--- title: Anytime Last-Iterate Guarantee url: https://www.emergentmind.com/topics/anytime-last-iterate-guarantee type: topic --- # Anytime Last-Iterate Guarantee An anytime last-iterate guarantee is a non-asymptotic guarantee on the quality of the current iterate, current policy, current action, or current released model at every time index, rather than on an ergodic average, a suffix average, or a horizon-specific output selected after the run. Across recent work, the object being controlled varies—objective suboptimality, residual norms, primal–dual gaps, policy suboptimality, action gaps, fairness deficits, or Rényi divergence—but the unifying requirement is prefix-wise validity: one may stop at the present time and retain an explicit guarantee on the present output [2507.11274], [2402.12711], [2605.11694], [2407.05237]. ## 1. Conceptual scope and formalizations In optimization, the canonical form is a bound such as $$ f(x_K)-f(x_*)\le \Phi(K), $$ holding for every iterate $K$, not only for an average $\bar x_K=\tfrac1K\sum_{k=1}^K x_k$ [2403.06873]. In smooth interpolation SGD, the guarantee is placed directly on the raw last iterate $x_t$, with no special averaging or output selection [2507.11274]. In monotone inclusion and saddle-point problems, the controlled quantity is often a residual, such as $\Res_{F,A}(z_T)$ or $\|F(z^T)\|$ [2604.12235], [2002.00057]. In bandits and reinforcement learning, the notion is formalized as Uniform Last-Iterate (ULI): $$ \Pr\bigl[\forall t\ge1:\;\Delta_{A_t}\le f(\delta,t)\bigr]\ge 1-\delta, $$ where $\Delta_{A_t}$ is the suboptimality gap of the action played at round $t$ [2402.12711]. A central distinction is between prefix-wise control and cumulative or ergodic control. Regret, PAC bounds, uniform-PAC, and averaged-iterate analyses may permit arbitrarily poor finite-time behavior of the currently deployed action or iterate, even when cumulative performance is near-optimal. Several papers therefore treat anytime last-iterate guarantees as strictly stronger than standard cumulative criteria, especially in settings where deployment must use the current policy or model rather than a mixture or an average [2402.12711], [2605.11694]. The usage of “anytime” is not fully uniform across the literature. Some papers reserve it for horizon-free procedures that do not require prior knowledge of $T$ and whose bounds are uniform over prefixes [2507.11274]. Others emphasize that a deterministic last-iterate bound is available for every finite horizon, even when the tuning contains the horizon explicitly [2604.10728]. This suggests that the invariant notion is prefix-wise validity, while horizon-freeness is an additional algorithmic property rather than the sole defining feature. | Setting | Controlled quantity | Representative prefix-wise guarantee | |---|---|---| | Smooth convex SGD | $F(x_t)-F(x^*)$ | $\widetilde O(1/(\eta t^{1-\beta\eta/2})+\eta t^{\beta\eta/2}\sigma_*^2)$ [2507.11274] | | Bandits / RL | $\Delta_{A_t}$ | $\Pr[\forall t\ge1:\Delta_{A_t}\le f(\delta,t)]\ge1-\delta$ [2402.12711] | | Monotone inclusion | $\Res_{F,A}(z_T)$ | $O(1/\sqrt T)$ [2604.12235] | | CMDPs | reward gap and constraint violation | $O(1/\sqrt t)$ on the current policy [2605.11694] | | Cyclic DP-SGD | $D_\alpha(X_t\|X_t')$ | explicit RDP upper bound for any $t\le T$ [2407.05237] | | Online fairness | maximum deficit | $c_t=O\!