Tianrong Chen

Guan-Horng Liu, Tianrong Chen, Oswin So et al. · mit

Mean-Field Game (MFG) serves as a crucial mathematical framework in modeling the collective behavior of individual agents interacting stochastically with a large population. In this work, we aim at solving a challenging class of MFGs in which the differentiability of these interacting preferences may not be available to the solver, and the population is urged to converge exactly to some desired distribution. These setups are, despite being well-motivated for practical purposes, complicated enough to paralyze most (deep) numerical solvers. Nevertheless, we show that Schrödinger Bridge - as an entropy-regularized optimal transport model - can be generalized to accepting mean-field structures, hence solving these MFGs. This is achieved via the application of Forward-Backward Stochastic Differential Equations theory, which, intriguingly, leads to a computational framework with a similar structure to Temporal Difference learning. As such, it opens up novel algorithmic connections to Deep Reinforcement Learning that we leverage to facilitate practical training. We show that our proposed objective function provides necessary and sufficient conditions to the mean-field problem. Our method, named Deep Generalized Schrödinger Bridge (DeepGSB), not only outperforms prior methods in solving classical population navigation MFGs, but is also capable of solving 1000-dimensional opinion depolarization, setting a new state-of-the-art numerical solver for high-dimensional MFGs. Our code will be made available at https://github.com/ghliu/DeepGSB.

Guan-Horng Liu, Tianrong Chen, Evangelos A. Theodorou et al.

Modern successes of diffusion models in learning complex, high-dimensional data distributions are attributed, in part, to their capability to construct diffusion processes with analytic transition kernels and score functions. The tractability results in a simulation-free framework with stable regression losses, from which reversed, generative processes can be learned at scale. However, when data is confined to a constrained set as opposed to a standard Euclidean space, these desirable characteristics appear to be lost based on prior attempts. In this work, we propose Mirror Diffusion Models (MDM), a new class of diffusion models that generate data on convex constrained sets without losing any tractability. This is achieved by learning diffusion processes in a dual space constructed from a mirror map, which, crucially, is a standard Euclidean space. We derive efficient computation of mirror maps for popular constrained sets, such as simplices and $\ell_2$-balls, showing significantly improved performance of MDM over existing methods. For safety and privacy purposes, we also explore constrained sets as a new mechanism to embed invisible but quantitative information (i.e., watermarks) in generated data, for which MDM serves as a compelling approach. Our work brings new algorithmic opportunities for learning tractable diffusion on complex domains. Our code is available at https://github.com/ghliu/mdm

Tianrong Chen, Jiatao Gu, David Berthelot et al.

Normalizing Flows (NFs) are a classical family of likelihood-based methods that have received revived attention. Recent efforts such as TARFlow have shown that NFs are capable of achieving promising performance on image modeling tasks, making them viable alternatives to other methods such as diffusion models. In this work, we further advance the state of Normalizing Flow generative models by introducing iterative TARFlow (iTARFlow). Unlike diffusion models, iTARFlow maintains a fully end-to-end, likelihood-based objective during training. During sampling, it performs autoregressive generation followed by an iterative denoising procedure inspired by diffusion-style methods. Through extensive experiments, we show that iTARFlow achieves competitive performance across ImageNet resolutions of 64, 128, and 256 pixels, demonstrating its potential as a strong generative model and advancing the frontier of Normalizing Flows. In addition, we analyze the characteristic artifacts produced by iTARFlow, offering insights that may shed light on future improvements. Code is available at https://github.com/apple/ml-itarflow.

Tianrong Chen, Guan-Horng Liu, Molei Tao et al.

