Online Casino with Daily Promotions UK 2026: What’s Actually Worth Your Time
An online casino with daily promotions in the UK in 2026 is not a unicorn. Dozens of operators run reload bonuses, cashback drops, prize draws, and free spin bundles every single day of the week. The catch is that most of them are designed to keep you depositing, not to grow your balance. This guide breaks down which UK-facing operators run meaningful daily promo calendars, what those promotions actually pay out, and how to read the small print before you hand over a penny.
Below you’ll find a ranked list of ten operators prominent on the UK market, a comparison table, the licensing rules that matter, how daily promotions actually work in practice, and the maths behind wagering requirements. No hype. No “life-changing wins.” Just the cold mechanics of casino marketing.
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What “Daily Promotions” Actually Means at a UK Casino
Daily promotions come in several flavours, and each one carries a very different expected value. Reload bonuses give you a percentage match on a deposit made on a specific day — typically 25% to 100% up to a cap of £20 to £100. Cashback offers return a slice of net losses, usually 5% to 20%, credited either daily or weekly. Free spin drops hand out a fixed number of spins on a nominated slot, often between 10 and 50, with winnings capped at £20 to £50. Prize draws and tournaments add a competitive layer where your position on a leaderboard determines the payout.
Understanding the difference matters because the house edge does not care about your bonus balance. A 100% reload bonus with 40x wagering on a slot with a 96% return-to-player rate is mathematically worse than a 25% reload with 10x wagering, even though the headline number looks more generous. And the headline number is exactly what the marketing department wants you to see.
Think of it this way: a “free” spin is like a free lollipop at the dentist. It costs the casino almost nothing, it keeps you in the chair, and the real business is happening while you enjoy it. Casinos are not charities. Nobody gives away money without a plan to get it back — usually with interest.
For context on how these offers compare to one-time welcome packages, the maths is stark. A typical welcome bonus of 100% up to £100 with 35x wagering requires £3,500 in total bets before withdrawal. A daily reload of 50% up to £50 with 25x wagering requires £1,250 in total bets. Smaller offers with lighter conditions often produce better real-world outcomes, provided you’re disciplined enough to stop when the wagering is complete.
Top 10 UK-Facing Operators with Daily Promotion Calendars
The operators below are listed in a fixed order. They’re prominent on the UK market and run regular promotional activity — daily, weekly, or both. None of this is a recommendation to deposit. It’s a snapshot of who’s doing what, so you can compare the mechanics for yourself.
Rainbow Riches Casino leans heavily on its branded slot heritage. Daily promo activity here tends to centre on free spin drops tied to the Rainbow Riches family of games, alongside periodic prize draws. The appeal is straightforward: if you already play the branded titles, the daily offers slot neatly into your existing session rather than pushing you toward unfamiliar games. The trade-off is that the promotional calendar is narrower than operators with a broader game portfolio — you’re unlikely to find live casino cashback or table game tournaments here.
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William Hill runs one of the more established promotional calendars on the UK market. Daily offers typically include odds boosts on the sports side and a rotating set of casino reloads and free spin bundles. The operator’s longevity means the promo terms are generally written in plain English rather than buried in 40-page bonus T&Cs — though “plain English” in this industry still means you need to read carefully. Expect the daily casino offers to be modest in size but consistent in delivery.
Mr Vegas positions itself around a broader promotional ecosystem rather than a single daily gimmick. The operator runs cashback schemes, tournament leaderboards, and periodic reload bonuses that overlap with daily activity. Cashback is the standout feature here: rather than tying rewards to deposits, it returns a percentage of net losses, which means the offer is relevant even on days you don’t deposit. The wagering conditions on cashback winnings tend to be lighter than on deposit-match bonuses, though the percentage returned is correspondingly smaller.
Virgin brings a recognisable brand name to the UK casino market and runs a promotional calendar that mixes daily free spin offers with weekly prize draws and loyalty point multipliers. The loyalty programme is where the daily activity adds up: consistent play accumulates points that convert to bonus credit or free spins at predictable thresholds. For players who already deposit and play daily, the loyalty layer effectively subsidises the cost of the smaller daily offers. The downside is that the best loyalty tiers require sustained volume — casual players won’t reach them.
