Silver Rate Is (Almost) Optimal for Gradient Descent

1. Introduction

We consider the minimization of a smooth convex function \(f:\mathbb R^d\to\mathbb R\). A standard algorithm for this problem is gradient descent (GD), with a history dating back to Cauchy [Cau47]. Given a stepsize schedule \(H=(h_1,\ldots,h_n)\in[0,\infty)^n\) fixed before the optimization begins, GD proceeds via the iteration

In this paper, we study the limits of this method’s convergence rate. To formalize this question, let \(\mathcal F_L(\mathbb R^d)\) be the class of convex \(L\)-smooth functions on \(\mathbb R^d\) with a nonempty set of minimizers. We measure the convergence rate achieved by a schedule \(H\) using its worst-case (normalized) function suboptimality

Classical Results.

Classical convergence analyses of gradient methods can be traced to [LP66]. With the constant stepsize of \(1/L\), GD achieves the classical \(O(n^{-1})\) rate; see the textbooks [Pol87], [Nes04], [Bub15]. This rate is tight, even in one dimension [DT14].

To obtain faster rates, classical acceleration methods introduce momentum or auxiliary sequences, as in Polyak’s heavy-ball method [Pol64] and Nesterov’s accelerated method [Nes83]. The latter attains an \(O(n^{-2})\) rate for smooth convex objectives, matching the classical \(\Omega(n^{-2})\) lower bound for deterministic first-order methods established by Nemirovsky and Yudin [NY79].¹ These accelerated methods, however, modify the original GD update. For GD with predetermined stepsizes, the classical results therefore leave a gap between the \(O(n^{-1})\) upper bound and the \(\Omega(n^{-2})\) lower bound.

Breakthrough in Upper Bounds: Acceleration via Silver Stepsize Schedule.

For many years, the \(O(n^{-1})\) rate remained the best general upper bound known for GD, leaving open the question of whether stepsize choice alone could improve it. In recent years, computer-assisted analyses have opened up new opportunities for acceleration by helping identify better stepsize schedules. Drori and Teboulle [DT14] introduced performance estimation problems (PEPs), which express a method’s worst-case error after a prescribed number of iterations as an optimization problem. Exact interpolation conditions characterize when prescribed function values and gradients at finitely many points come from a single function in the specified class [THG17]. These conditions give exact semidefinite formulations of PEPs for fixed-step methods.² Using PEPs to analyze and optimize GD stepsize schedules over short horizons, Altschuler showed in his thesis [Alt18, Chapter 8] that coordinating short and long steps can improve convergence in the strongly convex setting. Such gains require a joint analysis of multiple steps, since separate bounds on individual steps fail to reflect this advantage.

Altschuler and Parrilo [AP25a] introduced the recursive silver stepsize schedule for smooth strongly convex objectives. Their schedule reduces the squared distance to the minimizer by a factor of \(\varepsilon\) in \(O(\kappa^{1/p_{\mathrm{sil}}}\log(1/\varepsilon))\) iterations, where \(\kappa\) is the condition number and \(p_{\mathrm{sil}}:=\log_2(1+\sqrt2)\approx1.2716\).³ They later extended their silver schedule to general smooth convex objectives, obtaining an \(O(n^{-p_{\mathrm{sil}}})\) rate [AP25b]. In the same work, they conjectured that \(p_{\mathrm{sil}}\) is the optimal polynomial convergence exponent, which is supported by numerical solutions at small horizons [DGVPR24]. Subsequent work improved the constants in the bound on function suboptimality and established accelerated gradient-norm guarantees [GSW25a]. Further work also gave rules for combining shorter schedules to obtain the silver rate at arbitrary prescribed horizons [GSW25b], [ZJ26]. For a broader introduction to stepsize-based acceleration, see the recent survey [AP26].

The smooth convex silver-rate results above concern the non-anytime (finite-horizon) setting, in which the schedule may be chosen separately for each fixed horizon \(n\). In the anytime setting, one infinite schedule \(H=(h_t)_{t\ge1}\) is used across all horizons, with \(H_n=(h_1,\ldots,h_n)\). By combining silver-schedule blocks, [ZLDC25] construct such a schedule with an \(O(n^{-p_{\mathrm{any}}})\) guarantee at every stopping time, where \(p_{\mathrm{any}}:=2p_{\mathrm{sil}}/(1+p_{\mathrm{sil}})\approx1.1195\). This resolves the open problem [KS24] of whether stepsize choice alone can accelerate GD uniformly over all stopping times.

Recent Progress on Lower Bounds.

Despite these improved upper bounds, the best-known lower bound for GD until recently was the classical \(\Omega(n^{-2})\) bound [NY79]. Nesterov [Nes04] provides a simple quadratic zero-chain construction for this lower bound. By initializing at the origin, a method satisfying the linear-span assumption can reach at most one new coordinate per iteration, while the minimizer has nonzero entries farther along the chain. Even exact minimization over the reached coordinates leaves an \(\Omega(n^{-2})\) error. Using PEP, [KF16] developed the optimized gradient method (OGM), whose bound on function suboptimality was later proved optimal in both rate and constant [Dro17]. The lower bound construction in [Dro17] combines smooth convex interpolation with a zero-chain property. These results sharpened the constants of the classical lower bound, but did not improve the convergence exponent for GD with predetermined stepsizes.

