A.1 Frequency ablation study We perform an ablation study on the coarse-to-fine parameter αd and the number of frequency bands L. In Figure 1, we show the surface reconstruction results of the DTUBuddha model under different frequency parameters. Each model is trained for 300K iterations. In the first row we show the results of surface reconstruction quality under different coarse-to-fine parameters αd. It can be seen that when the parameter is too small, the surface reconstruction tends to be oversmoothed. When the parameter is too large, many artifacts will appear in the reconstruction results.

In this paper, we develop a novel homotopy smoothing (HOPS) algorithm for solving a family of non-smooth problems that is composed of a non-smooth term with an explicit max-structure and a smooth term or a simple non-smooth term whose proximal mapping is easy to compute. The best known iteration complexity for solving such non-smooth optimization problems is O(1/) without any assumption on the strong convexity. In this work, we will show that the proposed HOPS achieved a lower iteration complexity of O(1/1 θ) 1with θ (0,1] capturing the local sharpness of the objective function around the optimal solutions. To the best of our knowledge, this is the lowest iteration complexity achieved so far for the considered non-smooth optimization problems without strong convexity assumption. The HOPS algorithm employs Nesterov's smoothing technique and Nesterov's accelerated gradient method and runs in stages, which gradually decreases the smoothing parameter in a stage-wise manner until it yields a sufficiently good approximation of the original function. We show that HOPS enjoys a linear convergence for many well-known non-smooth problems (e.g., empirical risk minimization with a piece-wise linear loss function and `1 norm regularizer, finding a point in a polyhedron, cone programming, etc). Experimental results verify the effectiveness of HOPS in comparison with Nesterov's smoothing algorithm and the primal-dual style of first-order methods.

We propose a new primal-dual homotopy smoothing algorithm for a linearly constrained convex program, where neither the primal nor the dual function has to be smooth or strongly convex. The best known iteration complexity solving such a non-smooth problem is $\mathcal{O}(\varepsilon^{-1})$. In this paper, we show that by leveraging a local error bound condition on the dual function, the proposed algorithm can achieve a better primal convergence time of $\mathcal{O}\l(\varepsilon^{-2/(2+\beta)}\log_2(\varepsilon^{-1})\r)$, where $\beta\in(0,1]$ is a local error bound parameter. As an example application, we show that the distributed geometric median problem, which can be formulated as a constrained convex program, has its dual function non-smooth but satisfying the aforementioned local error bound condition with $\beta=1/2$, therefore enjoying a convergence time of $\mathcal{O}\l(\varepsilon^{-4/5}\log_2(\varepsilon^{-1})\r)$. This result improves upon the $\mathcal{O}(\varepsilon^{-1})$ convergence time bound achieved by existing distributed optimization algorithms. Simulation experiments also demonstrate the performance of our proposed algorithm.

Neural Information Processing Systems http://nips.cc/

Below are details of our results.

We propose a new primal-dual homotopy smoothing algorithm for a linearly constrained convex program, where neither the primal nor the dual function has to be smooth or strongly convex. The best known iteration complexity solving such a non-smooth problem is $\mathcal{O}(\varepsilon^{-1})$. In this paper, we show that by leveraging a local error bound condition on the dual function, the proposed algorithm can achieve a better primal convergence time of $\mathcal{O}\l(\varepsilon^{-2/(2+\beta)}\log_2(\varepsilon^{-1})\r)$, where $\beta\in(0,1]$ is a local error bound parameter. As an example application, we show that the distributed geometric median problem, which can be formulated as a constrained convex program, has its dual function non-smooth but satisfying the aforementioned local error bound condition with $\beta=1/2$, therefore enjoying a convergence time of $\mathcal{O}\l(\varepsilon^{-4/5}\log_2(\varepsilon^{-1})\r)$. This result improves upon the $\mathcal{O}(\varepsilon^{-1})$ convergence time bound achieved by existing distributed optimization algorithms. Simulation experiments also demonstrate the performance of our proposed algorithm.

We propose a new primal-dual homotopy smoothing algorithm for a linearly constrained convex program, where neither the primal nor the dual function has to be smooth or strongly convex.

Masked Diffusion Models (MDMs) as language models generate by iteratively unmasking tokens, yet their performance crucially depends on the inference time order of unmasking. Prevailing heuristics, such as confidence based sampling, are myopic: they optimize locally, fail to leverage extra test-time compute, and let early decoding mistakes cascade. We propose Lookahead Unmasking (LookUM), which addresses these concerns by reformulating sampling as path selection over all possible unmasking orders without the need for an external reward model. Our framework couples (i) a path generator that proposes paths by sampling from pools of unmasking sets with (ii) a verifier that computes the uncertainty of the proposed paths and performs importance sampling to subsequently select the final paths. Empirically, erroneous unmasking measurably inflates sequence level uncertainty, and our method exploits this to avoid error-prone trajectories. We validate our framework across six benchmarks, such as mathematics, planning, and coding, and demonstrate consistent performance improvements. LookUM requires only two to three paths to achieve peak performance, demonstrating remarkably efficient path selection. The consistent improvements on both LLaDA and post-trained LLaDA 1.5 are particularly striking: base LLaDA with LookUM rivals the performance of RL-tuned LLaDA 1.5, while LookUM further enhances LLaDA 1.5 itself showing that uncertainty based verification provides orthogonal benefits to reinforcement learning and underscoring the versatility of our framework. Code will be publicly released.