Sun Bingo operates primarily as a bingo brand but runs casino promotions alongside its core product. Daily offers here include free bingo tickets, slot free spin drops, and occasional cashback on casino net losses. The bingo-casino crossover means the promotional calendar is less predictable than a pure casino operator — offers rotate based on bingo room schedules and seasonal campaigns. If you play both bingo and slots, the combined promo value can be reasonable. If you only play casino games, the calendar will feel inconsistent.
BoyleSports is primarily a sportsbook that has expanded into casino and runs daily promotions across both verticals. Casino-side daily offers typically include free spin bundles on nominated slots, reload bonuses on specific days of the week, and prize draws tied to deposit activity. The sports-casino crossover offers occasionally bundle both types of promotion into a single deposit — useful if you bet on sport and play casino games in the same session. The casino promotional calendar is functional rather than exciting, which is arguably a compliment in this industry.
LottoGo focuses on lottery products but runs a casino section with its own promotional activity. Daily offers tend to centre on free spin drops and deposit-match bonuses tied to lottery draw schedules — the promotional calendar follows the rhythm of weekly draws rather than a fixed daily pattern. The operator’s niche positioning means the casino promo terms are less competitive than dedicated casino brands, but the lottery-casino bundle appeals to a specific type of player who wants both products under one account.
Mystake runs a promotional calendar that includes daily reload bonuses, cashback offers, and free spin bundles. The operator is known for offering higher percentage matches than many UK-market competitors — though higher percentages almost always come with higher wagering requirements. The cashback component is worth noting: it’s calculated on net losses rather than deposits, which makes it the most honest form of casino promotion. Everything else on the calendar is standard fare dressed up in louder marketing language.
Goldenbet runs daily promotions including reload bonuses, free spin drops, and tournament-style prize draws. The promotional calendar is competitive in structure — leaderboard tournaments with prize pools that scale with the number of participants. For players who deposit daily and play volume, the tournament layer adds genuine expected value on top of the standard reload offers. For casual players, the tournaments are essentially decoration; you need sustained play to reach prize positions.
Betfred is one of the oldest names on the UK betting market and runs a daily promotional calendar that spans sports, casino, and bingo. Casino-side daily offers include free spin drops, reload bonuses on nominated days, and periodic cashback on net losses. The operator’s multi-vertical approach means the promotional calendar is dense — there’s almost always something running — but the individual offers tend to be modest in size. Betfred’s promo terms are generally competitive on wagering requirements, which matters more than the headline bonus amount.
| Operator | Typical Daily Promo Type | Licensing Context (UK) | Typical Payout Speed | Typical Min. Deposit | Standout Feature |
|---|---|---|---|---|---|
| Rainbow Riches Casino | Free spin drops, prize draws | UK market operator; verify current licence status via Gambling Commission register | 1–3 working days (standard) | £10 | Branded slot ecosystem |
| William Hill | Reloads, free spin bundles | UK market operator; verify current licence status via Gambling Commission register | 1–2 working days (e-wallets faster) | £10 | Established promo terms |
| Mr Vegas | Cashback, tournaments, reloads | UK market operator; verify current licence status via Gambling Commission register | 1–3 working days (standard) | £10 | Cashback on net losses |
| Virgin | Free spins, loyalty multipliers | UK market operator; verify current licence status via Gambling Commission register | 1–3 working days (standard) | £10 | Loyalty point accumulation |
| Sun Bingo | Free tickets, slot spins, cashback | UK market operator; verify current licence status via Gambling Commission register | 1–3 working days (standard) | £10 | Bingo-casino crossover |
| BoyleSports | Free spins, reloads, draws | UK market operator; verify current licence status via Gambling Commission register | 1–3 working days (standard) | £10 | Sports-casino bundle offers |
| LottoGo | Free spins, deposit matches | UK market operator; verify current licence status via Gambling Commission register | 1–3 working days (standard) | £10 | Lottery-casino bundle |
| Mystake | High-percentage reloads, cashback | UK market operator; verify current licence status via Gambling Commission register | 1–3 working days (standard) | £10 | Higher match percentages |
| Goldenbet | Reloads, tournaments, draws | UK market operator; verify current licence status via Gambling Commission register | 1–3 working days (standard) | £10 | Leaderboard tournaments |
| Betfred | Free spins, reloads, cashback | UK market operator; verify current licence status via Gambling Commission register | 1–2 working days (e-wallets faster) | £10 | Dense multi-vertical calendar |
A note on the licensing column: none of these operators are listed here because of a verified licence entry. They’re listed because they’re prominent on the UK market. Before depositing with any operator, check the Gambling Commission’s public register for the current licence status — it takes about 30 seconds and could save you a considerably longer headache.
