Farzan Farnia

Azim Ospanov, Jingwei Zhang, Mohammad Jalali et al.

While standard evaluation scores for generative models are mostly reference-based, a reference-dependent assessment of generative models could be generally difficult due to the unavailability of applicable reference datasets. Recently, the reference-free entropy scores, VENDI and RKE, have been proposed to evaluate the diversity of generated data. However, estimating these scores from data leads to significant computational costs for large-scale generative models. In this work, we leverage the random Fourier features framework to reduce the computational price and propose the Fourier-based Kernel Entropy Approximation (FKEA) method. We utilize FKEA's approximated eigenspectrum of the kernel matrix to efficiently estimate the mentioned entropy scores. Furthermore, we show the application of FKEA's proxy eigenvectors to reveal the method's identified modes in evaluating the diversity of produced samples. We provide a stochastic implementation of the FKEA assessment algorithm with a complexity $O(n)$ linearly growing with sample size $n$. We extensively evaluate FKEA's numerical performance in application to standard image, text, and video datasets. Our empirical results indicate the method's scalability and interpretability applied to large-scale generative models. The codebase is available at https://github.com/aziksh-ospanov/FKEA.

Farzan Farnia, Amirhossein Reisizadeh, Ramtin Pedarsani et al.

Federated learning is a distributed machine learning paradigm, which aims to train a model using the local data of many distributed clients. A key challenge in federated learning is that the data samples across the clients may not be identically distributed. To address this challenge, personalized federated learning with the goal of tailoring the learned model to the data distribution of every individual client has been proposed. In this paper, we focus on this problem and propose a novel personalized Federated Learning scheme based on Optimal Transport (FedOT) as a learning algorithm that learns the optimal transport maps for transferring data points to a common distribution as well as the prediction model under the applied transport map. To formulate the FedOT problem, we extend the standard optimal transport task between two probability distributions to multi-marginal optimal transport problems with the goal of transporting samples from multiple distributions to a common probability domain. We then leverage the results on multi-marginal optimal transport problems to formulate FedOT as a min-max optimization problem and analyze its generalization and optimization properties. We discuss the results of several numerical experiments to evaluate the performance of FedOT under heterogeneous data distributions in federated learning problems.

Haochuan Li, Farzan Farnia, Subhro Das et al.

Gradient Descent Ascent (GDA) methods are the mainstream algorithms for minimax optimization in generative adversarial networks (GANs). Convergence properties of GDA have drawn significant interest in the recent literature. Specifically, for $\min_{\mathbf{x}} \max_{\mathbf{y}} f(\mathbf{x};\mathbf{y})$ where $f$ is strongly-concave in $\mathbf{y}$ and possibly nonconvex in $\mathbf{x}$, (Lin et al., 2020) proved the convergence of GDA with a stepsize ratio $η_{\mathbf{y}}/η_{\mathbf{x}}=Θ(κ^2)$ where $η_{\mathbf{x}}$ and $η_{\mathbf{y}}$ are the stepsizes for $\mathbf{x}$ and $\mathbf{y}$ and $κ$ is the condition number for $\mathbf{y}$. While this stepsize ratio suggests a slow training of the min player, practical GAN algorithms typically adopt similar stepsizes for both variables, indicating a wide gap between theoretical and empirical results. In this paper, we aim to bridge this gap by analyzing the \emph{local convergence} of general \emph{nonconvex-nonconcave} minimax problems. We demonstrate that a stepsize ratio of $Θ(κ)$ is necessary and sufficient for local convergence of GDA to a Stackelberg Equilibrium, where $κ$ is the local condition number for $\mathbf{y}$. We prove a nearly tight convergence rate with a matching lower bound. We further extend the convergence guarantees to stochastic GDA and extra-gradient methods (EG). Finally, we conduct several numerical experiments to support our theoretical findings.

Yulai Zhao, Wenhao Zhan, Xiaoyan Hu et al. · princeton

We study risk-sensitive Reinforcement Learning (RL), where we aim to maximize the Conditional Value at Risk (CVaR) with a fixed risk tolerance $τ$. Prior theoretical work studying risk-sensitive RL focuses on the tabular Markov Decision Processes (MDPs) setting. To extend CVaR RL to settings where state space is large, function approximation must be deployed. We study CVaR RL in low-rank MDPs with nonlinear function approximation. Low-rank MDPs assume the underlying transition kernel admits a low-rank decomposition, but unlike prior linear models, low-rank MDPs do not assume the feature or state-action representation is known. We propose a novel Upper Confidence Bound (UCB) bonus-driven algorithm to carefully balance the interplay between exploration, exploitation, and representation learning in CVaR RL. We prove that our algorithm achieves a sample complexity of $\tilde{O}\left(\frac{H^7 A^2 d^4}{τ^2 ε^2}\right)$ to yield an $ε$-optimal CVaR, where $H$ is the length of each episode, $A$ is the capacity of action space, and $d$ is the dimension of representations. Computational-wise, we design a novel discretized Least-Squares Value Iteration (LSVI) algorithm for the CVaR objective as the planning oracle and show that we can find the near-optimal policy in a polynomial running time with a Maximum Likelihood Estimation oracle. To our knowledge, this is the first provably efficient CVaR RL algorithm in low-rank MDPs.

Azim Ospanov, Farzan Farnia, Roozbeh Yousefzadeh

We perform a thorough analysis of the formal and informal statements in the miniF2F benchmark from the perspective of an AI system that is tasked to participate in a math Olympiad consisting of the problems in miniF2F. In such setting, the model has to read and comprehend the problems in natural language, formalize them in Lean language, then proceed with proving the problems, and it will get credit for each problem if the formal proof corresponds to the original informal statement presented to the model. Our evaluation results reveal that the best accuracy of such pipeline can be about 36% using the SoTA models in the literature, considerably lower than the individual SoTA accuracies, 97% and 69% reported in the autoformalization and theorem proving literature. Analyzing the failure modes, we trace back a considerable portion of this drop to discrepancies between the formal and informal statements for more than half of the problems in miniF2F. We proceed with correcting all the errors, discrepancies and simplifications in formal and informal statements, and present the miniF2F-v2 with fully verified formal and informal statements and proofs. Evaluating the full theorem proving pipeline on miniF2F-v2 leads to the best accuracy of 70%, a significant improvement from the 40% on the original miniF2F, yet indicating considerable misalignment between the autoformalization models and theorem provers. Our deep analysis suggests that a higher quality benchmark can help the community better evaluate progress in the field of formal reasoning and also better diagnose the failure and success modes of autoformalization and theorem proving models. Our dataset is available at https://github.com/roozbeh-yz/miniF2F_v2.

Zixiao Wang, Yunheng Shen, Wenqian Zhao et al.

