# Instance-dependent uniform tail bounds for empirical processes

@inproceedings{Bahmani2022InstancedependentUT, title={Instance-dependent uniform tail bounds for empirical processes}, author={Sohail Bahmani}, year={2022} }

We formulate a uniform tail bound for empirical processes indexed by a class of functions, in terms of the individual deviations of the functions rather than the worst-case deviation in the considered class. The tail bound is established by introducing an initial “deﬂation” step to the standard generic chaining argument. The resulting tail bound has a main complexity component, a variant of Talagrand’s γ functional for the deﬂated function class, as well as an instance-dependent deviation term…

## References

SHOWING 1-10 OF 11 REFERENCES

### The Bernstein–Orlicz norm and deviation inequalities

- Mathematics
- 2011

We introduce two new concepts designed for the study of empirical processes. First, we introduce a new Orlicz norm which we call the Bernstein–Orlicz norm. This new norm interpolates sub-Gaussian and…

### Rademacher Processes and Bounding the Risk of Function Learning

- Computer Science, Mathematics
- 2000

The bounds are based on local norms of the Rademacher process indexed by the underlying function class, and they do not require prior knowledge about the distribution of training examples or any specific properties of the function class.

### Concentration inequalities and asymptotic results for ratio type empirical processes

- Mathematics
- 2006

Let ℱ be a class of measurable functions on a measurable space $(S,\mathcal{S})$ with values in [0,1] and let
Pn=n−1∑i=1nδXi
be the empirical measure based on an i.i.d. sample (X1,…,Xn) from a…

### ON CONVERGENCE OF STOCHASTIC PROCESSES

- Mathematics
- 1962

It is clear that for given I,un } and t, the better theorem of this kind would be the one in which (2) is proved for the larger class of functions f. In this paper we shall show that certain known…

### Ratio Limit Theorems for Empirical Processes

- Mathematics
- 2003

Concentration inequalities are used to derive some new inequalities for ratio-type suprema of empirical processes. These general inequalities are used to prove several new limit theorems for…

### The Bennett-Orlicz Norm.

- MathematicsSankhya. Series A.
- 2017

The new Bennett-Orlicz norm yields inequalities for expectations of maxima which are potentially somewhat tighter than those resulting from the Bernstein-Or Alicz norm when they are both applicable.

### Multivariate mean estimation with direction-dependent accuracy

- Mathematics
- 2020

We consider the problem of estimating the mean of a random vector based on $N$ independent, identically distributed observations. We prove the existence of an estimator that has a near-optimal error…

### Statistical learning theory

- Computer Science
- 1998

Presenting a method for determining the necessary and sufficient conditions for consistency of learning process, the author covers function estimates from small data pools, applying these estimations to real-life problems, and much more.

### Weak convergence and empirical processes

- Mathematics
- 2019

1 Review of metric topology 3 1.1 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Open and closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . .…

### Upper and Lower Bounds for Stochastic Processes

- MathematicsErgebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
- 2021