Uniform Argument Stability

• Uniform argument stability is a concept in learning theory that controls the change in the hypothesis in a Banach space when a single training sample is replaced.
• It leverages Banach-space concentration, Rademacher complexity, and localization arguments to derive high-probability generalization and fast-rate excess-risk bounds.
• Applications in regularized ERM and stochastic gradient descent demonstrate its role in linking stability analysis with practical algorithm performance guarantees.

to=arxiv_search 娱乐彩票注册 彩神争霸怎么json {"^^^^^^^^10query10 argument stability^^^^^^^^10\10 OR ^^^^^^^^10\10 stability^^^^^^^^10\10 learning algorithm generalization10^^^^^^^^", "^^^^^^^^10max_results10 ^^^^10\10query10^^^^, "^^^^^^^^10sort_by10 "^^^^^^^^10relevance10 to=arxiv_search 大发快三豹子json {"^^^^^^^^10query10 stability and hypothesis complexity^^^^^^^^10\10 OR abs:^^^^^^^^10\10 stability^^^^^^^^10\10 "^^^^^^^^10max_results10 ^^^^10\10query10^^^^, "^^^^^^^^10sort_by10 "^^^^^^^^10relevance10 to=arxiv_search 彩神争霸是json {"^^^^^^^^10query10 stability^^^^^^^^10\10 Banach martingale generalization stability of learning algorithms", "^^^^^^^^10max_results10 ^^^^10\10query10^^^^, "^^^^^^^^10sort_by10 "^^^^^^^^10relevance10 to=arxiv_search 天天中彩票足彩 娱乐开号json {"^^^^^^^^10query10 "^^^^^^^^10max_results10 ^^^^10sort_by10^^^^, "^^^^^^^^10sort_by10 "^^^^^^^^10relevance10 Uniform argument stability is a notion of algorithmic stability that controls the perturbation of the learned hypothesis in its ambient normed function space rather than only the perturbation of the loss. In the formulation introduced in "Algorithmic stability and hypothesis complexity" (&&&^^^^10query10^^^^&&&), a learning algorithm PRESERVED_PLACEHOLDER_^^^^10query10^^^^ taking an i.i.d. sample PRESERVED_PLACEHOLDER_^^^^10\10^^^^ into a hypothesis PRESERVED_PLACEHOLDER_^^^^10 OR \10^^^^ in a separable Banach space PRESERVED_PLACEHOLDER_^^^^10 learning algorithm generalization10^^^^ is PRESERVED_PLACEHOLDER_^^^^10max_results10^^^^-uniformly argument stable when replacing any single training example changes the output by at most PRESERVED_PLACEHOLDER_^^^^10sort_by10^^^^ almost surely. This notion is strictly stronger than classical uniform stability under standard Lipschitz assumptions, and it yields high-probability generalization bounds through Banach-space martingale concentration, Rademacher complexity of a data-dependent hypothesis class, and localization arguments (&&&^^^^10query10^^^^&&&).

^^^^10\10^^^^. Formal definition and basic comparison

Let PRESERVED_PLACEHOLDER_^^^^10relevance10^^^^ be an i.i.d. training sample from PRESERVED_PLACEHOLDER_^^^^10query10^^^^ on PRESERVED_PLACEHOLDER_^^^^10ti:\10^^^^, let PRESERVED_PLACEHOLDER_^^^^10 OR abs:\10^^^^ be the sample in which the PRESERVED_PLACEHOLDER_^^^^10\10query10^^^^-th point is replaced by an independent copy PRESERVED_PLACEHOLDER_^^^^10\10\10^^^^, and let PRESERVED_PLACEHOLDER_^^^^10\10 OR \10^^^^. The algorithm is called PRESERVED_PLACEHOLDER_^^^^10\10 learning algorithm generalization10^^^^-uniformly argument stable if, for every PRESERVED_PLACEHOLDER_^^^^10\10max_results10^^^^,

PRESERVED_PLACEHOLDER_^^^^10\10sort_by10^^^^

This definition measures the sensitivity of the output hypothesis itself to a one-sample perturbation (&&&^^^^10query10^^^^&&&).