Deep generative models dominate the existing literature in layout pattern generation. However, leaving the guarantee of legality to an inexplicable neural network could be problematic in several applications. In this paper, we propose \tool{DiffPattern} to generate reliable layout patterns. \tool{DiffPattern} introduces a novel diverse topology generation method via a discrete diffusion model with compute-efficiently lossless layout pattern representation. Then a white-box pattern assessment is utilized to generate legal patterns given desired design rules. Our experiments on several benchmark settings show that \tool{DiffPattern} significantly outperforms existing baselines and is capable of synthesizing reliable layout patterns.

Yilin Wang, Farzan Farnia

Black-box adversarial attacks designing adversarial examples for unseen neural networks (NNs) have received great attention over the past years. While several successful black-box attack schemes have been proposed in the literature, the underlying factors driving the transferability of black-box adversarial examples still lack a thorough understanding. In this paper, we aim to demonstrate the role of the generalization properties of the substitute classifier used for generating adversarial examples in the transferability of the attack scheme to unobserved NN classifiers. To do this, we apply the max-min adversarial example game framework and show the importance of the generalization properties of the substitute NN in the success of the black-box attack scheme in application to different NN classifiers. We prove theoretical generalization bounds on the difference between the attack transferability rates on training and test samples. Our bounds suggest that a substitute NN with better generalization behavior could result in more transferable adversarial examples. In addition, we show that standard operator norm-based regularization methods could improve the transferability of the designed adversarial examples. We support our theoretical results by performing several numerical experiments showing the role of the substitute network's generalization in generating transferable adversarial examples. Our empirical results indicate the power of Lipschitz regularization methods in improving the transferability of adversarial examples.

Zixiao Wang, Farzan Farnia, Zhenghao Lin et al.

The evaluation of deep generative models has been extensively studied in the centralized setting, where the reference data are drawn from a single probability distribution. On the other hand, several applications of generative models concern distributed settings, e.g. the federated learning setting, where the reference data for conducting evaluation are provided by several clients in a network. In this paper, we study the evaluation of generative models in such distributed contexts with potentially heterogeneous data distributions across clients. We focus on the widely-used distance-based evaluation metrics, Fréchet Inception Distance (FID) and Kernel Inception Distance (KID). In the case of KID metric, we prove that scoring a group of generative models using the clients' averaged KID score will result in the same ranking as that of a centralized KID evaluation over a collective reference set containing all the clients' data. In contrast, we show the same result does not apply to the FID-based evaluation. We provide examples in which two generative models are assigned the same FID score by each client in a distributed setting, while the centralized FID scores of the two models are significantly different. We perform several numerical experiments on standard image datasets and generative models to support our theoretical results on the distributed evaluation of generative models using FID and KID scores.

Haniyeh Ehsani Oskouie, Farzan Farnia

Interpreting neural network classifiers using gradient-based saliency maps has been extensively studied in the deep learning literature. While the existing algorithms manage to achieve satisfactory performance in application to standard image recognition datasets, recent works demonstrate the vulnerability of widely-used gradient-based interpretation schemes to norm-bounded perturbations adversarially designed for every individual input sample. However, such adversarial perturbations are commonly designed using the knowledge of an input sample, and hence perform sub-optimally in application to an unknown or constantly changing data point. In this paper, we show the existence of a Universal Perturbation for Interpretation (UPI) for standard image datasets, which can alter a gradient-based feature map of neural networks over a significant fraction of test samples. To design such a UPI, we propose a gradient-based optimization method as well as a principal component analysis (PCA)-based approach to compute a UPI which can effectively alter a neural network's gradient-based interpretation on different samples. We support the proposed UPI approaches by presenting several numerical results of their successful applications to standard image datasets.

Jingwei Zhang, Farzan Farnia

Explaining the predictions of deep neural nets has been a topic of great interest in the computer vision literature. While several gradient-based interpretation schemes have been proposed to reveal the influential variables in a neural net's prediction, standard gradient-based interpretation frameworks have been commonly observed to lack robustness to input perturbations and flexibility for incorporating prior knowledge of sparsity and group-sparsity structures. In this work, we propose MoreauGrad as an interpretation scheme based on the classifier neural net's Moreau envelope. We demonstrate that MoreauGrad results in a smooth and robust interpretation of a multi-layer neural network and can be efficiently computed through first-order optimization methods. Furthermore, we show that MoreauGrad can be naturally combined with $L_1$-norm regularization techniques to output a sparse or group-sparse explanation which are prior conditions applicable to a wide range of deep learning applications. We empirically evaluate the proposed MoreauGrad scheme on standard computer vision datasets, showing the qualitative and quantitative success of the MoreauGrad approach in comparison to standard gradient-based interpretation methods.

Ching Lam Choi, Farzan Farnia

Despite their great success in image recognition tasks, deep neural networks (DNNs) have been observed to be susceptible to universal adversarial perturbations (UAPs) which perturb all input samples with a single perturbation vector. However, UAPs often struggle in transferring across DNN architectures and lead to challenging optimization problems. In this work, we study the transferability of UAPs by analyzing equilibrium in the universal adversarial example game between the classifier and UAP adversary players. We show that under mild assumptions the universal adversarial example game lacks a pure Nash equilibrium, indicating UAPs' suboptimal transferability across DNN classifiers. To address this issue, we propose Universal Adversarial Directions (UADs) which only fix a universal direction for adversarial perturbations and allow the perturbations' magnitude to be chosen freely across samples. We prove that the UAD adversarial example game can possess a Nash equilibrium with a pure UAD strategy, implying the potential transferability of UADs. We also connect the UAD optimization problem to the well-known principal component analysis (PCA) and develop an efficient PCA-based algorithm for optimizing UADs. We evaluate UADs over multiple benchmark image datasets. Our numerical results show the superior transferability of UADs over standard gradient-based UAPs.

Jingwei Zhang, Cheuk Ting Li, Farzan Farnia

The massive developments of generative model frameworks require principled methods for the evaluation of a model's novelty compared to a reference dataset. While the literature has extensively studied the evaluation of the quality, diversity, and generalizability of generative models, the assessment of a model's novelty compared to a reference model has not been adequately explored in the machine learning community. In this work, we focus on the novelty assessment for multi-modal distributions and attempt to address the following differential clustering task: Given samples of a generative model $P_\mathcal{G}$ and a reference model $P_\mathrm{ref}$, how can we discover the sample types expressed by $P_\mathcal{G}$ more frequently than in $P_\mathrm{ref}$? We introduce a spectral approach to the differential clustering task and propose the Kernel-based Entropic Novelty (KEN) score to quantify the mode-based novelty of $P_\mathcal{G}$ with respect to $P_\mathrm{ref}$. We analyze the KEN score for mixture distributions with well-separable components and develop a kernel-based method to compute the KEN score from empirical data. We support the KEN framework by presenting numerical results on synthetic and real image datasets, indicating the framework's effectiveness in detecting novel modes and comparing generative models. The paper's code is available at: www.github.com/buyeah1109/KEN

Haoyu Lei, Chin Wa Lau, Kaiwen Zhou et al.

