A tightened convergence analysis of clipped gradient descent on $(L_0, L_1)-smooth functions, with quantitative constants, is presented, with quantitative constants matching the more traditional analysis of smooth functions.
Abstract
We present a tightened convergence analysis of clipped gradient descent on $(L_0, L_1)$-smooth functions, with quantitative constants. Building on the ideas of Koloskova et al (2023), we refactor several case disjunctions to reveal the central role of a control of the bias derived from fundamental properties of $\ell_2$-projection, simplifying proofs. We also extend the domain of validity from $\eta \leq 1 / (9 \beta)$ to $\eta<1 /\beta$ where $\beta = L_0 + c L_1$ for clipping constant $c$, which matches the more traditional analysis of smooth functions. We strengthen the convergence criterion from $\left( \min_{t<T} \mathbb{E}[\lVert \nabla f(x_t) \rVert_2] \right)$ to $\left( \frac{1}{T} \sum_{t<T} \mathbb{E}[\lVert \nabla f(x_t) \rVert_2] \right)$ with matching speed, and lower the final achievable loss from $\mathcal{O}(\min(\sigma^2/c, \sigma))$ to the more precise $6 \min(\sigma^2 /c, 3 \sigma)$.
The last-epoch rate is proved, matching the known quadratic lower bound in its $(n,K)$-dependence, and it is shown that the $\beta_\star^2/K^2$ splitting term is unavoidable and obtain a matching lower bound up to logarithms in the stated constant-stepsize regime.
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We determine the exact worst-case value, at every horizon $N\geq7$, of the smallest queried gradient norm generated by Nesterov's fast gradient method on smooth convex functions. Let $t_0=1$ and $t_{k+1}=(1+\sqrt{1+4t_k^2})/2$, and let $x_0,\ldots,x_N$ denote the points at which the method evaluates gradients. For ever...
This work presents a new lower bound of $\Omega(T^{-1.9319})$ for the last-iterate convergence rate of gradient descent with predetermined nonnegative stepsize schedules, and provides rigorous evidence that stepsize schedules alone cannot accelerate plain GD to the optimal $O(T^{-2})$ convergence rate.
It is shown that for this mismatched problem the minimax value of the gap is of order $LR^2/N^3$ up to logarithmic factors, for deterministic and for randomized methods alike, and already in dimension proportional to the number of queries.
Т. В. Логінов, A. Gasnikov, Yuriy Dorn et al.· 0 citations
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