BetBlast Casino Free Spins 2026: What UK Players Actually Need to Know
BetBlast Casino free spins in 2026 sit at the intersection of aggressive affiliate marketing and genuine player demand. The keyword pulls thousands of monthly searches from UK punters who want one thing: a clear picture of whether the free spin offers floating around are worth the click, or just another trap dressed up as generosity. This guide strips the fluff and walks through how free spins work across the UK online casino market, which operators are worth your time, how to spot a rigged deal, and what the regulatory landscape actually looks like heading into 2026.
The honest starting point: nobody hands out money. A “free” spin comes with wagering requirements, maximum withdrawal caps, game restrictions, and expiry windows that most players never read. Treat every spin offer as a maths problem rather than a gift, and you will already be ahead of roughly nine out of ten people clicking these banners.
What BetBlast Casino Free Spins Actually Mean in Practice
Free spins promotions in the UK market come in three broad shapes: no-deposit spins awarded simply for registering an account, deposit-triggered spins bundled with a welcome match bonus, and loyalty spins handed out to existing players as part of ongoing reward schemes. Each type carries different strings attached, and conflating them is where most new players lose money they did not know they were risking.
A no-deposit free spin offer sounds like free money on paper. Register, receive 10 or 20 spins on a nominated slot, keep whatever you win — up to a cap. In reality, those winnings almost always carry wagering requirements between 30x and 65x before you can withdraw a penny. Do the arithmetic: win £15 from twenty spins on a slot paying an average return-to-player of 96%, face a 40x wagering requirement, and you need to cycle through £600 in total bets before your £15 becomes withdrawable cash. At typical slot stakes of £0.20 per spin that is 3,000 individual spins — roughly two hours of continuous play if you are spinning at one spin every three seconds.
Deposit-triggered free spins work differently. You put in real money — usually £10 or more — and receive a batch of spins alongside (or instead of) a percentage match on your deposit. The upside is higher volume: operators often throw 50 to 150 spins at new depositors compared to the stingy 10-20 from no-deposit offers. The downside is obvious once you think about it for four seconds: you have committed your own cash before seeing whether the platform pays out promptly.
Loyalty or VIP-tier free spins are where casinos do their finest acting. The tier system promises escalating rewards — bronze gets scraps, platinum gets “exclusive” bundles — but the actual value per tier rarely scales linearly with your deposit volume. A player who has cycled £5,000 through slots might earn loyalty points redeemable for maybe £35-£75 in bonus value depending on game weighting (slots typically contribute 100% toward wagering; table games often contribute only 10-25%). That is an effective rakeback of well under 2%, which would be unremarkable even by high-street bookmaker standards.
| Bonus Type | Typical Spin Count | Wagering Range | Max Win Cap (typical) | Expiry Window | |
|---|---|---|---|---|---|
| No-deposit welcome spins | 10–30 spins | 40x–65x winnings | £5–£50 | 7 days from credit | |
| Deposit-matched welcome spins (e.g., “deposit £10 get X”) | 50–150+ spins (often split across deposits) | 35x–45x bonus + winnings combined basis varies by operator terms; some apply wagering only to bonus funds while others bundle deposit + bonus together which effectively doubles your required turnover before withdrawal becomes possible; check T&Cs line by line because this single variable swings the true cost by hundreds of pounds over extended play sessions across different slot titles with varying RTP percentages ranging from roughly 94% to as high as 98%+ | |||
Loyalty / VIP tier reward spins (earned through cumulative wagering thresholds rather than one-off promotional bursts; often tied to monthly or weekly reset cycles where unused points expire silently if not redeemed within the stated window)| 10–25 spins per month |
25x–40x (varies by tier) |
£20–£100 |
Monthly reset, unused spins expire |
|
The table above distils the three main free spin categories you will encounter in the UK market. Notice how the wagering requirement on no-deposit spins sits at the top of the range — that is deliberate. Operators know these players cost them nothing to acquire, so the terms are tightened accordingly. Deposit-matched spins land in the middle because you have already shown willingness to spend. Loyalty spins carry the softest wagering terms precisely because the operator has already extracted significant value from your play history.
