-weighting matrix, and thus the re-weighting strategy can be regarded as a data-dependent regularization for better understanding. Besides, our analysis provides asymptotic expansion of kernel functions/vectors under covariate shift, which has its own interest.

Large language models based on the Transformer architecture have demonstrated impressive capabilities to learn in context. However, existing theoretical studies on how this phenomenon arises are limited to the dynamics of a single layer of attention trained on linear regression tasks. In this paper, we study the optimization of a Transformer consisting of a fully connected layer followed by a linear attention layer. The MLP acts as a common nonlinear representation or feature map, greatly enhancing the power of in-context learning. We prove in the mean-field and two-timescale limit that the infinite-dimensional loss landscape for the distribution of parameters, while highly nonconvex, becomes quite benign. We also analyze the second-order stability of mean-field dynamics and show that Wasserstein gradient flow almost always avoids saddle points. Furthermore, we establish novel methods for obtaining concrete improvement rates both away from and near critical points. This represents the first saddle point analysis of mean-field dynamics in general and the techniques are of independent interest.

Most variance reduction methods require multiple times of full gradient computation, which is time-consuming and hence a bottleneck in application to distributed optimization. We present a single-loop variance-reduced gradient estimator named SILVER (SIngle-Loop VariancE-Reduction) for the finite-sum non-convex optimization, which does not require multiple full gradients but nevertheless achieves the optimal gradient complexity. Notably, unlike existing methods, SILVER provably reaches second-order optimality, with exponential convergence in the Polyak-Łojasiewicz (PL) region, and achieves further speedup depending on the data heterogeneity. Owing to these advantages, SILVER serves as a new base method to design communication-efficient federated learning algorithms: we combine SILVER with local updates which gives the best communication rounds and number of communicated gradients across all range of Hessian heterogeneity, and, at the same time, guarantees second-order optimality and exponential convergence in the PL region.

We approach designing a state-space model for deep learning applications through its dual representation, the transfer function, and uncover a highly efficient sequence parallel inference algorithm that is state-free: unlike other proposed algorithms, state-free inference does not incur any significant memory or computational cost with an increase in state size. We achieve this using properties of the proposed frequency domain transfer function parametrization, which enables direct computation of its corresponding convolutional kernel’s spectrum via a single Fast Fourier Transform. Our experimental results across multiple sequence lengths and state sizes illustrates, on average, a 35% training speed improvement over S4 layers – parametrized in time-domain – on the Long Range Arena benchmark, while delivering state-of-the-art downstream performances over other attention-free approaches. Moreover, we report improved perplexity in language modeling over a long convolutional Hyena baseline, by simply introducing our transfer function parametrization. Our code is available at https://github.com/ruke1ire/RTF.

The development of deep learning architectures is a resource-demanding process, due to a vast design space, long prototyping times, and high compute costs associated with at-scale model training and evaluation. We set out to simplify this process by grounding it in an end-to-end mechanistic architecture design (MAD) pipeline, encompassing small-scale capability unit tests predictive of scaling laws. Through a suite of synthetic token manipulation tasks such as compression and recall, designed to probe capabilities, we identify and test new hybrid architectures constructed from a variety of computational primitives. We experimentally validate the resulting architectures via an extensive compute-optimal and a new state-optimal scaling law analysis, training over 500 language models between 70M to 7B parameters. Surprisingly, we find MAD synthetics to correlate with compute-optimal perplexity, enabling accurate evaluation of new architectures via isolated proxy tasks. The new architectures found via MAD, based on simple ideas such as hybridization and sparsity, outperform state-of-the-art Transformer, convolutional, and recurrent architectures (Transformer++, Hyena, Mamba) in scaling, both at compute-optimal budgets and in overtrained regimes. Overall, these results provide evidence that performance on curated synthetic tasks can be predictive of scaling laws, and that an optimal architecture should leverage specialized layers via a hybrid topology.

Transformers have achieved state-of-the-art performance in language modeling tasks. However, the reasons behind their tremendous success are still unclear. In this paper, towards a better understanding, we train a Transformer model on a simple next token prediction task, where sequences are generated as a first-order autoregressive process $s_{t+1} = W s_t$. We show how a trained Transformer predicts the next token by first learning $W$ in-context, then applying a prediction mapping. We call the resulting procedure *in-context autoregressive learning*. More precisely, focusing on commuting orthogonal matrices $W$, we first show that a trained one-layer linear Transformer implements one step of gradient descent for the minimization of an inner objective function, when considering augmented tokens. When the tokens are not augmented, we characterize the global minima of a one-layer diagonal linear multi-head Transformer. Importantly, we exhibit orthogonality between heads and show that positional encoding captures trigonometric relations in the data. On the experimental side, we consider the general case of non-commuting orthogonal matrices and generalize our theoretical findings.

In this paper, we study the feature learning ability of two-layer neural networks in the mean-field regime through the lens of kernel methods. To focus on the dynamics of the kernel induced by the first layer, we utilize a two-timescale limit, where the second layer moves much faster than the first layer. In this limit, the learning problem is reduced to the minimization problem over the intrinsic kernel. Then, we show the global convergence of the mean-field Langevin dynamics and derive time and particle discretization error. We also demonstrate that two-layer neural networks can learn a union of multiple reproducing kernel Hilbert spaces more efficiently than any kernel methods, and neural networks aquire data-dependent kernel which aligns with the target function. In addition, we develop a label noise procedure, which converges to the global optimum and show that the degrees of freedom appears as an implicit reguralization.

This work explores the feature learning capabilities of deep reinforcement learning algorithms in the pursuit of optimal policy determination. We particularly examine an over-parameterized neural actor-critic framework within the mean-field regime, where both actor and critic components undergo updates via policy gradient and temporal-difference (TD) learning, respectively. We introduce the mean-field Langevin TD learning (MFLTD) method, enhancing mean-field Langevin dynamics with proximal TD updates for critic policy evaluation, and assess its performance against conventional approaches through numerical analysis. Additionally, for actor policy updates, we present the mean-field Langevin policy gradient (MFLPG), employing policy gradient techniques augmented by Wasserstein gradient flows for parameter space exploration. Our findings demonstrate that MFLTD accurately identifies the true value function, while MFLPG ensures linear convergence of actor sequences towards the globally optimal policy, considering a Kullback-Leibler divergence regularized framework. Through both time particle and discretized analysis, we substantiate the linear convergence guarantees of our neural actor-critic algorithms, representing a notable contribution to neural reinforcement learning focusing on global optimality and feature learning, extending the existing understanding beyond the conventional scope of lazy training.

We investigate graph convolution networks with efficient learning from higher-order graph convolutions and direct learning from adjacency matrices for node classification. We revisit the scaled graph residual network and remove ReLU activation from residual layers and apply a single weight matrix at each residual layer. We show that the resulting models lead to new graph polynomial convolution methods constructed as polynomials of the normalized adjacency matrix, the residual weight matrix, and the residual scaling parameter. Additionally, we propose adaptive learning between graph polynomial convolution models and direct learning from the adjacency matrix. Furthermore, we propose fully adaptive models that learn scaling parameters at each residual layer. We show that generalization bounds of proposed methods are bounded as a polynomial of eigenvalue spectrum, scaling parameters, and upper bounds of residual weights. By theoretical analysis, we argue that the proposed models can obtain improved generalization bounds by limiting the higher-orders of convolutions and direct learning from the adjacency matrix. Using a wide set of real-data, we demonstrate that the proposed methods obtain improved accuracy for node-classification of non-homophilous graphs,

Transformer-based large language models (LLMs) have displayed remarkable creative prowess and emergence capabilities. Existing empirical studies have revealed a strong connection between these LLMs' impressive emergence abilities and their in-context learning (ICL) capacity, allowing them to solve new tasks using only task-specific prompts without further fine-tuning. On the other hand, existing empirical and theoretical studies also show that there is a linear regularity of the multi-concept encoded semantic representation behind transformer-based LLMs. However, existing theoretical work fail to build up an understanding of the connection between this regularity and the innovative power of ICL. Additionally, prior work often focuses on simplified, unrealistic scenarios involving linear transformers or unrealistic loss functions, and they achieve only linear or sub-linear convergence rates. In contrast, this work provides a fine-grained mathematical analysis to show how transformers leverage the multi-concept semantics of words to enable powerful ICL and excellent out-of-distribution ICL abilities, offering insights into how transformers innovate solutions for certain unseen tasks encoded with multiple cross-concept semantics. Inspired by empirical studies on the linear latent geometry of LLMs, the analysis is based on a concept-based low-noise sparse coding prompt model. Leveraging advanced techniques, this work showcases the exponential 0-1 loss convergence over the highly non-convex training dynamics, which pioneeringly incorporates the challenges of softmax self-attention, ReLU-activated MLPs, and cross-entropy loss. Empirical simulations corroborate the theoretical findings.

