Monitoring for Shifts at Unknown Times: Parameter-Free Conditional Conformal Martingales

Tuo Liu; Drew Prinster; Francesco Orabona

stat.MLOct 12, 2026

Monitoring for Shifts at Unknown Times: Parameter-Free Conditional Conformal Martingales

Authors: Tuo Liu, Drew Prinster, Francesco Orabona

Organizations: King Abdullah University of Science and Technology, Saudi Arabia · Johns Hopkins University, USA

Abstract

Conditional conformal test martingales provide sequential tests for detecting distribution shifts, using a fixed reference dataset to avoid contamination from test-time samples. Prior work used a penalized betting function via online optimization with smoothing, clipping, and other hyperparameters, which can adapt slowly when a shift occurs after a long stationary period. In this work, we introduce an approach that requires no hyperparameters and provably adapts quickly to shifts occurring at unknown times. Our key idea is to frame the penalized betting problem as portfolio selection in a three-asset market. Then, we use a restarting universal portfolio, with a heavy-tailed wealth allocation over time. We establish anytime false-alarm control and derive optimal finite-sample bounds on detection delay under detectable alternatives. We also develop an FFT-based implementation that computes the portfolio sequence over $T$ rounds in $\mathcal{O}(T^2\ln T)$ time, instead of $\mathcal{O}(T^3)$ of the naive implementation. Synthetic and real-world experiments suggest that the proposed approach adapts rapidly after long periods of null samples, with detection delays depending substantially less on shift time than prior baselines.

Figures & tables

Figure 1: Median detection delay vs. shift time on logarithmic

y

axes: (a) sharp change to

\mathcal{N}(1,1)

and (b) gradual drift, with

100

runs per

t_{0}

, and (c) ImageNet-C with

n=1000

, with

150

runs per

t_{0}

Figure 2: Power given no false alarm , versus the number of rounds

t-t_{0}

after the shift.

Figure 3: ImageNet-C by corruption group. For each seed, we take the median ratio over the corruptions of the group; lines show the mean over

10

seeds and bands one standard error. Values above 1 indicate faster detection by our detector.

Appendix figures & tables26 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 4 — Figure 4: The grid for $N=5$ .

Panel of Figure 5 — Figure 5: (a) The pairs of start rounds $s$ and rounds $t\geq s$ . The split at $K$ handles the dark block at once, later splits handle the lighter blocks, and the gray cells are computed directly. (b) The portfolios started in $[L,K)$ are merged at round $K$ into one polynomial, whose integrals are needed at every round in $[K,R)$ (white dots). (c) Without a known horizon, doubling trick is used, and all earlier portfolios are merged at the start of each epoch.

$T$	null	shift	drift	$\tau$	reversal	$\ln M$ there	first difference
	largest allocation difference			reversal stream
$500$	$3.3\cdot 10^{-16}$	$5.6\cdot 10^{-16}$	$1.3\cdot 10^{-15}$	$218$	$333$	$25$	$360$
$1000$	$3.3\cdot 10^{-16}$	$1.2\cdot 10^{-15}$	$9.4\cdot 10^{-16}$	$401$	$666$	$84$	$694$
$2000$	$2.2\cdot 10^{-16}$	$7.4\cdot 10^{-16}$	$1.0\cdot 10^{-15}$	$712$	$1333$	$165$	$1350$
$4000$	$3.3\cdot 10^{-16}$	$1.6\cdot 10^{-15}$	$7.8\cdot 10^{-16}$	$1371$	$2666$	$330$	$2687$
$8000$	$3.3\cdot 10^{-16}$	$1.1\cdot 10^{-15}$	$5.8\cdot 10^{-16}$	$2721$	$5333$	$644$	$5348$

Appendix

Table 1: Agreement of the direct and the divide-and-conquer methods. Left: largest difference

\max_{t\leq T}\max_{i}\ \mathopen{}\mathclose{{\left|w_{t}^{i}-\tilde{w}_{t}^{i}}}\right|

between the allocations

\boldsymbol{w}_{t}

and

\tilde{\boldsymbol{w}}_{t}

of the two methods. Right, for the reversal stream: the alarm time

\tau

(the same for both methods), the round

\lfloor 2T/3\rfloor

of the reversal,

\ln M

at the reversal, and the first round at which the allocations differ by more than

10^{-9}

Figure 6: Cost of the direct and the divide-and-conquer methods on one core, against the horizon

T

: (a) wall-clock time, with reference lines through the last point of each method, and (b) peak memory of the process, including about

0.1

GiB for the Python interpreter.