The standard comparator is classical uniform stability in the sense of Bousquet–Elisseeff, which bounds the loss difference

PRESERVED_PLACEHOLDER_^^^^10\10relevance10^^^^

Under the assumptions that PRESERVED_PLACEHOLDER_^^^^10\10query10^^^^ is PRESERVED_PLACEHOLDER_^^^^10\10ti:\10^^^^-Lipschitz in the prediction PRESERVED_PLACEHOLDER_^^^^10\10 OR abs:\10^^^^ and PRESERVED_PLACEHOLDER_^^^^10 OR \10query10^^^^ almost surely, argument stability implies loss stability through

PRESERVED_PLACEHOLDER_^^^^10 OR \10\10^^^^

The reverse implication does not hold in general (&&&^^^^10query10^^^^&&&). A common misconception is therefore to treat the two notions as interchangeable. The stated implication shows that uniform argument stability is stronger: it controls the entire hypothesis perturbation in norm, not merely its effect on a particular bounded loss.

^^^^10 OR \10^^^^. Banach-space framework and concentration around the mean

The generalization theory built around uniform argument stability in (&&&^^^^10query10^^^^&&&) is formulated in a geometric-probabilistic framework. The Banach space PRESERVED_PLACEHOLDER_^^^^10 OR \10 OR \10^^^^ is assumed to be PRESERVED_PLACEHOLDER_^^^^10 OR \10 learning algorithm generalization10^^^^-smooth, meaning that for all PRESERVED_PLACEHOLDER_^^^^10 OR \10max_results10^^^^,

PRESERVED_PLACEHOLDER_^^^^10 OR \10sort_by10^^^^

with Hilbert spaces as the special case PRESERVED_PLACEHOLDER_^^^^10 OR \10relevance10^^^^. The dual space is assumed to be of type PRESERVED_PLACEHOLDER_^^^^10 OR \10query10^^^^ with constant PRESERVED_PLACEHOLDER_^^^^10 OR \10ti:\10^^^^, so that for any PRESERVED_PLACEHOLDER_^^^^10 OR \10 OR abs:\10^^^^,

PRESERVED_PLACEHOLDER_^^^^10 learning algorithm generalization10query10^^^^

In addition, one assumes PRESERVED_PLACEHOLDER_^^^^10 learning algorithm generalization10\10^^^^ almost surely, and that the loss PRESERVED_PLACEHOLDER_^^^^10 learning algorithm generalization10 OR \10^^^^ is bounded by PRESERVED_PLACEHOLDER_^^^^10 learning algorithm generalization10 learning algorithm generalization10^^^^ and PRESERVED_PLACEHOLDER_^^^^10 learning algorithm generalization10max_results10^^^^-Lipschitz in PRESERVED_PLACEHOLDER_^^^^10 learning algorithm generalization10sort_by10^^^^ (&&&^^^^10query10^^^^&&&).

A central step is a martingale concentration argument due to Pinelis. If PRESERVED_PLACEHOLDER_^^^^10 learning algorithm generalization10relevance10^^^^ is a martingale-difference sequence in a PRESERVED_PLACEHOLDER_^^^^10 learning algorithm generalization10query10^^^^-smooth Banach space and PRESERVED_PLACEHOLDER_^^^^10 learning algorithm generalization10ti:\10^^^^, then for every PRESERVED_PLACEHOLDER_^^^^10 learning algorithm generalization10 OR abs:\10^^^^,

PRESERVED_PLACEHOLDER_^^^^10max_results10query10^^^^

Applied to the Doob martingale of PRESERVED_PLACEHOLDER_^^^^10max_results10\10^^^^, this yields the concentration lemma

PRESERVED_PLACEHOLDER_^^^^10max_results10 OR \10^^^^

Thus uniform argument stability implies that the random hypothesis is localized, with high probability, in a ball centered at its mean (&&&^^^^10query10^^^^&&&).

This localization is the basis for what the paper calls the algorithmic hypothesis class,

PRESERVED_PLACEHOLDER_^^^^10max_results10 learning algorithm generalization10^^^^

The generalization problem is then reduced to controlling the complexity of PRESERVED_PLACEHOLDER_^^^^10max_results10max_results10^^^^ rather than of the full ambient class.