\bigl(\sigma\sqrt{t\ln m/n}\bigr)$ [2605.19844] | ## 2. Horizon-aware optimality and the horizon-free barrier in convex optimization A decisive early result in convex Lipschitz optimization showed that the last iterate can match the information-theoretically optimal rate if the step-size schedule is redesigned specifically for the final time $T$. For general convex problems, Jain et al. use the standard suffix-averaged schedule $\gamma_t=C\,t^{-1/2}$ and replace it by the dyadically modified last-iterate schedule $$ \alpha_t = C\,2^{-i}\,T^{-1/2}\qquad \text{for } T_i < t \le T_{i+1}, $$ while in the strongly convex case they replace $\gamma_t=(\lambda t)^{-1}$ by $$ \alpha_t=\frac{2^{-i}}{\lambda t}\qquad \text{for } T_i < t \le T_{i+1}. $$ Under Lipschitz $G$ and diameter $D$, this yields $$ E[F(x_T)]-F(x^*) \le 15\,G\,D/\sqrt T, $$ and with probability at least $1-\delta$, $$ F(x_T)-F(x^*) = O\bigl(G\,D\,\sqrt{\log(1/\delta)/T}\bigr). $$ Under $\lambda$-strong convexity, the corresponding bound is $$ E[F(x_T)]-F(x^*) \le 130\,G^2/(\lambda T), $$ with high-probability rate $$ F(x_T)-F(x^*) = O\bigl(G^2\log(1/\delta)/(\lambda T)\bigr). $$ These bounds remove the classical extra $\log T$ factor for the last iterate, but the construction is explicitly non-anytime because the dyadic breakpoints depend on the total horizon $T$ [1904.12443]. The proof mechanism is a modification scheme that transfers suffix-average guarantees under the original schedule to last-iterate guarantees under the modified schedule. Its key device is a one-shot “look-ahead” lemma, $$ \sum_{t=t_0}^{t_1} 2\alpha_t\,E[F(x_t)-F(x_{t_0})] \le \sum_{t=t_0}^{t_1} G^2\alpha_t^2, $$ which prevents significant post hoc deterioration in function value after a sufficiently good point has been reached. Intuitively, the dyadic halving “slows down” the method exactly where suffix averaging would have concentrated the mass [1904.12443]. A later impossibility result shows that this horizon dependence is not an artifact of the proof. For convex $G$-Lipschitz optimization over a domain of diameter $D$, if a deterministic step-size sequence is truly anytime, then its worst-case last-iterate error must obey $$ E_T=\Omega\!\Bigl(\frac{DG\,(\log T)^{1/8}}{\sqrt T}\Bigr), $$ and more precisely, if $E_T\le \phi(T)\,DG/\sqrt T$ for all $T$ and some non-decreasing $\phi$, then $$ \phi(T)\ge \Omega\!\bigl((\log T)^{1/8}\bigr). $$ This resolves the conjecture that no truly anytime schedule can attain the exact $1/\sqrt T$ last-iterate rate in this setting, and it does so even in the noiseless GD case [2604.13870]. The resulting picture is sharply bifurcated. Horizon-aware schedules can close the last-iterate gap to the lower bound, whereas horizon-free schedules must pay at least a polylogarithmic penalty. The classical textbook rule $\eta_t=D/(G\sqrt{t+1})$ remains anytime, but only with the familiar $O((\log T)/\sqrt T)$ last-iterate rate [2604.13870]. A plausible implication is that, in convex Lipschitz optimization, the exact optimal last-iterate rate is not compatible with genuine horizon-free operation. ## 3. Positive anytime guarantees for stochastic first-order methods The smooth interpolation regime admits a qualitatively different phenomenon. For convex $\beta$-smooth $F$ and fixed constant step-size $\eta\le1/\beta$, SGD satisfies $$ E[F(x_T)-F(x^*)] = \widetilde O\!\left(\frac{1}{\eta T^{1-\beta\eta/2}}+\eta T^{\beta\eta/2}\sigma_*^2\right), $$ where $\sigma_*^2=E_z\|\nabla f(x^*;z)\|^2$. The bound is established directly for the raw last iterate, uses no step decay, requires no knowledge of $T$, and in fact holds uniformly for every prefix $t\le T$. Balancing the bias and variance terms yields a near-optimal $O(1/T+\sigma_*/\sqrt T)$ rate in the low-noise regime, while in pure interpolation $(\sigma_*=0)$ the greedy choice $\eta=1/\beta$ gives $$ E[F(x_T)-F(x^*)]=O(1/\sqrt T). $$ The analysis relies on a regret-style decomposition with non-uniform weights $v_t\downarrow0$, a smoothness lower bound converting inner products into function-value gaps and gradient-difference terms, and a step-dependent Young inequality that isolates the coefficient of the final iterate [2507.11274]. For smooth quadratics in the interpolation regime, including randomized Kaczmarz, a different anytime guarantee is