In recent months, a sequence of results brought the lower bounds closer to the silver-rate upper bounds \(O(n^{-p_{\mathrm{sil}}})\) and \(O(n^{-p_{\mathrm{any}}})\) in the non-anytime and anytime settings, respectively. [MC26] proved an \(\Omega(n^{-1.932})\) non-anytime lower bound, and a subsequent blog post [Tsa26] sharpened it to \(\Omega(n^{-\sqrt3})\). For the anytime setting, [TFZH26] proved an \(\Omega(n^{-4/3})\) lower bound.⁴ In our previous work [YL26], we proved an \(\Omega(n^{-1.6342})\) non-anytime lower bound and an \(\Omega(n^{-1.2408})\) anytime lower bound for arbitrary real-valued stepsizes. To the best of our knowledge, these remain the strongest known lower bounds when negative stepsizes are allowed. For nonnegative schedules, [JCY26] further improved the non-anytime and anytime lower bounds to \(\Omega(n^{-1.4500})\) and \(\Omega(n^{-1.1837})\), respectively. Their hard instances form a chain of local one-sided Huber functions⁵ coupling adjacent coordinates, much like Nesterov’s quadratic chain.

1.1 Main Results

A key limitation of the local one-sided Huber functions in [JCY26] is that their fixed gradient directions restrict the tightness of the resulting lower bounds. We address this by refining these local functions to bend the gradient descent trajectory, which allows us to establish tighter lower bounds with exponents \(p_{\mathrm{sil}}+o(1)\) and \(p_{\mathrm{any}}+o(1)\).

Together with the corresponding upper bounds [AP25b], [ZLDC25], our results identify \(p_{\mathrm{sil}}\) and \(p_{\mathrm{any}}\) as the optimal polynomial convergence exponents in the non-anytime and anytime settings, respectively. This resolves the conjecture of Altschuler and Parrilo [AP25b] that the silver exponent is optimal. Sections 2–4 explains the idea behind our hard function construction.

2. Classical Lower Bound Constructions

We first review classical lower bound constructions and the recent hard functions of [MC26], [JCY26], highlighting the ideas that guide our construction. We then present the idea behind our hard function construction.¹

2.1 The Role of Huber and Quadratic Functions

To build intuition, consider two basic one-dimensional hard functions first. A one-sided version of the classical Huber function [Hub64] is defined by

Its gradient is constant on the affine region \(z\ge\delta\), so short steps make slow progress toward the minimum. Next, consider the quadratic objective \(f(x)=x^2/2\), where the update is \(x_t=(1-h_t)x_{t-1}\), so a large step (\(h_t>2\)) overshoots the minimizer and increases the distance to it. These examples yield the following elementary lower bounds.

Every constant schedule falls into one of these two cases and therefore has a worst-case error of \(\Omega(n^{-1})\). More precisely, the same examples give [DT14, Section 3.2]

For all \(0<h<2\), this lower bound matches the upper bound exactly, including the constant factor [Kim24].

These examples suggest that a hard function for GD should combine affine regions that delay progress during short steps with quadratic curvature that causes large steps to overshoot the minimizer. This insight also appears in PEP analyses [DT14] and motivates hedging between short and long steps in the silver schedule, which yields the accelerated upper bound [AP25a, Section 2.1].

2.2 The Role of Zero-Chains

As discussed above, using the Huber function alone yields a lower bound that weakens as the sum of stepsizes grows. A different approach uses higher-dimensional hard functions to ensure the minimizer has non-zero components in unexplored directions. Nemirovsky and Yudin [NY79] used this idea to establish the classical \(\Omega(n^{-2})\) lower bound for general deterministic first-order methods. Under the linear-span assumption¹, Nesterov [Nes04, Section 2.1.2] presents a simpler quadratic zero-chain that limits how quickly a method can explore new coordinates, regardless of its stepsizes. This chain couples adjacent coordinates \(x^{(j)}\) and \(x^{(j+1)}\) through quadratic terms, where \(x^{(j)}\) denotes the \(j\)th coordinate of \(x\).

The bound follows from the first-order zero-chain property of \(f\) [DT22, Definition 1]. This means that, for every \(0\le r<d\),

Starting from \(x_0=0\), each iteration of a method satisfying the linear-span assumption can therefore introduce at most one new coordinate, so \(x_t^{(j)}=0\) for \(j>t\). The minimizer, however, has \(x_*^{(j)}=1-j/(d+1)>0\) in every coordinate. Minimizing \(f\) with \(x^{(n+1)}=\cdots=x^{(d)}=0\) and comparing the result with \(f(x_*)\) gives the lower bound in Fact 2.2 [Nes04, Theorem 2.1.7].

3. Recent Lower Bound Breakthrough

The classical \(\Omega(n^{-2})\) lower bound discussed above holds for general deterministic first-order methods [NY79]. To obtain tighter lower bounds for GD, we must therefore tailor the hard function to the prescribed stepsize schedule. The construction in [JCY26] combines a coupled coordinate chain with changes in gradient direction after selected steps, which are chosen adversarially for each prescribed schedule.