^^^^10 learning algorithm generalization10^^^^. Rademacher bounds and high-probability generalization

The localization step leads to an explicit Rademacher-complexity estimate for PRESERVED_PLACEHOLDER_^^^^10max_results10sort_by10^^^^. Under the preceding assumptions,

PRESERVED_PLACEHOLDER_^^^^10max_results10relevance10^^^^

In the Hilbert case, where PRESERVED_PLACEHOLDER_^^^^10max_results10query10^^^^, PRESERVED_PLACEHOLDER_^^^^10max_results10ti:\10^^^^, and PRESERVED_PLACEHOLDER_^^^^10max_results10 OR abs:\10^^^^, this simplifies to

PRESERVED_PLACEHOLDER_^^^^10sort_by10query10^^^^

The proof shifts the class by PRESERVED_PLACEHOLDER_^^^^10sort_by10\10^^^^, uses

PRESERVED_PLACEHOLDER_^^^^10sort_by10 OR \10^^^^

and then invokes the type-PRESERVED_PLACEHOLDER_^^^^10sort_by10 learning algorithm generalization10^^^^ inequality (&&&^^^^10query10^^^^&&&).

For Hilbert spaces, this yields a direct high-probability excess-risk estimate. If the loss is bounded by PRESERVED_PLACEHOLDER_^^^^10sort_by10max_results10^^^^ and PRESERVED_PLACEHOLDER_^^^^10sort_by10sort_by10^^^^-Lipschitz, then with probability at least PRESERVED_PLACEHOLDER_^^^^10sort_by10relevance10^^^^,

PRESERVED_PLACEHOLDER_^^^^10sort_by10query10^^^^

The first term is the stability-driven term and the second is the standard sampling term. The significance of the result is structural: the dependence on the learning algorithm enters only through PRESERVED_PLACEHOLDER_^^^^10sort_by10ti:\10^^^^, while the remainder of the bound is mediated by the geometry of the hypothesis space and by boundedness and Lipschitz assumptions (&&&^^^^10query10^^^^&&&).

The same framework also supports a localized fast-rate statement. For any PRESERVED_PLACEHOLDER_^^^^10sort_by10 OR abs:\10^^^^, with probability PRESERVED_PLACEHOLDER_^^^^10relevance10query10^^^^,

PRESERVED_PLACEHOLDER_^^^^10relevance10\10^^^^

The proof uses Bartlett–Mendelson locality on a deformed class. In this sense, uniform argument stability is not merely a replacement for loss-stability estimates; it is a route to localized high-probability control through an algorithm-dependent neighborhood around PRESERVED_PLACEHOLDER_^^^^10relevance10 OR \10^^^^ (&&&^^^^10query10^^^^&&&).

^^^^10max_results10^^^^. Instantiations: regularized ERM and stochastic gradient descent

The abstract framework becomes concrete once PRESERVED_PLACEHOLDER_^^^^10relevance10 learning algorithm generalization10^^^^ is computed for specific algorithms. For regularized empirical risk minimization,

PRESERVED_PLACEHOLDER_^^^^10relevance10max_results10^^^^

assume the penalty PRESERVED_PLACEHOLDER_^^^^10relevance10sort_by10^^^^ satisfies, for some PRESERVED_PLACEHOLDER_^^^^10relevance10relevance10^^^^ and PRESERVED_PLACEHOLDER_^^^^10relevance10query10^^^^,

PRESERVED_PLACEHOLDER_^^^^10relevance10ti:\10^^^^

Then one obtains

PRESERVED_PLACEHOLDER_^^^^10relevance10 OR abs:\10^^^^

For PRESERVED_PLACEHOLDER_^^^^10query10query10^^^^, one takes PRESERVED_PLACEHOLDER_^^^^10query10\10^^^^, giving PRESERVED_PLACEHOLDER_^^^^10query10 OR \10^^^^ (&&&^^^^10query10^^^^&&&). Combined with the localized generalization theorem, this yields explicit high-probability rates driven by regularization strength and sample size.