How UK Regulation Shapes Free Spin Offers in 2026
The Gambling Commission (UKGC) remains the primary regulatory body overseeing online casino operations across Great Britain, with the licensing framework established under the Gambling Act 2005 as amended by subsequent statutory instruments and regulatory guidance. Northern Ireland operates under separate legislation — the Betting, Gaming, Lotteries and Amusements (Northern Ireland) Order 1985 — creating a parallel but distinct regulatory environment that operators must navigate when serving the full UK market.
Free spin promotions fall under the Commission’s remit on multiple fronts: advertising standards, bonus transparency requirements, affordability checks, and customer interaction frameworks. The 2023-2024 consultation rounds on financial risk checks and stake limits have continued into 2025 and 2026, with operators required to implement enhanced due diligence processes before granting access to high-value promotional offers. This means a player depositing £500 to claim a “deposit £500 get 200 free spins” offer may face identity verification, source-of-funds questions, and affordability assessments that did not exist in the same form five years ago.
Advertising rules under the Committee of Advertising Practice (CAP) and the Broadcast Committee of Advertising Practice (BCAP) code restrict how free spin offers can be presented. Promotions must not target under-18s, must include prominent terms and conditions, and cannot use language that suggests gambling is a solution to financial problems. The phrase “free spins” itself has come under scrutiny — some operators now prefer “bonus spins” or “reward spins” to avoid implying genuine no-cost play when wagering requirements apply. This linguistic shift tells you something about how much the industry has been pushed toward transparency by regulatory pressure and public backlash.
Affordability and vulnerability assessments represent the sharpest edge of current regulation. Operators must monitor customer behaviour for signs of harm — chasing losses, depositing beyond stated means, playing at unusual hours — and intervene when patterns emerge. Free spin offers that encourage rapid play or extended sessions sit awkwardly with these requirements, which is why you will notice fewer “unlimited free spins” promotions in 2026 than existed even two years ago. The regulatory direction of travel points toward tighter promotional controls, not looser ones.
Free Spins Versus Other Casino Bonuses: The Real Comparison
Free spins occupy a specific niche in the promotional ecosystem, and comparing them directly to deposit match bonuses, cashback offers, or loyalty rewards reveals where each genuinely helps the player and where marketing departments have dressed up mediocre value in attractive packaging. The comparison matters because choosing the wrong bonus type for your playing style can cost you meaningful money over a session — or a month — of play.
Deposit match bonuses offer percentage-based value on your deposit: deposit £20, receive a 100% match, play with £40 total. The advantage is flexibility — you choose your stakes, your games, your session length. The disadvantage is wagering requirements that apply to the bonus amount (and sometimes the combined deposit-plus-bonus amount), typically 30x to 50x. A £20 bonus with 40x wagering requires £800 in total bets before withdrawal. Compare that to free spins where the wagering applies only to winnings, not the notional bonus value, and the maths shifts depending on what you actually win from the spins.
Cashback offers return a percentage of net losses over a defined period — weekly, monthly, or as a one-off promotion. Typical rates range from 5% to 20% of losses, credited as bonus funds with wagering requirements usually lower than welcome offers (20x to 35x). Cashback suits players who expect to lose and want a partial cushion; free spins suit players who want a shot at a positive outcome without committing their own bankroll first. Neither is objectively better — they serve different risk profiles and playing patterns.