	Ours	CCTM	CTM	Ours	CCTM	CTM	Ours	CCTM	CTM	Oracle
	$t_{0}=0$			$t_{0}=600$			$t_{0}=4000$
Sharp change, $d=1$	22	24	31	48	73	84	54	216	227	11
Sharp change, $d=1.5$	14	13	21	32	52	61	36	150	168	7
Sharp change, $d=2$	12	11	17	26	43	51	30	122	141	6
Gradual drift, $\lambda=1.5$	65	64	67	81	98	104	88	179	197	73
Gradual drift, $\lambda=3$	41	43	47	56	70	77	60	142	161	42

Appendix

Table 2: Median detection delay

\tau-t_{0}

in the synthetic experiments, over the trials without a false alarm; Ours is the restarting universal portfolio. Bold : the fastest method for each configuration and shift time excluding oracle (not implementable). Note that the oracle’s delay depends very weakly on

t_{0}

(here we only show it for

t_{0}=4000

). Tables 3 – 8 give further statistics.

Figure 7: Type-I error against the reference size

n

: fraction of

100

null streams of

T=4500

rounds on which the test martingale (

M_{t}

for the restarting portfolio) reaches

1/\alpha

. The two lines coincide except at

n=2500

Figure 8: Synthetic experiments: median delay

\tau-t_{0}

against the shift time , over the trials without a false alarm, for every sharp change (top) and every drift (bottom), on logarithmic vertical axes.

Figure 9: Synthetic experiments: power given no false alarm ,

\mathbb{P}\{\tau\leq t\mid\tau>t_{0}\}

, against the number of rounds after the shift, for every configuration (rows) and

t_{0}\in\{0,600,4000\}

(columns).

Figure 10: Synthetic experiments: median log-wealth after the shift , for the sharp change with

d=1

(top) and the drift with

\lambda=1.5

(bottom), and

t_{0}\in\{200,600,4000\}

(columns). For the restarting portfolio, the wealth is its test martingale

M_{t}

. The dashed line is the rejection threshold

\ln(1/\alpha)

, and the vertical axes are linear between

-1

and

1

and logarithmic outside.

Figure 11: Synthetic null streams : fraction of the

100

streams whose wealth (

M_{t}

for the restarting portfolio) has reached

1/\alpha

by round

t

, for (a)

n=1000

and (b)

n=2000

. The CCTM never rejects, and its line is hidden under that of the restarting portfolio.

$t_{0}$	strategy	false al.	power	median	mean	median	mean	median	mean
				delay after $t_{0}$		$\ln\Wealth_{T}$		$\ln M_{T}$
$0$	restarting UP	0.00	1.00	22	22.6	1157.0	1159.2	1158.4	1160.7
	conditional CTM (ONS)	0.00	1.00	24	24.0	747.8	747.5	747.8	747.5
	standard CTM	0.00	1.00	31	31.6	171.8	172.0	171.8	172.0
	best CRP after shift (oracle)	0.00	1.00	10	13.6	1164.0	1166.1	1164.0	1166.1
$200$	restarting UP	0.00	1.00	43	45.6	1088.2	1090.7	1090.9	1093.1

Appendix

Table 3: Sharp change point,

d=1

X_{t}\sim\mathcal{N}(1,1)

after

t_{0}

. Reference set

n=1000

, stream length

T=4500

, 100 trials; details in Appendix F.1 . False al. : fraction rejecting at or before

t_{0}

; power : fraction of the trials without a false alarm that reject by

T

\mathbb{P}(\tau\leq T\mid\tau>t_{0})

. Delays are over the trials without a false alarm; the median counts misses as

\infty

, the mean counts them as

T-t_{0}+1

. The restarting portfolio rejects on its test martingale

M_{t}

;

\Wealth_{T}

is its portfolio wealth, and

\Wealth_{T}=M_{T}

for the others. Bold : best implementable strategy in the block.

