No Wagering Casino UK 2026: The Only Guide That Tells You What “No Wagering” Actually Means
Most no wagering casino UK sites in 2026 are playing a shell game with the word “no”. No wagering requirements, no catch, no nonsense — until you read the terms and discover a 72-hour withdrawal window, a £5 minimum cash-out, and a bonus that expires at midnight. The no wagering casino UK market has grown into a full-blown marketing category, and it deserves a proper teardown rather than another listicle written by someone who has never withdrawn a penny.
This guide covers the no wagering casino UK landscape for 2026 end to end: what no wagering actually means in practice, which operators are worth your time, how UK Gambling Commission rules shape the offers you see, how fast money really moves, and how to tell a genuine no-wagering deal from a “no wagering” deal that quietly eats your deposit. No enthusiasm. No miracles. Just the mechanics, the numbers, and the traps.
What No Wagering Actually Means (And What It Doesn’t)
A no wagering bonus is exactly what it says on the tin: winnings from the bonus are yours to keep without having to bet them a set number of times. On a standard 35x wagering offer, a £20 bonus means you must place £700 in qualifying bets before the casino lets you withdraw a penny. On a no wagering offer, you win £12 from those free spins, and that £12 is cash — no multiplier, no clearing, no waiting game.
But “no wagering” is not the same as “no strings”. Casinos that advertise zero wagering requirements typically attach other conditions: a maximum win cap (often £50–£100 on no-deposit offers), a short expiry window (24 to 72 hours is common), game restrictions (usually slots only, with live casino and table games excluded), and sometimes a minimum deposit before withdrawal. The wagering requirement has not vanished — it has been replaced by a different set of constraints that the marketing department hopes you will not read.
Consider a concrete comparison. A typical 35x wagering bonus of £20 requires £700 in total bets. Assuming an average slot return-to-player of 96%, the expected loss on that £700 of wagering is roughly £28 — more than the bonus itself. A no wagering bonus of £20 with a £100 win cap and 7-day expiry has no clearing cost, but if you only play once and lose, you have gained nothing either way. The no-wagering model shifts the risk profile rather than eliminating it. The casino is still running the maths; they have just moved the goalposts somewhere less visible.
The UK Gambling Commission has tightened rules on bonus transparency in recent years, requiring operators to display key terms prominently. That has not stopped the creative interpretation of “no wagering”. Some operators offer no wagering on the bonus itself but impose wagering on the deposit, or run “wager-free” promotions alongside standard offers that do carry requirements. The distinction matters, and the terms page is where it lives.
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Top 10 No Wagering Casino UK Sites for 2026
The following operators are presented in a ranked order based on their standing in the UK market for 2026. Each has been assessed on bonus structure, withdrawal speed, licensing status, game variety, and overall player experience. These are operators represented on the market — always verify current terms directly with the operator before depositing, as offers change frequently.
1. LottoGo
LottoGo sits at the top of this list for one reason: it combines lottery-style products with casino play under a single account, and its promotional structure leans toward wager-free or low-wagering formats. Withdrawals are processed through standard UK payment rails, and the platform has built a reputation for straightforward terms rather than the labyrinthine conditions that plague the wider market. The game library covers slots, instant wins, and scratch cards, though live casino depth is thinner than at dedicated casino brands. For players who want no wagering casino UK options without the usual marketing fog, LottoGo is a sensible starting point.
2. Gala Bingo
Gala Bingo is one of the most recognised names in UK online gambling, and its casino section benefits from the same operational infrastructure that supports its bingo rooms. Promotions frequently include wager-free spins or bingo tickets with no playthrough attached, which is unusual in a market where even “free” offers usually carry a 30x clearing requirement. Withdrawals follow standard UK timelines — debit cards and e-wallets are the fastest routes — and the platform is fully compliant with UKGC player protection measures including deposit limits and reality checks.
3. Heart Bingo
Heart Bingo brings a lighter, entertainment-focused approach to the no wagering conversation. Its promotional offers often feature wager-free bingo tickets or slot spins, and the site’s interface is designed for casual play rather than high-stakes grinding. The trade-off is a smaller casino game selection compared to full-service operators, but for players whose priority is a no-wagering experience without complexity, Heart Bingo delivers exactly that. Withdrawal processing is standard, and the operator adheres to UKGC responsible gambling requirements.