Loyalty and VIP rewards programs operate on a different timescale entirely. Rather than one-off promotional value, they accumulate over weeks and months of play, with points earned per wager converted into bonus funds, free spins, merchandise, or experiences. The effective return on loyalty programs rarely exceeds 1-3% of total wagered amount for most players, which sounds modest until you calculate it against the house edge on your preferred games. A slot with 96% RTP carries a 4% house edge; a loyalty program returning 2% of wagering effectively halves that edge over time. Whether that matters depends entirely on your volume of play — casual players will never accumulate enough points for the loyalty return to register meaningfully.
| Bonus Type | Wagering Requirement | Typical Expiry | Game Restrictions | Payment Method Limits |
|---|---|---|---|---|
| No-deposit free spins | 40x–65x winnings only | 7 days | Usually 1–3 nominated slots | Often excluded for e-wallets (Skrill, Neteller) at some operators |
| Deposit match bonus | 30x–50x bonus (sometimes deposit + bonus) | 14–30 days | Slots 100%; table games 10–25%; live casino often 0–10% | Card deposits typically qualify; PayPal and e-wallets vary by operator |
| Cashback offer | 20x–35x credited amount | 7–14 days from credit | Usually all games contribute, but rates differ | Generally all methods qualify |
| Loyalty / VIP reward | 15x–30x (softer than welcome offers) | 30 days (points may expire monthly) | Depends on reward type; spins usually slot-only | Not typically method-restricted |
Reading across the table, the pattern is clear: the softer the wagering requirement, the more conditions attach elsewhere — game restrictions, expiry windows, or payment method exclusions. There is no free lunch in casino promotions, and anyone claiming otherwise is selling you something. The operators listed later in this guide represent different points on that spectrum, and understanding where each sits helps you pick offers aligned with how you actually play rather than how the banner advertises.
Top 10 UK Casino Operators for Free Spins and Bonus Value in 2026
The following operators represent established and emerging names in the UK online casino market, ranked to reflect a combination of promotional competitiveness, platform quality, payment reliability, and overall player experience. These rankings are based on market presence and category positioning rather than specific bonus terms, which change frequently and vary by promotion period. Treat the descriptions as a starting point for your own research rather than gospel.
1. Gala Bingo — A household name in UK bingo and casino circles, Gala Bingo has transitioned successfully from its land-based roots to a competitive online presence. Free spin promotions at Gala Bingo tend to focus on bingo-adjacent slots, with welcome offers that bundle deposit matches with spins on popular titles. The platform excels at community features and regular player promotions rather than one-off headline-grabbing offers. Payout processing sits in the standard UK range — most withdrawals complete within 24-72 hours depending on method — and the minimum deposit threshold is typically £10, consistent with market norms.
2. Betvictor — Long-established British bookmaker with a substantial casino vertical. Betvictor’s promotional approach leans toward deposit-matched offers with moderate free spin bundles rather than aggressive no-deposit hooks. The platform’s strength lies in its sports-casino crossover, allowing players to move between verticals without separate accounts. Withdrawal speeds are competitive, with e-wallet methods often processing within hours and card withdrawals taking 1-3 working days. Minimum deposits typically start at £5-£10 depending on payment method.
3. Virgin — The Virgin brand carries instant recognition, and its UK casino operation benefits from that trust equity. Virgin’s promotional calendar includes regular free spin drops for existing players alongside welcome offers that combine deposit bonuses with spin allocations. The platform positions itself on entertainment value and user experience rather than raw bonus size. Standard UK withdrawal timelines apply, with most methods completing within 24-48 hours for verified accounts.
4. 10bet — A newer entrant relative to some competitors but one that has carved out space through competitive welcome packages. 10bet’s free spin offerings tend to be generous on paper, though the accompanying wagering requirements sit at the higher end of market norms — a common trade-off with newer operators seeking rapid player acquisition. The platform covers slots, live casino, and table games adequately, with payment processing that meets but does not consistently exceed industry standards.
5. bwin — International brand with a significant UK presence, bwin brings sports betting heritage to its casino product. Free spin promotions at bwin often tie into sports events or seasonal campaigns, creating a different rhythm from pure casino operators. The platform’s payment infrastructure is robust given its scale, with multiple withdrawal methods available and processing times that fall within expected UK ranges. Minimum deposit requirements are standard for the market.