$t_{0}$	strategy	false al.	power	median	mean	median	mean	median	mean
				delay after $t_{0}$		$\ln\Wealth_{T}$		$\ln M_{T}$
$0$	restarting UP	0.00	1.00	14	14.6	1905.2	1903.6	1905.7	1904.2
	conditional CTM (ONS)	0.00	1.00	13	14.4	1113.4	1112.8	1113.4	1112.8
	standard CTM	0.00	1.00	21	21.1	259.5	258.7	259.5	258.7
	best CRP after shift (oracle)	0.00	1.00	7	7.5	1914.2	1912.5	1914.2	1912.5
$200$	restarting UP	0.00	1.00	28	28.4	1801.7	1801.3	1803.2	1802.9

Appendix

Table 4: Sharp change point,

d=1.5

X_{t}\sim\mathcal{N}(1.5,1)

after

t_{0}

. Reference set

n=1000

, stream length

T=4500

, 100 trials; details in Appendix F.1 . False al. : fraction rejecting at or before

t_{0}

; power : fraction of the trials without a false alarm that reject by

T

\mathbb{P}(\tau\leq T\mid\tau>t_{0})

. Delays are over the trials without a false alarm; the median counts misses as

\infty

, the mean counts them as

T-t_{0}+1

. The restarting portfolio rejects on its test martingale

M_{t}

;

\Wealth_{T}

is its portfolio wealth, and

\Wealth_{T}=M_{T}

for the others. Bold : best implementable strategy in the block.

$t_{0}$	strategy	false al.	power	median	mean	median	mean	median	mean
				delay after $t_{0}$		$\ln\Wealth_{T}$		$\ln M_{T}$
$0$	restarting UP	0.00	1.00	12	12.4	2333.2	2333.5	2333.7	2333.9
	conditional CTM (ONS)	0.00	1.00	11	11.8	1344.8	1343.7	1344.8	1343.7
	standard CTM	0.00	1.00	17	17.6	315.5	315.0	315.5	315.0
	best CRP after shift (oracle)	0.00	1.00	6	6.2	2342.5	2342.8	2342.5	2342.8
$200$	restarting UP	0.00	1.00	23	23.2	2221.1	2221.6	2222.5	2223.1

Appendix

Table 5: Sharp change point,

d=2

X_{t}\sim\mathcal{N}(2,1)

after

t_{0}

. Reference set

n=1000

, stream length

T=4500

, 100 trials; details in Appendix F.1 . False al. : fraction rejecting at or before

t_{0}

; power : fraction of the trials without a false alarm that reject by

T

\mathbb{P}(\tau\leq T\mid\tau>t_{0})

. Delays are over the trials without a false alarm; the median counts misses as

\infty

, the mean counts them as

T-t_{0}+1

. The restarting portfolio rejects on its test martingale

M_{t}

;

\Wealth_{T}

is its portfolio wealth, and

\Wealth_{T}=M_{T}

for the others. Bold : best implementable strategy in the block.

$t_{0}$	strategy	false al.	power	median	mean	median	mean	median	mean
				delay after $t_{0}$		$\ln\Wealth_{T}$		$\ln M_{T}$
$0$	restarting UP	0.00	1.00	65	62.4	2814.1	2814.6	2815.0	2815.4
	conditional CTM (ONS)	0.00	1.00	64	63.7	1634.8	1634.5	1634.8	1634.5
	standard CTM	0.00	1.00	67	68.1	575.6	575.6	575.6	575.6
	best CRP after shift (oracle)	0.00	1.00	76	72.4	2819.0	2818.6	2819.0	2818.6
$200$	restarting UP	0.00	1.00	79	78.5	2682.9	2682.5	2684.2	2683.8

Appendix

Table 6: Gradual drift,

\lambda=1.5

X_{t}\sim\mathcal{N}(1.5\,(t-t_{0})/100,1)

after

t_{0}

. Reference set

n=2000

, stream length

T=4500

, 100 trials; details in Appendix F.1 . False al. : fraction rejecting at or before

t_{0}

; power : fraction of the trials without a false alarm that reject by

T

\mathbb{P}(\tau\leq T\mid\tau>t_{0})

. Delays are over the trials without a false alarm; the median counts misses as

\infty

, the mean counts them as

T-t_{0}+1

. The restarting portfolio rejects on its test martingale

M_{t}

;

\Wealth_{T}

is its portfolio wealth, and

\Wealth_{T}=M_{T}

for the others. Bold : best implementable strategy in the block.