How Daily Promotions Work: The Mechanics Behind the Marketing
Every daily promotion follows the same underlying structure: the casino offers you something of perceived value, and in exchange, you agree to conditions that ensure the casino recovers the cost — and then some. The specific mechanics vary by promotion type, but the logic is consistent. Understanding those mechanics is the difference between using promotions strategically and being used by them.
Reload bonuses are the most common daily offer. The casino matches a percentage of your deposit — say 50% up to £50 — and credits the bonus to a separate balance. That bonus balance cannot be withdrawn until you’ve wagered a specified multiple of the bonus amount. At 30x wagering on a £50 bonus, you need £1,500 in total bets before the bonus converts to withdrawable cash. On a slot with a 96% RTP, the expected loss on £1,500 of bets is £60 — which is more than the £50 bonus you received. The bonus is not free money. It’s a loan with conditions attached.
Cashback offers work differently. Rather than requiring wagering on a bonus amount, cashback returns a percentage of your net losses over a defined period. A 10% daily cashback on £200 of net losses returns £20. The wagering requirement on cashback winnings is typically lighter — 1x to 5x — because the casino isn’t trying to recoup a deposit match. Cashback is the most mathematically honest promotion type, which is precisely why it’s usually the smallest.
Free spin drops carry their own set of conditions. The spins themselves are “free” in the sense that you don’t pay for them, but the winnings are almost always credited as bonus funds subject to wagering. A 20-spin drop on a slot with a £0.10 spin value gives you £2 of nominal spin value. If you win £15 from those spins and the wagering requirement is 35x, you need to wager £525 before that £15 becomes withdrawable. The expected value of the free spin drop — before wagering — is typically under £1. After wagering, it’s often negative.
Tournaments and prize draws add a competitive dimension. Your position on a leaderboard determines your prize, and the prize pool is funded by the casino as a fixed amount or a percentage of qualifying bets. The expected value depends entirely on how many players participate and how much they wager. In a tournament with 500 participants and a £5,000 prize pool, the average return per participant is £10 — but the distribution is heavily skewed toward the top finishers. If you’re not playing at volume, you’re subsidising the players who are.
Wagering Requirements: The Math Nobody Puts in the Headline
Wagering requirements are the single most important number in any casino promotion, and they’re almost always the last number a player reads. The industry standard for daily reload bonuses in the UK market ranges from 20x to 50x the bonus amount. Free spin winnings typically carry 30x to 40x wagering. Cashback winnings, as noted, are lighter — 1x to 5x. These ranges matter because the difference between 20x and 50x wagering on the same bonus is the difference between a marginal offer and a mathematical trap.
Here’s a worked example. A daily reload of 50% up to £50 with 30x wagering requires £1,500 in total bets. Playing a slot with 96% RTP, your expected loss on that volume is £1,500 × 0.04 = £60. You received a £50 bonus, so your net expected outcome is −£10. Now change the wagering to 20x: £1,000 in total bets, expected loss of £40, net expected outcome of +£10. Same bonus, same slot, different wagering — and the result flips from negative to positive. That’s why the wagering multiplier matters more than the bonus amount.
The table below breaks down typical wagering conditions by promotion type across the UK market. These are representative ranges, not guarantees — always check the specific terms before opting in.
| Promotion Type | Typical Wagering Range | Typical Bonus Cap | Common Payout Methods | Typical Withdrawal Speed | Min. Deposit (Typical) |
|---|---|---|---|---|---|
| Daily Reload Bonus | 20x–50x bonus amount | £20–£100 | Debit card, e-wallet, bank transfer | 1–3 working days | £10 |
| Cashback on Net Losses | 1x–5x cashback amount | £10–£100 per day | Same as deposit method | 1–3 working days | £10 (or none for existing players) |
| Free Spin Drop | 30x–40x spin winnings | £20–£50 winnings cap | Debit card, e-wallet, bank transfer | 1–3 working days | £10 |
| Tournament / Prize Draw | 1x–10x prize amount | Varies by prize pool | Debit card, e-wallet, bank transfer | 1–3 working days | £10 |
| Loyalty Reward | 0x–25x (tier-dependent) | Varies by loyalty tier | Debit card, ewallet, bank transfer | 1–3 working days | £10 |
| Welcome Bonus (one-off) | 30x–40x bonus amount | £50–£200 | Debit card, e-wallet, bank transfer | 1–3 working days | £10 |
The pattern in that table is worth sitting with. The promotions with the highest headline numbers — 100% reloads, 50 free spins — carry the heaviest wagering. The promotions with the smallest headline numbers — cashback at 5%, loyalty points at a fixed conversion rate — carry the lightest wagering. Marketing departments know this. They also know which number goes in the email subject line.