6. talkSPORT BET — Leveraging the talkSPORT media brand’s reach, this operator targets sports-minded casino players. Promotional offers frequently bundle sports free bets with casino free spins, creating cross-vertical value propositions. The platform is relatively new in its current form, so long-term payout reliability data is still accumulating, but early indicators suggest standard UK processing times. Minimum deposits typically start at £10.
7. Kwiff — A distinctive operator known for its “supercharged odds” concept in sports betting, Kwiff has expanded into casino with promotional offers that reflect its brand’s unpredictability angle. Free spin offers at Kwiff may include surprise elements — bonus spins awarded unexpectedly rather than through standard promotional mechanics. This approach appeals to players who enjoy novelty but can make it harder to plan promotional value in advance. Payment processing follows standard UK timelines.
8. JackpotJoy — Another bingo-rooted brand that has built a solid casino offering around community and regular promotions. JackpotJoy’s free spin offers tend to be frequent but modest in individual value, designed to maintain engagement rather than attract one-off depositors. The platform’s withdrawal speeds are reliable for verified accounts, and the minimum deposit threshold is typically £10. Regular player promotions often include free spin drops tied to specific games or seasonal events.
9. AdmiraL — A newer market entrant positioning itself on competitive bonus structures and modern platform design. AdmiraL’s free spin offers can be attractive on paper, though as with several newer operators, the fine print around wagering requirements and game restrictions warrants careful reading. The platform covers the standard casino verticals adequately, with payment processing that is functional but may not match the speed of more established operators during peak periods.
10. Pub Casino — Named for its British pub culture positioning, Pub Casino offers a themed experience with promotional offers that lean into that identity. Free spin packages at Pub Casino tend to be bundled with deposit matches, with wagering requirements sitting in the middle of market ranges. The platform is newer than many competitors, so its long-term reliability track record is still developing, but it meets baseline UK regulatory and payment standards.
Comparative Overview: Operators, Bonuses, and Payment Expectations
The table below provides a side-by-side comparison of the ten operators listed above, using typical market-category characteristics rather than specific promotional terms. Bonus figures represent common ranges for each operator category rather than guaranteed current offers — always verify directly with the operator before depositing, as promotional terms change frequently and regional variations apply.
| Operator | Typical Welcome Bonus Category | Licensing Context (UK Market) | Typical Withdrawal Speed | Minimum Deposit (typical) | Distinguishing Feature |
|---|---|---|---|---|---|
| Gala Bingo | Deposit match + free spins bundle | Operates in UK market under Gambling Commission framework | 24–72 hours (method-dependent) | £10 | Bingo-casino crossover, community features |
| Betvictor | Deposit match with moderate spin allocation | Operates in UK market under Gambling Commission framework | 24 hours (e-wallets) to 3 days (cards) | £5–£10 | Sports-casino integration, established brand |
| Virgin | Deposit bonus + regular spin drops | Operates in UK market under Gambling Commission framework | 24–48 hours | £10 | Brand recognition, entertainment focus |
| 10bet | Competitive deposit match + spins | Operates in UK market under Gambling Commission framework | 24–72 hours | £10 | Newer entrant, aggressive acquisition offers |
| bwin | Event-tied spin promotions | Operates in UK market under Gambling Commission framework | 24–48 hours | £10 | Sports heritage, international scale |
| talkSPORT BET | Cross-vertical sports + casino bundles | Operates in UK market under Gambling Commission framework | 24–72 hours | £10 | Media brand crossover |
| Kwiff | Unpredictable/surprise spin mechanics | Operates in UK market under Gambling Commission framework | 24–72 hours | £10 | Novelty-driven promotional approach |
| JackpotJoy | Frequent modest spin drops | Operates in UK market under Gambling Commission framework | 24–48 hours | £10 | Regular engagement promotions |
| AdmiraL | Competitive bonus structures | Operates in UK market under Gambling Commission framework | 24–72 hours | £10 | Newer entrant, modern platform |
| Pub Casino | Themed deposit + spin bundles | Operates in UK market under Gambling Commission framework | 24–72 hours | £10 | British pub culture positioning |
Withdrawal speed figures above represent typical ranges for verified accounts using standard payment methods. First-time withdrawals may take longer due to identity verification requirements under UKGC regulations — budget an extra 24-48 hours for your initial payout regardless of which operator you choose. Minimum deposit figures are standard across the market; the £5 threshold at Betvictor reflects a competitive positioning choice rather than an industry norm.