3.1 Selecting the Checkpoints

To formalize this selection, fix a nonnegative schedule \(H=(h_1,\ldots,h_n)\) and select a set of update indices \(T=\{t_1<\cdots<t_k\}\) called checkpoints. The updates between successive checkpoints form a gap. Set \(t_0=0\), \(t_{k+1}=n+1\), and define

Here \(b_i\) is the stepsize at checkpoint \(t_i\), and \(s_i\) is the mass of the preceding gap, the sum of its stepsizes. The final gap has mass \(s_{k+1}\). Figure 2 illustrates this notation.

The construction in [MC26] yields an \(\Omega(n^{-1.932})\) non-anytime lower bound. For selected long steps, the hard function in [MC26, Theorem 4.1] keeps the gradient constant during each gap and changes it after each selected step. See also [YL26, Appendix A] for a geometric illustration of the trajectory. The hard function in [MC26] is defined globally on all coordinates. The construction in [JCY26, Section 2] instead couples adjacent coordinates through local hard functions, which we also call components. This chain resembles Nesterov’s quadratic construction and leads to the stronger \(\Omega(n^{-\log_2(1+\sqrt3)})\) non-anytime lower bound, where \(\log_2(1+\sqrt3)\approx1.4500\). During each gap, the current component contributes a constant gradient, as in the construction of [MC26], while all later components remain inactive.¹ For a fixed checkpoint set, this gives the following bound.

Specifically, their construction uses the one-sided Huber function \(H_\delta\) defined above, joining adjacent coordinates through

Here \(\ell_i\) is the activation threshold of component \(i\), and \(D_i:=x_{t_{i-1}}^{(i)}-\ell_i>0\) is the excess above it at the start of gap \(i\). Starting from \(x_0=e_1\), set \(\ell_1=0\) and \(D_1=1\). For this Huber construction, the parameters are chosen recursively as \(\delta_i=\frac{2D_i}{2+s_i}\), \(\ell_{i+1}=\frac{s_i\delta_i}{4}\), and \(D_{i+1}=\frac{b_i\delta_i}{4}\) for \(1\le i\le k\). During gap \(i\), coordinate \(i+1\) never exceeds \(\ell_{i+1}\) and equals it just before the checkpoint. The checkpoint then raises this coordinate to \(\ell_{i+1}+D_{i+1}\) and activates component \(i+1\), while all later components remain inactive. Thus each factor \(b_i/[2(2+s_i)]\) in (5) is the transfer ratio \(D_{i+1}/D_i\). Since \(D_1=1\), their product is the final excess \(D_{k+1}\). The final one-dimensional component is \(\frac12H_{\delta_{k+1}}(x^{(k+1)}-\ell_{k+1})\). Choosing \(\delta_{k+1}=D_{k+1}/(1+s_{k+1})\) gives a lower bound proportional to \(D_{k+1}^2/(1+s_{k+1})\).

For a prescribed schedule \(H\), this Huber construction is available for every checkpoint set \(T\). The checkpoints can therefore be chosen to maximize the right-hand side of (5). The recursive argument in [JCY26, Section 3] bounds this maximum from below by \(\Omega(n^{-\log_2(1+\sqrt3)})\), uniformly over schedules \(H\).

3.2 The Bottleneck

For fixed \(H\) and \(T\), improving the local functions to give larger transfer ratios \(D_{i+1}/D_i\) would further strengthen the lower bound. In general, the argument in [JCY26, Section 3 and Appendix C] suggests a lower bound of order \(n^{-\log_2(1+\sqrt{1+c})}\) when the coefficient of \(s_i\) in the denominator of the transfer ratio \(D_{i+1}/D_i\) is \(c\). The transfer ratio \(b_i/[2(2+s_i)]\) in (5) has \(c=2\), whereas reducing \(c\) toward \(1\) would bring the exponent toward \(p_{\mathrm{sil}}\). During gap \(i\), the current Huber component contributes equally to the decrease in \(x^{(i)}\) and the increase in \(x^{(i+1)}\). Both changes reduce the margin \(x^{(i)}-x^{(i+1)}-\ell_i\), producing the coefficient \(2\) of \(s_i\) in the denominator \(4+2s_i\) in (5) [JCY26, Section 2].

4. Our Hard Function: Bending the Local Trajectory

Keeping the chain structure and notation above, we seek local hard functions \(\Phi_i\) for an objective of the form

For \(1\le i\le k\), each \(\Phi_i\) should depend only on \(x^{(i)}\) and \(x^{(i+1)}\) and be nonnegative, convex, and \(1\)-smooth. The final component \(\Phi_{k+1}\) depends only on \(x^{(k+1)}\). For \(1\le i\le k\), we impose the following transfer property so that only the current component \(\Phi_i\) contributes to the updates of \(x^{(i)}\) and \(x^{(i+1)}\) during gap \(i\) and in its checkpoint update¹

The first condition keeps later components inactive while their first coordinates stay at or below their thresholds and their second coordinates stay nonnegative. For \(i>1\), applying the second condition to \(\Phi_{i-1}\) gives \(\partial_{x^{(i)}}\Phi_{i-1}=0\) whenever \(x^{(i)}\ge\ell_i\), so the preceding component does not affect the update of \(x^{(i)}\). The shaded region in Figure 3 illustrates the second condition for \(\Phi_i\). For \(i<k\), the iterates remain in this region during gap \(i+1\), including just before the checkpoint update at \(t_{i+1}\).