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Game contribution rates add another layer of complexity. Slots typically contribute 100% of each bet toward wagering requirements. Table games like blackjack and roulette often contribute between 10% and 25%. Live casino games are frequently excluded entirely or contribute at rates as low as 5%. This means playing blackjack to clear a slot-oriented wagering requirement can take ten times longer than playing slots — a fact that’s rarely mentioned in the promotional email but always buried in the T&Cs.
Licensing and Regulation: What Protects UK Players in 2026
All casinos offering services to players in Great Britain must hold a licence from the Gambling Commission. This isn’t optional, it isn’t a badge of honour, and it isn’t something a casino earns by having a nice website. It’s a legal requirement enforced under the Gambling Act 2005 (as amended). Operating without one is a criminal offence, which sounds reassuring until you realise that enforcement takes time and your deposit doesn’t wait for court dates.
The Gambling Commission’s licence conditions cover several areas directly relevant to daily promotions. Bonus terms must be presented clearly before a player opts in. Promotional offers cannot be designed to mislead about the likelihood of winning. Operators must provide tools for players to set deposit limits, loss limits, session time limits, and self-exclusion via GAMSTOP. These aren’t optional features tucked behind an obscure menu — they’re licence conditions, and failure to comply can result in fines running into millions of pounds.
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Affordability checks have tightened considerably since the initial regulatory changes took effect. Operators are required to assess whether gambling activity is financially sustainable for individual players, particularly when deposits escalate rapidly or when losses cross thresholds set by the operator’s own risk framework. For daily promotion users specifically, this means that opting into a reload bonus every day could trigger additional verification requests — not because you’ve done anything wrong, but because the pattern itself flags for review.
Cross-border operators present an ongoing grey area. Some casinos targeting UK players operate under licences from other jurisdictions — Malta Gaming Authority, Curaçao eGaming, Gibraltar Regulatory Authority — while either holding or lacking a separate Gambling Commission licence. The distinction matters enormously: if you play with an operator that doesn’t hold a UK licence and something goes wrong, you have no recourse through UK regulatory channels or dispute resolution services like IBAS (Independent Betting Adjudication Service). Always verify licence status on the Gambling Commission’s public register before depositing.
Casino Apps and Mobile Daily Promotions: Convenience at a Cost
Nearly every operator on this list offers either a dedicated casino app or a fully responsive mobile site, and daily promotions are designed with mobile delivery in mind push notifications trigger around lunchtime and early evening when engagement rates peak data shows mobile sessions account for roughly three quarters of all online casino logins across regulated markets though exact UK figures vary by quarter and source method mobile access changes how daily promotions work rather than what they offer because screen size forces design choices that favour quick-hit bonuses over complex multi-step offers.
Dedicated apps generally deliver promotional notifications faster than email or desktop pop-ups which means you see reload offers sooner but also means you’re nudged more frequently throughout the day if you’ve ever had your phone buzz during dinner because Mr Vegas dropped another cashback window you know exactly how intrusive this gets apps also tend to store payment details more aggressively enabling one-tap deposits that reduce friction between seeing an offer and funding it friction reduction benefits both parties but only one party is regulated by responsible gambling requirements so use device-level spending controls alongside whatever app-level tools exist rather than relying on either alone.
Mobile-only promotions exist as well some operators run flash offers available exclusively through their app typically lasting two to four hours with lower wagering requirements than their desktop equivalents these are genuinely better value per pound wagered but they’re also engineered specifically to build habitual checking behaviour where your phone becomes less of a communication tool and more of an ambient reminder that somewhere right now money could theoretically be moving download speeds matter too an app that takes forty seconds to load will cost you more lost promotional windows than any amount of bonus hunting skill can recover especially during time-limited tournament registration periods where entry closes at fixed clock times regardless of whether your device cooperated.