For stochastic gradient descent in PRESERVED_PLACEHOLDER_^^^^10query10 learning algorithm generalization10^^^^, with PRESERVED_PLACEHOLDER_^^^^10query10max_results10^^^^ both PRESERVED_PLACEHOLDER_^^^^10query10sort_by10^^^^-Lipschitz and PRESERVED_PLACEHOLDER_^^^^10query10relevance10^^^^-smooth, the paper imports a stability estimate of the form

PRESERVED_PLACEHOLDER_^^^^10query10query10^^^^

after PRESERVED_PLACEHOLDER_^^^^10query10ti:\10^^^^ updates with suitably decaying PRESERVED_PLACEHOLDER_^^^^10query10 OR abs:\10^^^^. Substituting this into the general theorem gives a corresponding high-probability deformed bound. Variants for convex or strongly convex losses yield the simpler rates PRESERVED_PLACEHOLDER_^^^^10ti:\10query10^^^^ or PRESERVED_PLACEHOLDER_^^^^10ti:\10\10^^^^, and hence PRESERVED_PLACEHOLDER_^^^^10ti:\10 OR \10^^^^ generalization (&&&^^^^10query10^^^^&&&).

These examples clarify the operational role of uniform argument stability. It is not a property attached only to a hypothesis class; rather, it is an algorithm-level sensitivity parameter that can be derived from optimization or regularization structure and then inserted into a generalization theorem.

^^^^10sort_by10^^^^. Relation to classical uniform stability and worst-case limits

Classical uniform stability is formulated directly at the level of losses. In "Stability of Multi-Task Kernel Regression Algorithms" (&&&^^^^10\10 learning algorithm generalization10^^^^&&&), if PRESERVED_PLACEHOLDER_^^^^10ti:\10 learning algorithm generalization10^^^^, PRESERVED_PLACEHOLDER_^^^^10ti:\10max_results10^^^^, and PRESERVED_PLACEHOLDER_^^^^10ti:\10sort_by10^^^^, then PRESERVED_PLACEHOLDER_^^^^10ti:\10relevance10^^^^-uniform stability means

PRESERVED_PLACEHOLDER_^^^^10ti:\10query10^^^^

for all PRESERVED_PLACEHOLDER_^^^^10ti:\10ti:\10^^^^, all PRESERVED_PLACEHOLDER_^^^^10ti:\10 OR abs:\10^^^^, all training sets, and all test points PRESERVED_PLACEHOLDER_^^^^10 OR abs:\10query10^^^^ independent of PRESERVED_PLACEHOLDER_^^^^10 OR abs:\10\10^^^^. In an operator-valued RKHS PRESERVED_PLACEHOLDER_^^^^10 OR abs:\10 OR \10^^^^ with regularized empirical risk

PRESERVED_PLACEHOLDER_^^^^10 OR abs:\10 learning algorithm generalization10^^^^

the minimizer PRESERVED_PLACEHOLDER_^^^^10 OR abs:\10max_results10^^^^ is uniformly stable under the bounded-kernel, well-posedness, convexity, and Lipschitz assumptions (H^^^^10\10^^^^)–(H^^^^10 learning algorithm generalization10^^^^), with

PRESERVED_PLACEHOLDER_^^^^10 OR abs:\10sort_by10^^^^

This leads, under bounded loss PRESERVED_PLACEHOLDER_^^^^10 OR abs:\10relevance10^^^^, to the standard Bousquet–Elisseeff high-probability estimate

PRESERVED_PLACEHOLDER_^^^^10 OR abs:\10query10^^^^

(&&&^^^^10\10 learning algorithm generalization10^^^^&&&). The contrast with uniform argument stability is conceptual: the latter first controls PRESERVED_PLACEHOLDER_^^^^10 OR abs:\10ti:\10^^^^, then derives loss stability and generalization from geometry and Lipschitz structure.

The broader theory of uniform stability has sharp worst-case limitations. "A Tight Lower Bound for Uniformly Stable Algorithms" proves that for any uniformly PRESERVED_PLACEHOLDER_^^^^10 OR abs:\10 OR abs:\10^^^^-stable algorithm and PRESERVED_PLACEHOLDER_^^^^10\10query10query10^^^^-bounded loss, recent upper bounds give

PRESERVED_PLACEHOLDER_^^^^10\10query10\10^^^^

more precisely

PRESERVED_PLACEHOLDER_^^^^10\10query10 OR \10^^^^

and the paper matches this up to logarithmic factors with a lower bound: for any PRESERVED_PLACEHOLDER_^^^^10\10query10 learning algorithm generalization10^^^^ and integer PRESERVED_PLACEHOLDER_^^^^10\10query10max_results10^^^^, there exist a domain, a distribution, an PRESERVED_PLACEHOLDER_^^^^10\10query10sort_by10^^^^-bounded loss, and a PRESERVED_PLACEHOLDER_^^^^10\10query10relevance10^^^^-stable algorithm such that with constant probability,