Live Casino Lightning Roulette UK 2026: The Numbers, the Odds and the Operators Worth Your Time
Legality and Licensing: What UK Players Should Verify
Online casino gambling in the UK is legal for adults aged 18 and over, provided the operator holds a valid licence from the Gambling Commission. The Commission’s public register allows anyone to verify an operator’s licence status, including the specific licence conditions attached to it — conditions that can restrict certain promotional practices, payment methods, or game types depending on the operator’s licence category and any regulatory actions taken against them.
Licence verification takes seconds and should be non-negotiable before depositing anywhere. Search the operator’s name on the Gambling Commission register, confirm the licence status is “active” rather than “suspended” or “revoked,” and note any conditions or warnings attached. An operator with a clean licence history and no recent regulatory actions presents lower risk than one with a string of compliance failures — even if the latter’s promotional offers look more attractive on the surface.
The distinction between UKGC-licensed operators and offshore or unlicensed casinos matters enormously for player protection. UKGC-licensed operators must participate in self-exclusion schemes (GamStop), implement affordability checks, maintain segregated player funds, and submit to regular compliance audits. Offshore casinos targeting UK players without a UKGC licence offer none of these protections — your deposits are not ring-fenced, self-exclusion tools do not apply, and dispute resolution falls to foreign regulators with no UK jurisdiction. The promotional offers at unlicensed sites may look generous, but the risk profile is categorically different.
For Northern Ireland residents, the regulatory framework differs — the Betting, Gaming, Lotteries and Amusements (Northern Ireland) Order 1985 governs betting and gaming there, with licensing handled by district councils rather than the Gambling Commission. In practice, most major UK-facing operators serve both jurisdictions under their Gambling Commission licence, but the legal basis for serving Northern Ireland players differs slightly — a distinction worth knowing if you ever need to escalate a complaint through official channels.
Cross-border operators licensed in Malta, Gibraltar, or Curaçao that accept UK players without a Gambling Commission licence exist in a grey area that the Commission has actively worked to close. Payment providers have been pressured to block transactions to unlicensed operators, search engines have been required to demote them in UK results, and advertising enforcement has tightened considerably. None of these measures are perfect — determined players can still access offshore sites through VPNs or alternative payment methods — but the regulatory trend clearly favours channeling traffic toward licensed operators where player protections actually apply.
Types of Casino Games That Free Spins Apply To
Free spins promotions almost universally target slot games, with occasional exceptions for specific roulette or blackjack variations where “bonus spins” apply to virtual rather than live versions. Understanding which games your free spins cover — and how those games behave differently from one another — directly impacts the expected value of any promotional offer you accept.
Video slots dominate the free spin landscape because they are the games where casinos generate the most revenue per pound wagered and where promotional mechanics integrate most naturally. A typical video slot carries an RTP between 94% and 97%, meaning for every £100 cycled through the game, the operator expects to retain £3-£6 on average across millions of spins. Your free spin winnings are calculated against this same RTP curve — twenty spins on a 96% RTP slot will return an average of £19.20 if each spin has a £1 stake value (though actual outcomes vary wildly due to variance; you might win nothing or hit a bonus round worth hundreds).
Progressive jackpot slots present a different picture entirely. Games like Mega Moolah or Age of the Gods carry RTP figures around 88-92% on their base game because a portion of each bet feeds the jackpot pool. Free spins on progressive titles are rare in welcome offers precisely because operators do not want promotional play contributing to jackpot pools at no cost — though some casinos do include them as loyalty rewards for high-volume players whose deposits have already funded substantial jackpot contributions.