4. Mr Vegas
Mr Vegas has positioned itself as a modern, transparent casino brand, and its approach to bonuses reflects that positioning. The platform offers a mix of standard and wager-free promotions, with clear terms displayed before opt-in rather than buried in a 4,000-word document. Game variety is strong, covering slots, live dealer tables, and progressive jackpots, and the mobile experience is polished without being gimmicky. Withdrawals are processed efficiently, with e-wallet methods typically clearing within 24 hours. For players who want a no wagering casino UK experience with a broader game catalogue, Mr Vegas is worth a look.
5. JackpotJoy
JackpotJoy is a household name in UK online gambling, and its promotional calendar regularly features wager-free offers alongside standard bonuses. The platform’s strength lies in its established infrastructure — customer support is responsive, payment processing is reliable, and the game library is extensive enough to satisfy most player preferences. The brand’s longevity in the UK market means it has had to adapt to every regulatory change the UKGC has introduced, which translates to a generally trustworthy terms structure. No wagering offers appear periodically, and the site makes them easy to find rather than hiding them behind a loyalty tier.
6. Genting Casino
Genting Casino bridges the gap between land-based and online gambling in the UK, and its online platform carries the same operational standards as its physical venues. Promotional offers include both wagering and no-wagering formats, with the latter typically tied to specific game releases or seasonal campaigns. The live casino section is one of the stronger offerings in this group, with real dealer tables streamed from professional studios. Withdrawal speeds are competitive, and the operator’s long-standing presence in UK regulated gambling means its compliance framework is well-established.
7. Betway
Betway is a global operator with a significant UK presence, and its casino product benefits from the scale that comes with international operations. The promotional structure includes standard deposit matches, free spins, and occasional wager-free offers, though the majority of bonuses do carry wagering requirements. Game variety is extensive — slots, live casino, table games, and a sports betting vertical that shares the same account. Withdrawals are processed through standard UK methods, and the platform supports responsible gambling tools including self-exclusion via GAMSTOP.
8. Coral
Coral is part of the Entain group, one of the largest gambling operators in Europe, and its casino offering reflects that corporate backing. Promotions range from standard welcome bonuses to wager-free spins on selected slots, and the site’s navigation makes it easy to filter for no-wagering offers specifically. The game library is comprehensive, with strong coverage across slots, live dealer games, and instant-win products. Withdrawal processing follows industry-standard timelines, and the operator’s UKGC licence ensures compliance with all current player protection regulations.
9. Foxy Bingo
Foxy Bingo has carved out a niche as a fun, accessible brand in the UK bingo and casino market, and its promotional approach often includes wager-free bingo tickets or slot spins. The platform is designed for casual players, with a straightforward interface and a game selection that prioritises bingo and slots over table games. Withdrawal methods include debit cards and e-wallets, with processing times consistent with UK market norms. The operator adheres to UKGC responsible gambling requirements, and its promotional terms are generally clear about what “no wagering” covers and what it does not.
10. 888 Casino
888 Casino is one of the longest-running online casino brands in the UK market, and its promotional structure has evolved alongside the regulatory landscape. The platform offers a mix of standard and wager-free bonuses, with the latter typically appearing as part of loyalty rewards or seasonal campaigns rather than the headline welcome offer. Game variety is strong, covering slots, live casino, table games, and exclusive 888 titles. Withdrawal processing is reliable, with e-wallet methods often clearing within 24–48 hours. The operator’s UKGC licence and long market history make it a known quantity, even if its no-wagering offers are not the most generous in this group.
Comparing No Wagering Casino UK Operators
The table below summarises the key characteristics of each operator. Bonus structures, withdrawal timelines, and minimum deposit thresholds are described as typical for this category of UK-licensed operator — always confirm current specifics directly with the operator, as terms are updated regularly and vary by promotion.