Error Correction Codes (ECC) are fundamental to reliable digital communication, yet designing neural decoders that are both accurate and computationally efficient remains challenging. Recent denoising diffusion decoders with transformer backbones achieve state-of-the-art performance, but their iterative sampling limits practicality in low-latency settings. We introduce the Error Correction Consistency Flow Model (ECCFM), an architecture-agnostic training framework for high-fidelity one-step decoding. By casting the reverse denoising process as a Probability Flow Ordinary Differential Equation (PF-ODE) and enforcing smoothness through a differential time regularization, ECCFM learns to map noisy signals along the decoding trajectory directly to the original codeword in a single inference step. Across multiple decoding benchmarks, ECCFM attains lower bit-error rates (BER) than autoregressive and diffusion-based baselines, with notable improvements on longer codes, while delivering inference speeds up from 30x to 100x faster than denoising diffusion decoders.

Youqi Wu, Farzan Farnia

Von Neumann entropy (VNE) is a fundamental quantity in quantum information theory and has recently been adopted in machine learning as a spectral measure of diversity for kernel matrices and kernel covariance operators. While maximizing VNE under constraints is well known in quantum settings, a principled analogue of the classical maximum entropy framework, particularly its decision theoretic and game theoretic interpretation, has not been explicitly developed for VNE in data driven contexts. In this paper, we extend the minimax formulation of the maximum entropy principle due to Grünwald and Dawid to the setting of von Neumann entropy, providing a game-theoretic justification for VNE maximization over density matrices and trace-normalized positive semidefinite operators. This perspective yields a robust interpretation of maximum VNE solutions under partial information and clarifies their role as least committed inferences in spectral domains. We then illustrate how the resulting Maximum VNE principle applies to modern machine learning problems by considering two representative applications, selecting a kernel representation from multiple normalized embeddings via kernel-based VNE maximization, and completing kernel matrices from partially observed entries. These examples demonstrate how the proposed framework offers a unifying information-theoretic foundation for VNE-based methods in kernel learning.

Hossein Goli, Farzan Farnia

A reliable application of deep neural network classifiers requires robustness certificates against adversarial perturbations. Gaussian smoothing is a widely analyzed approach to certifying robustness against norm-bounded perturbations, where the certified prediction radius depends on the variance of the Gaussian noise and the confidence level of the neural net's prediction under the additive Gaussian noise. However, in application to high-dimensional image datasets, the certified radius of the plain Gaussian smoothing could be relatively small, since Gaussian noise with high variances can significantly harm the visibility of an image. In this work, we propose the Pixel Partitioning-based Randomized Smoothing (PPRS) methodology to boost the neural net's confidence score and thus the robustness radius of the certified prediction. We demonstrate that the proposed PPRS algorithm improves the visibility of the images under additive Gaussian noise. We discuss the numerical results of applying PPRS to standard computer vision datasets and neural network architectures. Our empirical findings indicate a considerable improvement in the certified accuracy and stability of the prediction model to the additive Gaussian noise in randomized smoothing.

Parham Rezaei, Farzan Farnia, Cheuk Ting Li

The availability of multiple training algorithms and architectures for generative models requires a selection mechanism to form a single model over a group of well-trained generation models. The selection task is commonly addressed by identifying the model that maximizes an evaluation score based on the diversity and quality of the generated data. However, such a best-model identification approach overlooks the possibility that a mixture of available models can outperform each individual model. In this work, we numerically show that a mixture of generative models on benchmark image datasets can indeed achieve a better evaluation score (based on FID and KID scores), compared to the individual models. This observation motivates the development of efficient algorithms for selecting the optimal mixture of the models. To address this, we formulate a quadratic optimization problem to find an optimal mixture model achieving the maximum of kernel-based evaluation scores including kernel inception distance (KID) and Rényi kernel entropy (RKE). To identify the optimal mixture of the models using the fewest possible sample queries, we view the selection task as a multi-armed bandit (MAB) problem and propose the Mixture Upper Confidence Bound (Mixture-UCB) algorithm that provably converges to the optimal mixture of the involved models. More broadly, the proposed Mixture-UCB can be extended to optimize every convex quadratic function of the mixture weights in a general MAB setting. We prove a regret bound for the Mixture-UCB algorithm and perform several numerical experiments to show the success of Mixture-UCB in finding the optimal mixture of text and image generative models. The project code is available at https://github.com/Rezaei-Parham/Mixture-UCB.

Jingwei Zhang, Mohammad Jalali, Cheuk Ting Li et al.

A fine-grained comparison of generative models requires the identification of sample types generated differently by each of the involved models. While quantitative scores have been proposed in the literature to rank different generative models, score-based evaluation and ranking do not reveal the nuanced differences between the generative models in producing different sample types. In this work, we propose solving a differential clustering problem to detect sample types generated differently by two generative models. To solve the differential clustering problem, we develop a spectral method called Fourier-based Identification of Novel Clusters (FINC) to identify modes produced by a generative model with a higher frequency in comparison to a reference distribution. FINC provides a scalable algorithm based on random Fourier features to estimate the eigenspace of kernel covariance matrices of two generative models and utilize the principal eigendirections to detect the sample types present more dominantly in each model. We demonstrate the application of the FINC method to large-scale computer vision datasets and generative modeling frameworks. Our numerical results suggest the scalability of the developed Fourier-based method in highlighting the sample types produced with different frequencies by generative models. The project code is available at https://github.com/buyeah1109/FINC.

Mohammad Jalali, Azim Ospanov, Amin Gohari et al.

Text-conditioned generation models are commonly evaluated based on the quality of the generated data and its alignment with the input text prompt. On the other hand, several applications of prompt-based generative models require sufficient diversity in the generated data to ensure the models' capability of generating image and video samples possessing a variety of features. However, most existing diversity metrics are designed for unconditional generative models, and thus cannot distinguish the diversity arising from variations in text prompts and that contributed by the generative model itself. In this work, our goal is to quantify the prompt-induced and model-induced diversity in samples generated by prompt-based models. We propose an information-theoretic approach for internal diversity quantification, where we decompose the kernel-based entropy $H(X)$ of the generated data $X$ into the sum of the conditional entropy $H(X|T)$, given text variable $T$, and the mutual information $I(X; T)$ between the text and data variables. We introduce the \emph{Conditional-Vendi} score based on $H(X|T)$ to quantify the internal diversity of the model and the \emph{Information-Vendi} score based on $I(X; T)$ to measure the statistical relevance between the generated data and text prompts. We provide theoretical results to statistically interpret these scores and relate them to the unconditional Vendi score. We conduct several numerical experiments to show the correlation between the Conditional-Vendi score and the internal diversity of text-conditioned generative models. The codebase is available at \href{https://github.com/mjalali/conditional-vendi}{https://github.com/mjalali/conditional-vendi}.

