We study nonmonotone games satisfying the weak Minty variational inequality (MVI) with parameter $\rho \in (-\tfrac{1}{L}, \infty)$, where $L$ is the Lipschitz constant of the gradient operator. An error corrected version of the inexact proximal point algorithm is proposed, with which we establish the first $\mathcal O(1/\epsilon)$ rate for the entire range $\rho \in (-\tfrac{1}{L}, \infty)$, thus removing a logarithmic factor compared with the complexity of existing methods. The scheme automatically selects the needed accuracy for the proximal computation, and can recover the relaxed extragradient method when $\rho > -\tfrac{1}{2L}$ and the relaxed proximal point algorithm (rPPA) when $\rho > -\tfrac{1}{L}$. Due to the error correction, the scheme inherits the strong properties of the _exact_ rPPA. Specifically, we show that linear convergence is automatically achieved under appropriate conditions. Tightness for the range of $\rho$ is established through a lower bound for rPPA. Central to the algorithmic construction is a halfspace projection, where the key insight is that the allowed error tolerance can both be used to correct for the proximal approximation and to enlarge the problem class.

Contrary to common belief, we show that gradient ascent-based unconstrained optimization methods frequently fail to perform machine unlearning, a phenomenon we attribute to the inherent statistical dependence between the forget and retain data sets. This dependence, which can manifest itself even as simple correlations, undermines the misconception that these sets can be independently manipulated during unlearning. We provide empirical and theoretical evidence showing these methods often fail precisely due to this overlooked relationship. For random forget sets, this dependence means that degrading forget set metrics (which, for a retrained model, should mirror test set metrics) inevitably harms overall test performance. Going beyond random sets, we consider logistic regression as an instructive example where a critical failure mode emerges: inter-set dependence causes gradient descent-ascent iterations to progressively diverge from the ideal retrained model. Strikingly, these methods can converge to solutions that are not only far from the retrained ideal but are potentially even further from it than the original model itself, rendering the unlearning process actively detrimental. A toy example further illustrates how this dependence can trap models in inferior local minima, inescapable via finetuning. Our findings highlight that the presence of such statistical dependencies, even when manifest only as correlations, can be sufficient for ascent-based unlearning to fail. Our theoretical insights are corroborated by experiments on complex neural networks, demonstrating that these methods do not perform as expected in practice due to this unaddressed statistical interplay.

Given a sequence of functions $f_1,\ldots,f_n$ with $f_i:\mathcal{D}\mapsto \mathbb{R}$, finite-sum minimization seeks a point ${x}^\star \in \mathcal{D}$ minimizing $\sum_{j=1}^nf_j(x)/n$. In this work, we propose a key twist into the finite-sum minimization, dubbed as *continual finite-sum minimization*, that asks for a sequence of points $x_1^\star, \ldots, x_n^\star \in D$ such that each ${x}^\star_i \in D$ minimizes the prefix-sum $\sum_{j=1}^if_j(x)/i$. Assuming that each prefix-sum is strongly convex, we develop a first-order continual stochastic variance reduction gradient method ($\mathrm{CSVRG}$) producing an $\epsilon$-optimal sequence with $\tilde{\mathcal{O}}(n/\epsilon^{1/3} + 1/\sqrt{\epsilon})$ overall *first-order oracles* (FO). An FO corresponds to the computation of a single gradient $\nabla f_j(x)$ at a given $x \in \mathcal{D}$ for some $j \in [n]$. Our approach significantly improves upon the $\mathcal{O}(n/\epsilon)$ FOs that $\mathrm{StochasticGradientDescent}$ requires and the $\mathcal{O}(n^2 \log (1/\epsilon))$ FOs that state-of-the-art variance reduction methods such as $\mathrm{Katyusha}$ require. We also prove that there is no natural first-order method with $\mathcal{O}\left(n/\epsilon^\alpha\right)$ gradient complexity for $\alpha < 1/4$, establishing that the first-order complexity of our method is nearly tight.