The Iterative Hard Thresholding (IHT) algorithm has been considered extensively as an effective deterministic algorithm for solving sparse optimizations. The IHT algorithm benefits from the information of the batch (full) gradient at each point and this information is a crucial key for the convergence analysis of the generated sequence. However, this strength becomes a weakness when it comes to machine learning and high dimensional statistical applications because calculating the batch gradient at each iteration is computationally expensive or impractical. Fortunately, in these applications the objective function has a summation structure that can be taken advantage of to approximate the batch gradient by the stochastic mini-batch gradient. In this paper, we study the mini-batch Stochastic IHT (SIHT) algorithm for solving the sparse optimizations. As opposed to previous works where increasing and variable mini-batch size is necessary for derivation, we fix the mini-batch size according to a lower bound that we derive and show our work. To prove stochastic convergence of the objective value function we first establish a critical sparse stochastic gradient descent property. Using this stochastic gradient descent property we show that the sequence generated by the stochastic mini-batch SIHT is a supermartingale sequence and converges with probability one. Unlike previous work we do not assume the function to be a restricted strongly convex. To the best of our knowledge, in the regime of sparse optimization, this is the first time in the literature that it is shown that the sequence of the stochastic function values converges with probability one by fixing the mini-batch size for all steps.

Rising global food demand and harsh working conditions make fruit harvest an important domain to automate. Peduncle localization is an important step for any automated fruit harvesting system, since fruit separation techniques are highly sensitive to peduncle location. Most work on peduncle localization has focused on computer vision, but peduncles can be difficult to visually access due to the cluttered nature of agricultural environments. Our work proposes an alternative method which relies on mechanical -- rather than visual -- perception to localize the peduncle. To estimate the location of this important plant feature, we fit wrench measurements from a wrist force/torque sensor to a physical model of the fruit-plant system, treating the fruit's attachment point as a parameter to be tuned. This method is performed inline as part of the fruit picking procedure. Using our orchard proxy for evaluation, we demonstrate that the technique is able to localize the peduncle within a median distance of 3.8 cm and median orientation error of 16.8 degrees.

We address the problem of \emph{convex optimization with dueling feedback}, where the goal is to minimize a convex function given a weaker form of \emph{dueling} feedback. Each query consists of two points and the dueling feedback returns a (noisy) single-bit binary comparison of the function values of the two queried points. The translation of the function values to the single comparison bit is through a \emph{transfer function}. This problem has been addressed previously for some restricted classes of transfer functions, but here we consider a very general transfer function class which includes all functions that can be approximated by a finite polynomial with a minimal degree $p$. Our main contribution is an efficient algorithm with convergence rate of $\smash{\widetilde O}(\epsilon^{-4p})$ for a smooth convex objective function, and an optimal rate of $\smash{\widetilde O}(\epsilon^{-2p})$ when the objective is smooth and strongly convex.

The cubic regularization method (CR) is a popular algorithm for unconstrained non-convex optimization. At each iteration, CR solves a cubically regularized quadratic problem, called the cubic regularization subproblem (CRS). One way to solve the CRS relies on solving the secular equation, whose computational bottleneck lies in the computation of all eigenvalues of the Hessian matrix. In this paper, we propose and analyze a novel CRS solver based on an approximate secular equation, which requires only some of the Hessian eigenvalues and is therefore much more efficient. Two approximate secular equations (ASEs) are developed. For both ASEs, we first study the existence and uniqueness of their roots and then establish an upper bound on the gap between the root and that of the standard secular equation. Such an upper bound can in turn be used to bound the distance from the approximate CRS solution based ASEs to the true CRS solution, thus offering a theoretical guarantee for our CRS solver. A desirable feature of our CRS solver is that it requires only matrix-vector multiplication but not matrix inversion, which makes it particularly suitable for high-dimensional applications of unconstrained non-convex optimization, such as low-rank recovery and deep learning. Numerical experiments with synthetic and real data-sets are conducted to investigate the practical performance of the proposed CRS solver. Experimental results show that the proposed solver outperforms two state-of-the-art methods.