available under the greedy step-size $\eta_t=1/\beta$. The analysis introduces stochastic contraction processes $v_{t+1}=(I-M_t)v_t$ with common mean $\overline M$, and proves $$ \mathbb E[\|v_t\|_{\overline M}^2] \le \frac{C\|v_0\|^2}{t^{3/4+\theta}} \qquad\text{for some absolute }C>0,\;\theta\ge0.001. $$ Consequently, $$ \mathbb E[f(x_t)-f(x_\star)] = O\bigl(\beta\|x_0-x_\star\|^2\,t^{-3/4-\theta}\bigr), $$ and for randomized Kaczmarz, $$ \mathbb E[\|x_t-x_\star\|^2] = O\Bigl(\frac{\|A\|_F^2\|x_0-x_\star\|^2}{t^{3/4+\theta}}\Bigr). $$ This improves the previously known $O(1/\sqrt t)$ guarantee in that regime [2604.09909]. Adaptive methods do not automatically inherit strong last-iterate guarantees. For scalar AdaGrad-Norm in convex non-smooth optimization, the deterministic final-iterate bound depends on an exponent parameter $$ \delta=\frac12\log_N\!\Bigl(1+\frac1{G^2}\sum_{i=1}^{N-1}\|g^i\|^2\Bigr)\in[0,1/2], $$ which captures the growth of the cumulative squared subgradients. Optimizing the base parameter $h=R/N^\gamma$ over the worst case leads to $$ f(x^{N+1})-f(x^*) = O(N^{-1/4}), $$ and matching lower bounds show that this rate is tight. The proof is built around a weighted last-iterate inequality of Zamani and Glineur and a backward choice of weights that annihilates all coefficients except the one attached to the final iterate [2604.10728]. These results show that positive anytime guarantees arise from different structural mechanisms: low noise at the optimum, interpolation, stochastic contraction, or finely tuned weighted inequalities. They also show that there is no universal last-iterate rate even within first-order convex optimization: rates from $O(1/\sqrt T)$ to $O(T^{-3/4-\theta})$ to $O(T^{-1/4})$ all occur, depending on geometry, noise, and algorithm class [2507.11274], [2604.09909], [2604.10728]. ## 4. Variational inequalities, saddle-point problems, and games In smooth convex–concave saddle-point problems, the last iterate can be provably slower than the averaged iterate. For Extragradient (EG) with constant step-size $\eta\le\min\{1/(30L),\,5/(\Lambda D)\}$, one has for every $T\ge1$ $$ \|F(z^T)\|\le \frac{2D}{\eta\sqrt T}, \qquad \Gap_f^{\mathcal Z}(x^T,y^T)\le \frac{2\sqrt2\,D^2}{\eta\sqrt T}. $$ Choosing $\eta=\Theta(1/L)$ gives $O(DL/\sqrt T)$ and $O(D^2L/\sqrt T)$, whereas the classical ergodic EG guarantee is $O(D^2L/T)$. A matching lower bound of $\Omega(1/\sqrt T)$ for the last iterate establishes a quadratic separation between ergodic and last-iterate convergence rates [2002.00057]. Monotone inclusion problems admit anytime last-iterate bounds for anchoring-only schemes. Proximal Anchored Gradient Descent (P-AGD), $$ z_{t+1} = J_{\alpha_tA}\Bigl((1-\beta_t)z_t+\beta_t z_0-\alpha_tF(z_t)\Bigr), $$ with $$ \alpha_t=\frac1{L\sqrt{t+\gamma}}, \qquad \beta_t=\frac{\gamma}{t+\gamma}, $$ satisfies $$ \Res_{F,A}(z_T)\le \|F(z_T)+c_T\| \le \frac{L(2E+\gamma D)}{\sqrt{T-1+\gamma}}, $$ hence $\Res_{F,A}(z_T)=O(1/\sqrt T)$ for every $T\ge1$. The proof combines a resolvent-based rewrite with a boundedness argument and a decay estimate for consecutive differences $\|d_t\|\le E/(t+\gamma)$ [2604.12235]. Other operator-splitting and game-theoretic settings exhibit linear anytime behavior. For min–max optimization, Hamiltonian Gradient Descent (HGD) minimizes $H(x)=\tfrac12\|\xi(x)\|^2$ and, under a PL inequality for $H$, yields $$ \|\xi(x^{(k)})\| \le \Bigl(1-\frac{\alpha}{L_H}\Bigr)^{k/2}\|\xi(x^{(0)})\|. $$ The same paper proves analogous linear last-iterate convergence for Consensus Optimization under a suitable parameter choice, and extends the theory beyond strongly convex–strongly concave problems to sufficiently bilinear regimes [1906.02027]. Under noisy feedback in $\lambda$-co-coercive games, vanilla stochastic gradient ascent with $$ \beta_t=\frac1{(t+T_0)^b},\qquad 0.5