The target's distribution is right skewed with some fairly high values compared to the mean: The Root Mean Squared Error (RMSE) or Mean Squared Error (MSE, which is basically the same as RMSE without the squared root) is the most popular regression metric. If there was a king/queen of regression metrics, this would have been it! Where y i is the prediction and yi the actual target value. In other words, you square all the errors (or residuals as they call them) per sample/row, then sum them, divide by the total number of observations and take the squared root to bring the metric back to the original space (or you don't in MSE). It is also one of the oldest regression metrics. Smaller errors (that are for example less than 1.) will have an even lower contribution to the overall error after being squared, whereas bigger errors will have much more weight after being squared. A large error in a given sample can have huge impact on the overall results and make an optimizer focus on reducing the error for that single sample, making the prediction for every other sample worse. This is because of the "squared" attribute, it makes it easily differentiable, something that gradient-based algorithms (like Stochastic Gradient Descent) can leverage.

We analyze (stochastic) gradient descent (SGD) with delayed updates on smooth quasi-convex and non-convex functions and derive concise, non-asymptotic, convergence rates. We show that the rate of convergence in all cases consists of two terms: (i) a stochastic term which is not affected by the delay, and (ii) a higher order deterministic term which is only linearly slowed down by the delay. Thus, in the presence of noise, the effects of the delay become negligible after a few iterations and the algorithm converges at the same optimal rate as standard SGD. This result extends a line of research that showed similar results in the asymptotic regime or for strongly-convex quadratic functions only. We further show similar results for SGD with more intricate form of delayed gradients---compressed gradients under error compensation and for localSGD where multiple workers perform local steps before communicating with each other. In all of these settings, we improve upon the best known rates. These results show that SGD is robust to compressed and/or delayed stochastic gradient updates. This is in particular important for distributed parallel implementations, where asynchronous and communication efficient methods are the key to achieve linear speedups for optimization with multiple devices.

We investigate implicit regularization schemes for gradient descent methods applied to unpenalized least squares regression to solve the problem of reconstructing a sparse signal from an underdetermined system of linear measurements under the restricted isometry assumption. For a given parametrization yielding a non-convex optimization problem, we show that prescribed choices of initialization, step size and stopping time yield a statistically and computationally optimal algorithm that achieves the minimax rate with the same cost required to read the data up to poly-logarithmic factors. Beyond minimax optimality, we show that our algorithm adapts to instance difficulty and yields a dimension-independent rate when the signal-to-noise ratio is high enough. Key to the computational efficiency of our method is an increasing step size scheme that adapts to refined estimates of the true solution. We validate our findings with numerical experiments and compare our algorithm against explicit $\ell_{1}$ penalization. Going from hard instances to easy ones, our algorithm is seen to undergo a phase transition, eventually matching least squares with an oracle knowledge of the true support.

We study optimal regret bounds for control in linear dynamical systems under adversarially changing strongly convex cost functions, given the knowledge of transition dynamics. This includes several well studied and fundamental frameworks such as the Kalman filter and the linear quadratic regulator. State of the art methods achieve regret which scales as $O(\sqrt{T})$, where $T$ is the time horizon. We show that the optimal regret in this setting can be significantly smaller, scaling as $O(\text{poly}(\log T))$. This regret bound is achieved by two different efficient iterative methods, online gradient descent and online natural gradient.

Neural networks have been proven to be vulnerable to a variety of adversarial attacks. From a safety perspective, highly sparse adversarial attacks are particularly dangerous. On the other hand the pixelwise perturbations of sparse attacks are typically large and thus can be potentially detected. We propose a new black-box technique to craft adversarial examples aiming at minimizing $l_0$-distance to the original image. Extensive experiments show that our attack is better or competitive to the state of the art. Moreover, we can integrate additional bounds on the componentwise perturbation. Allowing pixels to change only in region of high variation and avoiding changes along axis-aligned edges makes our adversarial examples almost non-perceivable. Moreover, we adapt the Projected Gradient Descent attack to the $l_0$-norm integrating componentwise constraints. This allows us to do adversarial training to enhance the robustness of classifiers against sparse and imperceivable adversarial manipulations.