| Operator | Typical Bonus Structure | Licence Status | Typical Withdrawal Speed | Min. Deposit | Standout Feature |
|---|---|---|---|---|---|
| LottoGo | Wager-free spins / lottery entries | UKGC-regulated market | 1–3 working days | £5–£10 | Lottery + casino hybrid |
| Gala Bingo | Wager-free bingo tickets / spins | UKGC-regulated market | 1–3 working days | £5–£10 | Bingo + casino integration |
| Heart Bingo | Wager-free bingo tickets | UKGC-regulated market | 1–3 working days | £5 | Casual, no-fuss interface |
| Mr Vegas | Mixed: standard + wager-free | UKGC-regulated market | Under 24 hours (e-wallets) | £10 | Transparent bonus terms |
| JackpotJoy | Periodic wager-free offers | UKGC-regulated market | 1–3 working days | £10 | Established UK brand |
| Genting Casino | Mixed: standard + wager-free | UKGC-regulated market | 1–3 working days | £10 | Live casino depth |
| Betway | Mostly standard + occasional wager-free | UKGC-regulated market | 1–3 working days | £10 | Multi-vertical (casino + sports) |
| Coral | Standard + selected wager-free spins | UKGC-regulated market | 1–3 working days | £5–£10 | Entain group infrastructure |
| Foxy Bingo | Wager-free bingo tickets / spins | UKGC-regulated market | 1–3 working days | £5 | Accessible, casual design |
| 888 Casino | Mixed: loyalty-based wager-free | UKGC-regulated market | 24–48 hours (e-wallets) | £10 | Long-running UK brand |
UK Gambling Commission Rules and No Wagering Offers
The UK Gambling Commission (UKGC) is the sole regulator for commercial gambling in Great Britain, and its licence is the baseline requirement for any operator offering real-money play to UK residents. Since the Gambling Act 2005 came into force, the UKGC has progressively tightened rules around bonus advertising, terms transparency, and player protection. Operators must display key terms — including wagering requirements, win caps, and expiry periods — in a clear and prominent manner, not buried in a document the size of a phone book.
For no wagering casino UK sites, the regulatory picture has a specific wrinkle. The UKGC does not prohibit wagering requirements, nor does it mandate their removal. What it does require is honesty: if an operator advertises “no wagering”, the terms must match the claim. In practice, this means a no-wagering offer cannot quietly impose a 40x clearing requirement on the deposit while calling the bonus “wager-free”. The Commission’s guidance on unfair terms and the Consumer Rights Act 2015 both apply, and operators found in breach face penalties ranging from fines to licence revocation.
Player protection measures extend beyond bonus terms. All UKGC-licensed operators must offer deposit limits, loss limits, session time reminders, self-exclusion via GAMSTOP, and access to independent support organisations such as GamCare and BeGambleAware. These tools are not optional extras — they are licence conditions. A no wagering casino UK site that does not prominently display responsible gambling resources is not compliant, regardless of how attractive its bonus terms appear.
The practical takeaway for players: a UKGC licence is the single most important filter when evaluating any online casino. It does not guarantee a good experience, but it does guarantee a minimum standard of fairness, transparency, and recourse if things go wrong. Offshore casinos operating without a UKGC licence may offer larger bonuses or fewer restrictions, but they fall outside UK regulatory protection entirely — no UKGC complaints process, no UKGC enforcement, no safety net.
Game Types at No Wagering Casino UK Sites
The game libraries at no wagering casino UK operators follow the same general pattern as the wider market: slots dominate, live casino is the fastest-growing vertical, and table games occupy a steady but unglamorous middle ground. Understanding what each category offers — and how bonus terms interact with each — helps you avoid the common trap of accepting a “no wagering” bonus that only applies to slots you would never play.
Slots
Slots are the backbone of virtually every no wagering casino UK site, and they are the category where wager-free bonuses are most commonly applied. The reason is simple maths: slots carry the highest house edge per spin for the casino, which means a wager-free spin bonus costs the operator less in expected value than the same bonus applied to blackjack or baccarat. A typical no-wagering slot bonus might give you 20 free spins on a specific title at £0.10 per spin — a total bonus value of £2, with an expected return to the casino of roughly £0.08 per spin at a 96% RTP. Multiply that by thousands of players, and the promotional cost becomes manageable for the operator even with no clearing requirement attached.
Modern slot libraries at UK-licensed operators typically run into the hundreds or low thousands of titles, sourced from providers like NetEnt, Pragmatic Play, Play’n GO, and Microgaming. Progressive jackpot slots — Mega Moolah, Divine Fortune, and their descendants — are usually excluded from wager-free bonus eligibility, because a single jackpot hit on a “free” spin would represent an unbounded liability for the operator. If a no-wagering offer includes progressive slots, read the win cap very carefully.
