Stochastic Gradient Hamiltonian Monte Carlo (SGHMC) is a momentum version of stochastic gradient descent with properly injected Gaussian noise to find a global minimum. In this paper, non-asymptotic convergence analysis of SGHMC is given in the context of non-convex optimization, where subsampling techniques are used over an i.i.d dataset for gradient updates. Our results complement those of [RRT17] and improve on those of [GGZ18].
Finding an approximate minimizer is similar to sampling distributions concentrate around the true minimizer. This well known connection give rise to the study of simulated annealing algorithms, see @cite_22 , @cite_2 , @cite_13 , @cite_18 , @cite_19 , @cite_7 , @cite_6 . Recently, there are many studies further investigate this connection by means of non asymptotic convergence of Langevin based algorithms and in stochastic non-convex optimization and large-scale data analysis, @cite_24 , @cite_30 .
@cite_30 @cite_18 @cite_22 @cite_7 @cite_6 @cite_24 @cite_19 @cite_2 @cite_13
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We consider the problem of finding the minimizer of a convex function @math of the form @math where a low-rank factorization of @math is readily available. We consider the regime where @math . As second-order methods prove to be effective in finding the minimizer to a high-precision, in this work, we propose randomized Newton-type algorithms that exploit sub-sampling of @math , as well as inexact updates, as means to reduce the computational complexity. Two non-uniform sampling distributions based on block norm squares and block partial leverage scores are considered in order to capture important terms among @math . We show that at each iteration non-uniformly sampling at most @math terms from @math is sufficient to achieve a linear-quadratic convergence rate in @math when a suitable initial point is provided. In addition, we show that our algorithms achieve a lower computational complexity and exhibit more robustness and better dependence on problem specific quantities, such as the condition number, compared to similar existing methods, especially the ones based on uniform sampling. Finally, we empirically demonstrate that our methods are at least twice as fast as Newton's methods with ridge logistic regression on several real datasets. In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly overdamped) Langevin diffusion and establish guarantees on its error measured in the Wasserstein-2 distance. Our guarantees improve or extend the state-of-the-art results in three directions. First, we provide an upper bound on the error of the first-order Langevin Monte Carlo (LMC) algorithm with optimized varying step-size. This result has the advantage of being horizon free (we do not need to know in advance the target precision) and to improve by a logarithmic factor the corresponding result for the constant step-size. Second, we study the case where accurate evaluations of the gradient of the log-density are unavailable, but one can have access to approximations of the aforementioned gradient. In such a situation, we consider both deterministic and stochastic approximations of the gradient and provide an upper bound on the sampling error of the first-order LMC that quantifies the impact of the gradient evaluation inaccuracies. Third, we establish upper bounds for two versions of the second-order LMC, which leverage the Hessian of the log-density. We nonasymptotic guarantees on the sampling error of these second-order LMCs. These guarantees reveal that the second-order LMC algorithms improve on the first-order LMC in ill-conditioned settings. We introduce a generalization of a deterministic relaxation algorithm for edge-preserving regularization in linear inverse problems. This algorithm transforms the original (possibly nonconvex) optimization problem into a sequence of quadratic optimization problems, and has been shown to converge under certain conditions when the original cost functional being minimized is strictly convex. We prove that our more general algorithm is globally convergent (i.e., converges to a local minimum from any initialization) under less restrictive conditions, even when the original cost functional is nonconvex. We apply this algorithm to tomographic reconstruction from limited-angle data by formulating the problem as one of regularized least-squares optimization. The results demonstrate that the constraint of piecewise smoothness, applied through the use of edge-preserving regularization, can provide excellent limited-angle tomographic reconstructions. Two edge-preserving regularizers-one convex, the other nonconvex-are used in numerous simulations to demonstrate the effectiveness of the algorithm under various limited-angle scenarios, and to explore how factors, such as the choice of error norm, angular sampling rate and amount of noise, affect the reconstruction quality and algorithm performance. These simulation results show that for this application, the nonconvex regularizer produces consistently superior results. We present a new efficient algorithm for the search version of the approximate Closest Vector Problem with Preprocessing (CVPP). Our algorithm achieves an approximation factor of O(n sqrt log n ), improving on the previous best of O(n^ 1.5 ) due to Lag arias, Lenstra, and Schnorr hkzbabai . We also show, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve the problem (with the slightly worse approximation factor of O(n)). We remark that this still leaves a large gap with respect to the decisional version of CVPP, where the best known approximation factor is O(sqrt n log n ) due to Aharonov and Regev AharonovR04 . To achieve these results, we show a reduction to the same problem restricted to target points that are close to the lattice and a more efficient reduction to a harder problem, Bounded Distance Decoding with preprocessing (BDDP). Combining either reduction with the previous best-known algorithm for BDDP by Liu, Lyubashevsky, and Micciancio LiuLM06 gives our main result. In the setting of CVP without preprocessing, we also give a reduction from (1+eps)gamma approximate CVP to gamma approximate CVP where the target is at distance at most 1+1 eps times the minimum distance (the length of the shortest non-zero vector) which relies on the lattice sparsification techniques of Dadush and Kun DadushK13 . As our final and most technical contribution, we present a substantially more efficient variant of the LLM algorithm (both in terms of run-time and amount of preprocessing advice), and via an improved analysis, show that it can decode up to a distance proportional to the reciprocal of the smoothing parameter of the dual lattice MR04 . We show that this is never smaller than the LLM decoding radius, and that it can be up to an wide tilde Omega (sqrt n ) factor larger.