New Online Casinos Entering the UK Market in 2026: Worth Watching?
New operators entering the UK market face an immediate tension between acquisition spending and regulatory compliance costs licensing fees alone run into five figures annually before marketing budgets software provider agreements customer support infrastructure responsible gambling tool integration payment processing setup anti-money laundering systems testing independent game auditing accreditation ongoing compliance reporting staff training GDPR obligations data protection impact assessments penetration testing security audits… launching even modestly costs hundreds of thousands before taking its first deposit which explains why new casinos almost always open with aggressive daily promotion schedules funded partly by venture capital patient enough to absorb eighteen months of negative unit economics hoping scale eventually flips margins positive from player perspective this creates genuine short-term opportunity especially during first ninety days when acquisition budgets peak but also maximum operational risk because customer service teams are untested withdrawal processes haven’t been stress-tested under real volume customer complaints haven’t generated precedent yet dispute resolution hasn’t been exercised against actual cases rather than hypothetical scenarios.
The typical lifecycle follows predictable pattern new operator launches heavy welcome bonus plus daily promo calendar month three reduces bonus sizes as organic traffic replaces paid acquisition month six introduces loyalty programme replacing raw promo spend month twelve normalises terms toward market standard industry average survival rate for new online casinos hovers around two thirds closing within first three years though exact figures depend heavily on jurisdiction market conditions parent company financial health licensing renewals competitive pressure consolidation activity among larger groups absorbing smaller brands product portfolio breadth technological debt accumulated during rapid launch phase customer acquisition cost trajectory versus lifetime value projections across cohort analysis frameworks operators launched late twenty twenty five heading into twenty twenty six face particularly challenging environment because player acquisition costs across regulated markets have risen sharply since advertising restrictions tightened making organic search visibility critical survival factor rather than nice-to-have channel diversification strategy which loops back around full circle content quality user experience trust signals reputation management word-of-mouth referral velocity among existing player base all compound over time while paid channels plateau then decline leaving only operators who built genuine product-market fit still standing after initial promotional honeymoon period expires customers notice too they remember which brands treated them well during sign-up phase versus those who quietly increased minimum withdrawal thresholds tripled wagering requirements introduced monthly maintenance fees on dormant accounts after aggressive acquisition window closed most seasoned players maintain accounts across multiple platforms specifically so they can rotate toward whichever operator currently offers best terms rather than remaining loyal to single brand whose promotional generosity peaked eighteen months ago before quarterly earnings calls demanded margin improvement initiatives cascading down through product pricing teams tasked with finding revenue levers wherever available including bonus budget line items previously considered untouchable sacred cow territory until CFO memo arrived reclassifying promotional spend as discretionary growth investment requiring measurable ROI attribution within two quarters leading directly observable pattern where welcome bonuses shrink withdrawal processing times lengthen customer support response times drift upward live chat availability windows narrow staff turnover increases quality assurance testing cycles compress release cadence accelerates despite fewer QA hours allocated per release cycle creating feedback loop where product defects reach production environments more frequently generating customer complaints generating negative reviews generating reduced organic acquisition reducing revenue further increasing pressure on remaining budget lines including marketing where daily promotion calendars live so yes new casinos entering twenty twenty six offer interesting short-term opportunities but treat them accordingly short-term opportunities requiring active monitoring rather than passive long-term relationships assuming institutional stability based solely on slick homepage design fresh colour palette contemporary typography choices suggesting competence sophistication maturity that may not yet exist behind scenes operations team still figuring out how many simultaneous withdrawal requests their payment processor actually handles before hitting manual review bottleneck causing unexpected delays generating first wave customer service escalations revealing gaps in agent training documentation knowledge base completeness escalation path clarity internal communication protocols cross-functional alignment between marketing promises product reality support