It is a crucial challenge to reconstruct population dynamics using unlabeled samples from distributions at coarse time intervals. Recent approaches such as flow-based models or Schrödinger Bridge (SB) models have demonstrated appealing performance, yet the inferred sample trajectories either fail to account for the underlying stochasticity or are $\underline{D}$eep $\underline{M}$omentum Multi-Marginal $\underline{S}$chrödinger $\underline{B}$ridge(DMSB), a novel computational framework that learns the smooth measure-valued spline for stochastic systems that satisfy position marginal constraints across time. By tailoring the celebrated Bregman Iteration and extending the Iteration Proportional Fitting to phase space, we manage to handle high-dimensional multi-marginal trajectory inference tasks efficiently. Our algorithm outperforms baselines significantly, as evidenced by experiments for synthetic datasets and a real-world single-cell RNA sequence dataset. Additionally, the proposed approach can reasonably reconstruct the evolution of velocity distribution, from position snapshots only, when there is a ground truth velocity that is nevertheless inaccessible.

Tianrong Chen, Jiatao Gu, Laurent Dinh et al.

Diffusion models (DMs) represent state-of-the-art generative models for continuous inputs. DMs work by constructing a Stochastic Differential Equation (SDE) in the input space (ie, position space), and using a neural network to reverse it. In this work, we introduce a novel generative modeling framework grounded in \textbf{phase space dynamics}, where a phase space is defined as {an augmented space encompassing both position and velocity.} Leveraging insights from Stochastic Optimal Control, we construct a path measure in the phase space that enables efficient sampling. {In contrast to DMs, our framework demonstrates the capability to generate realistic data points at an early stage of dynamics propagation.} This early prediction sets the stage for efficient data generation by leveraging additional velocity information along the trajectory. On standard image generation benchmarks, our model yields favorable performance over baselines in the regime of small Number of Function Evaluations (NFEs). Furthermore, our approach rivals the performance of diffusion models equipped with efficient sampling techniques, underscoring its potential as a new tool generative modeling.

Valentin De Bortoli, Guan-Horng Liu, Tianrong Chen et al.

Flow and bridge matching are a novel class of processes which encompass diffusion models. One of the main aspect of their increased flexibility is that these models can interpolate between arbitrary data distributions i.e. they generalize beyond generative modeling and can be applied to learning stochastic (and deterministic) processes of arbitrary transfer tasks between two given distributions. In this paper, we highlight that while flow and bridge matching processes preserve the information of the marginal distributions, they do \emph{not} necessarily preserve the coupling information unless additional, stronger optimality conditions are met. This can be problematic if one aims at preserving the original empirical pairing. We show that a simple modification of the matching process recovers this coupling by augmenting the velocity field (or drift) with the information of the initial sample point. Doing so, we lose the Markovian property of the process but preserve the coupling information between distributions. We illustrate the efficiency of our augmentation in learning mixture of image translation tasks.

Tianrong Chen, Ziyi Wang, Evangelos A. Theodorou

In this paper, we present a scalable deep learning approach to solve opinion dynamics stochastic optimal control problems with mean field term coupling in the dynamics and cost function. Our approach relies on the probabilistic representation of the solution of the Hamilton-Jacobi-Bellman partial differential equation. Grounded on the nonlinear version of the Feynman-Kac lemma, the solutions of the Hamilton-Jacobi-Bellman partial differential equation are linked to the solution of Forward-Backward Stochastic Differential Equations. These equations can be solved numerically using a novel deep neural network with architecture tailored to the problem in consideration. The resulting algorithm is tested on a polarized opinion consensus experiment. The large-scale (10K) agents experiment validates the scalability and generalizability of our algorithm. The proposed framework opens up the possibility for future applications on extremely large-scale problems.

Shuangfei Zhai, Ruixiang Zhang, Preetum Nakkiran et al. · apple-ml

Normalizing Flows (NFs) are likelihood-based models for continuous inputs. They have demonstrated promising results on both density estimation and generative modeling tasks, but have received relatively little attention in recent years. In this work, we demonstrate that NFs are more powerful than previously believed. We present TarFlow: a simple and scalable architecture that enables highly performant NF models. TarFlow can be thought of as a Transformer-based variant of Masked Autoregressive Flows (MAFs): it consists of a stack of autoregressive Transformer blocks on image patches, alternating the autoregression direction between layers. TarFlow is straightforward to train end-to-end, and capable of directly modeling and generating pixels. We also propose three key techniques to improve sample quality: Gaussian noise augmentation during training, a post training denoising procedure, and an effective guidance method for both class-conditional and unconditional settings. Putting these together, TarFlow sets new state-of-the-art results on likelihood estimation for images, beating the previous best methods by a large margin, and generates samples with quality and diversity comparable to diffusion models, for the first time with a stand-alone NF model. We make our code available at https://github.com/apple/ml-tarflow.