$t_{0}$	strategy	false al.	power	median	mean	median	mean	median	mean
				delay after $t_{0}$		$\ln\Wealth_{T}$		$\ln M_{T}$
$0$	restarting UP	0.00	1.00	41	40.8	2842.2	2842.4	2842.9	2843.0
	conditional CTM (ONS)	0.00	1.00	43	42.3	1649.9	1649.9	1649.9	1649.9
	standard CTM	0.00	1.00	47	47.0	575.6	575.6	575.6	575.6
	best CRP after shift (oracle)	0.00	1.00	43	42.4	2848.6	2848.5	2848.6	2848.5
$200$	restarting UP	0.00	1.00	54	53.2	2709.5	2709.6	2710.7	2710.8

Appendix

Table 7: Gradual drift,

\lambda=3

X_{t}\sim\mathcal{N}(3\,(t-t_{0})/100,1)

after

t_{0}

. Reference set

n=2000

, stream length

T=4500

, 100 trials; details in Appendix F.1 . False al. : fraction rejecting at or before

t_{0}

; power : fraction of the trials without a false alarm that reject by

T

\mathbb{P}(\tau\leq T\mid\tau>t_{0})

. Delays are over the trials without a false alarm; the median counts misses as

\infty

, the mean counts them as

T-t_{0}+1

. The restarting portfolio rejects on its test martingale

M_{t}

;

\Wealth_{T}

is its portfolio wealth, and

\Wealth_{T}=M_{T}

for the others. Bold : best implementable strategy in the block.

$t_{0}$	strategy	false al.	power	median	mean	median	mean	median	mean
				delay after $t_{0}$		$\ln\Wealth_{T}$		$\ln M_{T}$
$0$	restarting UP	0.00	1.00	31	31.1	2852.6	2852.9	2853.2	2853.4
	conditional CTM (ONS)	0.00	1.00	33	33.0	1655.8	1655.7	1655.8	1655.7
	standard CTM	0.00	1.00	38	37.8	575.6	575.6	575.6	575.6
	best CRP after shift (oracle)	0.00	1.00	29	29.2	2860.5	2860.5	2860.5	2860.5
$200$	restarting UP	0.00	1.00	41	41.4	2720.3	2720.3	2721.6	2721.5

Appendix

Table 8: Gradual drift,

\lambda=5

X_{t}\sim\mathcal{N}(5\,(t-t_{0})/100,1)

after

t_{0}

. Reference set

n=2000

, stream length

T=4500

, 100 trials; details in Appendix F.1 . False al. : fraction rejecting at or before

t_{0}

; power : fraction of the trials without a false alarm that reject by

T

\mathbb{P}(\tau\leq T\mid\tau>t_{0})

. Delays are over the trials without a false alarm; the median counts misses as

\infty

, the mean counts them as

T-t_{0}+1

. The restarting portfolio rejects on its test martingale

M_{t}

;

\Wealth_{T}

is its portfolio wealth, and

\Wealth_{T}=M_{T}

for the others. Bold : best implementable strategy in the block.

	$n=1000$		$n=2000$
strategy	rejected	median final log-wealth	rejected	median final log-wealth
restarting UP	0.00	-0.88	0.00	-0.97
conditional CTM (ONS)	0.00	-2.56	0.00	-2.75
standard CTM	0.02	-0.68	0.02	-0.76

Appendix

Table 9: Type-I error. Fraction of 100 null streams (

X_{t}\sim\mathcal{N}(0,1)

throughout,

T=4500

) whose wealth ever reaches

1/\alpha=20

(

M_{t}

for the restarting portfolio, the martingale itself for the CTMs), and the median final log-wealth, for the reference sizes of the sharp change point (

n=1000

) and the drift (

n=2000

	$t_{0}=0$					$t_{0}=1000$					$t_{0}=8000$
	All	Noise	Blur	Weather	Digital	All	Noise	Blur	Weather	Digital	All	Noise	Blur	Weather	Digital
Ours	17	18	16.5	12	27.5	42	41.5	38	27.5	88.5	50.5	54.5	45	31	130.5
CCTM	67	69	65.5	61	77.5	84.5	85	74.5	59.5	143	244.5	250	220.5	165	425.5
CTM	75	76	72.5	66	85	78	82	74	58	106.5	264	270	241	192.5	390.5
Oracle	8	7	8.5	6	19	8	7	8.5	6	19	8	7	8.5	6	19