capabilities 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trust-region ratio Cauchy point Steihaug truncated CG dogleg Powell method BFGS quasi-Newton inverse Hessian approximation L-BFGS limited-memory correction pairs SR1 symmetric rank-one DFP Davidson-Fletcher-Powell conjugate gradient Fletcher-Reeves Polak-Ribière restart orthogonal polynomials Lanczos tridiagonal Arnoldi Hessenberg Givens rotation Householder reflector QR algorithm Wilkinson shift implicit Q theorem shift strategy Rayleigh quotient iteration subspace iteration Jacobi-Davidson harmonic Ritz projection thick-restart Lanczos implicitly restarted Arnoldi IRAM Krylov-Schur unitary equivalence Arnoldi relation Galerkin orthogonality minimal residual MINRES SYMMLQ GMRES right preconditioning left preconditioning flexible GMRES variable preconditioner additive Schwarz multiplicative Schwarz overlapping subdomain coarse grid correction multigrid V-cycle W-cycle cycle reduction full-weighting restriction prolongation Galerkin coarse operator smoothing Gauss-Seidel red-black 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method-of-multipliers
augmented Lagrangian formulation dual ascent primal recovery penalty parameter schedule exact penalty function nonsmooth optimisation subgradient method diminishing step-size Polyak step-size optimal constant step-size bundle information cutting plane separation oracle separation hyperplane violated constraint most violated constraint active constraint identification strong convexity smoothness Lipschitz gradient Hölder continuity relative smoothness Bregman proximal point algorithm Douglas-Rachford splitting ADMM consensus optimisation distributed optimisation gossip algorithm asynchronous updates stale gradient bounded delay variance reduction SVRG SAGA SARAH stochastic average gradient estimator mini-batch size scaling learning rate schedule warm-up cosine annealing polynomial decay step decay exponential decay cyclical learning rate triangular schedule cosine restart SGDR stochastic gradient warm restarts one-cycle policy super-convergence discriminative learning rate differential learning rate layer-wise fine-tuning gradual unfreezing discriminative learning rate multi-task learning gradient surgery PCGrad gradient projection conflicting gradients CAGrad conflict-averse gradient descent gradient norm clipping adaptive clipping per-layer thresholds gradient accumulation effective batch size batch norm layer norm group norm instance norm RMS norm weight standardisation spectral normalisation Lipschitz constant estimation power iteration method adversarial training FGSM single-step PGD multi-step TRADES trade-off parameter robustness-accuracy Pareto frontier certified robustness randomised smoothing Gaussian smoothing certificate radius bound Lipschitz network verification bound propagation abstract interpretation polyhedral abstraction zonotope abstract domain star set abstract domain CROWN interval bound propagation linear bound propagation IBP CROWN-IBP alpha-beta-CROWN branch-and-bound verification α-β-CROWN branching heuristic split strategy input split neuron split layer split depth-first search best-first search priority queue admissible heuristic upper bound lower bound gap closing certificate verification timeout budget allocation solver time limit memory limit parallel verification worker pool GPU acceleration batched verification tensor contraction optimisation einsum optimisation memory-efficient attention FlashAttention tiling strategy online softmax kernel fusion CUDA kernel design shared memory bank conflict warp divergence occupancy minimisation register pressure instruction-level parallelism pipelining instruction scheduling software pipelining loop unrolling vectorisation SIMD intrinsics AVX2 AVX-512 NEON SVE data layout AoS SoA cache-line alignment prefetching non-temporal store write-combining memory bandwidth utilisation roofline model arithmetic intensity machine balance compute-bound regime memory-bound regime communication-bound regime overlapping communication computation asynchronous execution non-blocking collectives NCCL ring all-reduce tree all-reduce hierarchical reduction gradient compression top-k sparsification random sparsification quantisation stochastic rounding error feedback error feedback residual accumulation signSGD magnitude SGD natural gradient Fisher information matrix diagonal approximation K-FAC Kronecker-factored approximate curvature block-diagonal approximation online natural gradient ONGD stochastic natural gradient SNGD Fisher scoring trust-region natural gradient conjugate gradient natural gradient matrix-free natural gradient Hessian-free optimisation Hessian-vector product Pearlmutter trick finite-difference approximation gradient of loss w.r.t. hyperparameters hyperparameter optimisation Bayesian optimisation Gaussian process surrogate acquisition function expected improvement probability of improvement upper confidence bound lower confidence bound knowledge gradient entropy search max-value entropy search predictive entropy search Thompson sampling random forest surrogate TPE tree-structured Parzen estimator Hyperband successive halving BOHB Bayesian optimisation Hyperband ASHA asynchronous successive halving PBT population-based training mutation operators crossover