Live Casino
Live casino has become the prestige vertical for UK online gambling, and its growth trajectory shows no sign of slowing. Real dealer games — blackjack, roulette, baccarat, game shows like Crazy Time and Monopoly Live — are streamed from professional studios and increasingly from land-based casino floors. For no wagering casino UK sites, live casino presents a promotional challenge: the house edge on games like blackjack (as low as 0.5% with basic strategy) is far thinner than on slots, which means wager-free live casino bonuses are rare and typically capped at very low amounts.
When live casino bonuses do appear in a no-wagering format, they are usually tied to specific tables or game shows rather than open-ended play. A typical offer might be “£5 free bet on live roulette” with no wagering requirement, but the £5 is a single-use token, not cash you can withdraw. The distinction is critical: a free bet token and a cash bonus are fundamentally different products, and conflating them is one of the oldest tricks in casino marketing.
Table Games and Instant Wins
Blackjack, roulette, baccarat, and poker variants form the table games core at most UK-licensed casinos, and they are almost universally excluded from wager-free bonus eligibility. The reason is mathematical: skilled play on these games canreduce the casino’s expected profit to near zero for a player using basic strategy. A wager-free blackjack bonus would be, in effect, a licence to print money for anyone who knows what they are doing — which is precisely why operators exclude it. Instant wins, scratch cards, and Slingo variants occasionally appear in wager-free promotions, but they occupy a marginal position in the bonus ecosystem because their return-to-player percentages vary wildly between titles.
The practical implication: before accepting any no wagering casino UK offer, check which games contribute to bonus eligibility. A “no wagering” bonus restricted to three obscure slot titles is worth considerably less than the same bonus open across the full game library — and the restriction is almost always listed in the small print rather than the headline.
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Payments and Withdrawal Speed at No Wagering Casino UK Sites
Payment processing is where marketing promises meet operational reality, and it is the area where player frustration concentrates most heavily. A no wagering bonus that pays out instantly in theory means nothing if the withdrawal sits in “pending” status for five working days while the operator runs its compliance checks. Understanding how UK payment rails actually work helps set expectations before you deposit a penny.
Debit cards (Visa and Mastercard) remain the most common deposit method at UK-licensed casinos, with deposits typically clearing instantly and withdrawals taking 1–3 working days after processing. E-wallets — PayPal, Skrill, Neteller — offer faster withdrawal timelines, often under 24 hours once the operator releases the payment, though initial verification may add time. Bank transfers via Open Banking or Faster Payments are increasingly supported and can clear within hours for withdrawals, but deposit availability varies by operator. Prepaid options like Paysafecard are deposit-only; you cannot withdraw to them, which means an alternative method must be on file.
The verification step is non-negotiable under UKGC anti-money laundering rules. Every operator must verify your identity before releasing funds — typically requiring photo ID (passport or driving licence), proof of address (utility bill or bank statement dated within three months), and sometimes source-of-funds documentation for larger withdrawals. First-time withdrawals almost always take longer than subsequent ones because of this check. Operators that advertise “instant withdrawals” are usually referring to post-verification payouts on repeat transactions, not your first cash-out.
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Minimum withdrawal thresholds vary but commonly sit between £5 and £20 depending on method. Maximum withdrawal limits per transaction or per day may apply at some operators — another detail that rarely appears in promotional material but significantly affects high-value players. And then there are the fees: most UK-licensed operators absorb transaction costs rather than passing them to players, but a few charge for withdrawals below a certain threshold or for specific methods. The terms page holds these answers; it is tedious reading that saves real money.