As before, we write \(D_{i+1}:=x_{t_i}^{(i+1)}-\ell_{i+1}>0\) for the excess above \(\ell_{i+1}\) immediately after the checkpoint update at \(t_i\). Our goal is to construct \(\Phi_i\) with transfer ratio

We aim to make \(c\) arbitrarily close to \(1\).

Notice that growth in \(x^{(i+1)}\) during gap \(i\) raises the threshold \(\ell_{i+1}\) needed to keep \(\Phi_{i+1}\) inactive. To leave a larger excess \(D_{i+1}\) after checkpoint \(t_i\), we limit this early growth while ensuring a large increase at the checkpoint update. When \(s_i\) is sufficiently large, the trajectory in gap \(i\) initially moves mainly in coordinate \(x^{(i)}\) and turns toward coordinate \(x^{(i+1)}\) before checkpoint \(t_i\). We call this change in direction bending, the key new ingredient in our construction.

More concretely, we construct the GD trajectory in gap \(i\) with gradient vectors on a circular arc. For parameters \(\epsilon,\gamma>0\), consider the circle centered at \((0,\gamma)\) with radius \(R:=\sqrt{\epsilon^2+(1+\gamma)^2}\), which passes through \((\epsilon,-1)\). Its intersection with the positive horizontal axis is \((c,0)\), where \(c=\sqrt{1+2\gamma+\epsilon^2}\). Up to a common positive scale, we place the gradients on the arc joining these two points, as shown on the left of Figure 3. They turn toward \((\epsilon,-1)\) during gap \(i\). At \(x_{t_i-1}\), just before the checkpoint update, the gradient is proportional to \((\epsilon,-1)\), so the update mainly increases \(x^{(i+1)}\) when \(\epsilon\) is small.

Given the stepsizes in gap \(i\), we construct the trajectory backward from the iterate just before checkpoint \(t_i\); see Lemma 4.2. We use the arc vectors along this trajectory, together with \(0\) and \((c,0)\), as slopes in a finite maximum of affine functions. After scaling and translation, the Moreau envelope of this function gives a convex, \(1\)-smooth local hard function \(\Phi_i\) whose gradient at each constructed iterate is the corresponding scaled arc vector; see Lemma 4.3 and (10).²

The Improved Hard-Function Bound.

Our local hard functions will give the transfer ratio \(D_{i+1}/D_i=(b_i-2)/[c(a+s_i)]\), where \(a\) depends only on \(\epsilon,\gamma\). To leave positive excess, we use checkpoints with \(b_i>2\), which we call admissible. A checkpoint set is admissible if all its checkpoints are admissible. The construction below gives the following lower bound for every admissible checkpoint set.

We prove this lemma in three steps: constructing the gap trajectory in Section 4.1, realizing it with a local hard function in Section 4.2, and combining the local functions into a global objective in Section 4.3.

4.1 Constructing the Gap Trajectory

Using the parameters \(\epsilon,\gamma,c,R\) from Lemma 4.1, we first construct a two-dimensional gap trajectory. After scaling and translation, this will give the GD trajectory generated by the local hard function \(\Phi_i\) during gap \(i\). We choose the gradient vectors on a circular arc with center \((0,\gamma)\) and radius \(R\). Let \(K\) be the convex hull of the origin and the shorter arc joining \((c,0)\) to \((\epsilon,-1)\). Lemma 4.2 constructs points \(r_j\) backward from \((\epsilon,-1)\) so that \(r_j=r_{j-1}-\alpha_jv_{j-1}\), where \(v_j=\Pi_K(r_j)\) is the Euclidean projection of \(r_j\) onto \(K\). Each update has the form of a GD step with stepsize \(\alpha_j\) and gradient \(v_{j-1}\). The projection condition later lets us realize the scaled vectors as gradients of a convex, \(1\)-smooth function \(\Phi_i\).

Here \(P\) is the decrease in the first coordinate and \(Q\) is the increase in the second from \(r_0\) to \(r_m\). When defining \(\Phi_i\), we scale and translate these two coordinates to \(x^{(i)}\) and \(x^{(i+1)}\). The same \(c\) in \(P\le cS\) will appear in the denominator \(c(a+s_i)\) of the transfer ratio \(D_{i+1}/D_i\).

Proof of Lemma 4.2

We first construct the gap iterates backward, then verify the projection identity \(v_j=\Pi_K(r_j)\) and bound the displacements \(P\) and \(Q\).

Step 1. Constructing the gap trajectory.

