New bounds for LDP with heterogeneous privacy levels guaranteeing high probability of accuracy.
problem Statistical estimation under LDP with users having varying privacy levels.
method Developed finite sample upper bounds in ℓ_2-norm with high probability, complemented by lower bounds.
result Optimal guarantees for heterogeneous LDP in terms of probability and constants.
Sharp bounds for high-probability estimation of discrete distributions.
problem Estimating discrete distributions with high probability under χ 2 χ^2 χ 2 -divergence. method Sharp upper and lower bounds for the classical Laplace estimator, and characterization of minimax high-probability risk for any estimator.
result Sharp bounds for high-probability estimation of discrete distributions can be achieved through a simple smoothing strategy.
New approach for online learning with adaptive adversaries, simpler and more effective.
problem Online learning with adaptive adversaries, especially in bandits and MDPs.
method Uses standard unbiased estimators and a simple increasing learning rate schedule, aided by logarithmically homogeneous self-concordant barriers and strengthened Freedman's inequality.
result First high-probability regret bounds for adversarial bandits and MDPs, resolving open problems.
New algorithm reduces high-probability regret for time-varying feedback graphs.
problem High-probability regret bounds for adversarial bandits with time-varying feedback graphs.
method Online mirror descent framework with innovative techniques for pessimistic loss estimators.
result Achieves optimal high-probability regret bound for general and weakly observable graphs.
New method bounds stochastic subgradient methods with heavy-tailed noise.
problem Bounding stochastic subgradient methods under heavy-tailed noise.
method Clipped version of projected stochastic subgradient method.
result Near optimal any-time and finite horizon bounds for averaging schemes.
The paper provides bounds for LSA with fixed stepsizes under random estimates.
problem Analyzing the performance of LSA algorithms with fixed stepsize.
method Non-asymptotic analysis based on new results about matrix moments and high probability bounds.
result Derives high probability bounds on LSA performance under weaker conditions than previous works.
The study optimizes distribution estimation from samples with relative entropy error, adapting to sparse distributions.
problem Estimating discrete distributions with high-probability accuracy in relative entropy.
method Analysis of Laplace estimator and confidence-dependent smoothing techniques, including data-dependent smoothing.
result Optimal high-probability risk bounds for various estimators, including a new data-dependent smoothing method.
Unified high-probability regret bounds for online convex optimisation with randomised gradient estimators.
problem Online convex optimisation with randomised gradient estimators for ℓ q \ell_q ℓ q -Lipschitz losses. method FTRL with randomised two-point finite-difference gradient estimators based on cone-measure sampling from ℓ r \ell_r ℓ r -spheres. result Unified high-probability regret bounds for all p , q , r ∈ [ 1 , ∞ ] p,q,r \in [1,\infty] p , q , r ∈ [ 1 , ∞ ] . Method solves complex optimization problems with high probability bounds.
problem Nonlinear equality constrained stochastic optimization problems.
method Step-search sequential quadratic programming method.
result High-probability bound on iteration complexity for first-order stationarity.
We derive concentration inequalities for differentially private median and mean estimators building on the "Propose, Test, Release" (PTR) mechanism introduced by Dwork and Lei (2009). We introduce a new general version of the PTR mechanism that allows us to derive high probability error bounds for differentially privat…
High-probability bound for distributed stochastic approximation tracking error.
problem Analyzing the convergence of distributed stochastic approximation schemes.
method Analysis using ODE approach to stochastic approximation.
result High probability bound for tracking error between iterates and limiting differential equation.
Wavelet-based online learning adapts to noisy Besov spaces with high probability.
problem Minimizing integrated squared error in Besov spaces with noisy observations.
method Adaptive wavelet-based online learning algorithm that dynamically adjusts to gradient noise.
result Achieves minimax-optimal integrated squared error with high probability.
