A new logarithmic-free upper bound shows that a $\gamma$-uniformly stable algorithm with loss in $[0,L]$ has generalization gap at most, which determines the optimal high-probability and moment dependence of uniform stability up to universal constants.
Abstract
Uniform stability controls how much one training example can change the loss at any test point. A new logarithmic-free upper bound shows that a $\gamma$-uniformly stable algorithm with loss in $[0,L]$ has generalization gap at most $O \left(\gamma\log(1/\delta) +L\sqrt{\frac{\log(1/\delta)}{n}}\right)$ with probability $1-\delta$. Whether an actual bounded-loss learning algorithm can realize the linear dependence on $\log(1/\delta)$ has remained open. The known construction realizes it only for auxiliary weakly dependent random variables whose pointwise range grows with $n$. The known learning lower bound holds only at constant probability. We close this gap. For every $n$, stability level $\gamma$, and loss bound $L$, we construct one deterministic $\gamma$-uniformly stable learning problem whose tail satisfies, simultaneously for $1\le p\le c n$, $\mathbb P \left( R(A_S)-R_S(A_S) \ge c'\min \left\{L,\gamma p+L\sqrt{p/n}\right\} \right)\ge e^{-p}.$ The construction is ordinary bounded absolute-loss regression with constant labels. Its key is a multiscale collection of rare Rademacher features. A coordinatewise ramp is stable in sup norm, while an odd symmetrized maximum converts a unique extreme feature into a gap of order $\gamma p$ without violating the loss bound. Geometrically spaced ramps put all confidence levels into the same problem. Together with the logarithmic-free upper bound, this determines the optimal high-probability and moment dependence of uniform stability up to universal constants.
Uniform stability is a classical tool for controlling the generalization error of a learning algorithm. Bousquet, Klochkov, and Zhivotovskiy (2020) showed that the problem can be reduced to a moment inequality for a sum of weakly interacting functions of independent random variables. Their bound contains an additional factor $\log n$, and they asked whether this factor can be removed. We answer this upper-bound question affirmatively. More specifically, let $Z=(Z_1,\ldots,Z_n)$ have independent coordinates and let $g_i(Z)$ satisfy $$ \mathbb E[g_i(Z)\mid Z_{-i}]=0, \qquad \left| \mathbb E[g_i(Z)\mid Z_i]\right|\le M, \qquad \forall i = \overline{1, n} $$ while changing any coordinate $Z_j$, $j\neq i$, changes $g_i$ by at most $\beta$ and $Z_{-i}$ denotes all coordinates except $Z_i$. We prove that, for every $p\ge2$, $$ \left\| \sum_{i=1}^n g_i(Z)\right\|_p \le 16pn\beta+M\sqrt{2pn}. $$ This removes the $\log n$ factor from the previous bound and matches the lower bound of Bousquet, Klochkov, and Zhivotovskiy up to universal constants in the range covered by their construction. Our proof first establishes the required estimate on the Rademacher cube, then transfers it to arbitrary product distributions by a two-copy randomization argument.
We prove a strict space separation between static and ordinary dynamic approximate membership at every fixed error rate. For each fixed $\varepsilon\in(0,1)$, a capacity-$n$ ordinary dynamic filter over a universe of size $u$, with zero false negatives, pointwise false-positive probability at most $\varepsilon$, arbitrary history dependence, a free public random tape, and at most $H$ bits of persistent state, satisfies \[ H\ge \bigl(\log_2(1/\varepsilon)+a_\varepsilon^{\rm c}\bigr)n-o(n), \] under only $u/n\to\infty$. The constant $a_\varepsilon^{\rm c}$ is an explicit variational threshold obtained by preserving the dependence between the parent accepted mass and the successor reservoir. The structural step is a common-continuation transport lemma. A joint posterior KL bound gives a branch-specific survivor support; the same legal delete--insert word transports that support to one successor state, forcing an accepted reservoir. We then keep the parent outside mass $1-X$ in the conditional-entropy argument instead of replacing it by $1-\varepsilon$. This yields a two-variable analytic envelope, with no selected thresholds, dyadic witnesses, or numerical assumptions.
We determine exactly what a kurtosis bound buys for one-sided tail control. For the class $\mathcal{C}(\kappa)$ of real random variables with mean $0$, variance $1$, and fourth moment at most $\kappa$, the skewness left free, we compute the worst-case tail probability $V_1(t,\kappa)=\sup_{X\in\mathcal{C}(\kappa)}\mathbb{P}(X\geq t)$ for every threshold $t>0$ and every $\kappa\geq 1$. The answer is a four-regime map: a Cantelli tongue $b(\kappa)\le t\le c(\kappa)$ on which the two-moment bound $1/(1+t^2)$ remains tight and the kurtosis constraint is worthless; a tail regime $t\geq c(\kappa)$ with the closed form $V_1=(\kappa-1)/((t^2-1)^2+\kappa-1)$; a plateau regime, present only for $\kappa\le 3/2$, on which the worst case freezes and the value does not depend on $t$; and a central regime described exactly by an explicit algebraic system, provably admitting no closed form in nested square roots. Beyond $c(\kappa)$ the one-sided and two-sided worst cases coincide: Cantelli's improvement over Chebyshev is annihilated by fourth-moment information. The minimal degree of a sum-of-squares proof of the tight bound is $2$ on the closed tongue and $4$ everywhere else, an exact phase diagram of proof degree. Every closed-form regime carries an explicit dual certificate and an explicit extremal distribution, re-verified on parameter grids by an independent checker in exact arithmetic. The closed forms invert to exact worst-case quantiles, sharpen a median-of-means constant, and give the exact per-direction tail available to degree-4 reasoning under certifiable kurtosis. We found the map through an AI-guided search around the certifying pipeline, LemmaForge, which is validated on classical benchmarks, independently reproduces the symmetric-slice bound of Zelen (1954), and recovers the $2\sqrt{3}-3$ constant of He, Zhang, and Zhang (2010) at $t=0$.
