Machine unlearning work assumes a static, i.i.d training environment that doesn't truly exist. Modern ML pipelines need to learn, unlearn, and predict continuously on production streams of data. We translate batch unlearning to the online setting using notions of regret, sample complexity, and deletion capacity. We tighten regret bounds to a logarithmic $\mathcal{O}(\ln{T})$, a first for a certified unlearning algorithm. When fitted with an online variant of L-BFGS optimization, the algorithm achieves state of the art regret with a constant memory footprint. Such changes extend the lifespan of an ML model before expensive retraining, making for a more efficient unlearning process.

Non-asymptotic convergence analysis of quasi-Newton methods has gained attention with a landmark result establishing an explicit local superlinear rate of O$((1/\sqrt{t})^t)$. The methods that obtain this rate, however, exhibit a well-known drawback: they require the storage of the previous Hessian approximation matrix or all past curvature information to form the current Hessian inverse approximation. Limited-memory variants of quasi-Newton methods such as the celebrated L-BFGS alleviate this issue by leveraging a limited window of past curvature information to construct the Hessian inverse approximation. As a result, their per iteration complexity and storage requirement is O$(\tau d)$ where $\tau\le d$ is the size of the window and $d$ is the problem dimension reducing the O$(d^2)$ computational cost and memory requirement of standard quasi-Newton methods. However, to the best of our knowledge, there is no result showing a non-asymptotic superlinear convergence rate for any limited-memory quasi-Newton method. In this work, we close this gap by presenting a Limited-memory Greedy BFGS (LG-BFGS) method that can achieve an explicit non-asymptotic superlinear rate. We incorporate displacement aggregation, i.e., decorrelating projection, in post-processing gradient variations, together with a basis vector selection scheme on variable variations, which greedily maximizes a progress measure of the Hessian estimate to the true Hessian. Their combination allows past curvature information to remain in a sparse subspace while yielding a valid representation of the full history. Interestingly, our established non-asymptotic superlinear convergence rate demonstrates an explicit trade-off between the convergence speed and memory requirement, which to our knowledge, is the first of its kind. Numerical results corroborate our theoretical findings and demonstrate the effectiveness of our method.

Incorporating second order curvature information in gradient based methods have shown to improve convergence drastically despite its computational intensity. In this paper, we propose a stochastic (online) quasi-Newton method with Nesterov's accelerated gradient in both its full and limited memory forms for solving large scale non-convex optimization problems in neural networks. The performance of the proposed algorithm is evaluated in Tensorflow on benchmark classification and regression problems. The results show improved performance compared to the classical second order oBFGS and oLBFGS methods and popular first order stochastic methods such as SGD and Adam. The performance with different momentum rates and batch sizes have also been illustrated. Keywords: Neural networks · stochastic method · online training · Nesterov's accelerated gradient · quasi-Newton method · limited memory · Tensorflow 1 Introduction Neural networks have shown to be effective in innumerous real-world applications.

A common problem in training neural networks is the vanishing and/or exploding gradient problem which is more prominently seen in training of Recurrent Neural Networks (RNNs). Thus several algorithms have been proposed for training RNNs. This paper proposes a novel adaptive stochastic Nesterov accelerated quasiNewton (aSNAQ) method for training RNNs. The proposed method aSNAQ is an accelerated method that uses the Nesterov's gradient term along with second order curvature information. The performance of the proposed method is evaluated in Tensorflow on benchmark sequence modeling problems. The results show an improved performance while maintaining a low per-iteration cost and thus can be effectively used to train RNNs.

In this paper, we present a scalable distributed implementation of the sampled LSR1 (S-LSR1) algorithm. First, we show that a naive distributed implementation of S-LSR1 requires multiple rounds of expensive communications at every iteration and thus is inefficient. We then propose DS-LSR1, a communication-efficient variant of the S-LSR1 method, that drastically reduces the amount of data communicated at every iteration, that has favorable work-load balancing across nodes and that is matrix-free and inverse-free. The proposed method scales well in terms of both the dimension of the problem and the number of data points. Finally, we illustrate the performance of DS-LSR1 on standard neural network training tasks.

We present two sampled quasi-Newton methods for deep learning: sampled LBFGS (S-LBFGS) and sampled LSR1 (S-LSR1). Contrary to the classical variants of these methods that sequentially build Hessian or inverse Hessian approximations as the optimization progresses, our proposed methods sample points randomly around the current iterate at every iteration to produce these approximations. As a result, the approximations constructed make use of more reliable (recent and local) information, and do not depend on past iterate information that could be significantly stale. Our proposed algorithms are efficient in terms of accessed data points (epochs) and have enough concurrency to take advantage of parallel/distributed computing environments. We provide convergence guarantees for our proposed methods. Numerical tests on a toy classification problem as well as on popular benchmarking neural network training tasks reveal that the methods outperform their classical variants and are competitive with state-of-the-art first-order methods such as ADAM.