Multi-modal contrastive learning with language supervision has presented a paradigm shift in modern machine learning. By pre-training on a web-scale dataset, multi-modal contrastive learning can learn high-quality representations that exhibit impressive robustness and transferability. Despite its empirical success, the theoretical understanding is still in its infancy, especially regarding its comparison with single-modal contrastive learning. In this work, we introduce a feature learning theory framework that provides a theoretical foundation for understanding the differences between multi-modal and single-modal contrastive learning. Based on a data generation model consisting of signal and noise, our analysis is performed on a ReLU network trained with the InfoMax objective function. Through a trajectory-based optimization analysis and generalization characterization on downstream tasks, we identify the critical factor, which is the signal-to-noise ratio (SNR), that impacts the generalizability in downstream tasks of both multi-modal and single-modal contrastive learning. Through the cooperation between the two modalities, multi-modal learning can achieve better feature learning, leading to improvements in performance in downstream tasks compared to single-modal learning. Our analysis provides a unified framework that can characterize the optimization and generalization of both single-modal and multi-modal contrastive learning. Empirical experiments on both synthetic and real-world datasets further consolidate our theoretical findings.

Transformers have demonstrated great power in the recent development of large foundational models. In particular, the Vision Transformer (ViT) has brought revolutionary changes to the field of vision, achieving significant accomplishments on the experimental side. However, their theoretical capabilities, particularly in terms of generalization when trained to overfit training data, are still not fully understood. To address this gap, this work delves deeply into the \textit{benign overfitting} perspective of transformers in vision. To this end, we study the optimization of a Transformer composed of a self-attention layer with softmax followed by a fully connected layer under gradient descent on a certain data distribution model. By developing techniques that address the challenges posed by softmax and the interdependent nature of multiple weights in transformer optimization, we successfully characterized the training dynamics and achieved generalization in post-training. Our results establish a sharp condition that can distinguish between the small test error phase and the large test error regime, based on the signal-to-noise ratio in the data model. The theoretical results are further verified by experimental simulation. To the best of our knowledge, this is the first work to characterize benign overfitting for Transformers.

In-context learning (ICL) of large language models has proven to be a surprisingly effective method of learning a new task from only a few demonstrative examples. In this paper, we shed light on the efficacy of ICL from the viewpoint of statistical learning theory. We develop approximation and generalization error analyses for a transformer model composed of a deep neural network and one linear attention layer, pretrained on nonparametric regression tasks sampled from general function spaces including the Besov space and piecewise $\gamma$-smooth class. In particular, we show that sufficiently trained transformers can achieve -- and even improve upon -- the minimax optimal estimation risk in context by encoding the most relevant basis representations during pretraining. Our analysis extends to high-dimensional or sequential data and distinguishes the \emph{pretraining} and \emph{in-context} generalization gaps, establishing upper and lower bounds w.r.t. both the number of tasks and in-context examples. These findings shed light on the effectiveness of few-shot prompting and the roles of task diversity and representation learning for ICL.

We study the problem of gradient descent learning of a single-index target function $f_*(\boldsymbol{x}) = \textstyle\sigma_*\left(\langle\boldsymbol{x},\boldsymbol{\theta}\rangle\right)$ under isotropic Gaussian data in $\mathbb{R}^d$, where the unknown link function $\sigma_*:\mathbb{R}\to\mathbb{R}$ has information exponent $p$ (defined as the lowest degree in the Hermite expansion). Prior works showed that gradient-based training of neural networks can learn this target with $n\gtrsim d^{\Theta(p)}$ samples, and such complexity is predicted to be necessary by the correlational statistical query lower bound. Surprisingly, we prove that a two-layer neural network optimized by an SGD-based algorithm (on the squared loss) learns $f_*$ with a complexity that is not governed by the information exponent. Specifically, for arbitrary polynomial single-index models, we establish a sample and runtime complexity of $n \simeq T = \Theta(d\cdot\mathrm{polylog} d)$, where $\Theta(\cdot)$ hides a constant only depending on the degree of $\sigma_*$; this dimension dependence matches the information theoretic limit up to polylogarithmic factors. More generally, we show that $n\gtrsim d^{(p_*-1)\vee 1}$ samples are sufficient to achieve low generalization error, where $p_* \le p$ is the \textit{generative exponent} of the link function. Core to our analysis is the reuse of minibatch in the gradient computation, which gives rise to higher-order information beyond correlational queries.

Transformers can efficiently learn in-context from example demonstrations. Most existing theoretical analyses studied the in-context learning (ICL) ability of transformers for linear function classes, where it is typically shown that the minimizer of the pretraining loss implements one gradient descent step on the least squares objective. However, this simplified linear setting arguably does not demonstrate the statistical efficiency of ICL, since the pretrained transformer does not outperform directly solving linear regression on the test prompt. In this paper, we study ICL of a nonlinear function class via transformer with nonlinear MLP layer: given a class of \textit{single-index} target functions $f_*(\boldsymbol{x}) = \sigma_*(\langle\boldsymbol{x},\boldsymbol{\beta}\rangle)$, where the index features $\boldsymbol{\beta}\in\mathbb{R}^d$ are drawn from a $r$-dimensional subspace, we show that a nonlinear transformer optimized by gradient descent (with a pretraining sample complexity that depends on the \textit{information exponent} of the link functions $\sigma_*$) learns $f_*$ in-context with a prompt length that only depends on the dimension of the distribution of target functions $r$; in contrast, any algorithm that directly learns $f_*$ on test prompt yields a statistical complexity that scales with the ambient dimension $d$. Our result highlights the adaptivity of the pretrained transformer to low-dimensional structures of the function class, which enables sample-efficient ICL that outperforms estimators that only have access to the in-context data.

Differential private optimization for nonconvex smooth objective is considered. In the previous work, the best known utility bound is $\widetilde O(\sqrt{d}/(n\varepsilon_\mathrm{DP}))$ in terms of the squared full gradient norm, which is achieved by Differential Private Gradient Descent (DP-GD) as an instance, where $n$ is the sample size, $d$ is the problem dimensionality and $\varepsilon_\mathrm{DP}$ is the differential privacy parameter. To improve the best known utility bound, we propose a new differential private optimization framework called DIFF2 (DIFFerential private optimization via gradient DIFFerences) that constructs a differential private global gradient estimator with possibly quite small variance based on communicated gradient differences rather than gradients themselves. It is shown that DIFF2 with a gradient descent subroutine achieves the utility of $\widetilde O(d^{2/3}/(n\varepsilon_\mathrm{DP})^{4/3})$, which can be significantly better than the previous one in terms of the dependence on the sample size $n$. To the best of our knowledge, this is the first fundamental result to improve the standard utility $\widetilde O(\sqrt{d}/(n\varepsilon_\mathrm{DP}))$ for nonconvex objectives. Additionally, a more computational and communication efficient subroutine is combined with DIFF2 and its theoretical analysis is also given. Numerical experiments are conducted to validate the superiority of DIFF2 framework.

The entropic fictitious play (EFP) is a recently proposed algorithm that minimizes the sum of a convex functional and entropy in the space of measures — such an objective naturally arises in the optimization of a two-layer neural network in the mean-field regime. In this work, we provide a concise primal-dual analysis of EFP in the setting where the learning problem exhibits a finite-sum structure. We establish quantitative global convergence guarantees for both the continuous-time and discrete-time dynamics based on properties of a proximal Gibbs measure introduced in Nitanda et al. (2022). Furthermore, our primal-dual framework entails a memory-efficient particle-based implementation of the EFP update, and also suggests a connection to gradient boosting methods. We illustrate the efficiency of our novel implementation in experiments including neural network optimization and image synthesis.

While efficient distribution learning is no doubt behind the groundbreaking success of diffusion modeling, its theoretical guarantees are quite limited. In this paper, we provide the first rigorous analysis on approximation and generalization abilities of diffusion modeling for well-known function spaces. The highlight of this paper is that when the true density function belongs to the Besov space and the empirical score matching loss is properly minimized, the generated data distribution achieves the nearly minimax optimal estimation rates in the total variation distance and in the Wasserstein distance of order one. Furthermore, we extend our theory to demonstrate how diffusion models adapt to low-dimensional data distributions. We expect these results advance theoretical understandings of diffusion modeling and its ability to generate verisimilar outputs.

Recent studies have experimentally shown that we can achieve in non-Euclidean metric space effective and efficient graph embedding, which aims to obtain the vertices’ representations reflecting the graph’s structure in the metric space. Specifically, graph embedding in hyperbolic space has experimentally succeeded in embedding graphs with hierarchical-tree structure, e.g., data in natural languages, social networks, and knowledge bases. However, recent theoretical analyses have shown a much higher upper bound on non-Euclidean graph embedding’s generalization error than Euclidean one’s, where a high generalization error indicates that the incompleteness and noise in the data can significantly damage learning performance. It implies that the existing bound cannot guarantee the success of graph embedding in non-Euclidean metric space in a practical training data size, which can prevent non-Euclidean graph embedding’s application in real problems. This paper provides a novel upper bound of graph embedding’s generalization error by evaluating the local Rademacher complexity of the model as a function set of the distances of representation couples. Our bound clarifies that the performance of graph embedding in non-Euclidean metric space, including hyperbolic space, is better than the existing upper bounds suggest. Specifically, our new upper bound is polynomial in the metric space’s geometric radius $R$ and can be $O(\frac{1}{S})$ at the fastest, where $S$ is the training data size. Our bound is significantly tighter and faster than the existing one, which can be exponential to $R$ and $O(\frac{1}{\sqrt{S}})$ at the fastest. Specific calculations on example cases show that graph embedding in non-Euclidean metric space can outperform that in Euclidean space with much smaller training data than the existing bound has suggested.