Xiaoyu Li, Andi Han, Jiaojiao Jiang et al.· 0 citations
For stochastic gradient descent (SGD) with a constant stepsize $\alpha$, the invariant law of the iterates, centered at a minimizer, describes the behavior of the algorithm over long time horizons. In the strongly convex case, this invariant law has the familiar $\sqrt{\alpha}$ scaling and a Gaussian limit as $\alpha\downarrow 0$. We show that this behavior changes fundamentally for convex objectives $H$ with flat minima and (sub)quadratic tails. More specifically, we study SGD with Markovian noise generated by a contractive driving chain. For every sufficiently small constant stepsize $\alpha$, we prove existence, uniqueness, and geometric convergence to an augmented invariant law in a Wasserstein distance induced by an $\alpha$-dependent metric. When the minimizer $x_\star$ has local flatness exponent $m\ge2$, meaning that $\nabla^2 H(x)\asymp \lVert x-x_\star\rVert^{m-2} I_d$ as $x\to x_\star$, we obtain a contraction bound with factor $1-c\alpha^{m-1}$, where $c>0$ is a constant. This recovers the factor $1-c\alpha$ in the quadratic case $m=2$. We then analyze the small-stepsize scaling limit. We show that the invariant law concentrates on the scale $\alpha^{1/m}$ and that the rescaled iterates converge weakly to the stationary distribution of the stochastic differential equation $$ dY_t=-h_0(Y_t)\,dt+\Sigma^{1/2}\,dB_t , $$ where $h_0$ is the limiting drift at the minimizer and $\Sigma$ denotes the asymptotic covariance. This recovers the Gaussian limit when $m=2$ and gives generally non-Gaussian stationary limits in the flat case $m>2$. Finally, we give corresponding results for coordinate-separable objectives with unequal flatness exponents.
Whether exact scalar feedback intrinsically incurs the additional dimension $d$ paid by known zeroth-order methods remains open even for Lipschitz convex optimization. For a universal Lipschitz scale, the value only bound $O(d^2\log(d+1)\log(1/\epsilon))$ and two-point bound $O(d\epsilon^{-2})$ yield the upper bound $\widetilde O\left(d\min\{d,\epsilon^{-2}\}\right)$. By contrast, prior lower bounds for arbitrary randomized algorithms give only $\Omega(\min\{d,\epsilon^{-2}\})$, leaving a factor $d$ unexplained. We close this gap, up to logarithmic factors, for arbitrary adaptive randomized algorithms minimizing a convex objective with a universal Lipschitz scale over the $d$-dimensional Euclidean unit ball, where each query returns only the exact scalar value. Let $T_\epsilon$ denote the minimum number of queries required to return an $\epsilon$-suboptimal point with probability at least $1/2$, uniformly over the function class. We prove that \[T_\epsilon\ge c\,\frac{d\min\{d,\epsilon^{-2}\}}{\log\!\bigl(\min\{d,\epsilon^{-2}\}\bigr)},\] for $d\ge d_0$ and $0<\epsilon\le\epsilon_0$, where $c,\epsilon_0>0$ and $d_0\in\mathbb N$ are universal constants. This gives $\Omega\left(\frac{d}{\epsilon^2\log(1/\epsilon)}\right)$ in the low-accuracy regime $\epsilon\ge d^{-1/2}$ and $\Omega\left(\frac{d^2}{\log d}\right)$ in the high-accuracy regime $\epsilon\le d^{-1/2}$ with the latter independent of $\epsilon$. These bounds match the corresponding upper bound up to logarithmic factors. To our knowledge, this is the first near-optimal lower bound for arbitrary adaptive randomized algorithms throughout both accuracy regimes of exact value Lipschitz convex optimization. The proof uses a random support function hard family and develops a posterior mean energy method for adaptive exact max observations, in place of first-order zero chain constructions and noise based transcript inequalities.
Haihan Zhang, Chenheng Zhang, Zhiquan Qi et al.· 0 citations
Consider $n$ independent, non-negative, mean at most one random variables, $X_1,X_2,\ldots$. We show the following bound on the probability of their sum exceeding a threshold $t$: \[ \mathbb{P}\left[\sum_{i=1}^n X_i\ge t\right] \leq 1-\left(1-\frac{1}{t}\right)^n \text{ for all } t\ge 2n+1 \,. \] To prove this, we consider a relaxed optimization problem over a set of sequences of ordered, but non-independent random variables. This allows us to reformulate it recursively as dynamic programming problem. The bound becomes an equality for the binary i.i.d.~random variables satisfying $\mathbb{P}\left[X_i=0\right]= 1-\frac{1}{t}$ and $\mathbb{P}\left[X_i=t\right]=\frac{1}{t}$, which remains the maximizer in the relaxed problem.
