We characterize weakly regularized continual classification in homogeneous models as sequential projections onto task margin sets. This result generalizes prior analyses restricted to either stationary (single-task) deep models or continual linear models. We show that global convergence generally fails, even for simple models linear in data but nonlinear in parameters. Nevertheless, by leveraging results from nonconvex projection theory, we identify regularity properties of homogeneous deep networks that guarantee local linear convergence under random and cyclic task sequences. Finally, we extend our analysis to continual regression, unifying the framework for homogeneous models.

We present LrcSSM, a non-linear recurrent model that processes long sequences as fast as today's linear state-space layers. By forcing its Jacobian matrix to be diagonal, the full sequence can be solved in parallel, giving O(TD) computational work and memory and only O(logT) sequential depth, for input-sequence length T and a state dimension D. Moreover, LrcSSM offers a formal gradient-stability guarantee that other input-varying systems such as Liquid-S4 and Mamba do not provide. Importantly, the diagonal Jacobian structure of our model results in no performance loss compared to the original model with dense Jacobian, and the approach can be generalized to other non-linear recurrent models, demonstrating broader applicability. On a suite of long-range forecasting tasks, we demonstrate that LrcSSM outperforms Transformers, LRU, S5, and Mamba.

Discrete diffusion models offer a flexible, controllable approach to structured sequence generation, yet they still lag behind causal language models in expressive power. A key limitation lies in their reliance on the Markovian assumption, which restricts each step to condition only on the current state, leading to potential uncorrectable error accumulation. In this paper, we introduce CaDDi (Causal Discrete Diffusion Model), a discrete diffusion model that conditions on the entire generative trajectory, thereby lifting the Markov constraint and allowing the model to revisit and improve past states. By unifying sequential (causal) and temporal (diffusion) reasoning in a single non-Markovian transformer, CaDDi also treats standard causal language models as a special case and permits the direct reuse of pretrained LLM weights with no architectural changes. Empirically, CaDDi outperforms state-of-the-art discrete diffusion baselines on natural-language benchmarks, substantially narrowing the remaining gap to large autoregressive transformers.

Recent advances in foundational video generators (33; 38; 28), particularly through large diffusion transformers, have significantly improved video generation capabilities. This progress naturally suggests leveraging these powerful foundation models to advance video inpainting and editing. However, effectively utilizing their conditional generation abilities for these tasks would typically demand substantial computational resources for training, given their massive scale. Furthermore, as foundation models continue to evolve, traditional approaches relying on extensive fine-tuning will face increasing challenges in adapting to new video generators. An alternative solution is to employ these video generators as data priors, enabling task resolution in a training-free manner. Recent research has extensively explored methods for enabling conditional generation in image diffusion models.

This paper is concerned with online time series forecasting, where unknown distribution shifts occur over time, i.e., latent variables influence the mapping from historical to future observations. To develop an automated way of online time series forecasting, we propose a Theoretical framework for Online Time-series forecasting (TOT in short) with theoretical guarantees. Specifically, we prove that supplying a forecaster with latent variables tightens the Bayes risk--the benefit endures under estimation uncertainty of latent variables and grows as the latent variables achieve a more precise identifiability. To better introduce latent variables into online forecasting algorithms, we further propose to identify latent variables with minimal adjacent observations. Based on these results, we devise a modelagnostic blueprint by employing a temporal decoder to match the distribution of observed variables and two independent noise estimators to model the causal inference of latent variables and mixing procedures of observed variables, respectively. Experiment results on synthetic data support our theoretical claims. Moreover, plugin implementations built on several baselines yield general improvement across multiple benchmarks, highlighting the effectiveness in real-world applications.

Classifier-Free Guidance (CFG) is an essential component of text-to-image diffusion models, and understanding and advancing its operational mechanisms remains a central focus of research. Existing approaches stem from divergent theoretical interpretations, thereby limiting the design space and obscuring key design choices. To address this, we propose a unified perspective that reframes conditional guidance as fixed point iterations, seeking to identify a golden path where latents produce consistent outputs under both conditional and unconditional generation. We demonstrate that CFG and its variants constitute a special case of single-step short-interval iteration, which is theoretically proven to exhibit inefficiency. To this end, we introduce Foresight Guidance (FSG), which prioritizes solving longer-interval subproblems in early diffusion stages with increased iterations.