Despite the great success of Transformer networks in various applications such as natural language processing and computer vision, their theoretical aspects are not well understood. In this paper, we study the approximation and estimation ability of Transformers as sequence-to-sequence functions with infinite dimensional inputs. Although inputs and outputs are both infinite dimensional, we show that when the target function has anisotropic smoothness, Transformers can avoid the curse of dimensionality due to their feature extraction ability and parameter sharing property. In addition, we show that even if the smoothness changes depending on each input, Transformers can estimate the importance of features for each input and extract important features dynamically. Then, we proved that Transformers achieve similar convergence rate as in the case of the fixed smoothness. Our theoretical results support the practical success of Transformers for high dimensional data.

Federated learning is a privacy-preserving training method which consists of training from a plurality of clients but without sharing their confidential data. However, previous work on federated learning do not explore suitable neural network architectures for clients with different input images sizes and different numbers of output categories. In this paper, we propose an effective federated learning method named ScalableFL, where the depths and widths of the local models for each client are ad-justed according to the clients' input image size and the numbers of output categories. In addition, we provide a new bound for the generalization gap of federated learning. In particular, this bound helps to explain the effectiveness of our scalable neural network approach. We demonstrate the effectiveness of ScalableFL in several heterogeneous client settings for both image classification and object detection tasks.

We propose a modularization method that decomposes a deep neural network (DNN) into small modules as binary classifiers and recomposes a new model of some of these modules for a different task. Our goal is to develop appropriate models by quickly reusing existing well-designed models. To decompose a DNN into modules, we use weight masks based on the lottery ticket hypothesis to extract small single-function winning tickets as modules. By ensuring that each module shares as many parameters as possible, the size of the recomposed model is not much larger than the modules and significantly smaller than the original DNN. Furthermore, since the recomposed model is specialized for fewer classes, its accuracy is superior to models of comparable size produced by model compression. This method can be applied to DNNs of any architecture. We demonstrate that the proposed method can decompose and recompose DNNs with a high compression ratio and high accuracy, and is superior to existing methods.

We consider the learning of a single-index target function $f_*: \mathbb{R}^d\to\mathbb{R}$ under spiked covariance data: $$f_*(\boldsymbol{x}) = \textstyle\sigma_*(\frac{1}{\sqrt{1+\theta}}\langle\boldsymbol{x},\boldsymbol{\mu}\rangle), ~~ \boldsymbol{x}\overset{\small\mathrm{i.i.d.}}{\sim}\mathcal{N}(0,\boldsymbol{I_d} + \theta\boldsymbol{\mu}\boldsymbol{\mu}^\top), ~~ \theta\asymp d^{\beta} \text{ for } \beta\in[0,1), $$ where the link function $\sigma_*:\mathbb{R}\to\mathbb{R}$ is a degree-$p$ polynomial with information exponent $k$ (defined as the lowest degree in the Hermite expansion of $\sigma_*$), and it depends on the projection of input $\boldsymbol{x}$ onto the spike (signal) direction $\boldsymbol{\mu}\in\mathbb{R}^d$. In the proportional asymptotic limit where the number of training examples $n$ and the dimensionality $d$ jointly diverge: $n,d\to\infty, n/d\to\psi\in(0,\infty)$, we ask the following question: how large should the spike magnitude $\theta$ (i.e., the strength of the low-dimensional component) be, in order for $(i)$ kernel methods, $(ii)$ neural networks optimized by gradient descent, to learn $f_*$? We show that for kernel ridge regression, $\beta\ge 1-\frac{1}{p}$ is both sufficient and necessary. Whereas for two-layer neural networks trained with gradient descent, $\beta>1-\frac{1}{k}$ suffices. Our results demonstrate that both kernel methods and neural networks benefit from low-dimensional structures in the data. Further, since $k\le p$ by definition, neural networks can adapt to such structures more effectively.

Recent works have demonstrated that the sample complexity of gradient-based learning of single index models, i.e. functions that depend on a 1-dimensional projection of the input data, is governed by their information exponent. However, these results are only concerned with isotropic data, while in practice the input often contains additional structure which can implicitly guide the algorithm. In this work, we investigate the effect of a spiked covariance structure and reveal several interesting phenomena. First, we show that in the anisotropic setting, the commonly used spherical gradient dynamics may fail to recover the true direction, even when the spike is perfectly aligned with the target direction. Next, we show that appropriate weight normalization that is reminiscent of batch normalization can alleviate this issue. Further, by exploiting the alignment between the (spiked) input covariance and the target, we obtain improved sample complexity compared to the isotropic case. In particular, under the spiked model with a suitably large spike, the sample complexity of gradient-based training can be made independent of the information exponent while also outperforming lower bounds for rotationally invariant kernel methods.

Neural network in the mean-field regime is known to be capable of \textit{feature learning}, unlike the kernel (NTK) counterpart. Recent works have shown that mean-field neural networks can be globally optimized by a noisy gradient descent update termed the \textit{mean-field Langevin dynamics} (MFLD). However, all existing guarantees for MFLD only considered the \textit{optimization} efficiency, and it is unclear if this algorithm leads to improved \textit{generalization} performance and sample complexity due to the presence of feature learning. To fill this gap, in this work we study the statistical and computational complexity of MFLD in learning a class of binary classification problems. Unlike existing margin bounds for neural networks, we avoid the typical norm control by utilizing the perspective that MFLD optimizes the \textit{distribution} of parameters rather than the parameter itself; this leads to an improved analysis of the sample complexity and convergence rate. We apply our general framework to the learning of $k$-sparse parity functions, where we prove that unlike kernel methods, two-layer neural networks optimized by MFLD achieves a sample complexity where the degree $k$ is ``decoupled'' from the exponent in the dimension dependence.

As an example of the nonlinear Fokker-Planck equation, the mean field Langevin dynamics recently attracts attention due to its connection to (noisy) gradient descent on infinitely wide neural networks in the mean field regime, and hence the convergence property of the dynamics is of great theoretical interest. In this work, we give a concise and self-contained convergence rate analysis of the mean field Langevin dynamics with respect to the (regularized) objective function in both continuous and discrete time settings. The key ingredient of our proof is a proximal Gibbs distribution $p_q$ associated with the dynamics, which, in combination with techniques in Vempala and Wibisono (2019), allows us to develop a simple convergence theory parallel to classical results in convex optimization. Furthermore, we reveal that $p_q$ connects to the duality gap in the empirical risk minimization setting, which enables efficient empirical evaluation of the algorithm convergence.

Gradient Langevin dynamics (GLD) and stochastic GLD (SGLD) have attracted considerable attention lately, as a way to provide convergence guarantees in a non-convex setting. However, the known rates grow exponentially with the dimension of the space under the dissipative condition. In this work, we provide a convergence analysis of GLD and SGLD when the optimization space is an infinite-dimensional Hilbert space. More precisely, we derive non-asymptotic, dimension-free convergence rates for GLD/SGLD when performing regularized non-convex optimization in a reproducing kernel Hilbert space. Amongst others, the convergence analysis relies on the properties of a stochastic differential equation, its discrete time Galerkin approximation and the geometric ergodicity of the associated Markov chains.

Although data-parallel learning can efficiently reduce training time, the accuracy becomes poor when the batch size is extremely large. Adam, a prominent adaptive gradient optimizer which uses second-order information, is known to have better convergence than SGD, but its performance in large batch training highly depends on actual problems and the reasons for this remain unclear. While attempting answer to this, we found there are two factors playing import roles: the estimation accuracy of Hessian and the number of classes in data. Aiming to improve the former one, we introduce a new method called data-parallel momentum diagonal empirical Fisher (DP-MDEF), which collects the variance of the gradients between parallel workers for gradient adaptation and obtains superior precision compared to Adam and Momentum-SGD in large batch training. To discuss the latter one, we found that when the number of classes in a data-set is large, the variances of gradients form a long-tailed distribution, in which case gradient adaptive methods give bad solutions.

In neural architecture search (NAS), differentiable architecture search (DARTS) has recently attracted much attention due to its high efficiency. However, this method finds a model with the weights converging faster than the others, and such a model with fastest convergence often leads to overfitting. Accordingly, the resulting model cannot always be well-generalized. To overcome this problem, we propose a method called minimum stable rank DARTS (MSR-DARTS), for finding a model with the best generalization error by replacing architecture optimization with the selection process using the minimum stable rank criterion. Specifically, a convolution operator is represented by a matrix, and MSR-DARTS selects the one with the smallest stable rank. We evaluated MSR-DARTS on CIFAR-10 and ImageNet datasets. It achieves an error rate of 2.54% with 4.0M parameters within 0.3 GPU-days on CIFAR-10, and a top-1 error rate of 23.9% on ImageNet.

