We study the problem of symmetric matrix completion, where the goal is to reconstruct a positive semidefinite matrix $\rm{X}^\star \in \mathbb{R}^{d\times d}$ of rank-$r$, parameterized by $\rm{U}\rm{U}^{\top}$, from only a subset of its observed entries. We show that the vanilla gradient descent (GD) with small initialization provably converges to the ground truth $\rm{X}^\star$ without requiring any explicit regularization. This convergence result holds true even in the over-parameterized scenario, where the true rank $r$ is unknown and conservatively over-estimated by a search rank $r'\gg r$. The existing results for this problem either require explicit regularization, a sufficiently accurate initial point, or exact knowledge of the true rank $r$. In the over-parameterized regime where $r'\geq r$, we show that, with $\widetilde\Omega(dr^9)$ observations, GD with an initial point $\|\rm{U}_0\| \leq \epsilon$ converges near-linearly to an $\epsilon$-neighborhood of $\rm{X}^\star$. Consequently, smaller initial points result in increasingly accurate solutions. Surprisingly, neither the convergence rate nor the final accuracy depends on the over-parameterized search rank $r'$, and they are only governed by the true rank $r$. In the exactly-parameterized regime where $r'=r$, we further enhance this result by proving that GD converges at a faster rate to achieve an arbitrarily small accuracy $\epsilon>0$, provided the initial point satisfies $\|\rm{U}_0\| = O(1/d)$. At the crux of our method lies a novel weakly-coupled leave-one-out analysis, which allows us to establish the global convergence of GD, extending beyond what was previously possible using the classical leave-one-out analysis.

The rise of IoT devices has prompted the demand for deploying machine learning at-the-edge with real-time, efficient, and secure data processing. In this context, implementing machine learning (ML) models with real-valued weight parameters can prove to be impractical particularly for large models, and there is a need to train models with quantized discrete weights. At the same time, these low-dimensional models also need to preserve privacy of the underlying dataset. In this work, we present RQP-SGD, a new approach for privacy-preserving quantization to train machine learning models for low-memory ML-at-the-edge. This approach combines differentially private stochastic gradient descent (DP-SGD) with randomized quantization, providing a measurable privacy guarantee in machine learning. In particular, we study the utility convergence of implementing RQP-SGD on ML tasks with convex objectives and quantization constraints and demonstrate its efficacy over deterministic quantization. Through experiments conducted on two datasets, we show the practical effectiveness of RQP-SGD.

Bandit problems are most commonly thought of in terms of sequential decision making or as an elementary version of reinforcement learning. This is certainly true, but they are also intimately connected with optimisation. These notes focus on the convex bandit problem, where the set of actions is a convex subset of euclidean space and the function mapping actions to losses is convex. The learner chooses actions and observes the corresponding loss, possibly with additive noise. The duty of the learner is to minimise the loss. When phrased like this the problem seems more like zeroth-order optimisation and these notes largely take that viewpoint. We do borrow two key notions from the bandit community that give the algorithms and analysis a unique flavour: We focus on the cumulative regret as a performance criterion; and We (mostly) consider the adversarial setting, where the loss function is changing from one query to the next. These differences mean the algorithms and theorems presented here are often not directly comparable to standard settings in the optimisation literature. Generally speaking, the theorems presented here hold with fewer assumptions and consequentially are sometimes weaker.

Private data analysis faces a significant challenge known as the curse of dimensionality, leading to increased costs. However, many datasets possess an inherent low-dimensional structure. For instance, during optimization via gradient descent, the gradients frequently reside near a low-dimensional subspace. If the low-dimensional structure could be privately identified using a small amount of points, we could avoid paying (in terms of privacy and accuracy) for the high ambient dimension. On the negative side, Dwork, Talwar, Thakurta, and Zhang (STOC 2014) proved that privately estimating subspaces, in general, requires an amount of points that depends on the dimension. But Singhal and Steinke (NeurIPS 2021) bypassed this limitation by considering points that are i.i.d. samples from a Gaussian distribution whose covariance matrix has a certain eigenvalue gap. Yet, it was still left unclear whether we could provide similar upper bounds without distributional assumptions and whether we could prove lower bounds that depend on similar eigenvalue gaps. In this work, we make progress in both directions. We formulate the problem of private subspace estimation under two different types of singular value gaps of the input data and prove new upper and lower bounds for both types. In particular, our results determine what type of gap is sufficient and necessary for estimating a subspace with an amount of points that is independent of the dimension.