Azim Ospanov, Farzan Farnia

Evaluating the diversity of generative models without reference data poses methodological challenges. The reference-free Vendi and RKE scores address this by quantifying the diversity of generated data using matrix-based entropy measures. Among these two, the Vendi score is typically computed via the eigendecomposition of an $n \times n$ kernel matrix constructed from n generated samples. However, the prohibitive computational cost of eigendecomposition for large $n$ often limits the number of samples used to fewer than 20,000. In this paper, we investigate the statistical convergence of the Vendi and RKE scores under restricted sample sizes. We numerically demonstrate that, in general, the Vendi score computed with standard sample sizes below 20,000 may not converge to its asymptotic value under infinite sampling. To address this, we introduce the $t$-truncated Vendi score by truncating the eigenspectrum of the kernel matrix, which is provably guaranteed to converge to its population limit with $n=\mathcal{O}(t)$ samples. We further show that existing Nyström and FKEA approximation methods converge to the asymptotic limit of the truncated Vendi score. In contrast to the Vendi score, we prove that the RKE score enjoys universal convergence guarantees across all kernel functions. We conduct several numerical experiments to illustrate the concentration of Nyström and FKEA computed Vendi scores around the truncated Vendi score, and we analyze how the truncated Vendi and RKE scores correlate with the diversity of image and text data. The code is available at https://github.com/aziksh-ospanov/truncated-vendi.

Xiwei Cheng, Kexin Fu, Farzan Farnia

While adversarial training methods have significantly improved the robustness of deep neural networks against norm-bounded adversarial perturbations, the generalization gap between their performance on training and test data is considerably greater than that of standard empirical risk minimization. Recent studies have aimed to connect the generalization properties of adversarially trained classifiers to the min-max optimization algorithm used in their training. In this work, we analyze the interconnections between generalization and optimization in adversarial training using the algorithmic stability framework. Specifically, our goal is to compare the generalization gap of neural networks trained using the vanilla adversarial training method, which fully optimizes perturbations at every iteration, with the free adversarial training method, which simultaneously optimizes norm-bounded perturbations and classifier parameters. We prove bounds on the generalization error of these methods, indicating that the free adversarial training method may exhibit a lower generalization gap between training and test samples due to its simultaneous min-max optimization of classifier weights and perturbation variables. We conduct several numerical experiments to evaluate the train-to-test generalization gap in vanilla and free adversarial training methods. Our empirical findings also suggest that the free adversarial training method could lead to a smaller generalization gap over a similar number of training iterations. The paper code is available at https://github.com/Xiwei-Cheng/Stability_FreeAT.

Xiaoyan Hu, Ho-fung Leung, Farzan Farnia

Selecting a sample generation scheme from multiple prompt-based generative models, including large language models (LLMs) and prompt-guided image and video generation models, is typically addressed by choosing the model that maximizes an averaged evaluation score. However, this score-based selection overlooks the possibility that different models achieve the best generation performance for different types of text prompts. An online identification of the best generation model for various input prompts can reduce the costs associated with querying sub-optimal models. In this work, we explore the possibility of varying rankings of text-based generative models for different text prompts and propose an online learning framework to predict the best data generation model for a given input prompt. The proposed PAK-UCB algorithm addresses a contextual bandit (CB) setting with shared context variables across the arms, utilizing the generated data to update kernel-based functions that predict the score of each model available for unseen text prompts. Additionally, we leverage random Fourier features (RFF) to accelerate the online learning process of PAK-UCB. Our numerical experiments on real and simulated text-to-image and image-to-text generative models show that RFF-UCB performs successfully in identifying the best generation model across different sample types. The code is available at: github.com/yannxiaoyanhu/dgm-online-select.

Xiaoyan Hu, Lauren Pick, Ho-fung Leung et al.

The rapid advancement of generative AI has provided users with a wide range of well-trained models to address diverse prompts. When selecting a model for a given prompt, users should weigh not only its performance but also its service cost. However, existing model-selection methods typically emphasize performance while overlooking cost differences. In this paper, we introduce PromptWise, an online learning framework that assigns prompts to generative models in a cost-aware manner. PromptWise estimates prompt-model compatibility to select the least expensive model expected to deliver satisfactory outputs. Unlike standard contextual bandits that make a one-shot decision per prompt, PromptWise employs a cost-aware bandit structure that allows sequential model assignments per prompt to reduce total service cost. Through numerical experiments on tasks such as code generation and translation, we demonstrate that PromptWise can achieve performance comparable to baseline selection methods while incurring substantially lower costs. The code is available at: github.com/yannxiaoyanhu/PromptWise.

Azim Ospanov, Mohammad Jalali, Farzan Farnia

The use of CLIP embeddings to assess the fidelity of samples produced by text-to-image generative models has been extensively explored in the literature. While the widely adopted CLIPScore, derived from the cosine similarity of text and image embeddings, effectively measures the alignment of a generated image, it does not quantify the diversity of images generated by a text-to-image model. In this work, we extend the application of CLIP embeddings to quantify and interpret the intrinsic diversity of text-to-image models, which are responsible for generating diverse images from similar text prompts, which we refer to as prompt-aware diversity. To achieve this, we propose a decomposition of the CLIP-based kernel covariance matrix of image data into text-based and non-text-based components. Using the Schur complement of the joint image-text kernel covariance matrix, we perform this decomposition and define the matrix-based entropy of the decomposed component as the Schur Complement ENtopy DIversity (Scendi) score, as a measure of the prompt-aware diversity for prompt-guided generative models. Additionally, we discuss the application of the Schur complement-based decomposition to nullify the influence of a given prompt on the CLIP embedding of an image, enabling focus or defocus of the embedded vectors on specific objects. We present several numerical results that apply our proposed Scendi score to evaluate text-to-image and LLM (text-to-text) models. Our numerical results indicate the success of the Scendi score in capturing the intrinsic diversity of prompt-guided generative models. The codebase is available at https://github.com/aziksh-ospanov/scendi-score.