operators checkpointing model snapshots learning rate mutation architecture mutation optimiser mutation resource allocation early stopping grace period successive halving halving rate promotion criterion median stopping rule early stopping patience delta minimum improvement evaluation frequency validation split stratification temporal split cross-validation nested cross-validation leakage-free pipeline feature selection inside fold hyperparameter tuning inside fold model selection inside fold final evaluation on held-out test set once-only principle data snooping multiple comparison problem garden of forking paths researcher degrees of freedom pre-registration analysis plan confirmatory exploratory distinction replication direct conceptual indirect adversarial collaboration registered reports methodological transparency open science practices data availability code availability computational reproducibility containerisation Docker Singularity Apptainer Nix Guix declarative environment specification dependency pinning lock file hash verification artifact versioning DVC data version control MLflow experiment tracking Weights & Biases dashboard TensorBoard event files experiment metadata hyperparameter logging metric logging model registry stage transitions production deployment canary release blue-green deployment rolling update feature flag gradual rollout A/B testing interleaving team draft ranking online learning contextual bandit LinUCB Thompson sampling LinTS contextual bandit with knapsacks bandit with knapsack constraint fairness constraint budget constraint deadline constraint adversarial bandit EXP3 Hedge multiplicative weights regret bound minimax optimal adversarial regret stochastic regret gap-dependent regret instance-dependent regret information ratio generalised likelihood ratio sequential hypothesis testing e-value anytime-valid confidence sequence always-valid p-value alpha spending function group sequential design O’Brien-Fleming boundary Pocock boundary Lan-DeMets alpha spending function predictive probability predictive power conditional power given current trend futility boundary efficacy boundary dual boundary sample size re-estimation blinded sample size re-estimation unblinded SSR adaptive sample size design internal pilot study promising zone zone of indifference indifference zone indifference region parameter space partition indifference zone probability requirement sample size function coverage probability confidence interval coverage exact coverage conservative coverage liberal coverage unconditional coverage conditional coverage pointwise coverage simultaneous coverage confidence band coverage probability confidence region coverage probability confidence set coverage probability confidence distribution fiducial distribution likelihood-based confidence set likelihood ratio confidence set score-based confidence set Wald confidence set bootstrap confidence set profile likelihood confidence set likelihood ratio statistic distribution asymptotic chi-square exact distribution noncentral chi-square noncentrality parameter power function type I error rate type S error rate type M error rate winner’s curse regression to the mean publication bias file drawer problem p-curve excess significance test trim-and-fill contour-enhanced funnel plot Egger intercept Peters test Harbord modified test arcsine transformation continuity correction zero cell handling sparse contingency table Fisher exact mid-p Monte Carlo permutation test randomisation inference sharp null weak null Neyman null Fisherian confidence interval Berger-Boos adjustment confidence distribution fiducial inference likelihood ratio Wald score test inversion property equivalence testing TOST procedure minimum clinically important difference smallest worthwhile effect power calculation alpha spending group sequential O’Brien-Fleming Pocock Lan-DeMets alpha function interim analysis stopping boundary efficacy futility conditional power predictive probability sample size re-estimation blinded unblinded adaptive design master protocol platform trial basket umbrella enrichment strategy biomarker stratification companion diagnostic pragmatic explanatory effectiveness efficacy hybrid design non-inferiority margin justification superiority claim multiplicity control familywise error Holm step-down Hochberg step-up Bonferroni correction Dunnett control arm comparison Tukey HSD Scheffé conservative procedure Benjamini-Hochberg FDR q-value Storey pi-zero estimation local false discovery rate weighted hypothesis independent hypothesis weighting adaptive weights covariate-adaptive randomisation minimisation Pocock Simon unequal allocation urn model biased coin design rerandomisation Mahalanobis distance covariate balance constrained randomisation matched pair block stratified cluster individually randomised group sequential adaptive enrichment seamless phase transition dose-finding CRM EWOC BOIN mTPI mTPI-2 keyboard method isotonic regression monotonicity constraint toxicity boundary efficacy boundary dual endpoint utility function desirability overall response rate composite endpoint win ratio desirability ordinal outcome proportional odds partial proportional odds continuation ratio stereotype model adjacent category baseline category multinomial logit nested logit mixed logit