Set

\[v(q):=(p(q),-q),\qquad p(q):=\sqrt{R^2-(q+\gamma)^2},\qquad 0\le q\le1.\tag{21}\]

The vectors \(v(q)\) lie on the circle of radius \(R\) centered at \((0,\gamma)\), and

\[v(0)=(c,0),\qquad v(1)=(\epsilon,-1),\qquad \epsilon\le p(q)\le c,\qquad \frac{q}{p(q)}\le\frac1\epsilon.\tag{22}\]

Let

\[K:=\operatorname{conv}\bigl(\{0\}\cup\{v(q):0\le q\le1\}\bigr).\]

The set \(K\) lies in the closed disk centered at \((0,\gamma)\) with radius \(R\), and every \(w\in K\) satisfies \(w^{(1)}\le c\). Fix stepsizes \(\alpha_1,\ldots,\alpha_m\ge0\) and set \(S=\sum_{j=1}^m\alpha_j\). Write \(r_j=(u_j,-w_j)\) and \(v_j=(p_j,-q_j)\) for \(0\le j\le m\). Start with \(r_m=v_m=(\epsilon,-1)\). For \(j=m,m-1,\ldots,1\), define

\[\Delta_j:=w_j+\gamma-\alpha_j\gamma.\]

If \(\Delta_j\le u_j\gamma/c\), choose \(v_{j-1}=(p_{j-1},-q_{j-1}):=(c,0)\). Otherwise, choose

\[d_j:=\sqrt{u_j^2+\Delta_j^2},\qquad q_{j-1}:=\frac{R\Delta_j}{d_j}-\gamma,\qquad p_{j-1}:=p(q_{j-1})=\frac{Ru_j}{d_j}.\tag{23}\]

In both cases, set \(r_{j-1}:=r_j+\alpha_jv_{j-1}\), so

\[u_{j-1}=u_j+\alpha_jp_{j-1},\qquad w_{j-1}=w_j+\alpha_jq_{j-1}.\]

Geometrically, when \(\Delta_j>u_j\gamma/c\), we choose \(v_{j-1}\) where the ray from \((0,\gamma)\) through \(r_j+\alpha_j(0,\gamma)\) meets the circle, as shown in Figure 4. If the ray passes at or above the corner \((c,0)\), we choose the corner instead.

Step 2. Verifying the projection identity.

We prove by backward induction that, for \(0\le j\le m\),

\[v_j=\Pi_K(r_j),\qquad u_j\ge\epsilon,\qquad w_j\ge1,\qquad \frac{w_j}{u_j}\le\frac1\epsilon.\tag{24}\]

The statements in (24) hold at \(r_m=v_m=(\epsilon,-1)\). Assume they hold at \(r_j\). We first check that the chosen vector lies on the arc and that the coordinate bounds hold at \(r_{j-1}\). If \(\Delta_j>u_j\gamma/c\), then

\[\frac\gamma c<\frac{\Delta_j}{u_j} \le\frac{w_j+\gamma}{u_j} \le\frac{1+\gamma}{\epsilon}.\]

By (23), \(q_{j-1}=R(\Delta_j/u_j)/\sqrt{1+(\Delta_j/u_j)^2}-\gamma\). The map \(s\mapsto Rs/\sqrt{1+s^2}-\gamma\) is increasing and takes the values \(0\) and \(1\) at \(s=\gamma/c\) and \(s=(1+\gamma)/\epsilon\), respectively. Hence \(q_{j-1}\in(0,1]\). If \(\Delta_j\le u_j\gamma/c\), then \((p_{j-1},q_{j-1})=(c,0)\). In both cases, \(p_{j-1}\ge\epsilon\) and \(q_{j-1}\ge0\), giving \(u_{j-1}\ge\epsilon\) and \(w_{j-1}\ge1\). The inductive ratio bound and (22) also give

\[w_{j-1}=w_j+\alpha_jq_{j-1} \le\frac{u_j+\alpha_jp_{j-1}}{\epsilon} =\frac{u_{j-1}}{\epsilon}.\]

It remains to verify the projection. For \(v\in K\), the identity \(v=\Pi_K(r)\) is equivalent to \(\langle r-v,w-v\rangle\le0\) for every \(w\in K\). If \(\Delta_j>u_j\gamma/c\), we show that \(r_{j-1}-v_{j-1}\) points along the outward radius at \(v_{j-1}\). By (23),

\[r_j+\alpha_j(0,\gamma)-(0,\gamma) =(u_j,-\Delta_j) =\frac{d_j}{R}\bigl(v_{j-1}-(0,\gamma)\bigr).\]

Substituting this into \(r_{j-1}=r_j+\alpha_jv_{j-1}\) gives

\[r_{j-1}-v_{j-1} =\left(\frac{d_j}{R}+\alpha_j-1\right) \bigl(v_{j-1}-(0,\gamma)\bigr).\tag{25}\]

The reverse triangle inequality and the coordinate bounds give

\[d_j \ge\sqrt{u_j^2+(w_j+\gamma)^2}-\alpha_j\gamma \ge R-\alpha_j\gamma.\]

Thus the coefficient in (25) is nonnegative, since

\[\frac{d_j}{R}+\alpha_j-1 \ge\alpha_j\left(1-\frac\gamma R\right)\ge0.\]

Because \(K\) lies inside the disk, every \(w\in K\) satisfies

\[\langle v_{j-1}-(0,\gamma),w-v_{j-1}\rangle \le R\|w-(0,\gamma)\|-R^2\le0.\]

This proves \(v_{j-1}=\Pi_K(r_{j-1})\).