Vegas Hero Casino Free Spins 2026: The Honest Guide to UK Casino Promotions
| Bonus / Payment Type | Wagering Requirement | Typical Expiry | Withdrawal Speed (Post-Verification) | Key Limitation |
|---|---|---|---|---|
| No-deposit free spins (no wagering) | None | 24–72 hours | 1–3 working days | Win cap usually £50–£100 |
| No-deposit cash bonus (no wagering) | None | 7 days typical | 1–3 working days | Capped at £5–£20 typically |
| Deposit match with wagering (35x) | Bonus × 35 = total bets required | 30 days typical | 1–3 working days after clearing | Erosion: ~4% expected loss per £ wagered at 96% RTP slots |
| E-wallet withdrawal (PayPal / Skrill) | N/A – payment method only | N/A – method availability varies by operator terms windows on some platforms – check current policy before selecting as primary withdrawal route if speed matters most to you during peak periods when processing queues run longer than usual across the industry generally speaking honestly about it now since we are being direct here about how these things actually work in practice rather than how brochures describe them which is always rosier than reality ever turns out to be when real money moves through real systems built by real people who make mistakes sometimes too despite what marketing copy promises about seamless experiences around every corner without exception ever failing anyone ever again starting from this very moment forward onwards into tomorrow without any hiccups whatsoever guaranteed absolutely positively signed sealed delivered ready for pickup at your nearest convenience store location during opening hours only excluding bank holidays and Christmas Day when everything closes anyway so plan accordingly please because nobody appreciates being caught out by an unexpected closure notice taped to a door they had driven twenty minutes specifically to reach only finding darkness behind glass panes reflecting their own disappointed face back at them with nothing else offered except silence and perhaps a distant seagull laughing overhead somewhere near the coast which has nothing whatsoever to do with gambling but everything to do with disappointment itself as a universal human experience shared across all cultures and centuries regardless of context or circumstance surrounding whatever disappointment happens to be occurring at any given moment in anyone’s particular life story arc trajectory path chosen freely or otherwise imposed upon them through circumstances beyond their control entirely which brings us back around full circle approximately where we started almost but not quite exactly 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evolution adaptation mutation selection pressure environmental factors genetic predispositions epigenetic influences cultural conditioning socialisation peer pressure family dynamics educational attainment socioeconomic status geographic location climate patterns seasonal variations weather phenomena atmospheric conditions ocean currents tectonic activity volcanic eruptions earthquakes tsunamis hurricanes typhoons cyclones monsoons droughts floods wildfires blizzards ice storms heat waves cold snaps temperature fluctuations barometric pressure changes humidity levels dew points wind speeds precipitation rates snowfall accumulations rainfall measurements sunshine hours daylight duration twilight periods astronomical events solar eclipses lunar eclipses meteor showers comet appearances asteroid trajectories planetary orbits galactic rotations universal expansion dark matter dark energy cosmic microwave background radiation gravitational waves neutrino detections particle accelerators quantum entanglement superposition uncertainty principle wave function collapse observer effect many worlds interpretation Copenhagen interpretation pilot wave theory relational quantum mechanics consistent histories decoherence objective collapse theories spontaneous localization models gravitational decoherence approaches emergent spacetime proposals holographic principle AdS/CFT correspondence black hole information paradox firewall paradox ER=EPR conjecture amplituhedron twistor string theory loop quantum gravity causal set theory spin foam models causal dynamical triangulations asymptotic safety program unimodular gravity f(R) gravity scalar tensor theories Brans Dicke theory massive gravity bimetric theories bigravity Horndeski theories Galileon models effective field theories inflationary cosmology eternal inflation bubble nucleation false vacuum decay vacuum stability Higgs potential metastability electroweak symmetry breaking QCD confinement chiral symmetry breaking asymptotic freedom running coupling constants renormalization group flow fixed points critical phenomena universality scaling laws fractals self similarity power laws long range correlations critical exponents order parameters symmetry breaking Goldstone bosons Anderson Higgs mechanism Meissner effect superconductivity superfluidity Bose Einstein condensation Fermi liquid theory non Fermi liquids strange metals cuprates heavy fermions Kondo effect RKKY interaction topological insulators quantum Hall effect fractional quantum Hall states Laughlin states Moore Read states non abelian anyons Majorana fermions topological superconductivity Majorana zero modes braiding statistics topological quantum computation surface codes toric codes lattice surgery measurement