New bounds show faster convergence for learning algorithms.
problem Improving risk bounds for learning algorithms.
method Using algorithmic stability and common assumptions like Polyak-Lojasiewicz condition, smoothness, and Lipschitz continuity.
result Achieves convergence rate of O ( log 2 ( n ) / n 2 ) O(\log^2(n)/n^2) O ( log 2 ( n ) / n 2 ) with high probability. RS-NSGD improves SGD convergence for heavy-tailed noise.
problem Nonconvex optimization with heavy-tailed noise.
method Integrates direction normalization into subspace updates.
result Achieves better oracle complexity than full-dimensional normalized SGD.
New bounds improve generalization in machine learning with high probability.
problem Erratic behavior of KL divergence limits practical applications.
method Replaced KL divergence with Wasserstein distance for better bounds.
result Proved high probability generalization bounds for i.i.d. and non-i.i.d. data.
This work addresses the problem of regret minimization in non-stochastic multi-armed bandit problems, focusing on performance guarantees that hold with high probability. Such results are rather scarce in the literature since proving them requires a large deal of technical effort and significant modifications to the sta…
The paper analyzes Adam and SGD in nonstationary optimization, revealing tradeoffs between noise and drift.
problem Analyzing Adam and SGD in nonstationary optimization problems.
method Theoretical analysis of Adam and SGD under non-stationary stochastic objectives, separating two regimes.
result Characterizes the tradeoff between noise and drift in Adam and SGD, revealing when adaptive step-sizing is beneficial or harmful.
Paper develops a TR-SSQP method for noisy optimization with heavy-tailed noise.
problem Optimization problems with stochastic objectives and heavy-tailed noise.
method Trust-Region Stochastic Sequential Quadratic Programming (TR-SSQP) method.
result Achieves high-probability first-order and second-order stationarity bounds for heavy-tailed noise.
The paper improves PAC-Bayes bounds for losses with finite moments.
problem Bounding generalization for losses with heavy tails and finite moments.
method Truncation method and PAC-Bayes bounds for unbounded losses with heavy tails and bounded variance.
result Bounds interpolate between slow and fast rates depending on the moment.
Develops high-probability minimax quantile bounds for statistical problems.
problem Statistical procedures often lose information about tail behavior when reduced to expectations.
method Introduces minimax quantiles, develops high-probability variants of minimax methods, and converts risk lower bounds to quantile lower bounds.
result Obtains high-probability minimax quantile lower bounds for various statistical problems.
Paper develops PAC-Bayes bounds for unknown linear systems.
problem Learning controllers for unknown stochastic linear discrete-time systems.
method PAC-Bayes framework for data-dependent high probability bounds.
result Proposes efficient learning algorithms with theoretical guarantees.
Study contextual bandits with stage-wise constraints, proving regret bounds and extending results.
problem Contextual bandits with stage-wise constraints in high probability and expectation settings.
method Upper-confidence bound algorithms for linear and non-linear reward/cost functions, extending to multiple constraints.
result Regret bounds for various settings, including non-linear reward/cost functions.
Paper develops bounds for stochastic approximation with averaging.
problem Establish high-probability bounds for averaged stochastic approximation.
method Develops a general framework for non-asymptotic concentration bounds.
result Derives sharp bounds for averaged iterates and tightens existing results.
Improved SGD bounds for machine learning models with Markovian noise.
problem Uniform high-probability bounds for SGD under PL condition with Markovian noise.
method Combining Poisson equation for Markovian noise and probabilistic induction for almost-sure bounds.
result Matching 1 / k 1/k 1/ k decay rate for expected suboptimality. New algorithms achieve high-probability parameter-free regret in online convex optimization with heavy-tailed data.
problem Achieving high-probability parameter-free regret in online convex optimization with heavy-tailed data.
method Developed new regularization techniques to handle exponentially large iterates and heavy-tailed subgradients.