If \(\Delta_j\le u_j\gamma/c\), then \(v_{j-1}=(c,0)\) and \(w_{j-1}=w_j\). Rearranging this condition gives

\[u_{j-1}-c=u_j+\alpha_jc-c\ge\frac{cw_j}{\gamma}.\]

For \(w\in K\), we have \(w^{(1)}\le c\) and \(\langle(c,-\gamma),w-(c,0)\rangle\le0\), since \((c,-\gamma)\) is the outward radius at \((c,0)\). Hence

\[\begin{aligned} \langle r_{j-1}-(c,0),w-(c,0)\rangle &=(u_{j-1}-c)(w^{(1)}-c)-w_jw^{(2)}\\ &\le\frac{w_j}{\gamma}\langle(c,-\gamma),w-(c,0)\rangle\le0. \end{aligned}\tag{26}\]

By (26), \(v_{j-1}=\Pi_K(r_{j-1})\), completing the induction.

Step 3. Bounding \(P\) and \(Q\).

We first identify the indices at which \(q_{j-1}=0\). If \(\Delta_j\le u_j\gamma/c\) and \(j\ge2\), the choice \(v_{j-1}=(c,0)\) gives \(w_{j-1}=w_j\) and \(u_{j-1}=u_j+\alpha_jc\). Hence

\[\Delta_{j-1} =w_{j-1}+\gamma-\alpha_{j-1}\gamma \le w_j+\gamma \le\frac{u_j\gamma}{c}+\alpha_j\gamma =\frac{u_{j-1}\gamma}{c}.\]

Thus the backward recursion also chooses \(v_{j-2}=(c,0)\). Repeating this argument shows that once \((c,0)\) is chosen, all earlier gradient vectors equal \((c,0)\). Thus there is an index \(j_\star\in\{0,\ldots,m\}\) such that, for \(1\le j\le m\),

\[\Delta_j>\frac{u_j\gamma}{c} \quad\Longleftrightarrow\quad j_\star<j\le m,\tag{27}\]

where \(j_\star=m\) if the strict inequality never holds.

The coordinate recurrences give

\[P=\sum_{j=1}^m\alpha_jp_{j-1},\qquad Q=\sum_{j=1}^m\alpha_jq_{j-1}.\]

Since \(p_{j-1}\le c\), we have \(P\le cS\). To bound \(Q\) independently of \(S\), use \(q_{j-1}=0\) for \(j\le j_\star\) and \(q_{j-1}\le p_{j-1}/\epsilon\) from (22). These give

\[0\le Q =\sum_{j=j_\star+1}^m\alpha_jq_{j-1} \le\frac1\epsilon\sum_{j=j_\star+1}^m\alpha_jp_{j-1} =\frac{u_{j_\star}}\epsilon-1.\]

The last equality follows from \(\alpha_jp_{j-1}=u_{j-1}-u_j\) and \(u_m=\epsilon\). It therefore suffices to bound \(u_{j_\star}/\epsilon\).

Set \(\sigma_j:=(w_j+\gamma)/u_j\), the magnitude of the slope from the circle center \((0,\gamma)\) to \(r_j=(u_j,-w_j)\). For \(j_\star<j\le m\), the recurrence and (23) give

\[(u_{j-1},w_{j-1}+\gamma) =\left(1+\frac{\alpha_jR}{d_j}\right)(u_j,\Delta_j),\]

so

\[\sigma_{j-1}=\frac{\Delta_j}{u_j} =\sigma_j-\frac{\alpha_j\gamma}{u_j}, \qquad p_{j-1}=\frac{R}{\sqrt{1+\sigma_{j-1}^2}}.\]

Thus the change in slope determines the increase in \(u_j\) under the backward recursion,

\[\frac{u_{j-1}}{u_j} =1+\frac{\alpha_j}{u_j}p_{j-1} =1+\frac R\gamma \frac{\sigma_j-\sigma_{j-1}} {\sqrt{1+\sigma_{j-1}^2}}.\tag{28}\]

To obtain an inequality that telescopes when multiplied over \(j\), set \(A(\sigma):=\sigma+\sqrt{1+\sigma^2}\). Its derivative satisfies \(A'(\sigma)/A(\sigma)=1/\sqrt{1+\sigma^2}\), which matches the denominator in (28). Convexity of \(A\) gives

\[\frac{A(\sigma_j)}{A(\sigma_{j-1})} \ge1+\frac{\sigma_j-\sigma_{j-1}}{\sqrt{1+\sigma_{j-1}^2}}.\]