Xiaoyan Hu, Ho-fung Leung, Farzan Farnia

Existing frameworks for evaluating and comparing generative models consider an offline setting, where the evaluator has access to large batches of data produced by the models. However, in practical scenarios, the goal is often to identify and select the best model using the fewest possible generated samples to minimize the costs of querying data from the sub-optimal models. In this work, we propose an online evaluation and selection framework to find the generative model that maximizes a standard assessment score among a group of available models. We view the task as a multi-armed bandit (MAB) and propose upper confidence bound (UCB) bandit algorithms to identify the model producing data with the best evaluation score that quantifies the quality and diversity of generated data. Specifically, we develop the MAB-based selection of generative models considering the Fréchet Distance (FD) and Inception Score (IS) metrics, resulting in the FD-UCB and IS-UCB algorithms. We prove regret bounds for these algorithms and present numerical results on standard image datasets. Our empirical results suggest the efficacy of MAB approaches for the sample-efficient evaluation and selection of deep generative models. The project code is available at https://github.com/yannxiaoyanhu/dgm-online-eval.

Zixiao Wang, Jingwei Zhang, Wenqian Zhao et al.

Few-shot gradient methods have been extensively utilized in existing model pruning methods, where the model weights are regarded as static values and the effects of potential weight perturbations are not considered. However, the widely used large language models (LLMs) have several billion model parameters, which could increase the fragility of few-shot gradient pruning. In this work, we experimentally show that one-shot gradient pruning algorithms could lead to unstable results under perturbations to model weights. And the minor error of switching between data formats bfloat16 and float16 could result in drastically different outcomes. To address such instabilities, we leverage optimization analysis and propose an LLM structural pruning method, called MoreauPruner, with provable robustness against weight perturbations. In MoreauPruner, the model weight importance is estimated based on the neural network's Moreau envelope, which can be flexibly combined with $\ell_1$-norm regularization techniques to induce the sparsity required in the pruning task. We extensively evaluate the MoreauPruner algorithm on several well-known LLMs, including LLaMA-7B, LLaMA-13B, LLaMA3-8B, and Vicuna-7B. Our numerical results suggest the robustness of MoreauPruner against weight perturbations, and indicate the MoreauPruner's successful accuracy-based scores in comparison to several existing pruning methods. We have released the code in \url{https://github.com/ShiningSord/MoreauPruner}.

Farzan Farnia, Jesse M. Zhang, David Tse

Deep neural networks (DNNs) have set benchmarks on a wide array of supervised learning tasks. Trained DNNs, however, often lack robustness to minor adversarial perturbations to the input, which undermines their true practicality. Recent works have increased the robustness of DNNs by fitting networks using adversarially-perturbed training samples, but the improved performance can still be far below the performance seen in non-adversarial settings. A significant portion of this gap can be attributed to the decrease in generalization performance due to adversarial training. In this work, we extend the notion of margin loss to adversarial settings and bound the generalization error for DNNs trained under several well-known gradient-based attack schemes, motivating an effective regularization scheme based on spectral normalization of the DNN's weight matrices. We also provide a computationally-efficient method for normalizing the spectral norm of convolutional layers with arbitrary stride and padding schemes in deep convolutional networks. We evaluate the power of spectral normalization extensively on combinations of datasets, network architectures, and adversarial training schemes. The code is available at https://github.com/jessemzhang/dl_spectral_normalization.

Azim Ospanov, Farzan Farnia, Roozbeh Yousefzadeh

Formal reasoning and automated theorem proving constitute a challenging subfield of machine learning, in which machines are tasked with proving mathematical theorems using formal languages like Lean. A formal verification system can check whether a formal proof is correct or not almost instantaneously, but generating a completely correct formal proof with large language models (LLMs) remains a formidable task. The usual approach in the literature is to prompt the LLM many times (up to several thousands) until one of the generated proofs passes the verification system. In this work, we present APOLLO (Automated PrOof repair viaLLM and Lean cOllaboration), a modular, model-agnostic agentic framework that combines the strengths of the Lean compiler with an LLM's reasoning abilities to achieve better proof-generation results at a low token and sampling budgets. Apollo directs a fully automated process in which the LLM generates proofs for theorems, a set of agents analyze the proofs, fix the syntax errors, identify the mistakes in the proofs using Lean, isolate failing sub-lemmas, utilize automated solvers, and invoke an LLM on each remaining goal with a low top-K budget. The repaired sub-proofs are recombined and reverified, iterating up to a user-controlled maximum number of attempts. On the miniF2F benchmark, we establish a new state-of-the-art accuracy of 84.9% among sub 8B-parameter models (as of August 2025) while keeping the sampling budget below one hundred. Moreover, Apollo raises the state-of-the-art accuracy for Goedel-Prover-SFT to 65.6% while cutting sample complexity from 25,600 to a few hundred. General-purpose models (o3-mini, o4-mini) jump from 3-7% to over 40% accuracy. Our results demonstrate that targeted, compiler-guided repair of LLM outputs yields dramatic gains in both efficiency and correctness, suggesting a general paradigm for scalable automated theorem proving.

Shizhan Gong, Yankai Jiang, Qi Dou et al.

Vision-language models, such as CLIP, have achieved significant success in aligning visual and textual representations, becoming essential components of many multi-modal large language models (MLLMs) like LLaVA and OpenFlamingo. However, numerous studies have identified CLIP's limited fine-grained perception as a critical drawback, leading to substantial failures in downstream MLLMs. In contrast, vision-centric foundation models like DINOv2 demonstrate remarkable capabilities in capturing fine details from images. In this work, we propose a novel kernel-based method to align CLIP's visual representation with that of DINOv2, ensuring that the resulting embeddings maintain compatibility with text embeddings while enhancing perceptual capabilities. Our alignment objective is designed for efficient stochastic optimization. Following this image-only alignment fine-tuning, the visual encoder retains compatibility with the frozen text encoder and exhibits significant improvements in zero-shot object recognition, fine-grained spatial reasoning, and localization. By integrating the aligned visual encoder, downstream MLLMs also demonstrate enhanced performance.

Shizhan Gong, Qi Dou, Farzan Farnia

Gradient-based saliency maps have been widely used to explain the decisions of deep neural network classifiers. However, standard gradient-based interpretation maps, including the simple gradient and integrated gradient algorithms, often lack desired structures such as sparsity and connectedness in their application to real-world computer vision models. A frequently used approach to inducing sparsity structures into gradient-based saliency maps is to alter the simple gradient scheme using sparsification or norm-based regularization. A drawback with such post-processing methods is their frequently-observed significant loss in fidelity to the original simple gradient map. In this work, we propose to apply adversarial training as an in-processing scheme to train neural networks with structured simple gradient maps. We show a duality relation between the regularized norms of the adversarial perturbations and gradient-based maps, based on which we design adversarial training loss functions promoting sparsity and group-sparsity properties in simple gradient maps. We present several numerical results to show the influence of our proposed norm-based adversarial training methods on the standard gradient-based maps of standard neural network architectures on benchmark image datasets.