Stochastic optimization of the Area Under the Precision-Recall Curve (AUPRC) is a crucial problem for machine learning. Although various algorithms have been extensively studied for AUPRC optimization, the generalization is only guaranteed in the multi-query case. In this work, we present the first trial in the single-query generalization of stochastic AUPRC optimization. For sharper generalization bounds, we focus on algorithm-dependent generalization. There are both algorithmic and theoretical obstacles to our destination. From an algorithmic perspective, we notice that the majority of existing stochastic estimators are biased only when the sampling strategy is biased, and is leave-one-out unstable due to the non-decomposability. To address these issues, we propose a sampling-rate-invariant unbiased stochastic estimator with superior stability. On top of this, the AUPRC optimization is formulated as a composition optimization problem, and a stochastic algorithm is proposed to solve this problem. From a theoretical perspective, standard techniques of the algorithm-dependent generalization analysis cannot be directly applied to such a listwise compositional optimization problem. To fill this gap, we extend the model stability from instancewise losses to listwise losses and bridge the corresponding generalization and stability. Additionally, we construct state transition matrices to describe the recurrence of the stability, and simplify calculations by matrix spectrum. Practically, experimental results on three image retrieval datasets on speak to the effectiveness and soundness of our framework.

In edge computing, suppressing data size is a challenge for machine learning models that perform complex tasks such as autonomous driving, in which computational resources (speed, memory size and power) are limited. Efficient lossy compression of matrix data has been introduced by decomposing it into the product of an integer and real matrices. However, its optimisation is difficult as it requires simultaneous optimisation of an integer and real variables. In this paper, we improve this optimisation by utilising recently developed black-box optimisation (BBO) algorithms with an Ising solver for integer variables. In addition, the algorithm can be used to solve mixed-integer programming problems that are linear and non-linear in terms of real and integer variables, respectively. The differences between the choice of Ising solvers (simulated annealing, quantum annealing and simulated quenching) and the strategies of the BBO algorithms (BOCS, FMQA and their variations) are discussed for further development of the BBO techniques.

In this discussion paper, we survey recent research surrounding robustness of machine learning models. As learning algorithms become increasingly more popular in data-driven control systems, their robustness to data uncertainty must be ensured in order to maintain reliable safety-critical operations. We begin by reviewing common formalisms for such robustness, and then move on to discuss popular and state-of-the-art techniques for training robust machine learning models as well as methods for provably certifying such robustness. From this unification of robust machine learning, we identify and discuss pressing directions for future research in the area.

For strongly convex objectives that are smooth, the classical theory of gradient descent ensures linear convergence relative to the number of gradient evaluations. An analogous nonsmooth theory is challenging. Even when the objective is smooth at every iterate, the corresponding local models are unstable and the number of cutting planes invoked by traditional remedies is difficult to bound, leading to convergences guarantees that are sublinear relative to the cumulative number of gradient evaluations. We instead propose a multipoint generalization of the gradient descent iteration for local optimization. While designed with general objectives in mind, we are motivated by a ``max-of-smooth'' model that captures the subdifferential dimension at optimality. We prove linear convergence when the objective is itself max-of-smooth, and experiments suggest a more general phenomenon.

The current energy transition promotes the convergence of operation between the power and natural gas systems. In that direction, it becomes paramount to improve the modeling of non-convex natural gas flow dynamics within the coordinated power and gas dispatch. In this work, we propose a neural-network-constrained optimization method which includes a regression model of the Weymouth equation, based on supervised machine learning. The Weymouth equation links gas flow to inlet and outlet pressures for each pipeline via a quadratic equality, which is captured by a neural network. The latter is encoded via a tractable mixed-integer linear program into the set of constraints. In addition, our proposed framework is capable of considering bidirectionality without having recourse to complex and potentially inaccurate convexification approaches. We further enhance our model by introducing a reformulation of the activation function, which improves the computational efficiency. An extensive numerical study based on the real-life Belgian power and gas systems shows that the proposed methodology yields promising results in terms of accuracy and tractability.

In this paper, we present a novel Model Predictive Control method for autonomous robots subject to arbitrary forms of uncertainty. The proposed Risk-Aware Model Predictive Path Integral (RA-MPPI) control utilizes the Conditional Value-at-Risk (CVaR) measure to generate optimal control actions for safety-critical robotic applications. Different from most existing Stochastic MPCs and CVaR optimization methods that linearize the original dynamics and formulate control tasks as convex programs, the proposed method directly uses the original dynamics without restricting the form of the cost functions or the noise. We apply the novel RA-MPPI controller to an autonomous vehicle to perform aggressive driving maneuvers in cluttered environments. Our simulations and experiments show that the proposed RA-MPPI controller can achieve about the same lap time with significantly fewer collisions compared to the baseline MPPI controller. The proposed controller performs on-line computation at an update frequency of up to 80Hz, utilizing modern Graphics Processing Units (GPUs) to multi-thread the generation of trajectories as well as the CVaR values.