Since \(\sigma_j\ge\sigma_{j-1}\) and \(R/\gamma>1\), Bernoulli’s inequality gives

\[\frac{u_{j-1}}{u_j} \le\left(1+\frac{\sigma_j-\sigma_{j-1}}{\sqrt{1+\sigma_{j-1}^2}}\right)^{R/\gamma} \le\left(\frac{A(\sigma_j)}{A(\sigma_{j-1})}\right)^{R/\gamma}.\tag{29}\]

Multiplying (29) over \(j=j_\star+1,\ldots,m\) cancels all intermediate terms, leaving

\[\frac{u_{j_\star}}{\epsilon} \le\left[\frac{A(\sigma_m)}{A(\sigma_{j_\star})}\right]^{R/\gamma}.\]

Here \(u_m=\epsilon\) and \(\sigma_m=(1+\gamma)/\epsilon\). If \(j_\star<m\), then \(\sigma_{j_\star}=\Delta_{j_\star+1}/u_{j_\star+1}>\gamma/c\) by (27); if \(j_\star=m\), then \(\sigma_{j_\star}=\sigma_m\ge\gamma/c\). Since \(A\) is increasing,

\[A(\sigma_m)=\frac{1+\gamma+R}{\epsilon}, \qquad A(\sigma_{j_\star})\ge A(\gamma/c)=\frac{\gamma+R}{c}.\]

Combining these estimates yields

\[0\le Q\le\frac{u_{j_\star}}\epsilon-1 \le H_0:=\left[\frac{c(1+\gamma+R)}{\epsilon(\gamma+R)}\right]^{R/\gamma}-1.\tag{30}\]

□

4.2 Constructing the Local Hard Function

For gap stepsizes \(h_1,\ldots,h_m\), we apply Lemma 4.2 with \(\alpha_j=h_j/2\), then scale and translate its points so that the trajectory starts at \((1,0)\). We then construct a nonnegative convex, \(1\)-smooth function \(f\) such that GD on \(f/2\) follows the scaled points. The following lemma ensures the transfer property (6) and gives the excess after the checkpoint update.

During the gap, the second coordinate stays below \(\tau\). The checkpoint update leaves excess \(\frac{b-2}{c(a+s)}\), which becomes the transfer ratio \(D_{i+1}/D_i\) in the chain.

4.3 Constructing the Global Hard Function

We now combine the local hard functions along the coordinate chain.

5. Proving (Almost) Tight Lower Bounds

It remains to choose checkpoints that make (7) give a uniform lower bound over all schedules. Following [JCY26, Section 3 and Appendix C], we derive a checkpoint cost from (7) and minimize it over admissible checkpoint sets for each schedule. The structure of a schedule that maximizes this minimum allows us to split at a checkpoint and apply induction to the two shorter sequences. The resulting recursion gives an \(\Omega(n^{-p})\) lower bound for every \(p>p_{\mathrm{sil}}\) by taking \(c\) sufficiently close to \(1\). The explicit constant \(a\) in Lemma 4.1 tends to infinity as \(c\downarrow1\). Balancing this growth against the improvement in \(p\) gives Theorem 1.1, as shown in Section 5.1.

For the anytime result, bounded stepsizes already give an \(\Omega(n^{-1})\) Huber lower bound. Otherwise, we examine horizons at which the last stepsize exceeds all earlier ones, following [TFZH26, Section 4.1] and [JCY26, Section 4]. The Huber bound is strong when the total stepsize is small, while selecting a large last step as a checkpoint gives a complementary bound from (7). Together with the estimate on the total stepsize at these horizons in Lemma 5.6, these two bounds yield \(\Omega(n^{-2p/(1+p)})\) infinitely often. Choosing \(p\) to approach \(p_{\mathrm{sil}}\) as the horizon grows yields Theorem 1.2; see Section 5.2.

5.1 Proofs for the Non-anytime Lower Bound

With these preparations in hand, we now prove Theorem 1.1. We first give a high-level overview of the argument. For a fixed stepsize schedule \(H\), each admissible checkpoint set gives a hard function through Lemma 4.1, and we choose checkpoints to make the resulting lower bound as large as possible. We seek a lower bound on this maximum that holds for every schedule \(H\).

Fix \(0<\epsilon<1/2\) and \(\gamma>0\), set \(c:=\sqrt{1+2\gamma+\epsilon^2}\), and take \(a\) as in Lemma 4.1. For a schedule \(H=(h_1,\ldots,h_n)\) and an admissible checkpoint set \(T\) (so \(b_i>2\) at every checkpoint), choose a parameter \(\Lambda\) and, following [JCY26, Section 3 and Appendix C], define the checkpoint cost by

Here \(\Lambda\) replaces the constant \(a\) in the factor for the gap after the last checkpoint. The lower bound (7) implies

For fixed \(H\) and \(\Lambda\), maximizing the right-hand side of (14) is equivalent to minimizing \(Q_\Lambda(T;H)\) over checkpoint sets. We therefore define

A checkpoint set attaining \(V_\Lambda(H)\) is called tight. For fixed \(H\), this minimum admits the following recursive decomposition.

To make the lower bound uniform over schedules of length \(n\), define

Since \(V_\Lambda(H)\le U_n(\Lambda)\), we have

Thus it suffices to bound \(U_n(\Lambda)\) from above. We call a schedule attaining \(U_n(\Lambda)\) a maximizing schedule.