Rui Wang, Qianguo Sun, Tianrong Chen et al.

The emergence of multi-codebook neutral audio codecs such as Residual Vector Quantization (RVQ) and Group Vector Quantization (GVQ) has significantly advanced Large-Language-Model (LLM) based Text-to-Speech (TTS) systems. These codecs are crucial in separating semantic and acoustic information while efficiently harnessing semantic priors. However, since semantic and acoustic information cannot be fully aligned, a significant drawback of these methods when applied to LLM-based TTS is that large language models may have limited access to comprehensive audio information. To address this limitation, we propose DistilCodec and UniTTS, which collectively offer the following advantages: 1) This method can distill a multi-codebook audio codec into a single-codebook audio codec with 32,768 codes while achieving a near 100\% utilization. 2) As DistilCodec does not employ a semantic alignment scheme, a large amount of high-quality unlabeled audio (such as audiobooks with sound effects, songs, etc.) can be incorporated during training, further expanding data diversity and broadening its applicability. 3) Leveraging the comprehensive audio information modeling of DistilCodec, we integrated three key tasks into UniTTS's pre-training framework: audio modality autoregression, text modality autoregression, and speech-text cross-modal autoregression. This allows UniTTS to accept interleaved text and speech/audio prompts while substantially preserving LLM's text capabilities. 4) UniTTS employs a three-stage training process: Pre-Training, Supervised Fine-Tuning (SFT), and Alignment. Source code and model checkpoints are publicly available at https://github.com/IDEA-Emdoor-Lab/UniTTS and https://github.com/IDEA-Emdoor-Lab/DistilCodec.

Jiatao Gu, Ying Shen, Tianrong Chen et al.

Normalizing flows (NFs) are end-to-end likelihood-based generative models for continuous data, and have recently regained attention with encouraging progress on image generation. Yet in the video generation domain, where spatiotemporal complexity and computational cost are substantially higher, state-of-the-art systems almost exclusively rely on diffusion-based models. In this work, we revisit this design space by presenting STARFlow-V, a normalizing flow-based video generator with substantial benefits such as end-to-end learning, robust causal prediction, and native likelihood estimation. Building upon the recently proposed STARFlow, STARFlow-V operates in the spatiotemporal latent space with a global-local architecture which restricts causal dependencies to a global latent space while preserving rich local within-frame interactions. This eases error accumulation over time, a common pitfall of standard autoregressive diffusion model generation. Additionally, we propose flow-score matching, which equips the model with a light-weight causal denoiser to improve the video generation consistency in an autoregressive fashion. To improve the sampling efficiency, STARFlow-V employs a video-aware Jacobi iteration scheme that recasts inner updates as parallelizable iterations without breaking causality. Thanks to the invertible structure, the same model can natively support text-to-video, image-to-video as well as video-to-video generation tasks. Empirically, STARFlow-V achieves strong visual fidelity and temporal consistency with practical sampling throughput relative to diffusion-based baselines. These results present the first evidence, to our knowledge, that NFs are capable of high-quality autoregressive video generation, establishing them as a promising research direction for building world models. Code and generated samples are available at https://github.com/apple/ml-starflow.