PRESERVED_PLACEHOLDER_^^^^10\10query10query10^^^^

(&&&^^^^10\10sort_by10^^^^&&&). This shows that the dependence on PRESERVED_PLACEHOLDER_^^^^10\10query10ti:\10^^^^ and PRESERVED_PLACEHOLDER_^^^^10\10query10 OR abs:\10^^^^ cannot be improved, beyond logarithmic factors, for uniformly stable algorithms.

Because argument stability implies uniform stability under PRESERVED_PLACEHOLDER_^^^^10\10\10query10^^^^-Lipschitzness and PRESERVED_PLACEHOLDER_^^^^10\10\10\10^^^^, a plausible implication is that any reduction from argument stability to classical loss stability inherits these worst-case barriers at the loss-stability layer. At the same time, the Banach-space localization method of (&&&^^^^10query10^^^^&&&) extracts additional structure from the norm perturbation itself rather than stopping at the induced loss difference.

^^^^10relevance10^^^^. Adjacent developments and methodological extensions

Subsequent work on optimization-driven stability has emphasized direct analysis of iterate dynamics. In the PRESERVED_PLACEHOLDER_^^^^10\10\10 OR \10^^^^-smooth, PRESERVED_PLACEHOLDER_^^^^10\10\10 learning algorithm generalization10^^^^-strongly convex regime, "A Unified Lyapunov-IQC Framework for Uniform Stability of Smooth Quadratic First-Order Accelerated Optimizers" defines PRESERVED_PLACEHOLDER_^^^^10\10\10max_results10^^^^-uniform stability for a first-order optimizer PRESERVED_PLACEHOLDER_^^^^10\10\10sort_by10^^^^ by

PRESERVED_PLACEHOLDER_^^^^10\10\10relevance10^^^^

where PRESERVED_PLACEHOLDER_^^^^10\10\10query10^^^^ and PRESERVED_PLACEHOLDER_^^^^10\10\10ti:\10^^^^ are the terminal iterates on neighboring samples. Under PRESERVED_PLACEHOLDER_^^^^10\10\10 OR abs:\10^^^^-Lipschitzness of PRESERVED_PLACEHOLDER_^^^^10\10 OR \10query10^^^^ in its first argument,

PRESERVED_PLACEHOLDER_^^^^10\10 OR \10\10^^^^

For SGD with fixed step size PRESERVED_PLACEHOLDER_^^^^10\10 OR \10 OR \10^^^^, the classical coupling argument produces PRESERVED_PLACEHOLDER_^^^^10\10 OR \10 learning algorithm generalization10^^^^. For Nesterov accelerated gradient, the presence of momentum breaks the simple one-step contraction argument, and the paper develops a quadratic Lyapunov and then a Lyapunov–IQC certification via an LMI solvable by SDP. In the smooth-quadratic regime this yields PRESERVED_PLACEHOLDER_^^^^10\10 OR \10max_results10^^^^ uniform stability for NAG (&&&^^^^10\10query10^^^^&&&).

These developments do not replace uniform argument stability; they delineate a neighboring methodological direction. Uniform argument stability analyzes the perturbation of the output hypothesis in a Banach or Hilbert norm and then leverages hypothesis-space geometry. Lyapunov–IQC methods analyze optimizer state trajectories directly and certify contraction through control-theoretic machinery. This suggests a broader taxonomy of stability analyses: hypothesis-level stability, loss-level stability, and state-space dynamical stability.

Within that taxonomy, uniform argument stability occupies a distinctive position. It is stronger than classical loss stability, it naturally interfaces with Banach-space concentration, and it supports localized generalization analyses that are sensitive to the geometry of the hypothesis space and to algorithm-specific perturbation bounds. Its principal limitation is equally clear: useful generalization theorems still require boundedness, Lipschitzness, and structural assumptions on the ambient space, and any passage through loss stability is subject to the sharp worst-case limits known for uniformly stable algorithms (&&&^^^^10query10^^^^&&&, &&&^^^^10\10sort_by10^^^^&&&).