We study the first gradient descent step on the first-layer parameters $\boldsymbol{W}$ in a two-layer neural network: $f(\boldsymbol{x}) = \frac{1}{\sqrt{N}}\boldsymbol{a}^\top\sigma(\boldsymbol{W}^\top\boldsymbol{x})$, where $\boldsymbol{W}\in\mathbb{R}^{d\times N}, \boldsymbol{a}\in\mathbb{R}^{N}$ are randomly initialized, and the training objective is the empirical MSE loss: $\frac{1}{n}\sum_{i=1}^n (f(\boldsymbol{x}_i)-y_i)^2$. In the proportional asymptotic limit where $n,d,N\to\infty$ at the same rate, and an idealized student-teacher setting where the teacher $f^*$ is a single-index model, we compute the prediction risk of ridge regression on the conjugate kernel after one gradient step on $\boldsymbol{W}$ with learning rate $\eta$. We consider two scalings of the first step learning rate $\eta$. For small $\eta$, we establish a Gaussian equivalence property for the trained feature map, and prove that the learned kernel improves upon the initial random features model, but cannot defeat the best linear model on the input. Whereas for sufficiently large $\eta$, we prove that for certain $f^*$, the same ridge estimator on trained features can go beyond this ``linear regime'' and outperform a wide range of (fixed) kernels. Our results demonstrate that even one gradient step can lead to a considerable advantage over random features, and highlight the role of learning rate scaling in the initial phase of training.

The stochastic gradient Langevin Dynamics is one of the most fundamental algorithms to solve sampling problems and non-convex optimization appearing in several machine learning applications. Especially, its variance reduced versions have nowadays gained particular attention. In this paper, we study two variants of this kind, namely, the Stochastic Variance Reduced Gradient Langevin Dynamics and the Stochastic Recursive Gradient Langevin Dynamics. We prove their convergence to the objective distribution in terms of KL-divergence under the sole assumptions of smoothness and Log-Sobolev inequality which are weaker conditions than those used in prior works for these algorithms. With the batch size and the inner loop length set to $\sqrt{n}$, the gradient complexity to achieve an $\epsilon$-precision is $\tilde{O}((n+dn^{1/2}\epsilon^{-1})\gamma^2 L^2\alpha^{-2})$, which is an improvement from any previous analyses. We also show some essential applications of our result to non-convex optimization.

In recent centralized nonconvex distributed learning and federated learning, local methods are one of the promising approaches to reduce communication time. However, existing work has mainly focused on studying first-order optimality guarantees. On the other side, second-order optimality guaranteed algorithms, i.e., algorithms escaping saddle points, have been extensively studied in the non-distributed optimization literature. In this paper, we study a new local algorithm called Bias-Variance Reduced Local Perturbed SGD (BVR-L-PSGD), that combines the existing bias-variance reduced gradient estimator with parameter perturbation to find second-order optimal points in centralized nonconvex distributed optimization. BVR-L-PSGD enjoys second-order optimality with nearly the same communication complexity as the best known one of BVR-L-SGD to find first-order optimality. Particularly, the communication complexity is better than non-local methods when the local datasets heterogeneity is smaller than the smoothness of the local loss. In an extreme case, the communication complexity approaches to $\widetilde \Theta(1)$ when the local datasets heterogeneity goes to zero. Numerical results validate our theoretical findings.

Analysis of neural network optimization in the mean-field regime is important as the setting allows for feature learning. Existing theory has been developed mainly for neural networks in finite dimensions, i.e., each neuron has a finite-dimensional parameter. However, the setting of infinite-dimensional input naturally arises in machine learning problems such as nonparametric functional data analysis and graph classification. In this paper, we develop a new mean-field analysis of two-layer neural network in an infinite-dimensional parameter space. We first give a generalization error bound, which shows that the regularized empirical risk minimizer properly generalizes when the data size is sufficiently large, despite the neurons being infinite-dimensional. Next, we present two gradient-based optimization algorithms for infinite-dimensional mean-field networks, by extending the recently developed particle optimization framework to the infinite-dimensional setting. We show that the proposed algorithms converge to the (regularized) global optimal solution, and moreover, their rates of convergence are of polynomial order in the online setting and exponential order in the finite sample setting, respectively. To our knowledge this is the first quantitative global optimization guarantee of neural network on infinite-dimensional input and in the presence of feature learning.

Labeling cost is often expensive and is a fundamental limitation of supervised learning. In this paper, we study importance labeling problem, in which we are given many unlabeled data and select a limited number of data to be labeled from the unlabeled data, and then a learning algorithm is executed on the selected one. We propose a new importance labeling scheme that can effectively select an informative subset of unlabeled data in least squares regression in Reproducing Kernel Hilbert Spaces (RKHS). We analyze the generalization error of gradient descent combined with our labeling scheme and show that the proposed algorithm achieves the optimal rate of convergence in much wider settings and especially gives much better generalization ability in a small noise setting than the usual uniform sampling scheme. Numerical experiments verify our theoretical findings.

Although kernel methods are widely used in many learning problems, they have poor scalability to large datasets. To address this problem, sketching and stochastic gradient methods are the most commonly used techniques to derive computationally efficient learning algorithms. We consider solving a binary classification problem using random features and stochastic gradient descent, both of which are common and widely used in practical large-scale problems. Although there are plenty of previous works investigating the efficiency of these algorithms in terms of the convergence of the objective loss function, these results suggest that the computational gain comes at expense of the learning accuracy when dealing with general Lipschitz loss functions such as logistic loss. In this study, we analyze the properties of these algorithms in terms of the convergence not of the loss function, but the classification error under the strong low-noise condition, which reflects a realistic property of real-world datasets. We extend previous studies on SGD to a random features setting, examining a novel analysis about the error induced by the approximation of random features in terms of the distance between the generated hypothesis to show that an exponential convergence of the expected classification error is achieved even if random features approximation is applied. We demonstrate that the convergence rate does not depend on the number of features and there is a significant computational benefit in using random features in classification problems under the strong low-noise condition.

Deep learning empirically achieves high performance in many applications, but its training dynamics has not been fully understood theoretically. In this paper, we explore theoretical analysis on training two-layer ReLU neural networks in a teacher-student regression model, in which a student network learns an unknown teacher network through its outputs. We show that with a specific regularization and sufficient over-parameterization, the student network can identify the parameters of the teacher network with high probability via gradient descent with a norm dependent stepsize even though the objective function is highly non-convex. The key theoretical tool is the measure representation of the neural networks and a novel application of a dual certificate argument for sparse estimation on a measure space. We analyze the global minima and global convergence property in the measure space.

Recently, local SGD has got much attention and been extensively studied in the distributed learning community to overcome the communication bottleneck problem. However, the superiority of local SGD to minibatch SGD only holds in quite limited situations. In this paper, we study a new local algorithm called Bias-Variance Reduced Local SGD (BVR-L-SGD) for nonconvex distributed optimization. Algorithmically, our proposed bias and variance reduced local gradient estimator fully utilizes small second-order heterogeneity of local objectives and suggests randomly picking up one of the local models instead of taking the average of them when workers are synchronized. Theoretically, under small heterogeneity of local objectives, we show that BVR-L-SGD achieves better communication complexity than both the previous non-local and local methods under mild conditions, and particularly BVR-L-SGD is the first method that breaks the barrier of communication complexity $\Theta(1/\varepsilon)$ for general nonconvex smooth objectives when the heterogeneity is small and the local computation budget is large. Numerical results are given to verify the theoretical findings and give empirical evidence of the superiority of our method.

Variational autoencoder (VAE) estimates the posterior parameters (mean and variance) of latent variables corresponding to each input data. While it is used for many tasks, the transparency of the model is still an underlying issue. This paper provides a quantitative understanding of VAE property through the differential geometric and information-theoretic interpretations of VAE. According to the Rate-distortion theory, the optimal transform coding is achieved by using an orthonormal transform with PCA basis where the transform space is isometric to the input. Considering the analogy of transform coding to VAE, we clarify theoretically and experimentally that VAE can be mapped to an implicit isometric embedding with a scale factor derived from the posterior parameter. As a result, we can estimate the data probabilities in the input space from the prior, loss metrics, and corresponding posterior parameters, and further, the quantitative importance of each latent variable can be evaluated like the eigenvalue of PCA.

Compressing DNNs is important for the real-world applications operating on resource-constrained devices. However, we typically observe drastic performance deterioration when changing model size after training is completed. Therefore, retraining is required to resume the performance of the compressed models suitable for different devices. In this paper, we propose Decomposable-Net (the network decomposable in any size), which allows flexible changes to model size without retraining. We decompose weight matrices in the DNNs via singular value decomposition and adjust ranks according to the target model size. Unlike the existing low-rank compression methods that specialize the model to a fixed size, we propose a novel backpropagation scheme that jointly minimizes losses for both of full- and low-rank networks. This enables not only to maintain the performance of a full-rank network {\it without retraining} but also to improve low-rank networks in multiple sizes. Additionally, we introduce a simple criterion for rank selection that effectively suppresses approximation error. In experiments on the ImageNet classification task, Decomposable-Net yields superior accuracy in a wide range of model sizes. In particular, Decomposable-Net achieves the top-1 accuracy of 73.2% with 0.27xMACs with ResNet-50, compared to Tucker decomposition (67.4% / 0.30x), Trained Rank Pruning (70.6% / 0.28x), and universally slimmable networks (71.4% / 0.26x).

