Robust Descent Research Articles

Recent Early-Spring Drying Trend over Southern China Associated with Changes in the Zonal Thermal Contrast over the Pacific

Abstract Southern China has experienced a discernible early-spring drying trend in recent decades. This drying trend is a direct result of a severe deficiency of water vapor, which is induced by local descent anomalies and strong anomalous northerlies from the northwestern Pacific. The predominant anomalous northerlies are directly associated with two anomalous warm highs over the North Pacific (NPH) and East Asia (EAH), respectively. Anomalous easterlies along the southern edge of the NPH divert southwestward over southern Japan and form a branch of the northerlies, which merges into strong northerlies east of the EAH to construct the prevalent and deep northerly anomalies along the coast of East Asia. The NPH is related to changes in the zonal contrast of the Pacific sea surface temperature (SST-Tri), whose linear trend is dominated by warming in the western Pacific. The change in SST-Tri causes more precipitation in the tropical North Pacific, where strong ascent occurs and further facilitates robust descent over the extratropical North Pacific, facilitating the formation of the NPH. The EAH results from the interaction of tropical and higher latitudinal forcing in the context of global warming. On the one hand, enhanced precipitation over the Maritime Continent associated with the change in SST-Tri induces descent anomalies of the EAH over southern China via a local Hadley cell circulation. On the other hand, a “positive–negative–positive” tripolar tendency over the North Atlantic forces a wave train, which propagates eastward, becomes intensified over the Ural Mountains, and then moves farther eastward to affect the EAH. In addition, changes in SST-Tri can also modulate the tripole pattern over the North Pacific to further contribute to the EAH.

High dimensional robust M-estimation : arbitrary corruption and heavy tails

We consider the problem of sparsity-constrained $M$-estimation when both explanatory and response variables have heavy tails (bounded 4-th moments), or a fraction of arbitrary corruptions. We focus on the $k$-sparse, high-dimensional regime where the number of variables $d$ and the sample size $n$ are related through $n \sim k \log d$. We define a natural condition we call the Robust Descent Condition (RDC), and show that if a gradient estimator satisfies the RDC, then Robust Hard Thresholding (IHT using this gradient estimator), is guaranteed to obtain good statistical rates. The contribution of this paper is in showing that this RDC is a flexible enough concept to recover known results, and obtain new robustness results. Specifically, new results include: (a) For $k$-sparse high-dimensional linear- and logistic-regression with heavy tail (bounded 4-th moment) explanatory and response variables, a linear-time-computable median-of-means gradient estimator satisfies the RDC, and hence Robust Hard Thresholding is minimax optimal; (b) When instead of heavy tails we have $O(1/\sqrt{k}\log(nd))$-fraction of arbitrary corruptions in explanatory and response variables, a near linear-time computable trimmed gradient estimator satisfies the RDC, and hence Robust Hard Thresholding is minimax optimal. We demonstrate the effectiveness of our approach in sparse linear, logistic regression, and sparse precision matrix estimation on synthetic and real-world US equities data.

Computation of optimal transport on discrete metric measure spaces

In this paper we investigate the numerical approximation of an analogue of the Wasserstein distance for optimal transport on graphs that is defined via a discrete modification of the Benamou–Brenier formula. This approach involves the logarithmic mean of measure densities on adjacent nodes of the graph. For this model a variational time discretization of the probability densities on graph nodes and the momenta on graph edges is proposed. A robust descent algorithm for the action functional is derived, which in particular uses a proximal splitting with an edgewise nonlinear projection on the convex subgraph of the logarithmic mean. Thereby, suitable chosen slack variables avoid a global coupling of probability densities on all graph nodes in the projection step. For the time discrete action functional $$\Gamma $$-convergence to the time continuous action is established. Numerical results for a selection of test cases show qualitative and quantitative properties of the optimal transport on graphs. Finally, we use our algorithm to implement a JKO scheme for the gradient flow of the entropy in discrete transportation distance, which is known to coincide with underlying Markov semigroup, and test our results against a classical backward Euler discretization of this discrete heat flow.

A note on robust descent in differentiable optimization

In this note, we show how to alleviate the catastrophic cancellations that occur when comparing function values in trust-region algorithms. The main original contribution is to successfully adapt the line search strategy Hager and Zhang (2005) for use within trust-region-like algorithms.

Robust descent in differentiable optimization using automatic finite differences

Descent algorithms use comparisons of a merit function, to ensure the global convergence of generated iterates to at least a stationary point, even to solutions satisfying the second order necessary optimality conditions. Unfortunately, the comparison of the merit function values may fail to be precise enough for estimates reaching the square root of the machine precision. We present a clever way to compare merit function values, ensuring that full machine precision may be reached.

A robust descent type algorithm for geophysical inversion through adaptive regularization

A robust descent type algorithm through adaptive regularization is designed to solve a geophysical inverse problem. The scheme uses a regularized descent direction which is obtained through minimization of smoothed functional in every iterative step. The step length factor is chosen using Armijo's rule. Numerical experiment conducted to invert synthetic and field geophysical data demonstrates a high order of robustness in retrieving the model parameters.

Comparison of learning strategies for adaptation of fuzzy controller parameters

For tuning fuzzy controllers, several parameter identification techniques are available, ranging from more robust descent methods to sophisticated optimisation. However, from an application point of view, it is not always clear that numerical sophistication wins over more pragmatic approaches to tuning. Obviously, the data sets play crucial roles in efforts to reach successful tuning. Especially data sets generated from real processes often contain not only noisy data and conflicting subsets, but also the connected problem of non-covering input spaces. In this paper we will compare several parameter identification techniques w.r.t. different data sets. We focus on selections of learning rates and on defining training sequences related to subclasses of parameters.