In this work, we study offline convex optimization with smooth objectives, where the classical Nesterov's Accelerated Gradient (NAG) method achieves the optimal accelerated convergence. Extensive research has aimed to understand NAG from various perspectives, and a recent line of work approaches this from the viewpoint of online learning and online-to-batch conversion, emphasizing the role of optimistic online algorithms for acceleration. In this work, we contribute to this perspective by proposing novel optimistic online-to-batch conversions that incorporate optimism theoretically into the analysis, thereby significantly simplifying the online algorithm design while preserving the optimal convergence rates. Specifically, we demonstrate the effectiveness of our conversions through the following results: (i) when combined with simple online gradient descent, our optimistic conversion achieves the optimal accelerated convergence; (ii) our conversion also applies to strongly convex objectives, and by leveraging both optimistic online-to-batch conversion and optimistic online algorithms, we achieve the optimal accelerated convergence rate for strongly convex and smooth objectives, for the first time through the lens of online-to-batch conversion; (iii) our optimistic conversion can achieve universality to smoothness -- applicable to both smooth and non-smooth objectives without requiring knowledge of the smoothness coefficient -- and remains efficient as non-universal methods by using only one gradient query in each iteration. Finally, we highlight the effectiveness of our optimistic online-to-batch conversions by a precise correspondence with NAG.

While Classifier-Free Guidance (CFG) has become standard for improving sample fidelity in conditional diffusion models, it can harm diversity and induce memorization by applying constant guidance regardless of whether a particular sample needs correction. We propose FeedBack Guidance (FBG), which uses a state-dependent coefficient to self-regulate guidance amounts based on need. Our approach is derived from first principles by assuming the learned conditional distribution is linearly corrupted by the unconditional distribution, contrasting with CFG's implicit multiplicative assumption. Our scheme relies on feedback of its own predictions about the conditional signal informativeness to adapt guidance dynamically during inference, challenging the view of guidance as a fixed hyperparameter. The approach is benchmarked on ImageNet512x512, where it significantly outperforms Classifier-Free Guidance and is competitive to Limited Interval Guidance (LIG) while benefitting from a strong mathematical framework. On Text-To-Image generation, we demonstrate that, as anticipated, our approach automatically applies higher guidance scales for complex prompts than for simpler ones and that it can be easily combined with existing guidance schemes such as CFG or LIG. Our code is available at this link.

Direct preference optimization (DPO) methods have shown strong potential in aligning text-to-image diffusion models with human preferences by training on paired comparisons. These methods improve training stability by avoiding the REINFORCE algorithm but still struggle with challenges such as accurately estimating image probabilities due to the non-linear nature of the sigmoid function and the limited diversity of offline datasets. In this paper, we introduce Diffusion Denoising Ranking Optimization (Diffusion-DRO), a new preference learning framework grounded in inverse reinforcement learning. Diffusion-DRO removes the dependency on a reward model by casting preference learning as a ranking problem, thereby simplifying the training objective into a denoising formulation and overcoming the non-linear estimation issues found in prior methods. Moreover, Diffusion-DRO uniquely integrates offline expert demonstrations with online policy-generated negative samples, enabling it to effectively capture human preferences while addressing the limitations of offline data. Comprehensive experiments show that Diffusion-DRO delivers improved generation quality across a range of challenging and unseen prompts, outperforming state-of-the-art baselines in both both quantitative metrics and user studies.

Inverse problems, where the goal is to recover an unknown signal from noisy or incomplete measurements, are central to applications in medical imaging, remote sensing, and computational biology. Diffusion models have recently emerged as powerful priors for solving such problems. However, existing methods either rely on projection-based techniques that enforce measurement consistency through heuristic updates, or they approximate the likelihood p(y | x), often resulting in artifacts and instability under complex or high-noise conditions. To address these limitations, we propose a novel framework called coupled data and measurement space diffusion posterior sampling (C-DPS), which eliminates the need for constraint tuning or likelihood approximation. C-DPS introduces a forward stochastic process in the measurement space {yt}, evolving in parallel with the data-space diffusion {xt}, which enables the derivation of a closed-form posterior p(xt 1 | xt,yt 1). This coupling allows for accurate and recursive sampling based on a well-defined posterior distribution. Empirical results demonstrate that C-DPS consistently outperforms existing baselines, both qualitatively and quantitatively, across multiple inverse problem benchmarks.