We detail a novel class of implicit neural models. Leveraging time-parallel methods for differential equations, Multiple Shooting Layers (MSLs) seek solutions of initial value problems via parallelizable root-finding algorithms. MSLs broadly serve as drop-in replacements for neural ordinary differential equations (Neural ODEs) with improved efficiency in number of function evaluations (NFEs) and wall-clock inference time. We develop the algorithmic framework of MSLs, analyzing the different choices of solution methods from a theoretical and computational perspective. MSLs are showcased in long horizon optimal control of ODEs and PDEs and as latent models for sequence generation. Finally, we investigate the speedups obtained through application of MSL inference in neural controlled differential equations (Neural CDEs) for time series classification of medical data.

We propose the particle dual averaging (PDA) method, which generalizes the dual averaging method in convex optimization to the optimization over probability distributions with quantitative runtime guarantee. The algorithm consists of an inner loop and outer loop: the inner loop utilizes the Langevin algorithm to approximately solve for a stationary distribution, which is then optimized in the outer loop. The method can be interpreted as an extension of the Langevin algorithm to naturally handle nonlinear functional on the probability space. An important application of the proposed method is the optimization of neural network in the mean field regime, which is theoretically attractive due to the presence of nonlinear feature learning, but quantitative convergence rate can be challenging to obtain. By adapting finite-dimensional convex optimization theory into the space of measures, we not only establish global convergence of PDA for two-layer mean field neural networks under more general settings and simpler analysis, but also provide quantitative polynomial runtime guarantee. Our theoretical results are supported by numerical simulations on neural networks with reasonable size.

Deep learning has exhibited superior performance for various tasks, especially for high-dimensional datasets, such as images. To understand this property, we investigate the approximation and estimation ability of deep learning on {\it anisotropic Besov spaces}.The anisotropic Besov space is characterized by direction-dependent smoothness and includes several function classes that have been investigated thus far.We demonstrate that the approximation error and estimation error of deep learning only depend on the average value of the smoothness parameters in all directions. Consequently, the curse of dimensionality can be avoided if the smoothness of the target function is highly anisotropic.Unlike existing studies, our analysis does not require a low-dimensional structure of the input data.We also investigate the minimax optimality of deep learning and compare its performance with that of the kernel method (more generally, linear estimators).The results show that deep learning has better dependence on the input dimensionality if the target function possesses anisotropic smoothness, and it achieves an adaptive rate for functions with spatially inhomogeneous smoothness.

Deep neural networks generalize well on unseen data though the number of parameters often far exceeds the number of training examples. Recently proposed complexity measures have provided insights to understanding the generalizability in neural networks from perspectives of PAC-Bayes, robustness, overparametrization, compression and so on. In this work, we advance the understanding of the relations between the network’s architecture and its generalizability from the compression perspective. Using tensor analysis, we propose a series of intuitive, data-dependent and easily-measurable properties that tightly characterize the compressibility and generalizability of neural networks; thus, in practice, our generalization bound outperforms the previous compression-based ones, especially for neural networks using tensors as their weight kernels (e.g. CNNs). Moreover, these intuitive measurements provide further insights into designing neural network architectures with properties favorable for better/guaranteed generalizability. Our experimental results demonstrate that through the proposed measurable properties, our generalization error bound matches the trend of the test error well. Our theoretical analysis further provides justifications for the empirical success and limitations of some widely-used tensor-based compression approaches. We also discover the improvements to the compressibility and robustness of current neural networks when incorporating tensor operations via our proposed layer-wise structure.

Recently, several studies have proposed progressive or sequential layer-wise training methods based on the boosting theory for deep neural networks. However, most studies lack the global convergence guarantees or require weak learning conditions that can be verified a posteriori after running methods. Moreover, generalization bounds usually have a worse dependence on network depth. In this paper, to resolve these problems, we propose a new functional gradient boosting for learning deep residual-like networks in a layer-wise fashion with its statistical guarantees on multi-class classification tasks. In the proposed method, each residual block is recognized as a functional gradient (i.e., weak learner), and the functional gradient step is performed by stacking it on the network, resulting in a strong optimization ability. In the theoretical analysis, we show the global convergence of the method under a standard margin assumption on a data distribution instead of a weak learning condition, and we eliminate a worse dependence on the network depth in a generalization bound via a fine-grained convergence analysis. %, unlike existing studies. Moreover, we show that the existence of a learnable function with a large margin on a training dataset significantly improves a generalization bound. Finally, we experimentally demonstrate that our proposed method is certainly useful for learning deep residual networks.

Compression techniques for deep neural network models are becoming very important for the efficient execution of high-performance deep learning systems on edge-computing devices. The concept of model compression is also important for analyzing the generalization error of deep learning, known as the compression-based error bound. However, there is still huge gap between a practically effective compression method and its rigorous background of statistical learning theory. To resolve this issue, we develop a new theoretical framework for model compression and propose a new pruning method called {\it spectral pruning} based on this framework. We define the ``degrees of freedom'' to quantify the intrinsic dimensionality of a model by using the eigenvalue distribution of the covariance matrix across the internal nodes and show that the compression ability is essentially controlled by this quantity. Moreover, we present a sharp generalization error bound of the compressed model and characterize the bias--variance tradeoff induced by the compression procedure. We apply our method to several datasets to justify our theoretical analyses and show the superiority of the the proposed method.

It is known that the current graph neural networks (GNNs) are difficult to make themselves deep due to the problem known as over-smoothing. Multi-scale GNNs are a promising approach for mitigating the over-smoothing problem. However, there is little explanation of why it works empirically from the viewpoint of learning theory. In this study, we derive the optimization and generalization guarantees of transductive learning algorithms that include multi-scale GNNs. Using the boosting theory, we prove the convergence of the training error under weak learning-type conditions. By combining it with generalization gap bounds in terms of transductive Rademacher complexity, we show that a test error bound of a specific type of multi-scale GNNs that decreases corresponding to the number of node aggregations under some conditions. Our results offer theoretical explanations for the effectiveness of the multi-scale structure against the over-smoothing problem. We apply boosting algorithms to the training of multi-scale GNNs for real-world node prediction tasks. We confirm that its performance is comparable to existing GNNs, and the practical behaviors are consistent with theoretical observations. Code is available at https://github.com/delta2323/GB-GNN.

We introduce a new theoretical framework to analyze deep learning optimization with connection to its generalization error. Existing frameworks such as mean field theory and neural tangent kernel theory for neural network optimization analysis typically require taking limit of infinite width of the network to show its global convergence. This potentially makes it difficult to directly deal with finite width network; especially in the neural tangent kernel regime, we cannot reveal favorable properties of neural networks {\it beyond kernel methods}. To realize more natural analysis, we consider a completely different approach in which we formulate the parameter training as a transportation map estimation and show its global convergence via the theory of the {\it infinite dimensional Langevin dynamics}. This enables us to analyze narrow and wide networks in a unifying manner. Moreover, we give generalization gap and excess risk bounds for the solution obtained by the dynamics. The excess risk bound achieves the so-called fast learning rate. In particular, we show an exponential convergence for a classification problem and a minimax optimal rate for a regression problem.

Sparse regularization such as [Formula: see text] regularization is a quite powerful and widely used strategy for high-dimensional learning problems. The effectiveness of sparse regularization has been supported practically and theoretically by several studies. However, one of the biggest issues in sparse regularization is that its performance is quite sensitive to correlations between features. Ordinary [Formula: see text] regularization selects variables correlated with each other under weak regularizations, which results in deterioration of not only its estimation error but also interpretability. In this letter, we propose a new regularization method, independently interpretable lasso (IILasso), for generalized linear models. Our proposed regularizer suppresses selecting correlated variables, so that each active variable affects the response independently in the model. Hence, we can interpret regression coefficients intuitively, and the performance is also improved by avoiding overfitting. We analyze the theoretical property of the IILasso and show that the proposed method is advantageous for its sign recovery and achieves almost minimax optimal convergence rate. Synthetic and real data analyses also indicate the effectiveness of the IILasso.

Deep learning has been applied to various tasks in the field of machine learning and has shown superiority to other common procedures such as kernel methods. To provide a better theoretical understanding of the reasons for its success, we discuss the performance of deep learning and other methods on a nonparametric regression problem with a Gaussian noise. Whereas existing theoretical studies of deep learning have been based mainly on mathematical theories of well-known function classes such as Hölder and Besov classes, we focus on function classes with discontinuity and sparsity, which are those naturally assumed in practice. To highlight the effectiveness of deep learning, we compare deep learning with a class of linear estimators representative of a class of shallow estimators. It is shown that the minimax risk of a linear estimator on the convex hull of a target function class does not differ from that of the original target function class. This results in the suboptimality of linear methods over a simple but non-convex function class, on which deep learning can attain nearly the minimax-optimal rate. In addition to this extreme case, we consider function classes with sparse wavelet coefficients. On these function classes, deep learning also attains the minimax rate up to log factors of the sample size, and linear methods are still suboptimal if the assumed sparsity is strong. We also point out that the parameter sharing of deep neural networks can remarkably reduce the complexity of the model in our setting.

