Figure S2: Number of iterations at each grid point for the Newton and gradient descent methods applying to the ℓ2-regularized logistic regression over simulated data generated in Example 2. We summarize the results in Figure S1-S3. Figure S1 presents the results for ridge regression. In this case, the number of iterations by gradient method first increases and then stays flat as tk grows. Newton method, however, only takes one 1.51.5 iteration at each grid point. Moreover, the level of approximation (i.e., ϵ) seems to have no impact onthe number of iterations at each grid point, which is highly desirable.

--Autonomy in Minimally Invasive Robotic Surgery (MIRS) has the potential to reduce surgeon cognitive and task load, thereby increasing procedural efficiency. However, implementing accurate autonomous control can be difficult due to poor end-effector proprioception, a limitation of their cable-driven mechanisms. Although the robot may have joint encoders for the end-effector pose calculation, various non-idealities make the entire kinematics chain inaccurate. Modern vision-based pose estimation methods lack real-time capability or can be hard to train and generalize. In this work, we demonstrate a real-time capable, vision transformer-based pose estimation approach that is trained using end-to-end differentiable kinematics and rendering in simulation. We demonstrate the potential of this method to correct for noisy pose estimates in simulation, with the longer term goal of verifying the sim-to-real transferability of our approach. The da Vinci Surgical System has been widely applied into different kinds of MIRS procedures in specializations such as, urologic [1], gynecologic [2], and cardiothoracic [3] surgery.

This paper introduces Tempered Fractional Gradient Descent (TFGD), a novel optimization framework that synergizes fractional calculus with exponential tempering to enhance gradient-based learning. Traditional gradient descent methods often suffer from oscillatory updates and slow convergence in high-dimensional, noisy landscapes. TFGD addresses these limitations by incorporating a tempered memory mechanism, where historical gradients are weighted by fractional coefficients $|w_j| = \binomα{j}$ and exponentially decayed via a tempering parameter $λ$. Theoretical analysis establishes TFGD's convergence guarantees: in convex settings, it achieves an $\mathcal{O}(1/K)$ rate with alignment coefficient $d_{α,λ} = (1 - e^{-λ})^{-α}$, while stochastic variants attain $\mathcal{O}(1/k^α)$ error decay. The algorithm maintains $\mathcal{O}(n)$ time complexity equivalent to SGD, with memory overhead scaling as $\mathcal{O}(d/λ)$ for parameter dimension $d$. Empirical validation on the Breast Cancer Wisconsin dataset demonstrates TFGD's superiority, achieving 98.25\% test accuracy (vs. 92.11\% for SGD) and 2$\times$ faster convergence. The tempered memory mechanism proves particularly effective in medical classification tasks, where feature correlations benefit from stable gradient averaging. These results position TFGD as a robust alternative to conventional optimizers in both theoretical and applied machine learning.

Regularization is a widely recognized technique in mathematical optimization. It can be used to smooth out objective functions, refine the feasible solution set, or prevent overfitting in machine learning models. Due to its simplicity and robustness, the gradient descent (GD) method is one of the primary methods used for numerical optimization of differentiable objective functions. However, GD is not well-suited for solving $\ell^1$ regularized optimization problems since these problems are non-differentiable at zero, causing iteration updates to oscillate or fail to converge. Instead, a more effective version of GD, called the proximal gradient descent employs a technique known as soft-thresholding to shrink the iteration updates toward zero, thus enabling sparsity in the solution. Motivated by the widespread applications of proximal GD in sparse and low-rank recovery across various engineering disciplines, we provide an overview of the GD and proximal GD methods for solving regularized optimization problems. Furthermore, this paper proposes a novel algorithm for the proximal GD method that incorporates a variable step size. Unlike conventional proximal GD, which uses a fixed step size based on the global Lipschitz constant, our method estimates the Lipschitz constant locally at each iteration and uses its reciprocal as the step size. This eliminates the need for a global Lipschitz constant, which can be impractical to compute. Numerical experiments we performed on synthetic and real-data sets show notable performance improvement of the proposed method compared to the conventional proximal GD with constant step size, both in terms of number of iterations and in time requirements.

In this article, we consider designs of simple analog artificial neural networks based on adiabatic Josephson cells with a sigmoid activation function. A new approach based on the gradient descent method is developed to adjust the circuit parameters, allowing efficient signal transmission between the network layers. The proposed solution is demonstrated on the example of the system implementing XOR and OR logical operations.

Symbolic regression aims to derive interpretable symbolic expressions from data in order to better understand and interpret data. %which plays an important role in knowledge discovery and interpretable machine learning. In this study, a symbolic network called PruneSymNet is proposed for symbolic regression. This is a novel neural network whose activation function consists of common elementary functions and operators. The whole network is differentiable and can be trained by gradient descent method. Each subnetwork in the network corresponds to an expression, and our goal is to extract such subnetworks to get the desired symbolic expression. Therefore, a greedy pruning algorithm is proposed to prune the network into a subnetwork while ensuring the accuracy of data fitting. The proposed greedy pruning algorithm preserves the edge with the least loss in each pruning, but greedy algorithm often can not get the optimal solution. In order to alleviate this problem, we combine beam search during pruning to obtain multiple candidate expressions each time, and finally select the expression with the smallest loss as the final result. It was tested on the public data set and compared with the current popular algorithms. The results showed that the proposed algorithm had better accuracy.

Fractional derivatives are a well-studied generalization of integer order derivatives. Naturally, for optimization, it is of interest to understand the convergence properties of gradient descent using fractional derivatives. Convergence analysis of fractional gradient descent is currently limited both in the methods analyzed and the settings analyzed. This paper aims to fill in these gaps by analyzing variations of fractional gradient descent in smooth and convex, smooth and strongly convex, and smooth and non-convex settings. First, novel bounds will be established bridging fractional and integer derivatives. Then, these bounds will be applied to the aforementioned settings to prove linear convergence for smooth and strongly convex functions and $O(1/T)$ convergence for smooth and convex functions. Additionally, we prove $O(1/T)$ convergence for smooth and non-convex functions using an extended notion of smoothness - H\"older smoothness - that is more natural for fractional derivatives. Finally, empirical results will be presented on the potential speed up of fractional gradient descent over standard gradient descent as well as the challenges of predicting which will be faster in general.