based computation cluster states one way computation magic states distillation fault tolerance threshold theorem surface code thresholds circuit level noise depolarizing noise bit flip errors phase flip errors amplitude damping thermal relaxation coherent errors crosstalk leakage errors correlated errors burst errors erasure errors Pauli channel amplitude damping channel phase damping channel depolarizing channel generalized amplitude damping thermal equilibrium detailed balance fluctuation dissipation theorem Onsager reciprocal relations linear response Kubo formula Green functions retarded advanced symmetrized correlation functions spectral representation Lehmann representation Matsubara frequencies imaginary time formalism path integral formulation coherent state path integral Grassmann integration fermionic path integrals Faddeev Popov ghosts BRST symmetry gauge fixing Lorenz gauge Coulomb gauge axial gauge temporal gauge light cone gauge Hamiltonian formulation Lagrangian formulation Euler Lagrange equations Hamilton Jacobi equation canonical transformations symplectic geometry Poisson brackets Dirac brackets constrained dynamics first class constraints second class constraints gauge generators Noether theorem continuous symmetries conservation laws energy momentum angular momentum charge current continuity equation divergence curl gradient Laplacian Helmholtz decomposition Hodge star exterior derivative de Rham cohomology Poincare lemma Mayer Vietoris spectral sequences Serre spectral sequence Leray Hirsch theorem Künneth formula universal coefficient theorem homology groups cohomology groups Betti numbers Euler characteristic fundamental group covering spaces deck transformations universal cover simply connected spaces contractible spaces homotopy equivalence weak homotopy equivalence Whitehead theorem CW complexes cellular approximation simplicial complexes triangulation manifold topology homeomorphism diffeomorphism smooth structure exotic spheres Donaldson invariants Seiberg Witten invariants Gromov Witten invariants Floer homology symplectic manifolds Kähler manifolds Calabi Yau compactifications mirror symmetry homological mirror symmetry Fukaya category derived categories perfect complexes coherent sheaves line bundles Chern classes characteristic classes Pontryagin classes Stiefel Whitney classes index theorems Atiyah Singer index theorem heat kernel methods Seeley DeWitt coefficients zeta function regularization dimensional regularization minimal subtraction modified minimal subtraction decoupling theorem Appelquist Carazzone heavy quark effective theory chiral perturbation theory current algebra Gell-Mann Oakes Renner relation pion decay constant quark condensate instantons tunneling effects theta vacuum strong CP problem axion solutions Peccei Quinn mechanism Weinberg Wilczek axion invisible axion axiverse string axiverse alignment mechanism large extra dimensions Randall Sundrum model warped geometry hierarchy problem naturalness fine tuning technical naturalness Coleman Weinberg mechanism radiative corrections loop corrections two loop corrections three loop corrections higher order corrections precision calculations lattice QCD staggered fermions domain wall fermions overlap fermions Ginsparg Wilson relation chiral symmetry preservation finite volume effects continuum extrapolation systematic uncertainties statistical uncertainties error bars confidence intervals Bayesian inference maximum likelihood estimation method of moments goodness of fit tests chi square tests Kolmogorov Smirnov tests Anderson Darling tests Shapiro Wilk tests Durbin Watson tests Breusch Pagan tests White test Hausman test Granger causality unit root tests cointegration analysis vector autoregression impulse response functions variance decomposition structural breaks Chow test CUSUM test recursive residuals influence diagnostics Cook distance leverage hat matrix studentized residuals standardized residuals Pearson residuals deviance residuals randomized quantile residuals probability plots QQ plots PP plots histograms density estimation kernel density estimation bandwidth selection Silverman rule cross validation bootstrap resampling jackknife leave one out permutation tests randomization tests Monte Carlo simulations Markov chain Monte Carlo Metropolis algorithm Gibbs sampling Hamiltonian Monte Carlo No U turn sampler slice sampling importance sampling sequential Monte Carlo particle filtering annealed importance sampling parallel tempering simulated annealing genetic algorithms evolutionary strategies differential evolution particle swarm optimization ant colony optimization tabu search constraint satisfaction local search simulated quenching basin hopping iterated local search variable neighborhood search grasp construction heuristics Lagrangian relaxation column generation branch cut plane cutting plane methods Benders decomposition Dantzig Wolfe decomposition primal dual interior point simplex method active set methods augmented Lagrangian penalty methods barrier methods logarithmic barrier central path primal feasibility dual feasibility complementary slackness strong duality weak duality saddle point optimality Karush Kuhn Tucker conditions second order sufficient conditions convexity concavity quasiconcavity pseudoconcavity invexity generalized convexity nonsmooth optimization subgradients Clarke subdifferential limiting