result Achieved regret bound of O ( ∥ u ∥ T 1 / p log ( 1 / δ ) ) O(\| \mathbf{u} \| T^{1/\mathfrak{p}} \log (1/δ)) O ( ∥ u ∥ T 1/ p log ( 1/ δ )) with high probability for subgradients with bounded p t h p^{th} p t h moments. Consider the problem of minimizing functions that are Lipschitz and strongly convex, but not necessarily differentiable. We prove that after T T T steps of stochastic gradient descent, the error of the final iterate is O ( log ( T ) / T ) O(\log(T)/T) O ( log ( T ) / T ) with high probability. We also construct a function from this class for which the error …
The paper provides high-probability bounds on false discovery proportions in conformal inference.
problem Existing methods fail to provide high-probability bounds on the realized false discovery proportion.
method Constructing a high-probability envelope for the empirical distribution function of null conformal p-values by sampling from their joint distribution.
result Establishes finite-sample, distribution-free upper bounds on the FDP that hold simultaneously over all possible rejection thresholds.
The paper offers precise bounds for averaged LSA iterates in linear systems.
problem Computing approximate solutions of linear systems with noisy observations.
method Finite-time analysis of LSA algorithms with Polyak-Ruppert averaging.
result Sharp high-probability bounds for averaged LSA iterates.
The Gibbs algorithm's generalization error is bounded, improving with prior volume in low temperatures.
problem Bounding the generalization error of the Gibbs algorithm in low temperature regimes.
method Analyzes the Gibbs algorithm's performance, extending known high-temperature bounds to low-temperature scenarios.
result With high probability, the generalization error decreases with the total prior volume of similar hypotheses.
New algorithm finds critical points in non-convex optimization with heavy-tailed gradients.
problem Non-convex stochastic optimization with heavy-tailed gradient estimates.
method Gradient clipping, momentum, and normalized gradient descent.
result High-probability convergence to critical points with best-known rates.
Algorithmic stability is a classical approach to understanding and analysis of the generalization error of learning algorithms. A notable weakness of most stability-based generalization bounds is that they hold only in expectation. Generalization with high probability has been established in a landmark paper of Bousque…
Study large deviations rates for SGD with strongly convex functions.
problem High probability metrics with SGD.
method Large deviations theory, generic gradient noise, strongly convex functions.
result Upper large deviations bound for SGD with strongly convex functions.
New algorithm FLUTE achieves uniform-PAC convergence in RL with linear approx.
problem RL with linear function approximation lacks uniform-PAC guarantees.
method FLUTE algorithm with minimax value function estimator and multi-level partition scheme.
result Uniform-PAC convergence to optimal policy with high probability.
Paper proposes a matrix optimization model for reliable Euclidean embedding from noisy data.
problem Challenges in Euclidean embedding from noisy observations containing outliers.
method Matrix optimization based embedding model to detect and remove outliers.
result The model provides high accuracy estimators and successfully identifies outliers.
New bounds quantify estimation error in kernel-based system identification with unknown hyperparameters.
problem Inaccurate error bounds for kernel-based system identification with unknown hyperparameters.
method Construct a high-probability set for true hyperparameters from marginal likelihood, then find worst-case posterior covariance.
result Proposed bounds contain true model with high probability and verified in simulations.
New stability framework relaxes boundedness assumptions for generalization bounds.
problem Overly restrictive assumptions for modern learning settings with heavy-tailed or unbounded losses.
method Develops a stability-based framework requiring only finite L p L_p L p moment conditions. result Sharp generalization bounds derived for various learning paradigms.
ES reduces high-probability regret in stochastic linear bandits.
problem High-probability regret in stochastic linear bandits.
method Linear ensemble sampling with standard Gaussian perturbations, analyzing m = Θ ( d log n ) m=Θ(d\log n) m = Θ ( d log n ) ensemble size. result ES achieves i l d e O ( d 3 / 2 n ) ilde O(d^{3/2}\sqrt n) i l d e O ( d 3/2 n ) high-probability regret, closing the gap to Thompson sampling. New bounds using samplewise evaluated CMI for deep neural networks.
problem Improving generalization bounds for deep neural networks.
method Introduced a new family of information-theoretic generalization bounds using samplewise evaluated conditional mutual information (CMI).
result The new bounds can be tighter than previous ones for deep neural networks.