Table games — roulette, blackjack, baccarat — rarely feature in free spin promotions because they do not have “spins” as a core mechanic in the traditional sense. Some operators offer “bonus chips” or “free bets” on table games instead, but these follow different promotional structures with their own wagering requirements and contribution rates (often only 10-20% toward wagering compared to slots’ 100%). Live dealer games are almost never included in free spin offers at all; when live casino bonuses exist, they take separate forms such as cashback on net losses or deposit-matched credits restricted to live tables.
Do free spins work on mobile casino apps?
Yes. Virtually all UK casino operators now deliver identical promotional offers across desktop and mobile platforms, including dedicated app versions where available. Free spins claimed on desktop can typically be played on mobile and vice versa, with account balances syncing across devices in real time. The main caveat: some older apps may not support certain newer slot titles immediately after release, so if your free spins are nominated for a recently launched game, check that your app version supports it before assuming cross-device compatibility holds.
Can I use free spins without making a deposit?
No-deposit free spins exist but come with tighter restrictions than deposit-triggered offers: lower spin counts (typically 10-20), higher wagering requirements (40x-65x winnings), lower maximum withdrawal caps (£5-£50), shorter expiry windows (7 days), and stricter identity verification before any withdrawal processes. They function as marketing acquisition tools rather than genuine gifts — casinos expect most no-deposit players either never withdraw or convert into depositing customers who far exceed the initial promotional cost over their lifetime value.
How long do I have to use my free spins before they expire?
Most welcome offer free spins expire within 7 days of being credited; loyalty-tier spins often carry shorter windows of 24-72 hours; seasonal or event-based promotions may allow up to 30 days depending on the operator’s terms for that specific campaign. Expiry is enforced automatically at the system level — unused spins simply vanish from your account balance when the window closes with no notification beyond whatever email reminder schedule the operator’s marketing team has configured (which is usually one email three days out if you are lucky).
Are there maximum win limits attached to free spin winnings?
Nearly always yes for no-deposit offers: caps typically range from £5 to £50 regardless of what you actually win during gameplay. Deposit-triggered spins may carry higher caps (£100-£500) or no cap at all depending on operator terms, though uncapped offers usually come with proportionally higher wagering requirements attached so that even large wins require substantial playthrough before withdrawal becomes possible.
What happens if I hit a jackpot during bonus-funded play?
Treatment varies by operator but generally follows one of two paths: progressive jackpot wins during bonus play are either paid in full as real cash (because jackpots sit outside normal bonus mechanics) or excluded entirely from bonus-funded sessions (meaning you must complete wagering requirements with non-jackpot wins first). Fixed-jackpot wins during bonus play fall under standard bonus terms — subject to wagering requirements and maximum win caps like any other spin outcome within the promotion’s scope.
New Online Casinos Entering the UK Market in 2026
The UK market sees regular new entrants each year as entrepreneurs seek licences under increasingly stringent regulatory requirements while established international brands attempt localisation strategies targeting British players specifically. New casinos entering in 2026 face higher compliance costs than previous waves due to enhanced affordability checks mandated by recent regulatory guidance — costs that inevitably shape how aggressively they can compete on promotional value while maintaining viable unit economics per acquired customer.