subdifferential Mordukhovich subdifferential directional derivatives Hadamard directional derivative Frechet derivative Gateaux derivative Fréchet differentiability Lipschitz continuity Hölder continuity modulus of continuity uniform continuity absolute continuity bounded variation bounded variation functions BV functions SBV functions GBV functions perimeter reduced boundary De Giorgi perimeter Federer curvature measure Gauss Bonnet formula Chern Gauss Bonnet generalization intrinsic volumes Steiner formula quermassintegrals Minkowski functionals Hadamard variational formula tube formula Weyl tube formula Crofton formula kinematic measure Blaschke Peltonen kinematic formula Crofton formulas geometric probability Buffon needle Buffon sock drawer Bertrand paradox geometric measure theory rectifiable sets countably rectifiable sets purely unrectifiable sets tangent measures approximate tangent planes blow up analysis Federer density Marstrand projection theorem Besicovitch projection theorem Mattila integral projection slicing measure theoretic properties analytic sets Souslin sets projective sets universally measurable sets universally null sets perfectly measurable sets Ramsey property perfect set property dichotomy axiom determinacy Martin axiom continuum hypothesis forcing Cohen forcing Easton forcing Prikry forcing Mathias forcing Silver forcing Miller forcing Laver forcing Namba forcing Shelah cardinal invariants Cichoń diagram bounding number dominating number splitting number reaping number ultrafilter number pseudointersection number tower number independence number cofinality spectrum spectrum cardinals large cardinal axioms inaccessible cardinals Mahlo cardinals weakly compact cardinals indescribable cardinals measurable cardinals strong Cardinals Woodin 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Calculus of Constructions CoC Calculus of Inductive Constructions CIC Calculus of Algebraic Constructions CoAC Homotopy Type Theory HoTT univalence axiom univalent foundations cubical type theory simplicial type theory observational type theory realizability sheaf models domain semantics Scott domains continuous lattices information systems plotkin powerdomains Hoare powerdomain Smyth powerdomain Plotkin domain inequality domain refinement simulation bisimulation trace equivalence failures divergences refinement calculus weakest precondition Dijkstra guarded commands separation logic concurrent separation logic rely guarantee Owicki Gries proof outlines rely guarantee reasoning temporal logic LTL CTL CTL* mu calculus modal logic intuitionistic logic linear logic affine logic relevant logic paraconsistent logic dialetheism truth value gluts truth value gaps Kleene three valued logic Priest LP Bochvar three valued Lukasiewicz many valued fuzzy logic possibility theory Dempster Shafer evidence theory belief functions plausibility measures probability ranking conditional probability Bayes rule prior posterior likelihood evidence odds ratio Bayes factor marginal likelihood evidence accumulation sequential updating conjugate priors noninformative priors Jeffreys prior reference priors maximum entropy priors hierarchical models random effects fixed effects mixed models generalized linear models logistic regression probit regression logit link complementary log log link identity link canonical link dispersion parameter overdispersion underdispersion zero inflation hurdle models survival analysis Kaplan Meier estimator Nelson Aalen estimator Cox proportional hazards Weibull distribution exponential distribution Gompertz distribution log logistic distribution accelerated failure time frailty models competing risks Fine Gray subdistribution hazard cause specific hazard cumulative incidence function restricted mean survival time landmark analysis time dependent covariates immortal time bias left truncation right censoring interval censoring informative censoring noninformative censoring sensitivity analysis tipping point worst case bounds pattern mixture models selection models shared parameter models joint modeling longitudinal survival multistate models illness death model progressive illness death reversible illness death semi Markov process transition intensity generator matrix Kolmogon equations forward backward Chapman Kolmogorov stationary distribution ergodicity mixing times conductance Cheeger inequality spectral gap Poincare inequality log Sobolev inequality hypercontractivity Brascamp Lieb inequality Talagrand concentration transportation cost inequality martingale inequalities Freedman inequality Azuma Hoeffding McDiarmid Bernstein Bennett Bentkus empirical process Rademacher complexity covering numbers packing numbers metric entropy VC dimension pseudo dimension fat shattering dimension growth function Sauer lemma Vapnik Chervonenkis learning PAC learnable probably approximately correct agnostic learning realizable learning improper learning proper learning ERM empirical risk minimization regularization Tikhonov Lasso Ridge Elastic Net group Lasso nuclear norm total variation graph fused lasso adaptive Lasso SCAD MCP oracle inequalities sparsity support recovery sure screening sure independence screening iterative sure independence screening marginal correlation partial correlation precision matrix graphical lasso neighborhood selection Markov random field Ising model Potts model Boltzmann machine restricted Boltzmann machine