New method minimizes regret in AMDP with high probability.
problem Pessimistic regret analysis in AMDP.
method Developed a new policy search method achieving optimistic regret.
result Achieved sublinear optimistic regret with high probability.
Improved SGD with AdaGrad stepsizes adapts to unknown parameters and unbounded gradients.
problem Adaptive optimization with unknown parameters and unbounded gradients.
method Stochastic Gradient Descent with AdaGrad stepsizes, without assuming problem parameters or strong global Lipschitz conditions.
result Sharp rates of convergence in both low-noise and high-noise regimes, supporting an affine variance noise model.
Deriving generalization bounds for stable algorithms is a classical question in learning theory taking its roots in the early works by Vapnik and Chervonenkis (1974) and Rogers and Wagner (1978). In a series of recent breakthrough papers by Feldman and Vondrak (2018, 2019), it was shown that the best known high probabi…
The small-ball method was introduced as a way of obtaining a high probability, isomorphic lower bound on the quadratic empirical process, under weak assumptions on the indexing class. The key assumption was that class members satisfy a uniform small-ball estimate: that P r ( ∣ f ∣ ≥ κ ∥ f ∥ L 2 ) ≥ δ Pr(|f| \geq κ\|f\|_{L_2}) \geq δ P r ( ∣ f ∣ ≥ κ ∥ f ∥ L 2 ) ≥ δ for given const…
In this paper, the problem of maximizing a black-box function f : X → R f:\mathcal{X} \to \mathbb{R} f : X → R is studied in the Bayesian framework with a Gaussian Process (GP) prior. In particular, a new algorithm for this problem is proposed, and high probability bounds on its simple and cumulative regret are established. The query po…
Improved online Q-learning for MDPs with concentration bounds.
problem Online Q-learning in infinite-horizon discounted MDPs with sublinear regret for large gaps.
method Smoothed ε n ε_n ε n -Greedy exploration scheme combining ε n ε_n ε n -greedy and Boltzmann exploration, analyzed using concentration bounds for contractive Markovian stochastic approximation. result Near- i l d e O ( N 9 / 10 ) ilde{O}(N^{9/10}) i l d e O ( N 9/10 ) regret bound for Smoothed ε n ε_n ε n -Greedy exploration scheme. TD(0) with Polyak-Ruppert averaging achieves robust and fast convergence rates
problem TD(0) learning under Markovian sampling
method Polyak-Ruppert averaging with a single stepsize
result Simultaneous high-probability convergence guarantees for TD(0) iterates and PR average
New bounds for heavy-tailed SDEs without info-theory terms.
problem Understanding generalization of heavy-tailed stochastic optimization.
method Fractional Fokker-Planck equation to estimate entropy flows.
result High-probability bounds with better dimension dependence.
New algorithms improve stopping time for best arm identification.
problem Efficiently identifying the best alternative in experiments.
method Proposed algorithms with exponential-tailed stopping time.
result Proved that some algorithms never stop, leading to new methods.
New algorithm accelerates optimization in non-convex problems with heavy-tailed noise.
problem Optimizing non-convex functions with heavy-tailed noise.
method Proposes a variance-reduced accelerated algorithm for optimization problems in the form of F ( x ) = E Ξ ∼ D [ f ( x , Ξ ) ] F(x) = \mathbb{E}_{Ξ\sim\mathcal{D}}[f(x,Ξ)] F ( x ) = E Ξ ∼ D [ f ( x , Ξ )] . result Achieves a high-probability convergence rate of O ( log ( T / δ ) T 1 − p 2 p − 1 ) O(\log(T/δ)T^{\frac{1-p}{2p-1}}) O ( log ( T / δ ) T 2 p − 1 1 − p ) , faster than the lower bound Ω ( T 1 − p 3 p − 2 ) Ω(T^{\frac{1-p}{3p-2}}) Ω ( T 3 p − 2 1 − p ) .