Appendix

Table 10: Median detection delay

\tau-t_{0}

on ImageNet-C with

n=1000

, by corruption group, over

10

seeds and the corruptions of each group. The delays of the CTMs include their warm-up of

50

rounds. Bold : the fastest implementable method. Table 14 gives the false-alarm rates, the power, and the mean delays, and Tables 11 – 15 give the other reference sizes.

Figure 12: ImageNet-C: median delay

\tau-t_{0}

against the length

t_{0}

of the clean phase , for every reference size

n

, over

15

corruptions and

10

seeds. Runs that miss the shift within

2000

rounds count as infinite delays. The vertical axes are logarithmic, and the horizontal axes are logarithmic beyond

t_{0}=0

Figure 13: ImageNet-C: power given no false alarm ,

\mathbb{P}\{\tau\leq t\mid\tau>t_{0}\}

, against the number of rounds after the shift, for every reference size

n

(rows) and

t_{0}\in\{0,1000,8000\}

(columns), over

15

corruptions and

10

seeds.

$t_{0}$	strategy	false al.	power	all	noise	blur	weather	digital	delay
				median delay					mean
$0$	restarting UP	0.00	0.93	30	30	26.5	16.5	177.5	235.5 †
	conditional CTM (ONS)	0.00	0.93	83	89	78	65.5	203	269.6 †
	standard CTM	0.00	0.96	93	97	88	75.5	126.5	194.2 †
	best CRP after shift (oracle)	0.00	0.93	18.5	18.5	16	9	136.5	189.4 †
$1000$	restarting UP	0.00	0.92	90.5	99.5	66.5	40.5	818	360.5 †

Appendix

Table 11: ImageNet-C,

n=100

. 15 severity-5 corruptions

\times

10 seeds after

t_{0}

clean images; every method runs

H=2000

rounds past the change. False al. : rejection at or before

t_{0}

; power :

\mathbb{P}(\tau\leq t_{0}+H\mid\tau>t_{0})

. Delays

\tau-t_{0}

(warm-up rounds included), medians over the trials without a false alarm, misses counted as

\infty

; the mean counts misses as

H+1

( † : some runs missed, so the mean is a lower bound). The restarting portfolio rejects on

M_{t}

. Groups as in Shaer et al. [2026] , Figure 4. Bold : best implementable strategy.

$t_{0}$	strategy	false al.	power	all	noise	blur	weather	digital	delay
				median delay					mean
$0$	restarting UP	0.00	0.98	21	21	19	14	67	118.0 †
	conditional CTM (ONS)	0.00	0.99	73	75	68.5	63	102	151.9 †
	standard CTM	0.00	1.00	79	84	75.5	69	102.5	92.7
	best CRP after shift (oracle)	0.00	0.99	11	11	11	7	29	78.5 †
$1000$	restarting UP	0.00	0.96	60.5	65.5	49	31.5	185	194.2 †

Appendix

Table 12: ImageNet-C,

n=250

. 15 severity-5 corruptions

\times

10 seeds after

t_{0}

clean images; every method runs

H=2000

rounds past the change. False al. : rejection at or before

t_{0}

; power :

\mathbb{P}(\tau\leq t_{0}+H\mid\tau>t_{0})

. Delays

\tau-t_{0}

(warm-up rounds included), medians over the trials without a false alarm, misses counted as

\infty

; the mean counts misses as

H+1

( † : some runs missed, so the mean is a lower bound). The restarting portfolio rejects on

M_{t}

. Groups as in Shaer et al. [2026] , Figure 4. Bold : best implementable strategy.