Haoyu Lei, Amin Gohari, Farzan Farnia

Fair supervised learning algorithms assigning labels with little dependence on a sensitive attribute have attracted great attention in the machine learning community. While the demographic parity (DP) notion has been frequently used to measure a model's fairness in training fair classifiers, several studies in the literature suggest potential impacts of enforcing DP in fair learning algorithms. In this work, we analytically study the effect of standard DP-based regularization methods on the conditional distribution of the predicted label given the sensitive attribute. Our analysis shows that an imbalanced training dataset with a non-uniform distribution of the sensitive attribute could lead to a classification rule biased toward the sensitive attribute outcome holding the majority of training data. To control such inductive biases in DP-based fair learning, we propose a sensitive attribute-based distributionally robust optimization (SA-DRO) method improving robustness against the marginal distribution of the sensitive attribute. Finally, we present several numerical results on the application of DP-based learning methods to standard centralized and distributed learning problems. The empirical findings support our theoretical results on the inductive biases in DP-based fair learning algorithms and the debiasing effects of the proposed SA-DRO method.

Shizhan Gong, Jingwei Zhang, Qi Dou et al.

Saliency maps are widely used in the computer vision community for interpreting neural network classifiers. However, due to the randomness of training samples and optimization algorithms, the resulting saliency maps suffer from a significant level of stochasticity, making it difficult for domain experts to capture the intrinsic factors that influence the neural network's decision. In this work, we propose a novel pixel partitioning strategy to boost the stability and generalizability of gradient-based saliency maps. Through both theoretical analysis and numerical experiments, we demonstrate that the grouping of pixels reduces the variance of the saliency map and improves the generalization behavior of the interpretation method. Furthermore, we propose a sensible grouping strategy based on super-pixels which cluster pixels into groups that align well with the semantic meaning of the images. We perform several numerical experiments on CIFAR-10 and ImageNet. Our empirical results suggest that the super-pixel-based interpretation maps consistently improve the stability and quality over the pixel-based saliency maps.

Xiaoyan Hu, Farzan Farnia, Ho-fung Leung

Reinforcement learning (RL) problems where the learner attempts to infer an unobserved reward from some feedback variables have been studied in several recent papers. The setting of Interaction-Grounded Learning (IGL) is an example of such feedback-based RL tasks where the learner optimizes the return by inferring latent binary rewards from the interaction with the environment. In the IGL setting, a relevant assumption used in the RL literature is that the feedback variable $Y$ is conditionally independent of the context-action $(X,A)$ given the latent reward $R$. In this work, we propose Variational Information-based IGL (VI-IGL) as an information-theoretic method to enforce the conditional independence assumption in the IGL-based RL problem. The VI-IGL framework learns a reward decoder using an information-based objective based on the conditional mutual information (MI) between $(X,A)$ and $Y$. To estimate and optimize the information-based terms for the continuous random variables in the RL problem, VI-IGL leverages the variational representation of mutual information to obtain a min-max optimization problem. Also, we extend the VI-IGL framework to general $f$-Information measures leading to the generalized $f$-VI-IGL framework for the IGL-based RL problems. We present numerical results on several reinforcement learning settings indicating an improved performance compared to the existing IGL-based RL algorithm.

Haoyu Lei, Shizhan Gong, Qi Dou et al.

Federated learning (FL) algorithms commonly aim to maximize clients' accuracy by training a model on their collective data. However, in several FL applications, the model's decisions should meet a group fairness constraint to be independent of sensitive attributes such as gender or race. While such group fairness constraints can be incorporated into the objective function of the FL optimization problem, in this work, we show that such an approach would lead to suboptimal classification accuracy in an FL setting with heterogeneous client distributions. To achieve an optimal accuracy-group fairness trade-off, we propose the Personalized Federated Learning for Client-Level Group Fairness (pFedFair) framework, where clients locally impose their fairness constraints over the distributed training process. Leveraging the image embedding models, we extend the application of pFedFair to computer vision settings, where we numerically show that pFedFair achieves an optimal group fairness-accuracy trade-off in heterogeneous FL settings. We present the results of several numerical experiments on benchmark and synthetic datasets, which highlight the suboptimality of non-personalized FL algorithms and the improvements made by the pFedFair method.

Zixiao Wang, Wenqian Zhao, Yunheng Shen et al.

Recent advancements in layout pattern generation have been dominated by deep generative models. However, relying solely on neural networks for legality guarantees raises concerns in many practical applications. In this paper, we present \tool{DiffPattern}-Flex, a novel approach designed to generate reliable layout patterns efficiently. \tool{DiffPattern}-Flex incorporates a new method for generating diverse topologies using a discrete diffusion model while maintaining a lossless and compute-efficient layout representation. To ensure legal pattern generation, we employ {an} optimization-based, white-box pattern assessment process based on specific design rules. Furthermore, fast sampling and efficient legalization technologies are employed to accelerate the generation process. Experimental results across various benchmarks demonstrate that \tool{DiffPattern}-Flex significantly outperforms existing methods and excels at producing reliable layout patterns.

Haoyu Lei, Kaiwen Zhou, Yinchuan Li et al.

Diffusion-based Neural Combinatorial Optimization (NCO) has demonstrated effectiveness in solving NP-complete (NPC) problems by learning discrete diffusion models for solution generation, eliminating hand-crafted domain knowledge. Despite their success, existing NCO methods face significant challenges in both cross-scale and cross-problem generalization, and high training costs compared to traditional solvers. While recent studies on diffusion models have introduced training-free guidance approaches that leverage pre-defined guidance functions for conditional generation, such methodologies have not been extensively explored in combinatorial optimization. To bridge this gap, we propose a training-free inference time adaptation framework (DIFU-Ada) that enables both the zero-shot cross-problem transfer and cross-scale generalization capabilities of diffusion-based NCO solvers without requiring additional training. We provide theoretical analysis that helps understanding the cross-problem transfer capability. Our experimental results demonstrate that a diffusion solver, trained exclusively on the Traveling Salesman Problem (TSP), can achieve competitive zero-shot transfer performance across different problem scales on TSP variants, such as Prize Collecting TSP (PCTSP) and the Orienteering Problem (OP), through inference time adaptation.