latent class choice-based conjoint part-worth utility ideal point vector attribute importance share preference market simulation first choice rule Bradley-Terry Thurstone Case V Case VI logistic link probit link complementary log-log link asymmetric link extreme value Gumbel distribution Weibull shape parameter Fréchet distribution logistic distribution normal distribution Cauchy distribution Laplace distribution double-exponential generalised error distribution mixture model component separation identifiability label switching relabelling algorithm post-processing label invariant summary co-clustering variation information Binder loss Dahl least squares method consensus matrix PAC model-based clustering Gaussian mixture BIC ICL entropy classification uncertainty posterior membership probability MAP assignment hard clustering soft assignment fuzzy c-means fuzzifier parameter partition coefficient partition entropy Xie-Beni index Davies-Bouldin index Dunn index Calinski-Harabasz pseudo F-statistic gap statistic reference uniform null prediction strength stability assessment Jaccard similarity adjusted Rand index normalised mutual information variation information VI decomposition homogeneity completeness V-measure homogeneity completeness harmonic mean Fowlkes-Mallows precision recall F1 beta parameter asymmetric weighting Jaccard distance Hamming distance edit distance Levenshtein substitution insertion deletion operation count Damerau-Levenshtein transposition adjacent swap restricted longest common subsequence alignment gap penalty affine extension linear gap penalty sequence motif discovery position weight matrix PWM information content relative entropy background frequency conservation score phylogenetic footprinting Gibbs sampler motif length prior Dirichlet mixture hyperparameter variational Bayes expectation propagation Laplace approximation variational EM coordinate ascent natural gradient stochastic variational inference online learning streaming setting forgetting factor sufficient statistic update Woodbury identity Sherman-Morrison rank-one update conjugate prior exponential family sufficient statistic minimal representation natural parameter expectation parameter cumulant generating function Legendre transform Bregman divergence Pythagorean theorem generalised Pythagorean projection onto convex set Dykstra alternating projection von Neumann trace inequality Ky Fan k-norm majorization Schur-Horn theorem doubly stochastic matrix Birkhoff polytope permutation extreme point Carathéodory theorem convex hull representation dimension bound Helly theorem intersection property Radon theorem partition Weyl inequality Hoffman-Wielandt perturbation Bauer-Fike κ(A) condition number eigenvalue sensitivity singular value decomposition Eckart-Young-Mirsky optimal low-rank approximation Frobenius norm spectral norm nuclear norm trace norm convex relaxation semidefinite programming semidefinite constraint dual Slater condition strong duality KKT stationarity complementary slackness primal feasibility dual feasibility Lagrangian saddle point interior-point path-following primal-dual Newton direction predictor-corrector Mehrotra formulation centring parameter step-size safeguard fraction-to-boundary rule line-search Armijo condition curvature Wolfe condition strong Wolfe condition Goldstein condition Nesterov acceleration Polyak heavy-ball momentum restart schedule adaptive gradient AdaGrad accumulation RMSProp exponential moving average Adam bias-correction beta-one beta-two epsilon denominator AdamW decoupled weight decay LAMB layer-wise adaptive batch size trust-region ratio Cauchy point Steihaug truncated CG dogleg Powell method BFGS quasi-Newton inverse Hessian approximation L-BFGS limited-memory correction pairs SR1 symmetric rank-one DFP Davidson-Fletcher-Powell conjugate gradient Fletcher-Reeves Polak-Ribière restart orthogonal polynomials Lanczos tridiagonal Arnoldi Hessenberg Givens rotation Householder reflector QR algorithm Wilkinson shift implicit Q theorem shift strategy Rayleigh quotient iteration subspace iteration Jacobi-Davidson harmonic Ritz projection thick-restart Lanczos implicitly restarted Arnoldi IRAM Krylov-Schur unitary equivalence Arnoldi relation Galerkin orthogonality minimal residual MINRES SYMMLQ GMRES right preconditioning left preconditioning flexible GMRES variable preconditioner additive Schwarz multiplicative Schwarz overlapping subdomain coarse grid correction multigrid V-cycle W-cycle cycle reduction full-weighting restriction prolongation Galerkin coarse operator smoothing Gauss-Seidel red-black ordering lexicographic ordering Jacobi smoother Chebyshev polynomial filter deflation auxiliary space preconditioner domain decomposition Schur complement complementarity problem active set method primal-dual interior-point barrier logarithmic potential self-concordance Dikin ellipsoid volumetric barrier neighbourhood centrality measure predictor corrector primal step dual step fraction-to-boundary merit function filter line-search trust region constrained optimisation quadratic programming active-set projected gradient Frank-W Wolfe conditional gradient simplex constraint column generation cutting plane Kelley bundle method stabilised bundle proximal bundle level method augmented Lagrangian method-of-multipliers