We consider stochastic gradient descent and its averaging variant for binary classification problems in a reproducing kernel Hilbert space. In traditional analysis using a consistency property of loss functions, it is known that the expected classification error converges more slowly than the expected risk even when assuming a low-noise condition on the conditional label probabilities. Consequently, the resulting rate is sublinear. Therefore, it is important to consider whether much faster convergence of the expected classification error can be achieved. In recent research, an exponential convergence rate for stochastic gradient descent was shown under a strong low-noise condition but provided theoretical analysis was limited to the square loss function, which is somewhat inadequate for binary classification tasks. In this paper, we show an exponential convergence of the expected classification error in the final phase of the stochastic gradient descent for a wide class of differentiable convex loss functions under similar assumptions. As for the averaged stochastic gradient descent, we show that the same convergence rate holds from the early phase of training. In experiments, we verify our analyses on the $L_2$-regularized logistic regression.

The minibatching technique has been extensively adopted to facilitate stochastic first-order methods because of their computational efficiency in parallel computing for large-scale machine learning and data mining. However, the optimal minibatch size determination for accelerated stochastic gradient methods is not completely understood. Actually, there appears trade-off between the iteration complexity and the total computational complexity; that is, the number of iterations (minibatch queries) can be decreased by increasing the minibatch size, but too large minibatch size would result in an unnecessarily large total computational cost. In this study, we give a sharp characterization of the minimax optimal minibatch size to achieve the optimal iteration complexity by providing a reachable lower bound for minimizing finite sum of convex functions and, surprisingly, show that the optimal method with the minimax optimal minibatch size can achieve both of the optimal iteration complexity and the optimal total computational complexity simultaneously. Finally, this feature is verified experimentally.

Convolutional neural networks (CNNs) have been shown to achieve optimal approximation and estimation error rates (in minimax sense) in several function classes. However, previous analyzed optimal CNNs are unrealistically wide and difficult to obtain via optimization due to sparse constraints in important function classes, including the Hölder class. We show a ResNet-type CNN can attain the minimax optimal error rates in these classes in more plausible situations – it can be dense, and its width, channel size, and filter size are constant with respect to sample size. The key idea is that we can replicate the learning ability of Fully-connected neural networks (FNNs) by tailored CNNs, as long as the FNNs have block-sparse structures. Our theory is general in a sense that we can automatically translate any approximation rate achieved by block-sparse FNNs into that by CNNs. As an application, we derive approximation and estimation error rates of the aformentioned type of CNNs for the Barron and Hölder classes with the same strategy.

We propose a new technique that boosts the convergence of training generative adversarial networks. Generally, the rate of training deep models reduces severely after multiple iterations. A key reason for this phenomenon is that a deep network is expressed using a highly non-convex finite-dimensional model, and thus the parameter gets stuck in a local optimum. Because of this, methods often suffer not only from degeneration of the convergence speed but also from limitations in the representational power of the trained network. To overcome this issue, we propose an additional layer called the gradient layer to seek a descent direction in an infinite-dimensional space. Because the layer is constructed in the infinite-dimensional space, we are not restricted by the specific model structure of finite-dimensional models. As a result, we can get out of the local optima in finite-dimensional models and move towards the global optimal function more directly. In this paper, this phenomenon is explained from the functional gradient method perspective of the gradient layer. Interestingly, the optimization procedure using the gradient layer naturally constructs the deep structure of the network. Moreover, we demonstrate that this procedure can be regarded as a discretization method of the gradient flow that naturally reduces the objective function. Finally, the method is tested using several numerical experiments, which show its fast convergence.

We develop a new theoretical framework to analyze the generalization error of deep learning, and derive a new fast learning rate for two representative algorithms: empirical risk minimization and Bayesian deep learning. The series of theoretical analyses of deep learning has revealed its high expressive power and universal approximation capability. Our point of view is to deal with the ordinary finite dimensional deep neural network as a finite approximation of the infinite dimensional one. Our formulation of the infinite dimensional model naturally defines a reproducing kernel Hilbert space corresponding to each layer. The approximation error is evaluated by the degree of freedom of the reproducing kernel Hilbert space in each layer. We derive the generalization error bound of both of empirical risk minimization and Bayesian deep learning and it is shown that there appears bias-variance trade-off in terms of the number of parameters of the finite dimensional approximation. We show that the optimal width of the internal layers can be determined through the degree of freedom and derive the optimal convergence rate that is faster than $O(1/\sqrt{n})$ rate which has been shown in the existing studies.

Sparse regularization such as l1 regularization is a quite powerful and widely used strategy for high dimensional learning problems. The effectiveness of sparse regularization has been supported practically and theoretically by several studies. However, one of the biggest issues in sparse regularization is that its performance is quite sensitive to correlations between features. Ordinary l1 regularization can select variables correlated with each other, which results in deterioration of not only its generalization error but also interpretability. In this paper, we pro- pose a new regularization method, “Independently Interpretable Lasso” (IILasso). Our proposed regularizer suppresses selecting correlated variables, and thus each active variable independently affects the objective variable in the model. Hence, we can interpret regression coefficients intuitively and also improve the performance by avoiding overfitting. We analyze theoretical property of IILasso and show that the proposed method is much advantageous for its sign recovery and achieves almost minimax optimal convergence rate. Synthetic and real data analyses also indicate the effectiveness of IILasso.

This paper proposes a very-short-term, i.e., less than 1-hour, local weather forecast method. In general, a short-term weather forecast within 3 hours is difficult due to lack of surface weather data and limitations of computation resources. However, such a short-term prediction is getting more and more anticipated in several industrial situations such as transportation, retailing business, agriculture, and energy management as well as our daily life. To keep up with this huge demands, services based on very-short-term weather forecast began to be provided. Data sources for such kind of forecast are private company-owned surface sensor networks in addition to nation-owned surface sensors. Some surface weather sensors of private companies are more densely distributed than nation-owned sensors. We call these surface weather sensor network as dense weather stations. Among them, for example, POTEKA sensors are located in roughly every 2 to 3 km and provide observed data every minute through mobile network. Those dense sensors are spreading its locations over the world. However, a data mining technique for such a device has not been well developed. In this paper, we propose a deep learning architecture specifically developed for the short-term weather forecasting based on the dense weather station device. Our proposal consists of two folds: point prediction model and tensor prediction model. The point prediction model is useful for forecasting exactly on the location of the dense weather station. The tensor prediction model interpolate the prediction of the point prediction model to cover whole range of locations around the interested area. It is shown that our model outperforms the existing state-of-the-art methods such as XGBoost and support vector machines using a large real observed data.

Residual Networks (ResNets) have become state-of-the-art models in deep learning and several theoretical studies have been devoted to understanding why ResNet works so well. One attractive viewpoint on ResNet is that it is optimizing the risk in a functional space by consisting of an ensemble of effective features. In this paper, we adopt this viewpoint to construct a new gradient boosting method, which is known to be very powerful in data analysis. To do so, we formalize the boosting perspective of ResNet mathematically using the notion of functional gradients and propose a new method called ResFGB for classification tasks by leveraging ResNet perception. Two types of generalization guarantees are provided from the optimization perspective: one is the margin bound and the other is the expected risk bound by the sample-splitting technique. Experimental results show superior performance of the proposed method over state-of-the-art methods such as LightGBM.

In recent years, deep neural networks (DNNs) have been applied to various machine leaning tasks, including image recognition, speech recognition, and machine translation. However, large DNN models are needed to achieve state-of-the-art performance, exceeding the capabilities of edge devices. Model reduction is thus needed for practical use. In this paper, we point out that deep learning automatically induces group sparsity of weights, in which all weights connected to an output channel (node) are zero, when training DNNs under the following three conditions: (1) rectified-linear-unit (ReLU) activations, (2) an L2-regularized objective function, and (3) the Adam optimizer. Next, we analyze this behavior both theoretically and experimentally, and propose a simple model reduction method: eliminate the zero weights after training the DNN. In experiments on MNIST and CIFAR-10 datasets, we demonstrate the sparsity with various training setups. Finally, we show that our method can efficiently reduce the model size and performs well relative to methods that use a sparsity-inducing regularizer.

We develop new stochastic gradient methods for efficiently solving sparse linear regression in a partial attribute observation setting, where learners are only allowed to observe a fixed number of actively chosen attributes per example at training and prediction times. It is shown that the methods achieve essentially a sample complexity of $O(1/\varepsilon)$ to attain an error of $\varepsilon$ under a variant of restricted eigenvalue condition, and the rate has better dependency on the problem dimension than existing methods. Particularly, if the smallest magnitude of the non-zero components of the optimal solution is not too small, the rate of our proposed {\it Hybrid} algorithm can be boosted to near the minimax optimal sample complexity of {\it full information} algorithms. The core ideas are (i) efficient construction of an unbiased gradient estimator by the iterative usage of the hard thresholding operator for configuring an exploration algorithm; and (ii) an adaptive combination of the exploration and an exploitation algorithms for quickly identifying the support of the optimum and efficiently searching the optimal parameter in its support. Experimental results are presented to validate our theoretical findings and the superiority of our proposed methods.

