Training-free conditional diffusion provides a flexible alternative to task-specific conditional model training, but existing samplers often allocate computation inefficiently: independent guided trajectories can vary widely in quality, and additional function evaluations along a single trajectory may not recover from poor early decisions. We propose Tempered Guided Diffusion (TGD), an annealed sequential Monte Carlo framework for training-free conditional sampling with diffusion priors. TGD targets tempered posterior distributions over the clean signal, using noisy diffusion states only as auxiliary variables for proposing reconstructions and propagating particles. Particles are reweighted by incremental likelihood ratios, resampled, and propagated across noise levels, concentrating computation on trajectories plausible under both the prior and observation. Under idealized exact-reconstruction assumptions, full TGD yields a consistent particle approximation to the posterior as the number of particles grows. For expensive reconstruction tasks, Accelerated TGD (A-TGD) retains early particle exploration but prunes to a single high-likelihood trajectory partway through sampling. Experiments on a controlled two-dimensional inverse problem and image inverse problems show improved posterior approximation and favorable wall-clock speed-quality tradeoffs over independent multi-trajectory baselines.

Bandits

We study a bandit version of phase retrieval where the learner chooses actions (At)nt=1 in the d-dimensional unit ball and the expected reward is hAt,?i2 with? 2 Rd an unknown parameter vector. We prove an upper bound on the minimax cumulative regret in this problem of (d p n), which matches known lower bounds up to logarithmic factors and improves on the best known upper bound by a factor of p d. We also show that the minimax simple regret is (d/ p n) and that this is only achievable by an adaptive algorithm. Our analysis shows that an apparently convincing heuristic for guessing lower bounds can be misleading and that uniform bounds on the information ratio for information-directed sampling [Russo and Van Roy, 2014] are not sufficient for optimal regret.

Based on this model, we propose a significant semi-parametric generalization called misspecified phase retrieval (MPR), in which $Y = f(\boldsymbol{X}^{\top}\boldsymbol{\beta}^*, \varepsilon)$ with unknown $f$ and $\operatorname{Cov}(Y, (\boldsymbol{X}^{\top}\boldsymbol{\beta}^*)^2) > 0$. In this paper, we propose an estimation procedure, which consists of solving a cascade of two convex programs and provably recovers the direction of $\boldsymbol{\beta}^*$.

We introduce a novel deep-learning inspired formulation of the \textit{phase retrieval problem}, which asks to recover a signal $y_0 \in \R^n$ from $m$ quadratic observations, under structural assumptions on the underlying signal. As is common in many imaging problems, previous methodologies have considered natural signals as being sparse with respect to a known basis, resulting in the decision to enforce a generic sparsity prior. However, these methods for phase retrieval have encountered possibly fundamental limitations, as no computationally efficient algorithm for sparse phase retrieval has been proven to succeed with fewer than $O(k^2\log n)$ generic measurements, which is larger than the theoretical optimum of $O(k \log n)$. In this paper, we sidestep this issue by considering a prior that a natural signal is in the range of a generative neural network $G: \R^k \rightarrow \R^n$. We introduce an empirical risk formulation that has favorable global geometry for gradient methods, as soon as $m = O(k)$, under the model of a multilayer fully-connected neural network with random weights. Specifically, we show that there exists a descent direction outside of a small neighborhood around the true $k$-dimensional latent code and a negative multiple thereof. This formulation for structured phase retrieval thus benefits from two effects: generative priors can more tightly represent natural signals than sparsity priors, and this empirical risk formulation can exploit those generative priors at an information theoretically optimal sample complexity, unlike for a sparsity prior. We corroborate these results with experiments showing that exploiting generative models in phase retrieval tasks outperforms both sparse and general phase retrieval methods.

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

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

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

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