$t_{0}$	strategy	false al.	power	all	noise	blur	weather	digital	delay
				median delay					mean
$0$	restarting UP	0.00	1.00	19	20	17.5	13	29.5	48.3
	conditional CTM (ONS)	0.00	1.00	70	71	67	61.5	82	91.5
	standard CTM	0.00	1.00	76	78.5	74	68	93	84.3
	best CRP after shift (oracle)	0.00	1.00	9	8	9	6.5	20	29.3
$1000$	restarting UP	0.00	1.00	46.5	44	41.5	29	134.5	106.9

Appendix

Table 13: ImageNet-C,

n=500

. 15 severity-5 corruptions

\times

10 seeds after

t_{0}

clean images; every method runs

H=2000

rounds past the change. False al. : rejection at or before

t_{0}

; power :

\mathbb{P}(\tau\leq t_{0}+H\mid\tau>t_{0})

. Delays

\tau-t_{0}

(warm-up rounds included), medians over the trials without a false alarm, misses counted as

\infty

; the mean counts misses as

H+1

. The restarting portfolio rejects on

M_{t}

. Groups as in Shaer et al. [2026] , Figure 4. Bold : best implementable strategy.

$t_{0}$	strategy	false al.	power	all	noise	blur	weather	digital	delay
				median delay					mean
$0$	restarting UP	0.00	1.00	17	18	16.5	12	27.5	31.8
	conditional CTM (ONS)	0.00	1.00	67	69	65.5	61	77.5	80.3
	standard CTM	0.00	1.00	75	76	72.5	66	85	81.3
	best CRP after shift (oracle)	0.00	1.00	8	7	8.5	6	19	17.3
$1000$	restarting UP	0.00	1.00	42	41.5	38	27.5	88.5	74.1

Appendix

Table 14: ImageNet-C,

n=1000

. 15 severity-5 corruptions

\times

10 seeds after

t_{0}

clean images; every method runs

H=2000

rounds past the change. False al. : rejection at or before

t_{0}

; power :

\mathbb{P}(\tau\leq t_{0}+H\mid\tau>t_{0})

. Delays

\tau-t_{0}

(warm-up rounds included), medians over the trials without a false alarm, misses counted as

\infty

; the mean counts misses as

H+1

. The restarting portfolio rejects on

M_{t}

. Groups as in Shaer et al. [2026] , Figure 4. Bold : best implementable strategy.

$t_{0}$	strategy	false al.	power	all	noise	blur	weather	digital	delay
				median delay					mean
$0$	restarting UP	0.00	1.00	16	16	16	12	25	23.9
	conditional CTM (ONS)	0.00	1.00	65	66	64	60.5	73.5	73.8
	standard CTM	0.00	1.00	74	76	71.5	66	83.5	79.9
	best CRP after shift (oracle)	0.00	1.00	7	7	7	6	17.5	13.9
$1000$	restarting UP	0.00	1.00	39.5	41.5	36	26	73.5	58.0

Appendix

Table 15: ImageNet-C,

n=4000

. 15 severity-5 corruptions

\times

10 seeds after

t_{0}

clean images; every method runs

H=2000

rounds past the change. False al. : rejection at or before

t_{0}

; power :

\mathbb{P}(\tau\leq t_{0}+H\mid\tau>t_{0})

. Delays

\tau-t_{0}

(warm-up rounds included), medians over the trials without a false alarm, misses counted as

\infty

; the mean counts misses as

H+1

. The restarting portfolio rejects on

M_{t}

. Groups as in Shaer et al. [2026] , Figure 4. Bold : best implementable strategy.

	restarting UP		conditional CTM		standard CTM
$n$	rejected	med. final	rejected	med. final	rejected	med. final
$100$	0.00	-0.56	0.00	-0.89	0.03	-0.80
$500$	0.00	-0.79	0.00	-1.54	0.02	-0.94
$4000$	0.01	-1.07	0.00	-2.33	0.01	-0.86

Appendix

Table 16: ImageNet null check. The stream is the rest of the clean holdout (

T=12400

for

n=100

T=12000

for

n=500

T=8500

for

n=4000

), 100 seeds per

n

: fraction of streams whose wealth ever reaches

1/\alpha

(

M_{t}

for the restarting portfolio, the martingale itself for the CTMs), and the median final log-wealth.

Distribution Shift Detection

Distribution shift detection identifies when data or model behavior changes between reference and deployment settings.