Difference of convex functions (DC) programming is an important approach to nonconvex optimization problems because these structures can be encountered in several fields. Effective optimization methods, called DC algorithms, have been developed in deterministic optimization literature. In machine learning, a lot of important learning problems such as the Boltzmann machines (BMs) can be formulated as DC programming. However, there is no DC-like algorithm guaranteed by convergence rate analysis for stochastic problems that are more suitable settings for machine learning tasks. In this paper, we propose a stochastic variant of DC algorithm and give computational complexities to converge to a stationary point under several situations. Moreover, we show our method includes expectation-maximization (EM) and Monte Carlo EM (MCEM) algorithm as special cases on training BMs. In other words, we extend EM/MCEM algorithm to more effective methods from DC viewpoint with theoretical convergence guarantees. Experimental results indicate that our method performs well for training binary restricted Boltzmann machines and deep Boltzmann machines without pre-training.

Density ratio estimation is a vital tool in both machine learning and statistical community. However, due to the unbounded nature of density ratio, the estimation proceudre can be vulnerable to corrupted data points, which often pushes the estimated ratio toward infinity. In this paper, we present a robust estimator which automatically identifies and trims outliers. The proposed estimator has a convex formulation, and the global optimum can be obtained via subgradient descent. We analyze the parameter estimation error of this estimator under high-dimensional settings. Experiments are conducted to verify the effectiveness of the estimator.

We develop a new accelerated stochastic gradient method for efficiently solving the convex regularized empirical risk minimization problem in mini-batch settings. The use of mini-batches has become a golden standard in the machine learning community, because the mini-batch techniques stabilize the gradient estimate and can easily make good use of parallel computing. The core of our proposed method is the incorporation of our new ``double acceleration'' technique and variance reduction technique. We theoretically analyze our proposed method and show that our method much improves the mini-batch efficiencies of previous accelerated stochastic methods, and essentially only needs size $\sqrt{n}$ mini-batches for achieving the optimal iteration complexities for both non-strongly and strongly convex objectives, where $n$ is the training set size. Further, we show that even in non-mini-batch settings, our method achieves the best known convergence rate for non-strongly convex and strongly convex objectives.

We investigate the statistical efficiency of a nonparametric Gaussian process method for a nonlinear tensor estimation problem. Low-rank tensor estimation has been used as a method to learn higher order relations among several data sources in a wide range of applications, such as multi-task learning, recommendation systems, and spatiotemporal analysis. We consider a general setting where a common linear tensor learning is extended to a nonlinear learning problem in reproducing kernel Hilbert space and propose a nonparametric Bayesian method based on the Gaussian process method. We prove its statistical convergence rate without assuming any strong convexity, such as restricted strong convexity. Remarkably, it is shown that our convergence rate achieves the minimax optimal rate. We apply our proposed method to multi-task learning and show that our method significantly outperforms existing methods through numerical experiments on real-world data sets.

We learn the structure of a Markov Network between two groups of random variables from joint observations. Since modelling and learning the full MN structure may be hard, learning the links between two groups directly may be a preferable option. We introduce a novel concept called the \emphpartitioned ratio whose factorization directly associates with the Markovian properties of random variables across two groups. A simple one-shot convex optimization procedure is proposed for learning the \emphsparse factorizations of the partitioned ratio and it is theoretically guaranteed to recover the correct inter-group structure under mild conditions. The performance of the proposed method is experimentally compared with the state of the art MN structure learning methods using ROC curves. Real applications on analyzing bipartisanship in US congress and pairwise DNA/time-series alignments are also reported.

We investigate the statistical performance and computational efficiency of the alternating minimization procedure for nonparametric tensor learning. Tensor modeling has been widely used for capturing the higher order relations between multimodal data sources. In addition to a linear model, a nonlinear tensor model has been received much attention recently because of its high flexibility. We consider an alternating minimization procedure for a general nonlinear model where the true function consists of components in a reproducing kernel Hilbert space (RKHS). In this paper, we show that the alternating minimization method achieves linear convergence as an optimization algorithm and that the generalization error of the resultant estimator yields the minimax optimality. We apply our algorithm to some multitask learning problems and show that the method actually shows favorable performances.

We study the problem of learning sparse structure changes between two Markov networks P and Q. Rather than fitting two Markov networks separately to two sets of data and figuring out their differences, a recent work proposed to learn changes directly via estimating the ratio between two Markov network models. Such a direct approach was demonstrated to perform excellently in experiments, although its theoretical properties remained unexplored. In this paper, we give sufficient conditions for successful change detection with respect to the sample size np, nq, the dimension of data m, and the number of changed edges d.

The anomaly localization task aims at detecting faulty sensors automatically by monitoring the sensor values. In this paper, we propose an anomaly localization algorithm with a consistency guarantee on its results. Although several algorithms were proposed in the last decade, the consistency of the localization results was not discussed in the literature. To the best of our knowledge, this is the first study that provides theoretical guarantees for the localization results. Our new approach is to formulate the task as solving the sparsest subgraph problem on a difference graph. Since this problem is NP-hard, we then use a convex quadratic programming approximation algorithm, which is guaranteed to be consistent under suitable conditions. Across the simulations on both synthetic and real world datasets, we verify that the proposed method achieves higher anomaly localization performance compared to existing methods.

We investigate the statistical convergence rate of a Bayesian low-rank tensor estimator, and derive the minimax optimal rate for learning a low-rank tensor. Our problem setting is the regression problem where the regression coefficient forms a tensor structure. This problem setting occurs in many practical applications, such as collaborative filtering, multi-task learning, and spatio-temporal data analysis. The convergence rate of the Bayes tensor estimator is analyzed in terms of both in-sample and out-of-sample predictive accuracies. It is shown that a fast learning rate is achieved without any strong convexity of the observation. Moreover, we show that the method has adaptivity to the unknown rank of the true tensor, that is, the near optimal rate depending on the true rank is achieved even if it is not known a priori. Finally, we show the minimax optimal learning rate for the tensor estimation problem, and thus show that the derived bound of the Bayes estimator is tight and actually near minimax optimal.

We propose a new stochastic dual coordinate ascent technique that can be applied to a wide range of regularized learning problems. Our method is based on alternating direction method of multipliers (ADMM) to deal with complex regularization functions such as structured regularizations. Although the original ADMM is a batch method, the proposed method offers a stochastic update rule where each iteration requires only one or few sample observations. Moreover, our method can naturally afford mini-batch update and it gives speed up of convergence. We show that, under mild assumptions, our method converges exponentially. The numerical experiments show that our method actually performs efficiently.

We propose a new method for detecting changes in Markov network structure between two sets of samples. Instead of naively fitting two Markov network models separately to the two data sets and figuring out their difference, we directly learn the network structure change by estimating the ratio of Markov network models. This density-ratio formulation naturally allows us to introduce sparsity in the network structure change, which highly contributes to enhancing interpretability. Furthermore, computation of the normalization term, a critical bottleneck of the naive approach, can be remarkably mitigated. We also give the dual formulation of the optimization problem, which further reduces the computation cost for large-scale Markov networks. Through experiments, we demonstrate the usefulness of our method.

We develop new stochastic optimization methods that are applicable to a wide range of structured regularizations. Basically our methods are combinations of basic stochastic optimization techniques and Alternating Direction Multiplier Method (ADMM). ADMM is a general framework for optimizing a composite function, and has a wide range of applications. We propose two types of online variants of ADMM, which correspond to online proximal gradient descent and regularized dual averaging respectively. The proposed algorithms are computationally efficient and easy to implement. Our methods yield O(1/\sqrtT) convergence of the expected risk. Moreover, the online proximal gradient descent type method yields O(\log(T)/T) convergence for a strongly convex loss. Numerical experiments show effectiveness of our methods in learning tasks with structured sparsity such as overlapped group lasso.

We propose a new class of structured Schatten norms for tensors that includes two recently proposed norms (overlapped'' and "latent'') for convex-optimization-based tensor decomposition. Based on the properties of the structured Schatten norms, we mathematically analyze the performance of "latent'' approach for tensor decomposition, which was empirically found to perform better than the "overlapped'' approach in some settings. We show theoretically that this is indeed the case. In particular, when the unknown true tensor is low-rank in a specific mode, this approach performs as well as knowing the mode with the smallest rank. Along the way, we show a novel duality result for structures Schatten norms, which is also interesting in the general context of structured sparsity. We confirm through numerical simulations that our theory can precisely predict the scaling behaviour of the mean squared error. "

In this study, the computational properties of a kernel-based least-squares density-ratio estimator are investigated from the viewpoint of condition numbers . The condition number of the Hessian matrix of the loss function is closely related to the convergence rate of optimization and the numerical stability. We use smoothed analysis techniques and theoretically demonstrate that the kernel least-squares method has a smaller condition number than other M-estimators. This implies that the kernel least-squares method has desirable computational properties. In addition, an alternate formulation of the kernel least-squares estimator that possesses an even smaller condition number is presented. The validity of the theoretical analysis is verified through numerical experiments.