Lemma 5.2 shows that, for every maximizing schedule, there exists a tight singleton checkpoint set. This structure lets us turn the recursive formula for \(V_\Lambda(H)\) in Lemma 5.1 into recursive bounds on \(U_n(\Lambda)\), which is the key step in the proof.

For a tight singleton \(\{t_*\}\) at a maximizing schedule, write \(cW=U_n(\Lambda)\) and let \(cx,cy\) be the empty-set costs of the left and right subproblems. With \(B=h_{t_*}-2\), equality of the empty-set and singleton costs gives

To see heuristically how these identities give rise to the silver exponent, consider a balanced split \(x=y\). Neglect the fixed term \(a-2\) in (16), eliminating \(B\) gives

Combining two equal subproblems therefore multiplies the cost by \(1+\sqrt{1+c}\). This suggests growth of order \(n^{p(c)}\), where \(p(c)=\log_2(1+\sqrt{1+c})\), tending to \(p_{\mathrm{sil}}\) as \(c\downarrow1\).¹

A rigorous argument in the proof of Lemma 5.4 based on (16) gives the following bound on \(U_n(\Lambda)\).

Now we are ready to prove Theorem 1.1.

5.2 Proofs for the Anytime Lower Bound

To prove Theorem 1.2, we combine two consequences of Lemma 4.1. Selecting no checkpoints gives the one-dimensional Huber bound, which forces the sum of stepsizes to be large when the error is small. Selecting only the last step as a checkpoint, when its stepsize exceeds \(2\), gives a second lower bound in terms of that stepsize and the preceding sum. Following [TFZH26, Section 4.1] and [JCY26, Section 4], we apply these bounds at iterations when the stepsize is the largest so far.

To make this argument precise, we first introduce some notation. Fix an infinite nonnegative schedule \(H=(h_t)_{t\ge1}\). For each prefix \(H_n=(h_1,\ldots,h_n)\), denote its total stepsize, largest stepsize, and worst-case error by

We call \(n\) a record time if \(h_n=M_n\), and a strict record time if \(n\ge2\) and \(h_n>M_{n-1}\).

For \(p\in(p_{\mathrm{sil}},p_{\mathrm{sil}}+1/10]\), set \(\nu:=1/p\in(0,1)\) and choose \(a,c\) as in the parameter calculation, with \(\eta=p-p_{\mathrm{sil}}\). The two lower bounds above still depend on the schedule through \(S_n\) and \(M_n\). To obtain a bound in terms of \(n\), the next lemma uses (17) to bound \(S_n\) in terms of \(n\) and \(M_n\) when \(r_n\) is small.

6. Concluding Remarks

We proved lower bounds for GD with predetermined nonnegative stepsizes that match the silver exponents in both the finite-horizon and anytime settings. Bending the gradient during each gap improves the transfer between local hard functions and closes the polynomial gap to the upper bounds.

The remaining open questions are:

1. Can the lower bounds match the silver exponents for schedules that may include negative stepsizes? To our knowledge, the best known lower bounds for such schedules are \(\Omega(n^{-1.6342})\) in the non-anytime case and \(\Omega(n^{-1.2408})\) in the anytime case, as established in our earlier paper [YL26]. Extending the lower bounds proved here to such schedules remains open, although we believe that our approach can be adapted to handle negative stepsizes.

2. Can the \(\sqrt{\log\log n/\log n}\) losses in the exponents of (2) and (3) be reduced or removed?¹ In the circular construction illustrated in Figure 4, setting \(\gamma=0\) would make every gradient vector \(v_j\) equal to \((\epsilon,-1)\). A local calculation similar to the Huber analysis in Section 3 gives the coefficient \(\epsilon+\epsilon^{-1}\ge2\) for \(s_i\). This yields a suboptimal lower bound as in [JCY26]. We therefore need \(\gamma>0\), so \(c=\sqrt{1+2\gamma+\epsilon^2}\) can approach \(1\) but cannot equal \(1\). For this construction, balancing the improvement as \(c\downarrow1\) against the growth of \(a\) leaves the \(o(1)\) loss in the exponent \(p_{\mathrm{sil}}+o(1)\). Removing this loss may require a more refined local construction, or even an entirely new approach to constructing the global hard function.

Bibliographic Information

@misc{ye2026silver,
  author = {Yuhan Ye and Kaizhao Liu},
  title  = {Silver Rate Is (Almost) Optimal for Gradient Descent},
  year   = {2026},
  note   = {Blog post},
  url    = {https://yeyuhanyyh.github.io/gd-silver-rate/}
}

For the full paper, cite the arXiv preprint:

@misc{ye2026silverrate,
  author        = {Yuhan Ye and Kaizhao Liu},
  title         = {Silver Rate Is (Almost) Optimal for Gradient Descent},
  year          = {2026},
  eprint        = {2609.09152},
  archivePrefix = {arXiv},
  primaryClass  = {math.OC},
  doi           = {10.48550/arXiv.2609.09152},
  url           = {https://arxiv.org/abs/2609.09152}
}

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