The framework separates optimizer design into gradient prediction and online preconditioner selection, providing a principled perspective on how adaptive optimization methods may be understood through static regret and applied in nonconvex optimization.
Abstract
We study whether stochastic nonconvex optimization can be reduced to ordinary static regret minimization in online convex optimization in a black-box manner. For smooth nonconvex objectives, our reduction maintains a predictable gradient tracker, while a black-box online learner selects a preconditioner that determines how this tracker is transformed into the update direction. The learner receives linear convex losses and is evaluated against a single fixed comparator over one undiscounted online game. For a $\beta$-smooth objective with range bounded by $M$ and an unbiased stochastic-gradient oracle with variance bounded by \(\sigma^2\), we establish $$\frac{1}{T}\sum_{t=1}^T \mathbb E\!\left[\|\nabla f(x_t)\|_2^2\right] \lesssim \frac{\sigma\sqrt{M\beta}}{\sqrt T} + \frac{\sqrt{M\beta}\, \mathscr R_T(\mathcal A,I_d)}{T} + \frac{M\beta}{T}.$$ Consequently, any black-box OCO algorithm with $\mathscr R_T(\mathcal A,I_d)=O(\sqrt T)$ recovers the classical $O(\frac{1}{\sqrt{T}})$ convergence rate. We further show that the same black-box framework extends beyond the smooth setting to Lipschitz nonconvex objectives without Lipschitz continuous gradients. Importantly, this extension continues to rely only on an ordinary static-regret guarantee and requires no stronger notion of online regret. When the OCO oracle admits square-root static regret, the resulting conversion achieves the optimal $O(T^{-2/7})$ convergence rate for the corresponding Goldstein stationary point. These results resolve the open problem posed by Chen and Hazan (2024). More broadly, our framework separates optimizer design into gradient prediction and online preconditioner selection, providing a principled perspective on how adaptive optimization methods may be understood through static regret and applied in nonconvex optimization.
We study constrained online convex optimization with adversarial constraints when constraint values and gradients are observed through unbiased noise. Gaussian value noise of standard deviation $\sigma$ yields a worst-case lower bound of $\Omega(\min\{\sigma,1\}T/\log^7T)$ on the maximum of expected regret and expected hard violation, even with known gradients. This rules out any jointly $O(T^{1-\delta})$ guarantee for fixed $\delta>0$ and fixed positive noise level. We therefore study budget violation: the largest cumulative overspend over any window within a fixed horizon. We introduce \LEDGER, which tracks observed net consumption in a nonnegative balance and sets constraint weights before the current feedback noise. Under common feasibility and conditional finite-variance feedback, for fixed problem parameters, \LEDGER\ achieves $O(\sqrt T/V)$ expected regret and $O(\sqrt V\,T^{3/4}+\sigma\sqrt T)$ expected budget violation for $V\in[T^{-1/2},1]$. This gives the pair $(O(\sqrt T),O(T^{3/4}))$ at $V=1$ and $(O(T^{2/3}),O(T^{2/3}))$ at $V=T^{-1/6}$, without a Slater condition. The budget-focused endpoint $V=T^{-1/2}$ gives $(O(T),O(\sqrt T))$. The same update yields $O((1+E[P_T])\sqrt T/V)$ expected dynamic regret for predictable feasible comparator paths, without common feasibility or path-length input. Its budget bound instead depends on the shortest feasible path, up to a dimension factor.
Constrained online convex optimization requires minimizing regret against adversarial convex costs while satisfying a convex constraint at every round, as needed in safety-critical applications. A computationally efficient method combines online gradient descent with a Polyak feasibility step, using one constraint evaluation and one subgradient per round. Although this method achieves O(sqrt(T)) regret with per-round feasibility, we derive a tighter, data-dependent analysis by retaining two quantities omitted by the standard worst-case argument. First, we replace the gradient envelope G_f^2 T with the observed accumulation G_T = sum_t ||grad f_t(x_t)||^2. Second, we identify a nonnegative Polyak correction P_T that measures the cumulative squared displacement caused by feasibility projections and enters the regret bound with a negative sign. The resulting improvement, Delta_T = (eta/2)(G_f^2 T - G_T) + P_T/(2 eta), is always nonnegative. We further propose AdaOGD-PFS, an adaptive-step-size method that achieves O(sqrt(G_T)) regret while preserving per-round feasibility. Experiments on ball- and halfspace-constrained problems improve the regret bound by 38 to 43 percent, with both data-dependent gradients and Polyak corrections contributing substantially.
Can the classical Heavy-Ball method, with arbitrary horizon-dependent parameters chosen in advance, achieve Nesterov's $O(T^{-2})$ last-iterate rate on every smooth convex objective? We provide a negative answer. For every horizon $T\ge2$ and every predetermined schedule with nonnegative step sizes and momenta in $[0,1)$, there exists a convex $1$-smooth objective, with initialization distance at most one and zero initial velocity, for which the last iterate of the Heavy-Ball method satisfies \[ f(x_T)-f^\star=\Omega\!\left(\frac{1}{T^\alpha\log T}\right), \qquad \alpha=\frac{1+\sqrt5}{2}. \] Thus even fully nonstationary, horizon-dependent tuning cannot give the classical Heavy-Ball method a Nesterov-rate guarantee on the smooth convex class.
A sharp lower bound is proved for smooth nonconvex stochastic optimization with uniformly bounded gradient noise with uniformly bounded gradient noise and resolves the question raised by whether almost-surely bounded oracle error permits a better rate than bounded variance.
We study the time-uniform convergence of the raw iterate of standard stochastic gradient descent (SGD) for unconstrained smooth convex objectives. We prove that, under standard noise assumptions, the time-uniform convergence rate gets arbitrarily close to $\sqrt{\log n / n}$ but never reaches it. More specifically, we prove that for every positive, eventually nondecreasing sequence $h$ satisfying $h(n) = o(\sqrt{n})$, a bound of order $h(n)/\sqrt{n}$, holding simultaneously for all $n$ with probability at least $1-\alpha$ and uniformly over the problem class, is achievable if and only if \[ \sum_{j = 1}^{\infty} \frac{1}{h(2^j)^2}<\infty. \] The constructive sufficiency result follows from a dyadic horizon-free schedule together with an additive conditional-restart inequality. The necessity counterpart applies to every deterministic nonnegative schedule and holds even for a one-dimensional analytic smooth convex objective with Gaussian noise.
The dense result substantially generalizes a theorem of Bansal and Spencer (2020) for Rademacher inputs and gives an efficient $O(\sqrt{n})$ bound for Gaussian inputs, as conjectured by Gamarnik et al. (2022).
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