Abstract--We consider a general family of problems of which the output space admits vector-valued structure, covering a broad family of important domains, e.g. By using local Rademacher complexity and unlabeled data, we derived novel data-dependent excess risk bounds for vector-valued functions in both linear space and kernel space. The proposed bounds are much sharper than existing bounds and can be applied into specific vector-valued tasks in terms of different hypotheses sets and loss functions. Theoretical analysis motivates us to devise a unified learning framework for vector-valued functions based which is solved by proximal gradient descent on the primal, achieving a much better tradeoff between accuracy and efficiency . Empirical results on several benchmark datasets show that the proposed algorithm outperforms compared methods significantly, which coincides with our theoretical analysis. Index Terms --Statistical Learning Theory, Local Rademacher Complexity, Vector-Valued Functions, Semi-Supervised Learning.null 1 I NTRODUCTION I N the supervised learning, learning vector-valued functions is to learn a predict model from training data with vector-valued labels instead of scalar-valued labels, including a wide range of important tasks, such as multi-task learning [1], [2], [3], multi-label learning [4], [5], multi-class classification [6], [7], ranking [8], [9] and so on. The first unified learning framework for vector-valued functions in reproducing kernel Hilbert space (RKHS) was proposed in [10]. Then, the unified framework was further developed in [11], [12] and extended to semi-supervised learning by manifold regularization [13], [14], [15]. While current research about vector-valued functions mainly focus on the algorithmic front, we study vector-valued functions from both theoretical perspective and algorithmic perspective. In this paper, we integrate our previous works in multi-classification [7], [16] and generalize the idea into vector-valued settings. W e make the paper a significant improvement based on those two conference papers, with clearer and more general theoretical results, additional technical details,and a unified learning framework. Wang are with Institute of Information Engineering, Chinese Academy of Sciences, Beijing, China. As the most common and successful data-dependent tool, Rademacher complexity was firstly used to analysis generalization performance of multi-class tasks in [22] and further studied in [23], [24]. The convergence rate of Rademacher complexity based error bounds are usually O ( K/ n), where K and n are the number of classes and the size of labeled samples, respectively .

In this post we will see how to implement Gradient Descent using TensorFlow. Next, we will define our variable \(\omega \) and we will initialize it with \(-3 \). With the following peace of code we will also define our cost function \(J(\omega) (\omega – 3) 2 \). With the next two lines of code, we specify the initialization of our variables (here we have just one variable \(\omega \) and the gradient descent for minimizing our cost function with the learning rate of \(0.01 \). Then we will define a session as sess and we will run the init so we will initialize the variable \(\omega \).

At the time I'm writing this, every single trending article in my Towards Data Science home page is talking about applying or learning a particular skill in data science. At the top are big-picture skills such as How to Work With Stakeholders as a Data Scientist and How to Become a Data Engineer, followed by a litany of very specific skills including technical primers on Batch Gradient Descent vs. Stochastic Gradient Descent, Multi-Class Text Classification, Faster R-CNN, et cetera. As a dedicated Medium platform for "sharing concepts, ideas, and codes" in data science, it is not surprising that such learning resources attain high popularity amongst Towards Data Science followers, who are probably navigating data-centric projects and professions. But to a novice looking to prioritize what is essential, it can quickly become daunting. Should one train to become a master Kaggler?

We shed new insights on the two commonly used updates for the online $k$-PCA problem, namely, Krasulina's and Oja's updates. We show that Krasulina's update corresponds to a projected gradient descent step on the Stiefel manifold of the orthonormal $k$-frames, while Oja's update amounts to a gradient descent step using the unprojected gradient. Following these observations, we derive a more \emph{implicit} form of Krasulina's $k$-PCA update, i.e. a version that uses the information of the future gradient as much as possible. Most interestingly, our implicit Krasulina update avoids the costly QR-decomposition step by bypassing the orthonormality constraint. We show that the new update in fact corresponds to an online EM step applied to a probabilistic $k$-PCA model. The probabilistic view of the updates allows us to combine multiple models in a distributed setting. We show experimentally that the implicit Krasulina update yields superior convergence while being significantly faster. We also give strong evidence that the new update can benefit from parallelism and is more stable w.r.t. tuning of the learning rate.

We revisit the local Stochastic Gradient Descent (local SGD) method and prove new convergence rates. We close the gap in the theory by showing that it works under unbounded gradients and extend its convergence to weakly convex functions. Furthermore, by changing the assumptions, we manage to get new bounds that explain in what regimes local SGD is faster that its non-local version. For instance, if the objective is strongly convex, we show that, up to constants, it is sufficient to synchronize $M$ times in total, where $M$ is the number of nodes. This improves upon the known requirement of Stich (2018) of $\sqrt{TM}$ synchronization times in total, where $T$ is the total number of iterations, which helps to explain the empirical success of local SGD.