Zhuorui Ye, Farzan Farnia

Saliency maps have been widely used to interpret the decisions of neural network classifiers and discover phenomena from their learned functions. However, standard gradient-based maps are frequently observed to be highly sensitive to the randomness of training data and the stochasticity in the training process. In this work, we study the role of Gaussian smoothing in the well-known Smooth-Grad algorithm in the stability of the gradient-based maps to the randomness of training samples. We extend the algorithmic stability framework to gradient-based interpretation maps and prove bounds on the stability error of standard Simple-Grad, Integrated-Gradients, and Smooth-Grad saliency maps. Our theoretical results suggest the role of Gaussian smoothing in boosting the stability of gradient-based maps to the randomness of training settings. On the other hand, we analyze the faithfulness of the Smooth-Grad maps to the original Simple-Grad and show the lower fidelity under a more intense Gaussian smoothing. We support our theoretical results by performing several numerical experiments on standard image datasets. Our empirical results confirm our hypothesis on the fidelity-stability trade-off in the application of Gaussian smoothing to gradient-based interpretation maps.

Xiwei Cheng, Kexin Fu, Farzan Farnia

The Langevin Dynamics (LD), which aims to sample from a probability distribution using its score function, has been widely used for analyzing and developing score-based generative modeling algorithms. While the convergence behavior of LD in sampling from a uni-modal distribution has been extensively studied in the literature, the analysis of LD under a mixture distribution with distinct modes remains underexplored in the literature. In this work, we analyze LD in sampling from a mixture distribution and theoretically study its convergence properties. Our theoretical results indicate that for general mixture distributions of sub-Gaussian components, LD could fail in finding all the components within a sub-exponential number of steps in the data dimension. Following our result on the complexity of LD in sampling from high-dimensional variables, we propose Chained Langevin Dynamics (Chained-LD), which divides the data vector into patches of smaller sizes and generates every patch sequentially conditioned on the previous patches. Our theoretical analysis of Chained-LD indicates its faster convergence speed to the components of a mixture distribution. We present the results of several numerical experiments on synthetic and real image datasets, validating our theoretical results on the iteration complexities of sample generation from mixture distributions using the vanilla and chained LD algorithms.

Jingwei Zhang, Farzan Farnia

Transportation of samples across different domains is a central task in several machine learning problems. A sensible requirement for domain transfer tasks in computer vision and language domains is the sparsity of the transportation map, i.e., the transfer algorithm aims to modify the least number of input features while transporting samples across the source and target domains. In this work, we propose Elastic Net Optimal Transport (ENOT) to address the sparse distribution transfer problem. The ENOT framework utilizes the $L_1$-norm and $L_2$-norm regularization mechanisms to find a sparse and stable transportation map between the source and target domains. To compute the ENOT transport map, we consider the dual formulation of the ENOT optimization task and prove that the sparsified gradient of the optimal potential function in the ENOT's dual representation provides the ENOT transport map. Furthermore, we demonstrate the application of the ENOT framework to perform feature selection for sparse domain transfer. We present the numerical results of applying ENOT to several domain transfer problems for synthetic Gaussian mixtures and real image and text data. Our empirical results indicate the success of the ENOT framework in identifying a sparse domain transport map.

Farzan Farnia, Amirali Aghazadeh, James Zou et al.

Robust training methods against perturbations to the input data have received great attention in the machine learning literature. A standard approach in this direction is adversarial training which learns a model using adversarially-perturbed training samples. However, adversarial training performs suboptimally against perturbations structured across samples such as universal and group-sparse shifts that are commonly present in biological data such as gene expression levels of different tissues. In this work, we seek to close this optimality gap and introduce Group-Structured Adversarial Training (GSAT) which learns a model robust to perturbations structured across samples. We formulate GSAT as a non-convex concave minimax optimization problem which minimizes a group-structured optimal transport cost. Specifically, we focus on the applications of GSAT for group-sparse and rank-constrained perturbations modeled using group and nuclear norm penalties. In order to solve GSAT's non-smooth optimization problem in those cases, we propose a new minimax optimization algorithm called GDADMM by combining Gradient Descent Ascent (GDA) and Alternating Direction Method of Multipliers (ADMM). We present several applications of the GSAT framework to gain robustness against structured perturbations for image recognition and computational biology datasets.

Theo Diamandis, Yonina C. Eldar, Alireza Fallah et al.

Multi-modal distributions are commonly used to model clustered data in statistical learning tasks. In this paper, we consider the Mixed Linear Regression (MLR) problem. We propose an optimal transport-based framework for MLR problems, Wasserstein Mixed Linear Regression (WMLR), which minimizes the Wasserstein distance between the learned and target mixture regression models. Through a model-based duality analysis, WMLR reduces the underlying MLR task to a nonconvex-concave minimax optimization problem, which can be provably solved to find a minimax stationary point by the Gradient Descent Ascent (GDA) algorithm. In the special case of mixtures of two linear regression models, we show that WMLR enjoys global convergence and generalization guarantees. We prove that WMLR's sample complexity grows linearly with the dimension of data. Finally, we discuss the application of WMLR to the federated learning task where the training samples are collected by multiple agents in a network. Unlike the Expectation Maximization algorithm, WMLR directly extends to the distributed, federated learning setting. We support our theoretical results through several numerical experiments, which highlight our framework's ability to handle the federated learning setting with mixture models.

Farzan Farnia, Asuman Ozdaglar

The success of minimax learning problems of generative adversarial networks (GANs) has been observed to depend on the minimax optimization algorithm used for their training. This dependence is commonly attributed to the convergence speed and robustness properties of the underlying optimization algorithm. In this paper, we show that the optimization algorithm also plays a key role in the generalization performance of the trained minimax model. To this end, we analyze the generalization properties of standard gradient descent ascent (GDA) and proximal point method (PPM) algorithms through the lens of algorithmic stability under both convex concave and non-convex non-concave minimax settings. While the GDA algorithm is not guaranteed to have a vanishing excess risk in convex concave problems, we show the PPM algorithm enjoys a bounded excess risk in the same setup. For non-convex non-concave problems, we compare the generalization performance of stochastic GDA and GDmax algorithms where the latter fully solves the maximization subproblem at every iteration. Our generalization analysis suggests the superiority of GDA provided that the minimization and maximization subproblems are solved simultaneously with similar learning rates. We discuss several numerical results indicating the role of optimization algorithms in the generalization of the learned minimax models.

Farzan Farnia, William Wang, Subhro Das et al.

Generative adversarial networks (GANs) learn the distribution of observed samples through a zero-sum game between two machine players, a generator and a discriminator. While GANs achieve great success in learning the complex distribution of image, sound, and text data, they perform suboptimally in learning multi-modal distribution-learning benchmarks including Gaussian mixture models (GMMs). In this paper, we propose Generative Adversarial Training for Gaussian Mixture Models (GAT-GMM), a minimax GAN framework for learning GMMs. Motivated by optimal transport theory, we design the zero-sum game in GAT-GMM using a random linear generator and a softmax-based quadratic discriminator architecture, which leads to a non-convex concave minimax optimization problem. We show that a Gradient Descent Ascent (GDA) method converges to an approximate stationary minimax point of the GAT-GMM optimization problem. In the benchmark case of a mixture of two symmetric, well-separated Gaussians, we further show this stationary point recovers the true parameters of the underlying GMM. We numerically support our theoretical findings by performing several experiments, which demonstrate that GAT-GMM can perform as well as the expectation-maximization algorithm in learning mixtures of two Gaussians.