The Stochastic Gradient Descent (SGD) is extensively used in Deep Learning (see [2]). A direct and simple proof of a convergence result of the SGD is given in [3]. Here we will go further and investigate the convergence rate of the SGD, giving some elementary proofs, for two choices of the "learning rate", both in the class of'inverse time decay' schedules.

Background. A main theoretical puzzle is why over-parameterized Neural Networks (NNs) generalize well when trained to zero loss (i.e., so they interpolate the data). Usually, the NN is trained with Stochastic Gradient Descent (SGD) or one of its variants. However, recent empirical work examined the generalization of a random NN that interpolates the data: the NN was sampled from a seemingly uniform prior over the parameters, conditioned on that the NN perfectly classifying the training set. Interestingly, such a NN sample typically generalized as well as SGD-trained NNs. Contributions. We prove that such a random NN interpolator typically generalizes well if there exists an underlying narrow ``teacher NN" that agrees with the labels. Specifically, we show that such a `flat' prior over the NN parametrization induces a rich prior over the NN functions, due to the redundancy in the NN structure. In particular, this creates a bias towards simpler functions, which require less relevant parameters to represent -- enabling learning with a sample complexity approximately proportional to the complexity of the teacher (roughly, the number of non-redundant parameters), rather than the student's.

In modern radar systems, precise target localization using azimuth and velocity estimation is paramount. Traditional unbiased estimation methods have utilized gradient descent algorithms to reach the theoretical limits of the Cramer Rao Bound (CRB) for the error of the parameter estimates. As an extension, we demonstrate on a realistic simulated example scenario that our earlier presented data-driven neural network model outperforms these traditional methods, yielding improved accuracies in target azimuth and velocity estimation. We emphasize, however, that this improvement does not imply that the neural network outperforms the CRB itself. Rather, the enhanced performance is attributed to the biased nature of the neural network approach. Our findings underscore the potential of employing deep learning methods in radar systems to achieve more accurate localization in cluttered and dynamic environments.

Exploding predictive AI has enabled fast yet effective evaluation and decision-making in modern chip physical design flows. State-of-the-art frameworks typically include the objective of minimizing the mean square error (MSE) between the prediction and the ground truth. We argue the averaging effect of MSE induces limitations in both model training and deployment, and good MSE behavior does not guarantee the capability of these models to assist physical design flows which are likely sabotaged due to a small portion of prediction error. To address this, we propose mini-pixel batch gradient descent (MPGD), a plug-and-play optimization algorithm that takes the most informative entries into consideration, offering probably faster and better convergence. Experiments on representative benchmark suits show the significant benefits of MPGD on various physical design prediction tasks using CNN or Graph-based models.

Machine learning models are susceptible to membership inference attacks (MIAs), which aim to infer whether a sample is in the training set. Existing work utilizes gradient ascent to enlarge the loss variance of training data, alleviating the privacy risk. However, optimizing toward a reverse direction may cause the model parameters to oscillate near local minima, leading to instability and suboptimal performance. In this work, we propose a novel method -- Convex-Concave Loss, which enables a high variance of training loss distribution by gradient descent. Our method is motivated by the theoretical analysis that convex losses tend to decrease the loss variance during training. Thus, our key idea behind CCL is to reduce the convexity of loss functions with a concave term. Trained with CCL, neural networks produce losses with high variance for training data, reinforcing the defense against MIAs. Extensive experiments demonstrate the superiority of CCL, achieving state-of-the-art balance in the privacy-utility trade-off.

We propose a novel family of decision-aware surrogate losses, called Perturbation Gradient (PG) losses, for the predict-then-optimize framework. These losses directly approximate the downstream decision loss and can be optimized using off-the-shelf gradient-based methods. Importantly, unlike existing surrogate losses, the approximation error of our PG losses vanishes as the number of samples grows. This implies that optimizing our surrogate loss yields a best-in-class policy asymptotically, even in misspecified settings. This is the first such result in misspecified settings and we provide numerical evidence confirming our PG losses substantively outperform existing proposals when the underlying model is misspecified and the noise is not centrally symmetric. Insofar as misspecification is commonplace in practice -- especially when we might prefer a simpler, more interpretable model -- PG losses offer a novel, theoretically justified, method for computationally tractable decision-aware learning.