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Jun 18, 2026

Betting on Moments: Legendre Jumper Martingales for Online Exchangeability Testing

A fundamental assumption in statistics and machine learning is that ``the future looks like the past,'' formalized as exchangeability: the joint data distribution is order-invariant. In practice, this assumption is often violated due to distribution shifts over time. Early detection of exchangeability violations is crucial to prevent performance degradation and enable timely interventions like model retraining. Conformal test martingales offer a flexible, distribution-free framework for sequential exchangeability testing with guaranteed false-alarm rate control by betting against the uniformity of conformal p-values. While alternatives such as plug-in martingales and mixture-based strategies exist, computationally efficient baselines like the Simple Jumper are limited to detecting mean location shifts. We propose a family of conformal test martingales based on shifted Legendre polynomials that extend the Simple Jumper to higher-order moments. The Simple Legendre Jumper replaces linear betting functions with polynomials of arbitrary degree, enabling rapid detection of variance, skewness, and other higher-order deviations. The Product Legendre Jumper combines multiple polynomial degrees into a single betting function but suffers from exponential state-space growth, termed the jumping tax. To resolve this, we introduce the Variational Legendre Jumper, which employs a mean-field approximation to reduce complexity to constant time per step with minimal power loss, providing an expressive, scalable framework for real-time distribution shift monitoring.

Johan Hallberg Szabadváry

Department of Mathematics, Stockholm University

Conformal Prediction Distribution Shift Detection Change-Point Detection Sequential Hypothesis Testing Temporal Distribution Shift Statistical Hypothesis Testing

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Aug 24, 2026

Change Detection in Probability Flow ODE: Online Testing in Diffusion Latent Spaces

A rapidly growing range of sequential data tasks, such as identifying trend reversals in financial markets, auto-segmenting video and audio recordings, detecting changes in movement direction from motion sensors cannot be fully addressed without detection of distributional shifts in time-ordered data. We consider a sequential change-point detection problem where the conditional density switches at an unknown time, yet neither the pre- nor post-change distribution admits a closed-form. Classical likelihood-ratio statistics are inapplicable in this settings. A conditional diffusion model, trained on pre-change-point data with a frozen context encoder, defines a deterministic bijection via the probability flow ODE. Pre-change observations are mapped onto standard Gaussian latent variables. Post-change observations, processed through the same frozen map, deviate from this reference. We employ the Maximum Mean Discrepancy as the test statistic, derive closed-form expressions for its components under the Gaussian null, and establish its asymptotic distribution as a degenerate U-statistic. Afterwards we apply an online detection procedure of Shiryaev--Roberts to the resulting statistic with exact threshold calibration. The method detects arbitrary distributional shifts, including covariance rotations and higher-order structural breaks, without parametric assumptions on either regime.

Artem Kraevskiy, Artem Prokhorov

National Research University Higher School of Economics Moscow, Russia · The University of Sydney Business School & CIREQ & CEBDA Sydney, Australia

Distribution Shift Detection Change-Point Detection Sequential Hypothesis Testing

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May 31, 2026

Distribution-free changepoint localization after sequential change detection

This paper introduces a distribution-free framework for constructing post-detection confidence sets for changepoints after stopping a sequential change detection procedure. It is well known that conformal test martingales can be used to sequentially detect changes in distribution, but by themselves provide no inference for the time at which a proclaimed change occurred. Past work on post-detection inference requires pre- and post-change classes of distributions to be known, but this paper accomplishes localization of the changepoint without any distributional assumptions. We establish finite-sample coverage guarantees (conditional on correct detection). We provide non-asymptotic bounds on the conditional expected size of the confidence sets. Under suitable asymptotic regimes, we prove that the conditional expected size of the confidence set remains uniformly bounded and demonstrate strong empirical performance on simulated and real data. To the best of our knowledge, this is the first general distribution-free framework for sequential changepoint localization with valid post-detection coverage.

Aytijhya Saha, Aaditya Ramdas

Massachusetts Institute of Technology · Carnegie Mellon University

Confidence Region Estimation Distribution Shift Detection Change-Point Detection

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Abstract

Distribution Shift Detection

Explore similar work

Betting on Moments: Legendre Jumper Martingales for Online Exchangeability Testing

Change Detection in Probability Flow ODE: Online Testing in Diffusion Latent Spaces

Distribution-free changepoint localization after sequential change detection