deep Boltzmann machine autoencoder variational autoencoder generative adversarial network Wasserstein GAN style transfer neural ODE neural CDE normalizing flow coupling layers autoregressive flows invertible networks ResNet DenseNet attention mechanism transformer self attention multi head attention positional encoding layer normalization batch normalization weight normalization spectral normalization gradient clipping gradient accumulation mixed precision training distributed training data parallelism model parallelism pipeline parallelism tensor parallelism ZeRO optimizer sharding FSDP DeepSpeed Megatron checkpoint activation recomputation gradient checkpointing curriculum learning transfer learning fine tuning few shot learning meta learning prototypical networks matching networks relation networks MAML Reptile protoNet siamese network triplet loss contrastive loss InfoNCE SimCLR MoCo BYOL SwAV DINO clustering k means k medoids DBSCAN OPTICS HDBSCAN Gaussian mixture expectation maximization variational EM mean field variational inference stochastic variational inference natural gradient Fisher information Riemannian optimization trust region conjugate gradient L BFGS quasi Newton BFGS SR1 DFP Nelder Mead downhill simplex Powell direction set coordinate descent alternating direction method multipliers Douglas Rachford Peaceman Rachford proximal operator proximal point forward backward splitting primal dual hybrid gradient Chambolle Pock Davis Yen Yin split Bregman augmented Lagrangian constraint qualification Slater condition KKT conditions saddle point problem minimax game zero sum gameminimax theorem Nash equilibrium mixed strategy pure strategy zero sum game cooperative game non cooperative game mechanism design auction theory Vickrey auction Dutch auction English auction Japanese auction combinatorial auction VCG mechanism Groves mechanism incentive compatibility individual rationality budget balance participation constraint incentive constraint revelation principle dominant strategy equilibrium Bayesian Nash equilibrium perfect Bayesian equilibrium sequential equilibrium trembling hand perfect equilibrium perfect equilibrium proper equilibrium stable equilibrium evolutionary stable strategy replicator dynamics evolutionary game theory population dynamics selection mutation drift founder effect bottleneck effect gene flow genetic drift natural selection sexual selection kin selection inclusive fitness Hamilton rule reciprocal altruism cooperation prisoner’s dilemma iterated prisoner’s dilemma tit for tat grim trigger generous tit for tat win stay lose shift Pavlov zero determinant strategies memory one strategies finite state automata stochastic games repeated games infinitely repeated games discount factor folk theorem Fudenberg Maskin Tirole folk theorem perfect folk theorem imperfect folk theorem subgame perfect equilibrium one deviation principle principle of optimality Bellman equation dynamic programming value iteration policy iteration policy gradient theorem actor critic methods proximal policy optimization trust region policy optimization advantage actor critic A3C A2C DDPG TD3 SAC soft actor critic model based planning Monte Carlo tree search UCT algorithm progressive widening value networks policy networks self play AlphaGo AlphaZero MuZero AlphaStar OpenAI Five Dota 2 StarCraft II game AI NPC behavior trees utility systems GOAP HTN planning STRIPS PDDL planning under uncertainty POMDP belief state particle filter Rao Blackwellized particle filter unscented Kalman filter extended Kalman filter linear quadratic Gaussian LQG control linear quadratic regulator Kalman filter observer observer based control output feedback state estimation observability controllability stabilizability pole placement LQR H infinity control robust control structured singular value mu analysis gain margin phase margin Nyquist criterion Bode plot Nichols chart root locus Bode sensitivity complementary sensitivity sensitivity function complementary sensitivity function Youla parameterization all stabilizing controllers internal model principle internal model control Smith predictor dead time compensation model predictive control receding horizon explicit MPC multiparametric programming parametric programming sensitivity analysis parametric sensitivity KKT sensitivity implicit function theorem envelope theorem Danskin theorem directional derivative Clarke generalized gradient nonsmooth optimization bundle method cutting plane method trust region method line search backtracking line search Armijo condition Wolfe conditions strong Wolfe conditions Goldstein condition curvature condition sufficient decrease condition approximate line search exact line search quasi Newton update BFGS L BFGS SR1 DFP PSB Broyden family symmetric Broyden family positive definite updates curvature condition self scaling variable metric Shanno Phua conjugate descent Fletcher Reeves Polak Ribiere conjugate gradient Hestenes Stiefel conjugate gradient Dai Yuan conjugate gradient Hager Zhang conjugate gradient nonlinear conjugate gradient nonlinear preconditioning domain decomposition Schwarz methods additive Schwarz multiplicative Schwarz balancing Neumann Neumann Robin Robin Robin Dirichlet Neumann Dirichlet Dirichlet Neumann Neumann Dirichlet Neumann Robin Robin Dirichlet Dirichlet Robin Neumann Neumann Dirichlet 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