Tianrong Chen, Huangjie Zheng, David Berthelot et al. · apple-ml

Diffusion models have demonstrated exceptional capabilities in generating high-fidelity images but typically suffer from inefficient sampling. Many solver designs and noise scheduling strategies have been proposed to dramatically improve sampling speeds. In this paper, we introduce a new sampling method that is up to $186\%$ faster than the current state of the art solver for comparative FID on ImageNet512. This new sampling method is training-free and uses an ordinary differential equation (ODE) solver. The key to our method resides in using higher-dimensional initial noise, allowing to produce more detailed samples with less function evaluations from existing pretrained diffusion models. In addition, by design our solver allows to control the level of detail through a simple hyper-parameter at no extra computational cost. We present how our approach leverages momentum dynamics by establishing a fundamental equivalence between momentum diffusion models and conventional diffusion models with respect to their training paradigms. Moreover, we observe the use of higher-dimensional noise naturally exhibits characteristics similar to stochastic differential equations (SDEs). Finally, we demonstrate strong performances on a set of representative pretrained diffusion models, including EDM, EDM2, and Stable-Diffusion 3, which cover models in both pixel and latent spaces, as well as class and text conditional settings. The code is available at https://github.com/apple/ml-tada.

Tianrong Chen, Guan-Horng Liu, Evangelos A. Theodorou

Schrödinger Bridge (SB) is an entropy-regularized optimal transport problem that has received increasing attention in deep generative modeling for its mathematical flexibility compared to the Scored-based Generative Model (SGM). However, it remains unclear whether the optimization principle of SB relates to the modern training of deep generative models, which often rely on constructing log-likelihood objectives.This raises questions on the suitability of SB models as a principled alternative for generative applications. In this work, we present a novel computational framework for likelihood training of SB models grounded on Forward-Backward Stochastic Differential Equations Theory - a mathematical methodology appeared in stochastic optimal control that transforms the optimality condition of SB into a set of SDEs. Crucially, these SDEs can be used to construct the likelihood objectives for SB that, surprisingly, generalizes the ones for SGM as special cases. This leads to a new optimization principle that inherits the same SB optimality yet without losing applications of modern generative training techniques, and we show that the resulting training algorithm achieves comparable results on generating realistic images on MNIST, CelebA, and CIFAR10. Our code is available at https://github.com/ghliu/SB-FBSDE.

Guan-Horng Liu, Tianrong Chen, Evangelos A. Theodorou

We propose a novel second-order optimization framework for training the emerging deep continuous-time models, specifically the Neural Ordinary Differential Equations (Neural ODEs). Since their training already involves expensive gradient computation by solving a backward ODE, deriving efficient second-order methods becomes highly nontrivial. Nevertheless, inspired by the recent Optimal Control (OC) interpretation of training deep networks, we show that a specific continuous-time OC methodology, called Differential Programming, can be adopted to derive backward ODEs for higher-order derivatives at the same O(1) memory cost. We further explore a low-rank representation of the second-order derivatives and show that it leads to efficient preconditioned updates with the aid of Kronecker-based factorization. The resulting method -- named SNOpt -- converges much faster than first-order baselines in wall-clock time, and the improvement remains consistent across various applications, e.g. image classification, generative flow, and time-series prediction. Our framework also enables direct architecture optimization, such as the integration time of Neural ODEs, with second-order feedback policies, strengthening the OC perspective as a principled tool of analyzing optimization in deep learning. Our code is available at https://github.com/ghliu/snopt.

Jiatao Gu, Tianrong Chen, Ying Shen et al.

Diffusion-based models decompose sampling into many small Gaussian denoising steps -- an assumption that breaks down when generation is compressed to a few coarse transitions. Existing few-step methods address this through distillation, consistency training, or adversarial objectives, but sacrifice the likelihood framework in the process. We introduce Normalizing Trajectory Models (NTM), which models each reverse step as an expressive conditional normalizing flow with exact likelihood training. Architecturally, NTM combines shallow invertible blocks within each step with a deep parallel predictor across the trajectory, forming an end-to-end network trainable from scratch or initializable from pretrained flow-matching models. Its exact trajectory likelihood further enables self-distillation: a lightweight denoiser trained on the model's own score produces high-quality samples in four steps. On text-to-image benchmarks, NTM matches or outperforms strong image generation baselines in just four sampling steps while uniquely retaining exact likelihood over the generative trajectory.

