AI 中文总结
研究解决了Yun等人提出的SS - RS - GD不等式猜想，通过具体反例证明SS - RS不等式不成立，又证明了RS - GD不等式成立，且借助GPT - 5.5 Pro找到证明过程。
AI 中文摘要
Yun、Sra和Jadbabaie（COLT 2021，开放问题）推测了SS - RS - GD不等式：对于良态对称矩阵\(A_1,\dots,A_n\)，编码单洗牌随机梯度下降（single - shuffle SGD）、随机重洗牌随机梯度下降（random - reshuffle SGD）和二次有限和上梯度下降预期迭代的算子\(W_{ss}\)、\(W_{rs}\)和\(W_{gd}\)应满足\(\|W_{ss}\|\leq \| W_{rs}\|\leq \|W_{gd}\|\)。本研究解决了该猜想，证明了SS - RS不等式不成立，对于\(n = 3\)，\(K = 2\)，\(d = 4\)时就存在反例；而RS - GD不等式成立，对于满足\(\bigl(1-\frac1{4n^2+1}\bigr)I\preceq A_i\preceq I\)的对称\(A_i\)，有\(\|W_{rs}\|\leq\|W_{gd}\|\)。证明是通过作者提示扩展的GPT - 5.5 Pro找到的。
英文摘要
Yun, Sra, and Jadbabaie (COLT 2021, open question) conjectured the SS--RS--GD inequalities: for well-conditioned symmetric matrices $A_1,\dots,A_n$, the operators $W_{ss}$, $W_{rs}$, and $W_{gd}$ that encode the expected iterate of single-shuffle SGD, random-reshuffle SGD, and gradient descent on a quadratic finite sum should satisfy \[ \|W_{ss}\|\le \| W_{rs}\|\le \|W_{gd}\|. \] The conjecture is resolved, $\bullet$ SS-RS inequality fails. Already for $n=3$, $K=2$, and $d=4$, we exhibit explicit PSD matrices whose condition number is arbitrarily close to $1$, yet $\|W_{ss}\|>\|W_{rs}\|$. $\bullet$ RS-GD inequality holds. For every symmetric $A_i$ with $\bigl(1-\frac1{4n^2+1}\bigr)I\preceq A_i\preceq I$, one has $\|W_{rs}\|\le\|W_{gd}\|$. The proof was found via GPT-5.5 Pro extended prompted by the author.
