Stein's method for Gaussian process approximation can be used to bound the differences between the expectations of smooth functionals $h$ of a c\`adl\`ag random process $X$ of interest and the expectations of the same functionals of a well understood target random process $Z$ with continuous paths. Unfortunately, the class of smooth functionals for which this is easily possible is very restricted. Here, we prove an infinite dimensional Gaussian smoothing inequality, which enables the class of functionals to be greatly expanded -- examples are Lipschitz functionals with respect to the uniform metric, and indicators of arbitrary events -- in exchange for a loss of precision in the bounds. Our inequalities are expressed in terms of the smooth test function bound, an expectation of a functional of $X$ that is closely related to classical tightness criteria, a similar expectation for $Z$, and, for the indicator of a set $K$, the probability $\mathbb{P}(Z \in K^\theta \setminus K^{-\theta})$ that the target process is close to the boundary of $K$.
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Satisfiability is considered the canonical NP-complete problem and is used as a starting point for hardness reductions in theory, while in practice heuristic SAT solving algorithms can solve large-scale industrial SAT instances very efficiently. This disparity between theory and practice is believed to be a result of inherent properties of industrial SAT instances that make them tractable. Two characteristic properties seem to be prevalent in the majority of real-world SAT instances, heterogeneous degree distribution and locality. To understand the impact of these two properties on SAT, we study the proof complexity of random k-SAT models that allow to control heterogeneity and locality. Our findings show that heterogeneity alone does not make SAT easy as heterogeneous random k-SAT instances have superpolynomial resolution size. This implies intractability of these instances for modern SAT-solvers. On the other hand, modeling locality with an underlying geometry leads to small unsatisfiable subformulas, which can be found within polynomial time. A key ingredient for the result on geometric random k-SAT can be found in the complexity of higher-order Voronoi diagrams. As an additional technical contribution, we show a linear upper bound on the number of non-empty Voronoi regions, that holds for points with random positions in a very general setting. In particular, it covers arbitrary p-norms, higher dimensions, and weights affecting the area of influence of each point multiplicatively. This is in stark contrast to quadratic lower bounds for the worst case.
We study the problem of density estimation for a random vector ${\boldsymbol X}$ in $\mathbb R^d$ with probability density $f(\boldsymbol x)$. For a spanning tree $T$ defined on the vertex set $\{1,\dots ,d\}$, the tree density $f_{T}$ is a product of bivariate conditional densities. The optimal spanning tree $T^*$ is the spanning tree $T$, for which the Kullback-Leibler divergence of $f$ and $f_{T}$ is the smallest. From i.i.d. data we identify the optimal tree $T^*$ and computationally efficiently construct a tree density estimate $f_n$ such that, without any regularity conditions on the density $f$, one has that $\lim_{n\to \infty} \int |f_n(\boldsymbol x)-f_{T^*}(\boldsymbol x)|d\boldsymbol x=0$ a.s. For Lipschitz continuous $f$ with bounded support, $\mathbb E\{ \int |f_n(\boldsymbol x)-f_{T^*}(\boldsymbol x)|d\boldsymbol x\}=O(n^{-1/4})$.
We present a new data structure to approximate accurately and efficiently a polynomial $f$ of degree $d$ given as a list of coefficients. Its properties allow us to improve the state-of-the-art bounds on the bit complexity for the problems of root isolation and approximate multipoint evaluation. This data structure also leads to a new geometric criterion to detect ill-conditioned polynomials, implying notably that the standard condition number of the zeros of a polynomial is at least exponential in the number of roots of modulus less than $1/2$ or greater than $2$.Given a polynomial $f$ of degree $d$ with $\|f\|_1 \leq 2^\tau$ for $\tau \geq 1$, isolating all its complex roots or evaluating it at $d$ points can be done with a quasi-linear number of arithmetic operations. However, considering the bit complexity, the state-of-the-art algorithms require at least $d^{3/2}$ bit operations even for well-conditioned polynomials and when the accuracy required is low. Given a positive integer $m$, we can compute our new data structure and evaluate $f$ at $d$ points in the unit disk with an absolute error less than $2^{-m}$ in $\widetilde O(d(\tau+m))$ bit operations, where $\widetilde O(\cdot)$ means that we omit logarithmic factors. We also show that if $\kappa$ is the absolute condition number of the zeros of $f$, then we can isolate all the roots of $f$ in $\widetilde O(d(\tau + \log \kappa))$ bit operations. Moreover, our algorithms are simple to implement. For approximating the complex roots of a polynomial, we implemented a small prototype in Python/NumPy that is an order of magnitude faster than the state-of-the-art solver MPSolve for high degree polynomials with random coefficients.
Approximate Bayesian Computation (ABC) enables statistical inference in complex models whose likelihoods are difficult to calculate but easy to simulate from. ABC constructs a kernel-type approximation to the posterior distribution through an accept/reject mechanism which compares summary statistics of real and simulated data. To obviate the need for summary statistics, we directly compare empirical distributions with a Kullback-Leibler (KL) divergence estimator obtained via classification. In particular, we blend flexible machine learning classifiers within ABC to automate fake/real data comparisons. We consider the traditional accept/reject kernel as well as an exponential weighting scheme which does not require the ABC acceptance threshold. Our theoretical results show that the rate at which our ABC posterior distributions concentrate around the true parameter depends on the estimation error of the classifier. We derive limiting posterior shape results and find that, with a properly scaled exponential kernel, asymptotic normality holds. We demonstrate the usefulness of our approach on simulated examples as well as real data in the context of stock volatility estimation.