Ying Shen, Tianrong Chen, Yuan Gao et al.

Deep generative models have advanced rapidly across text and vision, motivating unified multimodal systems that can understand, reason over, and generate interleaved text-image sequences. Most existing approaches combine autoregressive language modeling with diffusion-based image generators, inheriting a structural mismatch between causal text generation and iterative visual denoising. We observe that autoregressive normalizing flows are autoregressive Transformers--sharing the same causal mask, KV-cache mechanism, and left-to-right structure as LLMs--making them the most natural paradigm for true unified multimodal generation. We present STARFlow2, built on the Pretzel architecture that vertically interleaves a pretrained VLM stream with a TarFlow stream via residual skip connections, both operating under the same causal mask. Combined with a deep-shallow flow design and a unified FAE latent space, STARFlow2 enables cache-friendly interleaved generation where both text and visual outputs directly enter the KV-cache without re-encoding. Experiments demonstrate strong performance across image generation and multimodal understanding benchmarks, validating autoregressive flows as a viable foundation for unified multimodal modeling.

Yuan Gao, Chen Chen, Tianrong Chen et al.

Visual generative models (e.g., diffusion models) typically operate in compressed latent spaces to balance training efficiency and sample quality. In parallel, there has been growing interest in leveraging high-quality pre-trained visual representations, either by aligning them inside VAEs or directly within the generative model. However, adapting such representations remains challenging due to fundamental mismatches between understanding-oriented features and generation-friendly latent spaces. Representation encoders benefit from high-dimensional latents that capture diverse hypotheses for masked regions, whereas generative models favor low-dimensional latents that must faithfully preserve injected noise. This discrepancy has led prior work to rely on complex objectives and architectures. In this work, we propose FAE (Feature Auto-Encoder), a simple yet effective framework that adapts pre-trained visual representations into low-dimensional latents suitable for generation using as little as a single attention layer, while retaining sufficient information for both reconstruction and understanding. The key is to couple two separate deep decoders: one trained to reconstruct the original feature space, and a second that takes the reconstructed features as input for image generation. FAE is generic; it can be instantiated with a variety of self-supervised encoders (e.g., DINO, SigLIP) and plugged into two distinct generative families: diffusion models and normalizing flows. Across class-conditional and text-to-image benchmarks, FAE achieves strong performance. For example, on ImageNet 256x256, our diffusion model with CFG attains a near state-of-the-art FID of 1.29 (800 epochs) and 1.70 (80 epochs). Without CFG, FAE reaches the state-of-the-art FID of 1.48 (800 epochs) and 2.08 (80 epochs), demonstrating both high quality and fast learning.

Yuchen Zhu, Tianrong Chen, Evangelos A. Theodorou et al.

This article considers the generative modeling of the (mixed) states of quantum systems, and an approach based on denoising diffusion model is proposed. The key contribution is an algorithmic innovation that respects the physical nature of quantum states. More precisely, the commonly used density matrix representation of mixed-state has to be complex-valued Hermitian, positive semi-definite, and trace one. Generic diffusion models, or other generative methods, may not be able to generate data that strictly satisfy these structural constraints, even if all training data do. To develop a machine learning algorithm that has physics hard-wired in, we leverage mirror diffusion and borrow the physical notion of von Neumann entropy to design a new map, for enabling strict structure-preserving generation. Both unconditional generation and conditional generation via classifier-free guidance are experimentally demonstrated efficacious, the latter enabling the design of new quantum states when generated on unseen labels.

