The classical Langevin Monte Carlo method looks for i.i.d. samples from a target distribution by descending along the gradient of the target distribution. It is popular partially due to its fast convergence rate. However, the numerical cost is sometimes high because the gradient can be hard to obtain. One approach to eliminate the gradient computation is to employ the concept of "ensemble", where a large number of particles are evolved together so that the neighboring particles provide gradient information to each other. In this article, we discuss two algorithms that integrate the ensemble feature into LMC, and the associated properties. There are two sides of our discovery: 1. By directly surrogating the gradient using the ensemble approximation, we develop Ensemble Langevin Monte Carlo. We show that this method is unstable due to a potentially small denominator that induces high variance. We provide a counterexample to explicitly show this instability. 2. We then change the strategy and enact the ensemble approximation to the gradient only in a constrained manner, to eliminate the unstable points. The algorithm is termed Constrained Ensemble Langevin Monte Carlo. We show that, with a proper tuning, the surrogation takes place often enough to bring the reasonable numerical saving, while the induced error is still low enough for us to maintain the fast convergence rate, up to a controllable discretization and ensemble error. Such combination of ensemble method and LMC shed light on inventing gradient-free algorithms that produce i.i.d. samples almost exponentially fast.
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The main aim of this paper is to solve an inverse source problem for a general nonlinear hyperbolic equation. Combining the quasi-reversibility method and a suitable Carleman weight function, we define a map of which fixed point is the solution to the inverse problem. To find this fixed point, we define a recursive sequence with an arbitrary initial term by the same manner as in the classical proof of the contraction principle. Applying a Carleman estimate, we show that the sequence above converges to the desired solution with the exponential rate. Therefore, our new method can be considered as an analog of the contraction principle. We rigorously study the stability of our method with respect to noise. Numerical examples are presented.
Many meta-learning algorithms can be formulated into an interleaved process, in the sense that task-specific predictors are learned during inner-task adaptation and meta-parameters are updated during meta-update. The normal meta-training strategy needs to differentiate through the inner-task adaptation procedure to optimize the meta-parameters. This leads to a constraint that the inner-task algorithms should be solved analytically. Under this constraint, only simple algorithms with analytical solutions can be applied as the inner-task algorithms, limiting the model expressiveness. To lift the limitation, we propose an adaptation-agnostic meta-training strategy. Following our proposed strategy, we can apply stronger algorithms (e.g., an ensemble of different types of algorithms) as the inner-task algorithm to achieve superior performance comparing with popular baselines. The source code is available at //github.com/jiaxinchen666/AdaptationAgnosticMetaLearning.
Bayesian statistical inference for Generalized Linear Models (GLMs) with parameters lying on a constrained space is of general interest (e.g., in monotonic or convex regression), but often constructing valid prior distributions supported on a subspace spanned by a set of linear inequality constraints can be challenging, especially when some of the constraints might be binding leading to a lower dimensional subspace. For the general case with canonical link, it is shown that a generalized truncated multivariate normal supported on a desired subspace can be used. Moreover, it is shown that such prior distribution facilitates the construction of a general purpose product slice sampling method to obtain (approximate) samples from corresponding posterior distribution, making the inferential method computationally efficient for a wide class of GLMs with an arbitrary set of linear inequality constraints. The proposed product slice sampler is shown to be uniformly ergodic, having a geometric convergence rate under a set of mild regularity conditions satisfied by many popular GLMs (e.g., logistic and Poisson regressions with constrained coefficients). One of the primary advantages of the proposed Bayesian estimation method over classical methods is that uncertainty of parameter estimates is easily quantified by using the samples simulated from the path of the Markov Chain of the slice sampler. Numerical illustrations using simulated data sets are presented to illustrate the superiority of the proposed methods compared to some existing methods in terms of sampling bias and variances. In addition, real case studies are presented using data sets for fertilizer-crop production and estimating the SCRAM rate in nuclear power plants.
A standard way to move particles in a SMC sampler is to apply several steps of a MCMC (Markov chain Monte Carlo) kernel. Unfortunately, it is not clear how many steps need to be performed for optimal performance. In addition, the output of the intermediate steps are discarded and thus wasted somehow. We propose a new, waste-free SMC algorithm which uses the outputs of all these intermediate MCMC steps as particles. We establish that its output is consistent and asymptotically normal. We use the expression of the asymptotic variance to develop various insights on how to implement the algorithm in practice. We develop in particular a method to estimate, from a single run of the algorithm, the asymptotic variance of any particle estimate. We show empirically, through a range of numerical examples, that waste-free SMC tends to outperform standard SMC samplers, and especially so in situations where the mixing of the considered MCMC kernels decreases across iterations (as in tempering or rare event problems).
Distantly supervised models are very popular for relation extraction since we can obtain a large amount of training data using the distant supervision method without human annotation. In distant supervision, a sentence is considered as a source of a tuple if the sentence contains both entities of the tuple. However, this condition is too permissive and does not guarantee the presence of relevant relation-specific information in the sentence. As such, distantly supervised training data contains much noise which adversely affects the performance of the models. In this paper, we propose a self-ensemble filtering mechanism to filter out the noisy samples during the training process. We evaluate our proposed framework on the New York Times dataset which is obtained via distant supervision. Our experiments with multiple state-of-the-art neural relation extraction models show that our proposed filtering mechanism improves the robustness of the models and increases their F1 scores.