In this work, we consider the linear inverse problem $y=Ax+\epsilon$, where $A\colon X\to Y$ is a known linear operator between the separable Hilbert spaces $X$ and $Y$, $x$ is a random variable in $X$ and $\epsilon$ is a zero-mean random process in $Y$. This setting covers several inverse problems in imaging including denoising, deblurring, and X-ray tomography. Within the classical framework of regularization, we focus on the case where the regularization functional is not given a priori but learned from data. Our first result is a characterization of the optimal generalized Tikhonov regularizer, with respect to the mean squared error. We find that it is completely independent of the forward operator $A$ and depends only on the mean and covariance of $x$. Then, we consider the problem of learning the regularizer from a finite training set in two different frameworks: one supervised, based on samples of both $x$ and $y$, and one unsupervised, based only on samples of $x$. In both cases, we prove generalization bounds, under some weak assumptions on the distribution of $x$ and $\epsilon$, including the case of sub-Gaussian variables. Our bounds hold in infinite-dimensional spaces, thereby showing that finer and finer discretizations do not make this learning problem harder. The results are validated through numerical simulations.
Density ratio estimation (DRE) is a fundamental machine learning technique for comparing two probability distributions. However, existing methods struggle in high-dimensional settings, as it is difficult to accurately compare probability distributions based on finite samples. In this work we propose DRE-\infty, a divide-and-conquer approach to reduce DRE to a series of easier subproblems. Inspired by Monte Carlo methods, we smoothly interpolate between the two distributions via an infinite continuum of intermediate bridge distributions. We then estimate the instantaneous rate of change of the bridge distributions indexed by time (the "time score") -- a quantity defined analogously to data (Stein) scores -- with a novel time score matching objective. Crucially, the learned time scores can then be integrated to compute the desired density ratio. In addition, we show that traditional (Stein) scores can be used to obtain integration paths that connect regions of high density in both distributions, improving performance in practice. Empirically, we demonstrate that our approach performs well on downstream tasks such as mutual information estimation and energy-based modeling on complex, high-dimensional datasets.
In sparse estimation, such as fused lasso and convex clustering, we apply either the proximal gradient method or the alternating direction method of multipliers (ADMM) to solve the problem. It takes time to include matrix division in the former case, while an efficient method such as FISTA (fast iterative shrinkage-thresholding algorithm) has been developed in the latter case. This paper proposes a general method for converting the ADMM solution to the proximal gradient method, assuming that assumption that the derivative of the objective function is Lipschitz continuous. Then, we apply it to sparse estimation problems, such as sparse convex clustering and trend filtering, and we show by numerical experiments that we can obtain a significant improvement in terms of efficiency.
We consider the computation of free energy-like quantities for diffusions in high dimension, when resorting to Monte Carlo simulation is necessary. Such stochastic computations typically suffer from high variance, in particular in a low noise regime, because the expectation is dominated by rare trajectories for which the observable reaches large values. Although importance sampling, or tilting of trajectories, is now a standard technique for reducing the variance of such estimators, quantitative criteria for proving that a given control reduces variance are scarce, and often do not apply to practical situations. The goal of this work is to provide a quantitative criterion for assessing whether a given bias reduces variance, and at which order. We rely for this on a recently introduced notion of stochastic solution for Hamilton-Jacobi-Bellman equations. Based on this tool, we introduce the notion of k-stochastic viscosity approximation of a HJB equation. We next prove that such approximate solutions are associated with estimators having a relative variance of order k-1 at log-scale. Finally, in order to show that our definition is relevant, we provide examples of stochastic viscosity approximations of order one and two, with a numerical illustration confirming our theoretical findings.
We propose a method for finding approximate compilations of quantum circuits, based on techniques from policy gradient reinforcement learning. The choice of a stochastic policy allows us to rephrase the optimization problem in terms of probability distributions, rather than variational parameters. This implies that searching for the optimal configuration is done by optimizing over the distribution parameters, rather than over the circuit free angles. The upshot of this is that we can always compute a gradient, provided that the policy is differentiable. We show numerically that this approach is more competitive than those using gradient-free methods, even in the presence of depolarizing noise, and argue analytically why this is the case. Another interesting feature of this approach to variational compilation is that it does not need a separate register and long-range interactions to estimate the end-point fidelity. We expect these techniques to be relevant for training variational circuit in other contexts
This work considers the problem of provably optimal reinforcement learning for episodic finite horizon MDPs, i.e. how an agent learns to maximize his/her long term reward in an uncertain environment. The main contribution is in providing a novel algorithm --- Variance-reduced Upper Confidence Q-learning (vUCQ) --- which enjoys a regret bound of $\widetilde{O}(\sqrt{HSAT} + H^5SA)$, where the $T$ is the number of time steps the agent acts in the MDP, $S$ is the number of states, $A$ is the number of actions, and $H$ is the (episodic) horizon time. This is the first regret bound that is both sub-linear in the model size and asymptotically optimal. The algorithm is sub-linear in that the time to achieve $\epsilon$-average regret for any constant $\epsilon$ is $O(SA)$, which is a number of samples that is far less than that required to learn any non-trivial estimate of the transition model (the transition model is specified by $O(S^2A)$ parameters). The importance of sub-linear algorithms is largely the motivation for algorithms such as $Q$-learning and other "model free" approaches. vUCQ algorithm also enjoys minimax optimal regret in the long run, matching the $\Omega(\sqrt{HSAT})$ lower bound. Variance-reduced Upper Confidence Q-learning (vUCQ) is a successive refinement method in which the algorithm reduces the variance in $Q$-value estimates and couples this estimation scheme with an upper confidence based algorithm. Technically, the coupling of both of these techniques is what leads to the algorithm enjoying both the sub-linear regret property and the asymptotically optimal regret.
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