Huangjie Zheng, Shansan Gong, Ruixiang Zhang et al. · apple-ml

Standard discrete diffusion models treat all unobserved states identically by mapping them to an absorbing [MASK] token. This creates an 'information void' where semantic information that could be inferred from unmasked tokens is lost between denoising steps. We introduce Continuously Augmented Discrete Diffusion (CADD), a framework that augments the discrete state space with a paired diffusion in a continuous latent space. This yields graded, gradually corrupted states in which masked tokens are represented by noisy yet informative latent vectors rather than collapsed 'information voids'. At each reverse step, CADD may leverage the continuous latent as a semantic hint to guide discrete denoising. The design is clean and compatible with existing discrete diffusion training. At sampling time, the strength and choice of estimator for the continuous latent vector enables a controlled trade-off between mode-coverage (generating diverse outputs) and mode-seeking (generating contextually precise outputs) behaviors. Empirically, we demonstrate CADD improves generative quality over mask-based diffusion across text generation, image synthesis, and code modeling, with consistent gains on both qualitative and quantitative metrics against strong discrete baselines.

Chenru Duan, Guan-Horng Liu, Yuanqi Du et al.

Transition states (TSs) are transient structures that are key in understanding reaction mechanisms and designing catalysts but challenging to be captured in experiments. Alternatively, many optimization algorithms have been developed to search for TSs computationally. Yet the cost of these algorithms driven by quantum chemistry methods (usually density functional theory) is still high, posing challenges for their applications in building large reaction networks for reaction exploration. Here we developed React-OT, an optimal transport approach for generating unique TS structures from reactants and products. React-OT generates highly accurate TS structures with a median structural root mean square deviation (RMSD) of 0.053Å and median barrier height error of 1.06 kcal/mol requiring only 0.4 second per reaction. The RMSD and barrier height error is further improved by roughly 25\% through pretraining React-OT on a large reaction dataset obtained with a lower level of theory, GFN2-xTB. We envision that the remarkable accuracy and rapid inference of React-OT will be highly useful when integrated with the current high-throughput TS search workflow. This integration will facilitate the exploration of chemical reactions with unknown mechanisms.

Ruixiang Zhang, Shuangfei Zhai, Jiatao Gu et al. · apple-ml

Autoregressive models have driven remarkable progress in language modeling. Their foundational reliance on discrete tokens, unidirectional context, and single-pass decoding, while central to their success, also inspires the exploration of a design space that could offer new axes of modeling flexibility. In this work, we explore an alternative paradigm, shifting language modeling from a discrete token space to a continuous latent space. We propose a novel framework TarFlowLM, that employs transformer-based autoregressive normalizing flows to model these continuous representations. This approach unlocks substantial flexibility, enabling the construction of models that can capture global bi-directional context through stacked, alternating-direction autoregressive transformations, support block-wise generation with flexible token patch sizes, and facilitate a hierarchical multi-pass generation process. We further propose new mixture-based coupling transformations designed to capture complex dependencies within the latent space shaped by discrete data, and demonstrate theoretical connections to conventional discrete autoregressive models. Extensive experiments on language modeling benchmarks demonstrate strong likelihood performance and highlight the flexible modeling capabilities inherent in our framework.

Rui Wang, Qianguo Sun, Chao Song et al.

DPO (Direct Preference Optimization) has become a widely used offline preference optimization algorithm due to its simplicity and training stability. However, DPO is prone to overfitting and collapse. To address these challenges, we propose Linear Preference Optimization (LPO), a novel alignment framework featuring three key innovations. First, we introduce gradient decoupling by replacing the log-sigmoid function with an absolute difference loss, thereby isolating the optimization dynamics. Second, we improve stability through an offset constraint combined with a positive regularization term to preserve the chosen response quality. Third, we implement controllable rejection suppression using gradient separation with straightforward estimation and a tunable coefficient that linearly regulates the descent of the rejection probability. Through extensive experiments, we demonstrate that LPO consistently improves performance on various tasks, including general text tasks, math tasks, and text-to-speech (TTS) tasks. These results establish LPO as a robust and tunable paradigm for preference alignment, and we release the source code, models, and training data publicly.