Amirhossein Reisizadeh, Farzan Farnia, Ramtin Pedarsani et al.

Federated learning is a distributed paradigm that aims at training models using samples distributed across multiple users in a network while keeping the samples on users' devices with the aim of efficiency and protecting users privacy. In such settings, the training data is often statistically heterogeneous and manifests various distribution shifts across users, which degrades the performance of the learnt model. The primary goal of this paper is to develop a robust federated learning algorithm that achieves satisfactory performance against distribution shifts in users' samples. To achieve this goal, we first consider a structured affine distribution shift in users' data that captures the device-dependent data heterogeneity in federated settings. This perturbation model is applicable to various federated learning problems such as image classification where the images undergo device-dependent imperfections, e.g. different intensity, contrast, and brightness. To address affine distribution shifts across users, we propose a Federated Learning framework Robust to Affine distribution shifts (FLRA) that is provably robust against affine Wasserstein shifts to the distribution of observed samples. To solve the FLRA's distributed minimax problem, we propose a fast and efficient optimization method and provide convergence guarantees via a gradient Descent Ascent (GDA) method. We further prove generalization error bounds for the learnt classifier to show proper generalization from empirical distribution of samples to the true underlying distribution. We perform several numerical experiments to empirically support FLRA. We show that an affine distribution shift indeed suffices to significantly decrease the performance of the learnt classifier in a new test user, and our proposed algorithm achieves a significant gain in comparison to standard federated learning and adversarial training methods.

Farzan Farnia, Asuman Ozdaglar

Generative adversarial networks (GANs) represent a zero-sum game between two machine players, a generator and a discriminator, designed to learn the distribution of data. While GANs have achieved state-of-the-art performance in several benchmark learning tasks, GAN minimax optimization still poses great theoretical and empirical challenges. GANs trained using first-order optimization methods commonly fail to converge to a stable solution where the players cannot improve their objective, i.e., the Nash equilibrium of the underlying game. Such issues raise the question of the existence of Nash equilibrium solutions in the GAN zero-sum game. In this work, we show through several theoretical and numerical results that indeed GAN zero-sum games may not have any local Nash equilibria. To characterize an equilibrium notion applicable to GANs, we consider the equilibrium of a new zero-sum game with an objective function given by a proximal operator applied to the original objective, a solution we call the proximal equilibrium. Unlike the Nash equilibrium, the proximal equilibrium captures the sequential nature of GANs, in which the generator moves first followed by the discriminator. We prove that the optimal generative model in Wasserstein GAN problems provides a proximal equilibrium. Inspired by these results, we propose a new approach, which we call proximal training, for solving GAN problems. We discuss several numerical experiments demonstrating the existence of proximal equilibrium solutions in GAN minimax problems.

Farzan Farnia, David Tse

Generative adversarial network (GAN) is a minimax game between a generator mimicking the true model and a discriminator distinguishing the samples produced by the generator from the real training samples. Given an unconstrained discriminator able to approximate any function, this game reduces to finding the generative model minimizing a divergence measure, e.g. the Jensen-Shannon (JS) divergence, to the data distribution. However, in practice the discriminator is constrained to be in a smaller class $\mathcal{F}$ such as neural nets. Then, a natural question is how the divergence minimization interpretation changes as we constrain $\mathcal{F}$. In this work, we address this question by developing a convex duality framework for analyzing GANs. For a convex set $\mathcal{F}$, this duality framework interprets the original GAN formulation as finding the generative model with minimum JS-divergence to the distributions penalized to match the moments of the data distribution, with the moments specified by the discriminators in $\mathcal{F}$. We show that this interpretation more generally holds for f-GAN and Wasserstein GAN. As a byproduct, we apply the duality framework to a hybrid of f-divergence and Wasserstein distance. Unlike the f-divergence, we prove that the proposed hybrid divergence changes continuously with the generative model, which suggests regularizing the discriminator's Lipschitz constant in f-GAN and vanilla GAN. We numerically evaluate the power of the suggested regularization schemes for improving GAN's training performance.

Farzan Farnia, David Tse

Given a task of predicting $Y$ from $X$, a loss function $L$, and a set of probability distributions $Γ$ on $(X,Y)$, what is the optimal decision rule minimizing the worst-case expected loss over $Γ$? In this paper, we address this question by introducing a generalization of the principle of maximum entropy. Applying this principle to sets of distributions with marginal on $X$ constrained to be the empirical marginal from the data, we develop a general minimax approach for supervised learning problems. While for some loss functions such as squared-error and log loss, the minimax approach rederives well-knwon regression models, for the 0-1 loss it results in a new linear classifier which we call the maximum entropy machine. The maximum entropy machine minimizes the worst-case 0-1 loss over the structured set of distribution, and by our numerical experiments can outperform other well-known linear classifiers such as SVM. We also prove a bound on the generalization worst-case error in the minimax approach.

Meisam Razaviyayn, Farzan Farnia, David Tse

Consider the binary classification problem of predicting a target variable $Y$ from a discrete feature vector $X = (X_1,...,X_d)$. When the probability distribution $\mathbb{P}(X,Y)$ is known, the optimal classifier, leading to the minimum misclassification rate, is given by the Maximum A-posteriori Probability decision rule. However, estimating the complete joint distribution $\mathbb{P}(X,Y)$ is computationally and statistically impossible for large values of $d$. An alternative approach is to first estimate some low order marginals of $\mathbb{P}(X,Y)$ and then design the classifier based on the estimated low order marginals. This approach is also helpful when the complete training data instances are not available due to privacy concerns. In this work, we consider the problem of finding the optimum classifier based on some estimated low order marginals of $(X,Y)$. We prove that for a given set of marginals, the minimum Hirschfeld-Gebelein-Renyi (HGR) correlation principle introduced in [1] leads to a randomized classification rule which is shown to have a misclassification rate no larger than twice the misclassification rate of the optimal classifier. Then, under a separability condition, we show that the proposed algorithm is equivalent to a randomized linear regression approach. In addition, this method naturally results in a robust feature selection method selecting a subset of features having the maximum worst case HGR correlation with the target variable. Our theoretical upper-bound is similar to the recent Discrete Chebyshev Classifier (DCC) approach [2], while the proposed algorithm has significant computational advantages since it only requires solving a least square optimization problem. Finally, we numerically compare our proposed algorithm with the DCC classifier and show that the proposed algorithm results in better misclassification rate over various datasets.