We address the problem of estimating the difference between two probability densities. A naive approach is a two-step procedure of first estimating two densities separately and then computing their difference. However, this procedure does not necessarily work well because the first step is performed without regard to the second step, and thus a small estimation error incurred in the first stage can cause a big error in the second stage. In this letter, we propose a single-shot procedure for directly estimating the density difference without separately estimating two densities. We derive a nonparametric finite-sample error bound for the proposed single-shot density-difference estimator and show that it achieves the optimal convergence rate. We then show how the proposed density-difference estimator can be used in L²-distance approximation. Finally, we experimentally demonstrate the usefulness of the proposed method in robust distribution comparison such as class-prior estimation and change-point detection.

The goal of sufficient dimension reduction in supervised learning is to find the low-dimensional subspace of input features that contains all of the information about the output values that the input features possess. In this letter, we propose a novel sufficient dimension-reduction method using a squared-loss variant of mutual information as a dependency measure. We apply a density-ratio estimator for approximating squared-loss mutual information that is formulated as a minimum contrast estimator on parametric or nonparametric models. Since cross-validation is available for choosing an appropriate model, our method does not require any prespecified structure on the underlying distributions. We elucidate the asymptotic bias of our estimator on parametric models and the asymptotic convergence rate on nonparametric models. The convergence analysis utilizes the uniform tail-bound of a U-process, and the convergence rate is characterized by the bracketing entropy of the model. We then develop a natural gradient algorithm on the Grassmann manifold for sufficient subspace search. The analytic formula of our estimator allows us to compute the gradient efficiently. Numerical experiments show that the proposed method compares favorably with existing dimension-reduction approaches on artificial and benchmark data sets.

Divergence estimators based on direct approximation of density ratios without going through separate approximation of numerator and denominator densities have been successfully applied to machine learning tasks that involve distribution comparison such as outlier detection, transfer learning, and two-sample homogeneity test. However, since density-ratio functions often possess high fluctuation, divergence estimation is a challenging task in practice. In this letter, we use relative divergences for distribution comparison, which involves approximation of relative density ratios. Since relative density ratios are always smoother than corresponding ordinary density ratios, our proposed method is favorable in terms of nonparametric convergence speed. Furthermore, we show that the proposed divergence estimator has asymptotic variance independent of the model complexity under a parametric setup, implying that the proposed estimator hardly overfits even with complex models. Through experiments, we demonstrate the usefulness of the proposed approach.

We address the problem of estimating the difference between two probability densities. A naive approach is a two-step procedure of first estimating two densities separately and then computing their difference. However, such a two-step procedure does not necessarily work well because the first step is performed without regard to the second step and thus a small estimation error incurred in the first stage can cause a big error in the second stage. In this paper, we propose a single-shot procedure for directly estimating the density difference without separately estimating two densities. We derive a non-parametric finite-sample error bound for the proposed single-shot density-difference estimator and show that it achieves the optimal convergence rate. We then show how the proposed density-difference estimator can be utilized in L2-distance approximation. Finally, we experimentally demonstrate the usefulness of the proposed method in robust distribution comparison such as class-prior estimation and change-point detection.

A density ratio is defined by the ratio of two probability densities. We study the inference problem of density ratios and apply a semiparametric density-ratio estimator to the two-sample homogeneity test. In the proposed test procedure, the$f$-divergence between two probability densities is estimated using a density-ratio estimator. The$f$-divergence estimator is then exploited for the two-sample homogeneity test. We derive an optimal estimator of$f$-divergence in the sense of the asymptotic variance in a semiparametric setting, and provide a statistic for two-sample homogeneity test based on the optimal estimator. We prove that the proposed test dominates the existing empirical likelihood score test. Through numerical studies, we illustrate the adequacy of the asymptotic theory for finite-sample inference.

In this paper, we give a new generalization error bound of Multiple Kernel Learning (MKL) for a general class of regularizations. Our main target in this paper is dense type regularizations including ℓp-MKL that imposes ℓp-mixed-norm regularization instead of ℓ1-mixed-norm regularization. According to the recent numerical experiments, the sparse regularization does not necessarily show a good performance compared with dense type regularizations. Motivated by this fact, this paper gives a general theoretical tool to derive fast learning rates that is applicable to arbitrary monotone norm-type regularizations in a unifying manner. As a by-product of our general result, we show a fast learning rate of ℓp-MKL that is tightest among existing bounds. We also show that our general learning rate achieves the minimax lower bound. Finally, we show that, when the complexities of candidate reproducing kernel Hilbert spaces are inhomogeneous, dense type regularization shows better learning rate compared with sparse ℓ1 regularization.

We analyze the statistical performance of a recently proposed convex tensor decomposition algorithm. Conventionally tensor decomposition has been formulated as non-convex optimization problems, which hindered the analysis of their performance. We show under some conditions that the mean squared error of the convex method scales linearly with the quantity we call the normalized rank of the true tensor. The current analysis naturally extends the analysis of convex low-rank matrix estimation to tensors. Furthermore, we show through numerical experiments that our theory can precisely predict the scaling behaviour in practice.

Divergence estimators based on direct approximation of density-ratios without going through separate approximation of numerator and denominator densities have been successfully applied to machine learning tasks that involve distribution comparison such as outlier detection, transfer learning, and two-sample homogeneity test. However, since density-ratio functions often possess high fluctuation, divergence estimation is still a challenging task in practice. In this paper, we propose to use relative divergences for distribution comparison, which involves approximation of relative density-ratios. Since relative density-ratios are always smoother than corresponding ordinary density-ratios, our proposed method is favorable in terms of the non-parametric convergence speed. Furthermore, we show that the proposed divergence estimator has asymptotic variance independent of the model complexity under a parametric setup, implying that the proposed estimator hardly overfits even with complex models. Through experiments, we demonstrate the usefulness of the proposed approach.

Accurately evaluating statistical independence among random variables is a key element of independent component analysis (ICA). In this letter, we employ a squared-loss variant of mutual information as an independence measure and give its estimation method. Our basic idea is to estimate the ratio of probability densities directly without going through density estimation, thereby avoiding the difficult task of density estimation. In this density ratio approach, a natural cross-validation procedure is available for hyperparameter selection. Thus, all tuning parameters such as the kernel width or the regularization parameter can be objectively optimized. This is an advantage over recently developed kernel-based independence measures and is a highly useful property in unsupervised learning problems such as ICA. Based on this novel independence measure, we develop an ICA algorithm, named least-squares independent component analysis.

Methods for directly estimating the ratio of two probability density functions without going through density estimation have been actively explored recently since they can be used for various data processing tasks such as non-stationarity adaptation, outlier detection, conditional density estimation, feature selection, and independent component analysis. However, even the state-of-the-art density ratio estimation methods still perform rather poorly in high-dimensional problems. In this paper, we propose a new density ratio estimation method which incorporates dimensionality reduction into a density ratio estimation procedure. Our key idea is to identify a low-dimensional subspace in which the two densities corresponding to the denominator and the numerator in the density ratio are significantly different. Then the density ratio is estimated only within this low-dimensional subspace. Through numerical examples, we illustrate the effectiveness of the proposed method.

We propose a new method of approximating mutual information based on maximum likelihood estimation of a density ratio function. The proposed method, Maximum Likelihood Mutual Information (MLMI), possesses useful properties, e.g., it does not involve density estimation, the global optimal solution can be efficiently computed, it has suitable convergence properties, and model selection criteria are available. Numerical experiments show that MLMI compares favorably with existing methods.

BACKGROUND: Although microarray gene expression analysis has become popular, it remains difficult to interpret the biological changes caused by stimuli or variation of conditions. Clustering of genes and associating each group with biological functions are often used methods. However, such methods only detect partial changes within cell processes. Herein, we propose a method for discovering global changes within a cell by associating observed conditions of gene expression with gene functions. RESULTS: To elucidate the association, we introduce a novel feature selection method called Least-Squares Mutual Information (LSMI), which computes mutual information without density estimaion, and therefore LSMI can detect nonlinear associations within a cell. We demonstrate the effectiveness of LSMI through comparison with existing methods. The results of the application to yeast microarray datasets reveal that non-natural stimuli affect various biological processes, whereas others are no significant relation to specific cell processes. Furthermore, we discover that biological processes can be categorized into four types according to the responses of various stimuli: DNA/RNA metabolism, gene expression, protein metabolism, and protein localization. CONCLUSION: We proposed a novel feature selection method called LSMI, and applied LSMI to mining the association between conditions of yeast and biological processes through microarray datasets. In fact, LSMI allows us to elucidate the global organization of cellular process control.

Many studies of man-machine interaction using eye trackers have been tackled over recent decades. In this paper, we present a new learning system to estimate user interest with gaze sensory information. In short, a statistical learning scheme, especially the variational Bayes (VB), is incorporated for building probabilistic model parameters, dealing with the uncertainty of estimated user interest. Several computational results show how the VB can cope with user interest estimation, by selectively modeling their uncertainty.