Guan-Horng Liu, Tianrong Chen, Evangelos A. Theodorou

Optimization of deep neural networks (DNNs) has been a driving force in the advancement of modern machine learning and artificial intelligence. With DNNs characterized by a prolonged sequence of nonlinear propagation, determining their optimal parameters given an objective naturally fits within the framework of Optimal Control Programming. Such an interpretation of DNNs as dynamical systems has proven crucial in offering a theoretical foundation for principled analysis from numerical equations to physics. In parallel to these theoretical pursuits, this paper focuses on an algorithmic perspective. Our motivated observation is the striking algorithmic resemblance between the Backpropagation algorithm for computing gradients in DNNs and the optimality conditions for dynamical systems, expressed through another backward process known as dynamic programming. Consolidating this connection, where Backpropagation admits a variational structure, solving an approximate dynamic programming up to the first-order expansion leads to a new class of optimization methods exploring higher-order expansions of the Bellman equation. The resulting optimizer, termed Optimal Control Theoretic Neural Optimizer (OCNOpt), enables rich algorithmic opportunities, including layer-wise feedback policies, game-theoretic applications, and higher-order training of continuous-time models such as Neural ODEs. Extensive experiments demonstrate that OCNOpt improves upon existing methods in robustness and efficiency while maintaining manageable computational complexity, paving new avenues for principled algorithmic design grounded in dynamical systems and optimal control theory.

Guan-Horng Liu, Tianrong Chen, Evangelos A. Theodorou

Generalized Schrödinger Bridges (GSBs) are a fundamental mathematical framework used to analyze the most likely particle evolution based on the principle of least action including kinetic and potential energy. In parallel to their well-established presence in the theoretical realms of quantum mechanics and optimal transport, this paper focuses on an algorithmic perspective, aiming to enhance practical usage. Our motivated observation is that transportation problems with the optimality structures delineated by GSBs are pervasive across various scientific domains, such as generative modeling in machine learning, mean-field games in stochastic control, and more. Exploring the intrinsic connection between the mathematical modeling of GSBs and the modern algorithmic characterization therefore presents a crucial, yet untapped, avenue. In this paper, we reinterpret GSBs as probabilistic models and demonstrate that, with a delicate mathematical tool known as the nonlinear Feynman-Kac lemma, rich algorithmic concepts, such as likelihoods, variational gaps, and temporal differences, emerge naturally from the optimality structures of GSBs. The resulting computational framework, driven by deep learning and neural networks, operates in a fully continuous state space (i.e., mesh-free) and satisfies distribution constraints, setting it apart from prior numerical solvers relying on spatial discretization or constraint relaxation. We demonstrate the efficacy of our method in generative modeling and mean-field games, highlighting its transformative applications at the intersection of mathematical modeling, stochastic process, control, and machine learning.

Guan-Horng Liu, Tianrong Chen, Evangelos A. Theodorou

The connection between training deep neural networks (DNNs) and optimal control theory (OCT) has attracted considerable attention as a principled tool of algorithmic design. Despite few attempts being made, they have been limited to architectures where the layer propagation resembles a Markovian dynamical system. This casts doubts on their flexibility to modern networks that heavily rely on non-Markovian dependencies between layers (e.g. skip connections in residual networks). In this work, we propose a novel dynamic game perspective by viewing each layer as a player in a dynamic game characterized by the DNN itself. Through this lens, different classes of optimizers can be seen as matching different types of Nash equilibria, depending on the implicit information structure of each (p)layer. The resulting method, called Dynamic Game Theoretic Neural Optimizer (DGNOpt), not only generalizes OCT-inspired optimizers to richer network class; it also motivates a new training principle by solving a multi-player cooperative game. DGNOpt shows convergence improvements over existing methods on image classification datasets with residual and inception networks. Our work marries strengths from both OCT and game theory, paving ways to new algorithmic opportunities from robust optimal control and bandit-based optimization.

Tianrong Chen, Ziyi Wang, Ioannis Exarchos et al.