Parameters in climate models are usually calibrated manually, exploiting only small subsets of the available data. This precludes both optimal calibration and quantification of uncertainties. Traditional Bayesian calibration methods that allow uncertainty quantification are too expensive for climate models; they are also not robust in the presence of internal climate variability. For example, Markov chain Monte Carlo (MCMC) methods typically require $O(10^5)$ model runs and are sensitive to internal variability noise, rendering them infeasible for climate models. Here we demonstrate an approach to model calibration and uncertainty quantification that requires only $O(10^2)$ model runs and can accommodate internal climate variability. The approach consists of three stages: (i) a calibration stage uses variants of ensemble Kalman inversion to calibrate a model by minimizing mismatches between model and data statistics; (ii) an emulation stage emulates the parameter-to-data map with Gaussian processes (GP), using the model runs in the calibration stage for training; (iii) a sampling stage approximates the Bayesian posterior distributions by sampling the GP emulator with MCMC. We demonstrate the feasibility and computational efficiency of this calibrate-emulate-sample (CES) approach in a perfect-model setting. Using an idealized general circulation model, we estimate parameters in a simple convection scheme from synthetic data generated with the model. The CES approach generates probability distributions of the parameters that are good approximations of the Bayesian posteriors, at a fraction of the computational cost usually required to obtain them. Sampling from this approximate posterior allows the generation of climate predictions with quantified parametric uncertainties.
Despite pre-trained language models have proven useful for learning high-quality semantic representations, these models are still vulnerable to simple perturbations. Recent works aimed to improve the robustness of pre-trained models mainly focus on adversarial training from perturbed examples with similar semantics, neglecting the utilization of different or even opposite semantics. Different from the image processing field, the text is discrete and few word substitutions can cause significant semantic changes. To study the impact of semantics caused by small perturbations, we conduct a series of pilot experiments and surprisingly find that adversarial training is useless or even harmful for the model to detect these semantic changes. To address this problem, we propose Contrastive Learning with semantIc Negative Examples (CLINE), which constructs semantic negative examples unsupervised to improve the robustness under semantically adversarial attacking. By comparing with similar and opposite semantic examples, the model can effectively perceive the semantic changes caused by small perturbations. Empirical results show that our approach yields substantial improvements on a range of sentiment analysis, reasoning, and reading comprehension tasks. And CLINE also ensures the compactness within the same semantics and separability across different semantics in sentence-level.
This paper studies task adaptive pre-trained model selection, an \emph{underexplored} problem of assessing pre-trained models so that models suitable for the task can be selected from the model zoo without fine-tuning. A pilot work~\cite{nguyen_leep:_2020} addressed the problem in transferring supervised pre-trained models to classification tasks, but it cannot handle emerging unsupervised pre-trained models or regression tasks. In pursuit of a practical assessment method, we propose to estimate the maximum evidence (marginalized likelihood) of labels given features extracted by pre-trained models. The maximum evidence is \emph{less prone to over-fitting} than the likelihood, and its \emph{expensive computation can be dramatically reduced} by our carefully designed algorithm. The Logarithm of Maximum Evidence (LogME) can be used to assess pre-trained models for transfer learning: a pre-trained model with high LogME is likely to have good transfer performance. LogME is fast, accurate, and general, characterizing it as \emph{the first practical assessment method for transfer learning}. Compared to brute-force fine-tuning, LogME brings over $3000\times$ speedup in wall-clock time. It outperforms prior methods by a large margin in their setting and is applicable to new settings that prior methods cannot deal with. It is general enough to diverse pre-trained models (supervised pre-trained and unsupervised pre-trained), downstream tasks (classification and regression), and modalities (vision and language). Code is at \url{//github.com/thuml/LogME}.
Few-shot learning methods offer pre-training techniques optimized for easier later adaptation of the model to new classes (unseen during training) using one or a few examples. This adaptivity to unseen classes is especially important for many practical applications where the pre-trained label space cannot remain fixed for effective use and the model needs to be "specialized" to support new categories on the fly. One particularly interesting scenario, essentially overlooked by the few-shot literature, is Coarse-to-Fine Few-Shot (C2FS), where the training classes (e.g. animals) are of much `coarser granularity' than the target (test) classes (e.g. breeds). A very practical example of C2FS is when the target classes are sub-classes of the training classes. Intuitively, it is especially challenging as (both regular and few-shot) supervised pre-training tends to learn to ignore intra-class variability which is essential for separating sub-classes. In this paper, we introduce a novel 'Angular normalization' module that allows to effectively combine supervised and self-supervised contrastive pre-training to approach the proposed C2FS task, demonstrating significant gains in a broad study over multiple baselines and datasets. We hope that this work will help to pave the way for future research on this new, challenging, and very practical topic of C2FS classification.
This paper proposes a Reinforcement Learning (RL) algorithm to synthesize policies for a Markov Decision Process (MDP), such that a linear time property is satisfied. We convert the property into a Limit Deterministic Buchi Automaton (LDBA), then construct a product MDP between the automaton and the original MDP. A reward function is then assigned to the states of the product automaton, according to accepting conditions of the LDBA. With this reward function, our algorithm synthesizes a policy that satisfies the linear time property: as such, the policy synthesis procedure is "constrained" by the given specification. Additionally, we show that the RL procedure sets up an online value iteration method to calculate the maximum probability of satisfying the given property, at any given state of the MDP - a convergence proof for the procedure is provided. Finally, the performance of the algorithm is evaluated via a set of numerical examples. We observe an improvement of one order of magnitude in the number of iterations required for the synthesis compared to existing approaches.
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