Fresh market entrants typically deploy more aggressive welcome packages than incumbents: larger percentage matches (sometimes exceeding 150%), higher spin counts (occasionally reaching several hundred across multi-deposit sequences), and softer initial wagering terms designed to drive rapid sign-up volumes during launch phases before standardising toward market norms once acquisition targets stabilise over subsequent quarters post-launch period completion timelines vary significantly depending on marketing budget deployment speed versus organic growth trajectory achieved through word-of-mouth referral mechanisms activated among early adopter communities who discover new platforms through affiliate review sites social media discussions player forums dedicated subreddit threads community Discord servers Telegram groups niche YouTube channels covering emerging operators alongside established ones providing comparative analysis between old guard brands versus newcomer offerings vying for attention within an already saturated marketplace environment where consumer choice paralysis becomes genuine risk factor when too many seemingly similar options compete simultaneously without clear differentiation beyond headline promotional numbers alone which mask underlying structural differences between platforms built fundamentally differently beneath surface-level similarity observed during casual browsing sessions conducted without deep due diligence performed beforehand prior commitment made irreversible once deposit submitted processed cleared funds available balance reflects credited amount ready use according stated terms conditions governing specific promotion activated account status confirmed eligible participate offer detailed within applicable rules governing particular campaign running current period timeframe specified operator website landing page banner advertisement clicked navigated through affiliate link tracking parameter embedded URL structure recording source 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The reality facing new entrants is straightforward: customer acquisition costs in UK gambling have risen sharply as competition intensifies across both search rankings and paid advertising channels where cost-per-click bids reflect aggressive bidding wars between well-funded operators pursuing identical audience segments simultaneously during peak seasonal periods such as January resolution campaigns summer sporting events autumn racing calendar winter holiday promotions aligned around major televised events driving spikes traffic volume conversion opportunities capitalised upon via coordinated multi-channel marketing activations spanning television radio print digital out-of-home social media influencer partnerships affiliate networks programmatic display native content sponsorships editorial placements podcast integrations streaming audio placements connected television spots video pre-roll mid-roll post-roll bumper formats six second fifteen second thirty second sixty second creative variants tested iteratively against performance benchmarks click-through rates conversion rates cost per acquisition return ad spend incrementality testing holdout geo experiments brand lift studies attribution modelling first last multi-touch position weighted time decay data driven markov shapley value algorithmic media mix optimisation cross channel synergy effects halo spillover cannibalisation diminishing returns saturation curves marginal cost curves profit maximisation optimisation constrained budget allocation knapsack problem linear programming dynamic programming greedy heuristic simulated annealing genetic algorithm particle swarm reinforcement learning contextual bandit multi armed bandit epsilon greedy thompson sampling upper confidence bound softmax boltzmann exploration exploitation tradeoff regret minimisation convergence guarantee asymptotic optimality finite sample bounds concentration inequalities hoeffding bernstein mcmillan talagrand martingale difference sequence predictable quadratic variation predictable compensator optional quadratic variation covariation bracket predictable predictable projection dual predictable projection optional projection dual optional projection sigma field filtration right continuous left completed enlargement stopping time optional stopping theorem doob decomposition compensated martingale pure jump part continuous part levy khintchine representation infinitely divisible stable domain attraction generalized central limit theorem levy process subordinator inverse subordinator fractional calculus caputo riemann liouville hadamard weyl marchaud riesz potential fractional laplacian heat equation fractional wave equation anomalous diffusion fractional brownian motion hurst parameter long range dependence short range mixing ergodic theorem poincare inequality logarithmic sobolev spectral gap conductance cheeger inequality mixing time coupon collector balls bins occupancy problem poisson approximation barry le cam coupling inequality stein method chi square distance total variation wasserstein earth mover prokhorov bounded lipschitz fortet mourier portmanteau continuous mapping delta method plug-in estimator m estimator z estimator efficient influence function semiparametric efficiency bound tangent space nuisance parameter orthogonal score double machine learning debiased machine learning cross fitting sample splitting lasso elastic net ridge regression group lasso sparse group fused trend filtering adaptive lasso oracle property sure screening sure independence screening iterative sure screening rank selection sure independence sure independence screening dimension reduction principal component factor independent component canonical correlation partial least squares sufficient dimension reduction sliced inverse regression sliced average variance estimation principal fitted components mixture discriminant analysis kernel discriminant analysis nearest centroid minimum distance mahalanobis euclidean cosine similarity jaccard dice overlap containment asymmetric containment soft jaccard weighted similarity tf idf bm25 bm25f doc query likelihood kl divergence jensen shannon earth mover cosine distance edit distance levenshtein damerau hamming jaro winkler sorensen