In this paper we present a scalable deep learning framework for finding Markovian Nash Equilibria in multi-agent stochastic games using fictitious play. The motivation is inspired by theoretical analysis of Forward Backward Stochastic Differential Equations (FBSDE) and their implementation in a deep learning setting, which is the source of our algorithm's sample efficiency improvement. By taking advantage of the permutation-invariant property of agents in symmetric games, the scalability and performance is further enhanced significantly. We showcase superior performance of our framework over the state-of-the-art deep fictitious play algorithm on an inter-bank lending/borrowing problem in terms of multiple metrics. More importantly, our approach scales up to 3000 agents in simulation, a scale which, to the best of our knowledge, represents a new state-of-the-art. We also demonstrate the applicability of our framework in robotics on a belief space autonomous racing problem.

Guan-Horng Liu, Tianrong Chen, Evangelos A. Theodorou

Connections between Deep Neural Networks (DNNs) training and optimal control theory has attracted considerable attention as a principled tool of algorithmic design. Differential Dynamic Programming (DDP) neural optimizer is a recently proposed method along this line. Despite its empirical success, the applicability has been limited to feedforward networks and whether such a trajectory-optimization inspired framework can be extended to modern architectures remains unclear. In this work, we derive a generalized DDP optimizer that accepts both residual connections and convolution layers. The resulting optimal control representation admits a game theoretic perspective, in which training residual networks can be interpreted as cooperative trajectory optimization on state-augmented dynamical systems. This Game Theoretic DDP (GT-DDP) optimizer enjoys the same theoretic connection in previous work, yet generates a much complex update rule that better leverages available information during network propagation. Evaluation on image classification datasets (e.g. MNIST and CIFAR100) shows an improvement in training convergence and variance reduction over existing methods. Our approach highlights the benefit gained from architecture-aware optimization.

Guan-Horng Liu, Tianrong Chen, Evangelos A. Theodorou

Interpretation of Deep Neural Networks (DNNs) training as an optimal control problem with nonlinear dynamical systems has received considerable attention recently, yet the algorithmic development remains relatively limited. In this work, we make an attempt along this line by reformulating the training procedure from the trajectory optimization perspective. We first show that most widely-used algorithms for training DNNs can be linked to the Differential Dynamic Programming (DDP), a celebrated second-order method rooted in the Approximate Dynamic Programming. In this vein, we propose a new class of optimizer, DDP Neural Optimizer (DDPNOpt), for training feedforward and convolution networks. DDPNOpt features layer-wise feedback policies which improve convergence and reduce sensitivity to hyper-parameter over existing methods. It outperforms other optimal-control inspired training methods in both convergence and complexity, and is competitive against state-of-the-art first and second order methods. We also observe DDPNOpt has surprising benefit in preventing gradient vanishing. Our work opens up new avenues for principled algorithmic design built upon the optimal control theory.

Marcus A Pereira, Ziyi Wang, Tianrong Chen et al.

We present a deep recurrent neural network architecture to solve a class of stochastic optimal control problems described by fully nonlinear Hamilton Jacobi Bellmanpartial differential equations. Such PDEs arise when one considers stochastic dynamics characterized by uncertainties that are additive and control multiplicative. Stochastic models with the aforementioned characteristics have been used in computational neuroscience, biology, finance and aerospace systems and provide a more accurate representation of actuation than models with additive uncertainty. Previous literature has established the inadequacy of the linear HJB theory and instead rely on a non-linear Feynman-Kac lemma resulting in a second order forward-backward stochastic differential equations representation. However, the proposed solutions that use this representation suffer from compounding errors and computational complexity leading to lack of scalability. In this paper, we propose a deep learning based algorithm that leverages the second order Forward-Backward SDE representation and LSTM based recurrent neural networks to not only solve such Stochastic Optimal Control problems but also overcome the problems faced by previous approaches and scales well to high dimensional systems. The resulting control algorithm is tested on non-linear systems in robotics and biomechanics to demonstrate feasibility and out-performance against previous methods.